Uncertain project selection model considering sustainability and compatibility

Abstract

Project selection is a significant decision-making process in most companies. Most previous researches focused only on the investment profit and used random variables to determine project parameters. However, in some cases, the distributions of parameters cannot be obtained from similar projects due to the rapid changing environment. So experts’ estimations will be adopted. In order to handle inaccurate estimation, an uncertainty theory is applied to solve project selection decision-making in this situation. In addition, besides considering financial aspects, the non-financial aspects including compatibility, are taken into account. A new project selection model is developed and the deterministic form of the model is given. For the sake of illustration, an example has been presented. This paper aims at proposing a new project appraisal method comprehensively considering financial and non-financial aspects.

Keywords

Uncertain programming project selection sustainability compatibility

1 Introduction

Selection of the most appropriate projects from several candidate projects is an important investment decision to many companies. In the past, project selection method used to focus only on investment profit [1]. According to the classical theory of investment, project selection used to examine whether the net present value (NPV) or expected NPV is greater than zero. Obviously, this is the “golden rule” for investors to select the optimal projects from alternatives.

NPV only shows comprehensive result of the discounted streams of net cash flows without paying attention to the individual net cash flow. However, this consideration has some flaws. For example, the NPV value of a project is positive, but for some continuous years, the net cash flows are negative. It is clear if the company can have enough capital to support itself in those bad years, the company can still make profits from this project. Unfortunately, the company may not be able to have enough money to pull through the bad years so that it cannot finally benefit from the project. Obviously, this situation shows poor sustainability of the project. And in some extreme situations, the company may go bankruptcy due to the huge negative net cash flows. Therefore, we propose that both the aggregate result of NPV and the number of negative net cash flow years should be considered. If the number of negative net profit accounts for a small number, or almost zero, the project sustainability is considered better; otherwise not.

NPV and sustainability evaluate projects only from financial aspects without considering the important non-financial aspects, e.g. strategic compatibility. Non-financial aspects are argued to be taken into account when selecting projects. Sherer [17] argued that strategic viewpoints would affect the assessment of the type of project. Gerry et al. [7] pointed out that the strategic matching degree influenced decision making. They all stressed the importance of strategic matching degree, so the strategic matching degree can make a great difference to the decision of a project selection. For example, if Louis Vuitton whose target market is high-end consumers, decides to invest a project which produces ordinary students’ schoolbags, it still may be a disaster for the company despite the profit it might make. Therefore, we take the strategic matching degree into account when selecting projects.

Furthermore, whether the project has some effects to other businesses in the company is another point that needs consideration. Danese and Kalchschmidt [3] have provided evidence that synergistic effects among forecasting practices influence companies’ performances. For example, a large enterprise group with businesses in disparate fields such as car and drinking water invests a new project which manufactures engines. Obviously, this project will make synergistic effect to the existing car business. If the company chooses the project which produces shoes, this new project makes no difference to the existing businesses in this company. Therefore, we take the synergistic degree of the project to other businesses in the corporation into account when selecting projects.

Since project selection is done in complex economic environment, investors usually cannot predict parameters exactly. Therefore, indeterminate project parameters were considered. In the past, these indeterminate project parameters used to be considered as random variables [6 , 12]. However, economic environment is changing fast, so sometimes distributions of parameters of candidate projects cannot be given by the historical data of similar projects but have to be assessed by experts, such as R&D projects. It is difficult for us to predict the exact values of project parameters in real life. Since imprecise estimations are not random in nature, treating estimations as random variables can cause counter intuitive results [9, 18]. To better handle imprecise estimations, Liu [14] proposed an uncertain measure and developed an uncertainty theory and refined it in 2010 [13]. With uncertainty theory, no paradoxes appear in decision making. For this reason, uncertainty theory has been widely used in decision making problems in uncertain environment. An important application is by Huang [10] who founded the uncertainty portfolio selection theory by systematically applying uncertainty theory to portfolio selection. Other applications can also be found in shortest path [5], inventory [16] and facility location [8], etc. In project selection with parameter values estimated by experts, Zhang et al. [19] studied how to select domestic projects, Zhang et al. [18] studied how to select international projects, and Huang and Zhao [9] studied project selection and adjustment, based on uncertainty theory. However, in their selection method, only profit aspect from the project investment is considered. In this paper, we will go on studying project selection problem in such environment. Different from existing studies, uncertainty theory will be used to solve the project selection problem considering non-financial aspects. Furthermore, NPV will be improved considering sustainability of projects.

