Frontiers in Business, Economics and Management 
ISSN: 2766-824X | Vol. 13, No. 3, 2024 

 

333 

Project	Investment	Decision	Under	Uncertain	
Information	
Mengying Xiao1, * 

1 Department of Business, Hunan University of Science and Technology, Xiangtan, Hunan 411201, PR China 
* Corresponding author: Mengying Xiao (Email: mengyingxiao6@gmail.com) 

 

Abstract: With the change of environment and market, the current investment project decision is facing a more complex 
environment, the competition of enterprises intensifies, the decision complexity increases, and the demand for scientific decision 
is increasing. The advantage of multi-attribute decision making in investment project decision is increasingly prominent. This 
paper studies the application of multi-attribute decision making under triangular fuzzy uncertainty information in investment 
decision making. Considering the decision makers' risk aversion and preference attitude in multi-stage decision making process, 
a dynamic reference point setting method based on stage development characteristics is proposed. Considering that the attribute 
value and attribute value will produce compensation in the decision-making process, the attribute value difference formula is 
constructed to reduce the decision-making bias, and on this basis, an improved TODIM method is proposed. The triangular fuzzy 
number uncertainty information is introduced to evaluate the attributes of each scheme in each decision stage, and the information 
is aggregated by the aggregation operator. Combining the dynamic multi-attribute decision model, the practical problem is solved. 
The practicability and feasibility of this method are demonstrated by the example analysis and method comparison. 

Keywords: Dynamic reference point, TODIM, Triangular fuzzy number, Project investment decision. 
 

1. Introduction 
In today's constantly developing global economic 

environment, we are faced with complex and changeable 
market conditions and uncertainties. Project investment 
decision is a crucial part of enterprise strategic management, 
but because of information asymmetry and uncertainty, 
project investment decision is faced with great challenges. In 
this context, uncertain information and prospect theory has 
become one of the important theories to explain people's 
psychological behavior when facing uncertain decisions. In 
1965, Zadeh[1] introduced fuzzy sets into the field of decision 
making, combining fuzziness with mathematics and 
promoting the development of decision theory. In the actual 
decision-making process, the complexity of the decision-
making environment, the uncertainty of information itself and 
people's subjective cognition lead to the uncertainty of the 
decision maker in the decision-making process, and the 
generation of fuzzy set can solve the problem of fuzzy 
uncertainty. 

In 2009,Eddie et al. [2] introduced the fuzzy logic system 
into the evaluation of the real estate market, providing a 
systematic theoretical basis for the investment decision of real 
estate projects, and then helping the investment decision 
makers to better solve the difficulties in the investment 
decision of real estate projects. In 2011,William C. Waheato 
et al. [3] proposed the time series method to effectively 
quantify the risks in the real estate market, which is widely 
forward-looking. In 2013, based on selection 
experiments,Marmolejo et al. [4] combined commercial 
factors and social and political factors in the real estate market, 
and adopted the traditional polynomial regression model to 
comprehensively evaluate each decision attribute of the real 
estate project, and then selected the best real estate project. 
Based on the fuzziness of traditional real option values, Liu 
Pengyang [5] built a fuzzy real option pricing model for large-
scale real estate projects with the idea of fuzzy mathematics, 

and verified that the real option method is a relatively 
scientific and reasonable real estate investment method 
through examples. Wang Junwu [6] et al. built a hierarchical 
structure model of real estate project investment decision by 
conducting in-depth analysis of many influencing factors that 
affect real estate project investment decision, classifying them 
and using AHP method. Jia Yuanhui [7] believes that there are 
many factors affecting wind power decision making. He 
proposes to make investment decision on wind power projects 
from the perspective of risk, and establishes a multi-attribute 
decision model based on the four major risks of economy, 
technology, policy and environment, aiming to realize project 
decision making through risk control. Liu Min [8] pointed out 
that wind power projects are an effective way to control CO2 
emissions, calculated the impact of CO2 trading mechanism 
on wind power project decision-making, and considered the 
uncertainty of CDM trading price and the flexibility of 
delaying investment in the investment decision model. 
Through the above literature, it is found that with the 
complexity of the investment environment, domestic and 
foreign scholars no longer only study from a single economic 
perspective, but from the perspective of national policies, 
social preferences and green mechanisms, research on wind 
power project investment is gradually increasing, and many 
scholars also consider uncertain factors in the research of 
investment decision. 