The rest of the paper is organized as follows. Firstly, the compatibility and sustainability index as well as an uncertain project selection model will be proposed in Section 2. In addition, the equivalent form of the model is represented in Section 3. To illustrate the application of the model, an example is provided in Section 4. Finally, conclusion is provided in Section 5. What’s more, in order to understand the paper well, fundamentals of uncertainty theory will be briefly reviewed in the Appendix.

2 Uncertain project selection model

To solve the uncertain project selection problem, a corporation should select projects considering financial and non-financial aspects. Suppose all candidate projects begin to produce net cash flow at the end of each year. In this paper, we use the following notations for the convenience of expression:

$\tilde{{NCF}_{h, t}}$ : The uncertain annual net cash flow at the end of year t of project h;

IO_h: The initial outlay of project h;

N_h: The number of negative net cash flow years of project h. Define N_h,t = 1, when NCF_h,t satisfies $M {{NCF}_{h, t} | {NCF}_{h, t} \leq 0} \geq β$ , $M$ is the uncertain measure of the uncertain variable and β denotes the tolerance to negative profit, h = 1, 2, …, k, where k is the number of all the projects. Otherwise, N_h,t = 0. Then N_h is obtained $N_{h} = \sum_{t = 1}^{T_{h}} N_{h, t}$ ;

T_h: The number of the years that the project h will last;

$\tilde{a_{h, 1}}$ : The matching degree of corporative strategy of the project h. $\tilde{a_{h, 1}}$ should be a continuous value of uncertain variables from 0 to 10, h = 1, 2, …, k;

$\tilde{a_{h, 2}}$ : The synergistic degree of the project h to other businesses in the corporation. $\tilde{a_{h, 2}}$ should be a continuous value of uncertain variables from 0 to 10, h = 1, 2, …, k;

r: The discount rate;

w₁: The weight of $\tilde{a_{h, 1}}$ ;

w₂: The weight of $\tilde{a_{h, 2}}$ ;

x_h: Decision variables which are defined by

$x_{h} = {\begin{matrix} 1, if the project is selected, \\ 0, if the project is abandoned, \end{matrix} h = 1, 2, \dots, k$

2.1 NPV

Due to the time value of money, we should use the present value to describe the current cash value. Considering the meaning of IO_h, it can be described as a definite value. The net cash flows of the project h is demonstrated in Fig. 1. Therefore, the NPV_h in this case is $\tilde{{NPV}_{h}} = \sum_{t = 1}^{T_{h}} \frac{\tilde{{NCF}_{h, t}}}{{(1 + r)}^{t}} - {IO}_{h} .$ (1)

Fig.1

Cash flows of the project h.

Therefore, the total investment return in NPV form is $E [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{\tilde{{NCF}_{h, t}}}{{(1 + r)}^{t}} - {IO}_{h}) x_{h}]$ , where E is the expected value operator of uncertain variables. The company wants to gain the maximum profit in the project selection. According to Equation (1), we can maximize the investment profit gaining from project h in the NPV form. So, the company objective is shown below: $max E [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{\tilde{{NCF}_{h, t}}}{{(1 + r)}^{t}} - {IO}_{h}) x_{h}] .$ (2)

2.2 Sustainability index

Besides thinking about the NPV, we should also consider the sustainability of the projects. The sustainability index of project h is defined as $\frac{N_{h}}{T_{h}}, h = 1, 2, \dots, k$ .

The lower the $\frac{N_{h}}{T_{h}}$ value, the more sustainable the project is. When selecting the projects, we require that the sustainability index value should be equal to or less than a pre-given level C. To express it mathematically, we have $x_{h} \frac{N_{h}}{T_{h}} \leq C, h = 1, 2, \dots, k .$ (3)

2.3 Compatibility index

We integrate strategic matching degree and synergistic degree into one compatibility index, which is defined by $w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}$ .

Since $\tilde{a_{h, 1}}$ and $\tilde{a_{h, 2}}$ are uncertain variables, the $w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}$ is also an uncertain variable. When the $\tilde{a_{h, 1}}$ increases, the project matches better to the corporation’s strategy. The higher the $\tilde{a_{h, 2}}$ value, the higher the synergic degree is. When $w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}$ is equal to or higher than p_h, it means that the project is compatible. To measure the chance of good compatibility, we define the compatibility index as $M {x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \geq p_{h}}$ . Since ${x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \geq p_{h}}$ is a perceptible beneficial event, it is easy for the company to provide their tolerance of the chance of this benefit event. Then, the company can require that the compatibility index of the investment should be greater than a preset tolerable level α. Then, the constraint can be expressed as follows:

$\begin{matrix} M {x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \geq p_{h}} \\ \geq α, h = 1, 2, \dots, k . \end{matrix}$ (4)

2.4 Capital limitation

Since every corporation has a capital limitation, let b be the available maximum capital. Then the capital limitation can be expressed as follows: $\sum_{h = 1}^{k} {IO}_{h} x_{h} \leq b .$ (5)

2.5 Comprehensive model

Thus, if the company wants to obtain the maximum expected NPV, and requires that the selected projects should satisfy the sustainability index and compatibility index requirements, with the capital limitation, the company should select the project portfolio according to the following model: ${\begin{matrix} max E [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{{\tilde{NCF}}_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \\ subject to \\ E [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{{\tilde{NCF}}_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \geq 0, \\ h = 1, 2, \dots, k \\ x_{h} \frac{N_{h}}{T_{h}} \leq C, h = 1, 2, \dots, k \\ M {x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \geq p_{h}} \geq α, \\ h = 1, 2, \dots, k \\ \sum_{h = 1}^{k} {IO}_{h} x_{h} \leq b \\ x_{h} \in {0, 1}, h = 1, 2, \dots, k . \end{matrix}$ (6)

3 Equivalent of the model

Suppose ${\tilde{NCF}}_{h, t}$ are independent and the mean value of ${\tilde{NCF}}_{h, t}$ is e_h,t. According to Theorem 1 in Appendix, then we can have the formula as follows:

$\begin{matrix} E [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{\tilde{{NCF}_{h, t}}}{{(1 + r)}^{t}} - {IO}_{h}) x_{h}] \\ = [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{E [\tilde{{NCF}_{h, t}}]}{{(1 + r)}^{t}} - {IO}_{h}) x_{h}] \\ = [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{e_{h, t}}{{(1 + r)}^{t}} - {IO}_{h}) x_{h}] . \end{matrix}$ (7)

According to Definition 1 self-duality property of the uncertain variable, the formula (4) can be changed into the following form. $1 - M {x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \leq p_{h}} \geq α .$ (8)

Therefore, we can change constraint (8) into the standard form: $M {x_{h} (w_{1} \tilde{a_{h, 1}} + w_{2} \tilde{a_{h, 2}}) \leq p_{h}} \leq 1 - α .$ (9)

$\tilde{a_{h, 1}}$ and $\tilde{a_{h, 2}}$ obey continuous and strictly increasing uncertainty distribution function Φ_h,1 and Φ_h,2 respectively. The uncertainty distribution of a linear uncertain variable ( $\tilde{a_{h, 1}}$ ) is shown in Fig. 2, and distribution of $\tilde{a_{h, 2}}$ is similar. According to Theorem 2 in Appendix, we can change form (9) into the following expression:

$\begin{matrix} Ψ^{- 1} (1 - α) \\ = x_{h} [w_{1} Φ_{h, 1}^{- 1} (1 - α) + w_{2} Φ_{h, 2}^{- 1} (1 - α)] \geq p_{h} . \end{matrix}$ (10)

Fig.2

Uncertainty distribution of a linear uncertain variable ( $\tilde{a_{h, 1}}$ ).