2. Theoretical Knowledge  

2.1. Triangular fuzzy number 
Definition 1. [9] if 𝑎 𝑎 , 𝑎 , 𝑎  , where 0 𝑎

𝑎 𝑎 1,Angular fuzzy number𝑎, whose characteristic 
function (membership function) can be expressed as 

 



 

334 

   𝜇

⎩
⎨

⎧ ,        𝑎 𝑥 𝑎 ,

,        𝑎 𝑥 𝑎 ,

0,                            𝑜𝑡ℎ𝑒𝑟𝑠.

                                1   

                        
Where 𝑎  and 𝑎  are the lower and upper bounds of  𝑎, 

respectively, and 𝑎  is the median of 𝑎.           
Definition 2. [10] triangle fuzzy number 𝑎

𝑎 , 𝑎 , 𝑎 and 𝑏 𝑏 , 𝑏 , 𝑏  the distance between the d 
(a, b) can be defined as follows: 

 

𝑑 𝑎, 𝑏
1

3
𝑎 𝑏 2 𝑎 𝑏 2 𝑎 𝑏 2  2  

 
Where 0≤λ≤1, the risk situation pursued by the decision 

maker determines the choice of λ value. When λ>0.5, the 
decision maker is taking a risk. The decision maker does not 
have a great position when λ=0.5; A λ<0.5 indicates that the 
decision maker is cautious. In general λ=0.5. 

2.2. Multi-attribute decision method TODIM 
TODIM (Multi-objective decision making) is a multi-

objective decision making method proposed by Brazilian 
scholars J. Figueira, S. Greco and M. Ehrgott in 2005. 
TODIM method is a combination of judgment matrix and 
ranking technology, aiming to solve the multi-objective 
decision-making problem, and by introducing cognitive 
preferences and descriptive information into the decision-
making process, the decision is more in line with the 
subjective cognition of decision makers. On the basis of the 
prospect theory,TODIM method is based on pairwise 
comparison of schemes and calculates the dominance of each 
scheme under another scheme under each attribute to sort. 
The specific decision-making steps are as follows: 

Let 𝐴 𝐴 , … , 𝐴  is the set of m alternatives, 𝐶
𝐶 , … , 𝐶   is the set of n attributes, 𝑥  is the evaluation 

value of scheme 𝐴 under attribute𝐶 , 𝑊 𝑤 , … , 𝑤 is the 
attribute weight vector with 0 𝑤 1and∑ 𝑤 1. 

Step 1 to build decision matrix, and the decision matrix are 
normalized processing; 

𝑋

𝑥
𝑥

…
…

𝑥  
𝑥

⋮ ⋱ ⋮
𝑥 ⋯ 𝑥

 

Step 2 of relative attributes𝐶  attribute𝐶  and standardized 

weights, 𝑤  , among them, the 𝑤   is the reference 

weight, 𝑤 max 𝑤 , 𝑗, 𝑟 1,2, … , 𝑛; 
Step 3 to calculate the properties under𝐶  , plan 𝐴   and 

𝐴   of relative superiority degree ∅ 𝐴 , 𝐴   , calculated as 
follows: 

∅ 𝐴 , 𝐴   

⎩
⎪
⎨

⎪
⎧ 𝑤 𝑥 𝑥

∑ 𝑤
,            𝑥 𝑥     

1
𝜃

𝑤 𝑥 𝑥
∑ 𝑤

, 𝑥 𝑥

 

Among them, the∅ 𝐴 , 𝐴  properties under C solution𝐴  
relative to the scheme 𝐴  of dominance, 𝑗 1,2, … , 𝑛; 𝑖, 𝑘
1,2, … , 𝑚 ; 𝑤  is the weight of the reference attribute, 𝑙
1,2, … , 𝑛; 𝜃 is the loss decay coefficient, 𝜃 0; 

Step 4 calculating global dominance factor 𝜉 𝐴
∑ ∑ ∅ 𝐴 , 𝐴  ; 

Step 5 Sort based on the global preponsibility 𝜉 𝐴 . The 
greater the𝜉 𝐴  value, the better the 𝐴 solution is. 

3. Model Construction 

3.1. Multi-attribute decision model of TODIM 
method 

The basic principle of TODIM involves goal setting, 
scheme comparison and consideration of uncertainty factors. 
The introduction of dynamic reference points makes the 
model more close to the psychological and risk attitude of 
actual decision makers. The concept of prospect theory is 
considered to have good applicability in project investment 
because it can capture the decision maker's dynamic 
assessment of benefits and risks at different points in time. 
However, multi-attribute decision making still faces a series 
of challenges in project investment. These challenges include 
the complexity of the interrelationships among attributes, the 
existence of information uncertainty, and individual 
differences in decision makers' subjective preferences. 
TODIM method can effectively overcome these challenges 
and improve the accuracy and reliability of investment 
decisions. 

In this section, after improving TODIM, a TODIM multi-
attribute group decision-making model based on the dynamic 
reference point of prospect theory will be established and 
applied to project investment decision-making. The specific 
steps are as follows: 

Assume a dynamic multi-stage 𝑇 𝜎 1,2, … , 𝜍  under 
multi-attribute decision problem, where 𝐴 𝐴 , … , 𝐴  is 
the set of m alternatives 𝐶 𝐶 , … , 𝐶  is a set of n 
attributes, 𝐸 𝐸 , … , 𝐸 is a group of decision experts. At 
stage T_σ, decision expert 𝐸 𝑘 1,2, … , 𝑡  for scheme 
𝐴 𝑖 1,2, … , 𝑚  in the attribute 𝑐 𝑗 1,2, … , 𝑛  .The 
evaluation information under,n) can be represented by the 
decision matrix 𝑋 𝑥

∗
 as follows: 

𝑋 

⎝

⎛

𝑥
𝑥

⋯
⋯

𝑥
𝑥

⋮ ⋱ ⋮
𝑥 ⋯ 𝑥 ⎠

⎞ 

Step 1 Under stage 𝑇  , the expert's weight vector is 
𝜔 𝜔 , 𝜔 , … , 𝜔 , , with the number of uncertain 
fuzzy aggregation operator to aggregate each expert decision 
matrix, aggregate after each stage of matrix 𝑋 

∗

𝜔 𝑋 ⨁𝜔 𝑋 ⨁ … ⨁𝜔 𝑋 ; 
Step 2 calculate attribute 𝐶   and relative attributes 𝐶  

standardized weights 𝑤  , among them, the𝑤  is the 

reference weight, 𝑤 max 𝑤 , 𝑗, 𝑟 1,2, … , 𝑛; 
Step 3 to calculate the properties𝐶 , scheme𝐴 and scheme 

𝐴  
 the relative dominance ∅ 𝐴 , 𝐴  , calculated as follows: 

∅ 𝐴 , 𝐴   

⎩
⎪
⎨

⎪
⎧ 𝑤 𝑥 𝑥

∑ 𝑤
,          𝑥 𝑥     

1
𝜃

𝑤 𝑥 𝑥

∑ 𝑤
, 𝑥 𝑥

 

 



 

335 

Among them, the ∅ 𝐴 , 𝐴    under the stage 𝑇  
properties under C solution 𝐴  relative to the scheme𝐴   of 
dominance, 𝑗 1,2, … , 𝑛; 𝑖, 𝑘 1,2, … , 𝑚 ; 𝑤   is the 
weight of the reference attribute, 𝑙 1,2, … , 𝑛; 𝜃 is the loss 
decay coefficient,θ>0;  

Step4 Construct comprehensive dominance degree  
𝛿 𝐴 ∑ ∑ ∅ 𝐴 , 𝐴  , 𝑚𝑎𝑥∅ 𝐴 , 𝐴  

𝑚𝑖𝑛∅ 𝐴 , 𝐴  ; Combined with TOPSIS thought, to find 
the ideal solution to the and non-ideal solution of the𝛿
𝑚𝑖𝑛 ∑ ∑ ∅ 𝐴 , 𝐴  , max 𝑚𝑎𝑥∅ 𝐴 , 𝐴  

𝑚𝑖𝑛∅ 𝐴 , 𝐴   ; Ideal solution to different schemes are 
calculated distance 𝐷 𝛿 , 𝛿  , distance non-ideal 
solution 𝐷 𝛿 , 𝛿 ; 

Step 5 to calculate comprehensive score of each alternative, 

𝐶 ; 

Step 6 In the stage 𝑇 , the optimal expected level of each 
scheme under the same attribute can be obtained from 𝐶 . 
The dynamic expectation level programming model is 
expressed as: 

𝐶 𝑚𝑎𝑥𝐶  

𝑠. 𝑡.