Then the uncertain project selection model can be converted into the following form: ${\begin{matrix} max [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{e_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \\ subject to \\ [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{e_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \geq 0, \\ h = 1, 2, \dots, k \\ x_{h} \frac{N_{h}}{T_{h}} \leq C, h = 1, 2, \dots, k \\ x_{h} [w_{1} Φ_{h, 1}^{- 1} (1 - α) + w_{2} Φ_{h, 2}^{- 1} (1 - α)] \geq p_{h}, \\ h = 1, 2, \dots, k \\ \sum_{h = 1}^{k} {IO}_{h} x_{h} \leq b \\ x_{h} \in {0, 1}, h = 1, 2, \dots, k . \end{matrix}$ (11)

Furthermore, if $\tilde{a_{h, 1}}$ is a linear uncertain variable which has the following linear distribution function: $Φ_{h, 1} (t) = {\begin{matrix} 0, if t < c_{h, 1} \\ \frac{t - c_{h, 1}}{c_{h, 2} - c_{h, 1}}, if c_{h, 1} \leq t \leq c_{h, 2} \\ 1, if t > c_{h, 2} . \end{matrix}$ (12)

Table 1

The net cash flows of 10 candidate projects (Unit: million dollars)

Project h	1	2	3	4	5
Year
1	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–15,5)	$L$ (–15,5)	$L$ (–75,25)
2	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–0.6,5)	$L$ (–0.6,5)	$L$ (–26,5.4)
3	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–1.1,5.9)	$L$ (–1.1,5.9)	$L$ (–6.9,2.7)
4	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–4.6,1.4)	$L$ (–4.6,1.4)	$L$ (2,5.2)
5	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–2.2,14.6)	$L$ (–2.2,14.6)	$L$ (4,7.6)
6	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (–30.6,10)	$L$ (–2.2,14.6)	$L$ (4,14.6)
7	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (4,16.4)	$L$ (–2.2,14.6)	$L$ (10,25.2)
8	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (4,16.4)	$L$ (–2.2,14.6)	$L$ (10,49.8)
9	$L$ (–0.5,5.7)	$L$ (–2.3,13.7)	$L$ (4,16.4)	$L$ (–2.2,14.6)	$L$ (54,73.2)
10	$L$ (5,10.6)	$L$ (2.2,33.6)	$L$ (5,31.4)	$L$ (5,31.4)	$L$ (54,73.2)
Project h	6	7	8	9	10
Year
1	$L$ (–75,25)	$L$ (–20.6,10)	$L$ (–30.6,10)	$L$ (–0.5,2.7)	$L$ (–0.3,2.1)
2	$L$ (–37.4,10.2)	$L$ (0,4.4)	$L$ (–19.4,5)	$L$ (2,9.2)	$L$ (4,4.6)
3	$L$ (0.1,2.1)	$L$ (–0.3,2.9)	$L$ (–0.3,2.9)	$L$ (1,7.6)
4	$L$ (2,5.2)	$L$ (3,5.8)	$L$ (3,5.8)	$L$ (1,12.8)
5	$L$ (4,7.6)	$L$ (–8.4,2)	$L$ (1,11.4)	$L$ (1,12.8)
6	$L$ (4,14.6)	$L$ (1.1,6.1)	$L$ (10,11)
7	$L$ (10,25.2)	$L$ (1,12.4)	$L$ (–61.2,10)
8	$L$ (10,49.8)	$L$ (1,12.4)	$L$ (22,51.8)
9	$L$ (54,73.2)	$L$ (1,12.4)	$L$ (22,51.8)
10	$L$ (54,73.2)	$L$ (7.2,33.4)	$L$ (22,51.8)

According to experts’ estimations, the distributions of $\tilde{a_{h, 1}} and \tilde{a_{h, 2}}$ are given in Table 2.