⎩
⎪
⎨

⎪
⎧

𝐶 𝐶 ∗

0 𝜔 1,

 𝜔 1,
 

Step 7 Under stage 𝑇 , the value function of the solution is 
defined as: 

𝑣 ∆𝐶   
∆𝐶 ,              ∆𝐶 0    
𝜃 ∆𝐶 , ∆𝐶 0

 

As ∆Cσ Cσ C∗σ(C∗σ as a reference point), theta risk 
preference, 0 α、β 1. 

Step 8 Sort scheme 𝑣 ∆𝐶  . The larger the value, the 
better. 

Step 9 In phase 𝑇  , 𝐶 ∗  can obtain the optimal 
expected level of each scheme with the same attribute. The 
dynamic expectation level programming model is expressed 
as: 

𝐶 𝑚𝑎𝑥𝐶  

𝑠. 𝑡.

⎩
⎪
⎨

⎪
⎧

𝐶 𝐶 ∗ 𝐶 ∗

0 𝜔 1,

 𝜔 1,
 

To solve the programming model, if there is a feasible 
solution, then directly use the solution to proceed to the next 
step; If there is no feasible solution, the optimal attribute 
weight and optimal expectation level of the previous stage are 
directly used to carry out the next step. Repeat the previous 
steps until the final stage. 

4. Case Analysis 

4.1. Instance Background 
Project investment decision is a key part of the successful 

development of an enterprise, which is directly related to the 
company's future profitability and market competitiveness. In 
the current context of fierce competition and ever-changing 
market environment, precise and prudent investment 
decisions are particularly important. First, an in-depth 
analysis of the project context, including market demand, 
competitive landscape, technology trends and other factors, 
helps to fully understand the potential risks and opportunities 
of the project. In this process, information gathering, risk 
management and market forecasting capabilities are 
particularly critical. In summary, scientific and reasonable 
project investment decisions can not only effectively promote 
the innovation and development of enterprises, but also give 
enterprises more powerful competitiveness in the complex 
and changeable market. 

At present, a venture capital company intends to make 
investment decisions on 4 alternative enterprises (schemes) 
𝐴 𝑖 1,2,3,4  . Through the analysis of enterprise 
development, three evaluation attributes are selected, namely, 
economic benefit 𝑐  , social benefit 𝑐   and environmental 
pollution degree 𝑐 . The decision makers now evaluate the 
attributes of each enterprise in three different segments (the 
evaluation results are shown in Table 4.1) in order to 
determine the best investment plan. At the same time the 
decision-making time under the weight of each attribute 
vector respectively 𝑤 0.4,0.4,0.2 , 𝑤
0.4,0.35,0.25 , 𝑤 0.4,0.3,0.3 . 

 
Table 4.1 Triangular fuzzy number evaluation information in each stage 

𝑇  𝐴  𝑐  𝑐  𝑐  
 𝐴  [0.60,0.70,0.80] [0.80,0.85,0.90] [0.35,0.40,0.50] 

𝑇  𝐴  [0.85,0.90,0.95] [0.70,0.75,0.80] [0.40,0.45,0.50] 
 𝐴  [0.75,0.80,0.90] [0.80,0.85,0.95] [0.30,0.40,0.45] 
 𝐴  [0.80,0.85,0.90] [0.70,0.75,0.85] [0.45,0.50,0.65] 
 𝐴  [0.80,0.85,0.90] [0.85,0.90,0.95] [0.30,0.35,0.40] 

𝑇  𝐴  [0.85,0.90,0.95] [0.75,0.80,0.85] [0.35,0.40,0.45] 
 𝐴  [0.75,0.80,0.85] [0.65,0.70,0.80] [0.30,0.40,0.45] 
 𝐴  [0.85,0.90,0.95] [0.80,0.90,0.95] [0.55,0.60,0.65] 
 𝐴  [0.85,0.90,0.95] [0.80,0.85,0.95] [0.30,0.35,0.40] 

𝑇  𝐴  [0.80,0.85,0.90] [0.70,0.75,1.00] [0.25,0.30,0.40] 
 𝐴  [0.85,0.90,0.95] [0.80,0.85,0.85] [0.40,0.45,0.50] 
 𝐴  [0.80,0.85,0.95] [0.85,0.90,0.95] [0.35,0.40,0.50] 

 
The manuscript should include a conclusion. In this section, 

summarize what was described in your paper. Future 
directions may also be included in this section. Authors are 
strongly encouraged not to reference multiple figures or tables 



 

336 

in the conclusion; these should be referenced in the body of 
the paper. 