Table 2

The parameter information about the distribution of ${\tilde{a}}_{h, 1}$ , ${\tilde{a}}_{h, 2}$ and IO_h (Unit of IO_h: million dollars)

Parameters	${\tilde{a}}_{h, 1}$	${\tilde{a}}_{h, 2}$	IO _h	Parameters	${\tilde{a}}_{h, 1}$	${\tilde{a}}_{h, 2}$	IO _h
Project h				Project h
1	$L$ (6.3,6.7)	$L$ (2.7,3.6)	–20	6	$L$ (4.6,5.1)	$L$ (4.4,8.2)	–45.5
2	$L$ (2.1,2.6)	$L$ (4.4,4.9)	–35	7	$L$ (5.5,7.1)	$L$ (4.3,4.9)	–7.8
3	$L$ (7.9,8.2)	$L$ (9.1,9.6)	–16	8	$L$ (3.6,4.1)	$L$ (6.3,6.7)	–17.8
4	$L$ (3.4,4.1)	$L$ (5.2,5.6)	–16	9	$L$ (8.9,9.1)	$L$ (7.8,8.6)	–7.3
5	$L$ (4.9,5.6)	$L$ (1.1,1.8)	–45.5	10	$L$ (3.1,4.4)	$L$ (1.3,2.2)	–3.1

We express it by $\tilde{a_{h, 1}} \sim L (c_{h, 1}, c_{h, 2})$ . And suppose $\tilde{a_{h, 2}}$ is a linear uncertain variable as well, which can be expressed by $\tilde{a_{h, 2}} \sim L (d_{h, 1}, d_{h, 2})$ .

Then Equation (10) can be transformed into x_h (w₁c_h,1 + w₂d_h,1) α + (1 - α) (w₁c_h,2 + w₂d_h,2) x_h ≥ p_h.

Let x_h (w₁c_h,1 + w₂d_h,1) α + (1 - α) (w₁c_h,2 + w₂d_h,2) x_h = CI_h. Then model (11) can be changed into the following equivalent:

${\begin{matrix} max [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{e_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \\ subject to \\ [\sum_{h = 1}^{k} (\sum_{t = 1}^{T_{h}} \frac{e_{h, t}}{(1 + r)^{t}} - {IO}_{h}) x_{h}] \geq 0, \\ h = 1, 2, \dots, k \\ x_{h} \frac{N_{h}}{T_{h}} \leq C, h = 1, 2, \dots, k \\ {CI}_{h} \geq p_{h}, h = 1, 2, \dots, k \\ \sum_{h = 1}^{k} {IO}_{h} x_{h} \leq b \\ x_{h} \in {0, 1}, h = 1, 2, \dots, k . \end{matrix}$ (13)

4 An example

To illustrate the application of the proposed model, an example is presented here. Table 1 shows the net cash flows of 10 candidate projects.

Suppose that the investor only has 40 million dollars, i.e., b = 40, and sets p_h = 4, C = 0.2, w₁ = 0.44 and w₂ = 0.56. If α is determined at 0.95 and β at 0.65, according to model (12), the investor should select projects 4, 7 and 9. The objective value is $23.70 million.

5 Conclusions

This paper has solved a project selection problem in which project parameters are given by experts’ evaluations for the lack of historical data from similar projects. Uncertain variables have been used to describe these parameters. Based on uncertainty theory, a project selection model has been presented considering not only NPV and capital limitation, but also compatibility index and sustainability index. Aiming at solving the problem, a deterministic equivalent of the model has been put forward. At last, an example has been provided to illustrate the application of the model.

Footnotes

Appendix

Acknowledgments

This work has been supported by National Natural Science Foundation of China Grants No. 71302164 and National Social Science Foundation of China Grants No. CFA130152.

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