4.2. Calculation process 
The specific arithmetic steps of TODIM method based on 

fuzzy information of triangular fuzzy numbers in this section 

are as follows: 

The evaluation information of the above three stages is 
integrated and processed to obtain the decision matrix, as 
shown in the following table: 

 

Table 4.2 Decision matrix of each stage 

𝑇  𝐴  𝑐  𝑐  𝑐  
 𝐴  [0.60,0.70,0.80] [0.80,0.85,0.90] [0.35,0.40,0.50] 

𝑇  𝐴  [0.85,0.90,0.95] [0.70,0.75,0.80] [0.40,0.45,0.50] 
 𝐴  [0.75,0.80,0.90] [0.80,0.85,0.95] [0.30,0.40,0.45] 
 𝐴  [0.80,0.85,0.90] [0.70,0.75,0.85] [0.45,0.50,0.65] 
 𝐴  [0.80,0.85,0.90] [0.85,0.90,0.95] [0.30,0.35,0.40] 

𝑇  𝐴  [0.85,0.90,0.95] [0.75,0.80,0.85] [0.35,0.40,0.45] 
 𝐴  [0.75,0.80,0.85] [0.65,0.70,0.80] [0.30,0.40,0.45] 
 𝐴  [0.85,0.90,0.95] [0.80,0.90,0.95] [0.55,0.60,0.65] 
 𝐴  [0.85,0.90,0.95] [0.80,0.85,0.95] [0.30,0.35,0.40] 

𝑇  𝐴  [0.80,0.85,0.90] [0.70,0.75,1.00] [0.25,0.30,0.40] 
 𝐴  [0.85,0.90,0.95] [0.80,0.85,0.85] [0.40,0.45,0.50] 
 𝐴  [0.80,0.85,0.95] [0.85,0.90,0.95] [0.35,0.40,0.50] 

 

The results obtained by calculating each step are as follows 

 

Table 4.3 Decision-making results of each stage 

𝑇  𝐴  
Expected 

value 
reference 

poin 
value 

function 
 𝐴  0.127 0.065 0.086 

𝑇  𝐴  1.268 0.934 0.381 
 𝐴  1.176 0.597 0.618 
 𝐴  1.000 0.344 0.689 
 𝐴  -0.504 0.876 -2.989 

𝑇  𝐴  0 0.123 -0.356 
 𝐴  0.140 0.099 0.060 
 𝐴  1 1 0 
 𝐴  0.127 1 -1.99 

𝑇  𝐴  1.268 0 1.232 
 𝐴  1.176 0.009 1.146 
 𝐴  1.000 1 0 

 
From the analysis of the final results, it can be seen that in 

the 𝑇   stage, the scheme order is 𝐴 𝐴 𝐴 𝐴  ; In 
the 𝑇  stage, the sequence is 𝐴 𝐴 𝐴 𝐴 . In the 𝑇  
phase, the scheme is ordered as 𝐴 𝐴 𝐴 𝐴  . If an 
optimal project investment needs to be selected in each stage, 
the corresponding optimal project in these three stages is: 
𝐴 → 𝐴 → 𝐴 . 

5. Research Conclusions 
In today's rapidly developing economic environment, 

project investment decision is an important link in the 
development of enterprises. Therefore, when making project 
investment decisions, it is necessary to conduct adequate 
market research and risk assessment to ensure the 
effectiveness and sustainability of the investment. Multi-
attribute decision making is an important part of modern 
decision science, system engineering and management 
science, and its theory and method are widely used in many 
fields such as economy, management, engineering, military 
and social life. However, both theoretical research and 
method application, especially in fuzzy multi-attribute 

decision theory and method research, are still not perfect, still 
facing new challenges, need further research. This paper 
mainly focuses on fuzzy information multi-attribute (group) 
decision-making problem. The main content of this paper 
considers the risk aversion and preference attitude of decision 
makers in the multi-stage decision-making process, 
establishes a multi-stage decision-making analysis 
framework based on prospect theory, and proposes a dynamic 
reference point setting method based on stage development 
characteristics. A multi-attribute decision making method 
based on improved TODIM method is proposed. The 
practicability and feasibility of the proposed method are 
demonstrated by example analysis and method comparison. 
To sum up, the investment decision model based on prospect 
theory under uncertain information proposed in this paper 
provides a new decision method for decision makers and has 
important significance in solving practical problems. 

Acknowledgment  
The author would like to take this opportunity to express 

his heartfelt thanks and gratitude to the master and reviewers 
for their valuable suggestions and suggestions for the 
improvement and perfection of this paper. 

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