###### Abstract  
To successfully finalize projects and attain their determined purposes, it is indispensable to control all success criteria of a project. The time-cost trade-off (TCT) is known as a prevalent and efficient approach applied when the planned finish date of a project is not admitted by stakeholders, and consequently, the project duration must be decreased. This paper proposes a new mathematical model under fuzzy uncertainty to deal with the project cost-risk-quality trade-off problem (CRQT) under time constraints. Because of the unique nature of projects and their uncertain circumstances, applying crisp values for some project parameters does not seem appropriate. Hence, this paper employs fuzzy sets to resolve these weaknesses. In this study, two approaches are presented to handle proposed fuzzy multi-objective mathematical model. First, fuzzy credibility theory and then goal attainment method are used. Secondly, the model is solved by a fuzzy method based on expected interval and value and augmented \(\varepsilon\)-constraint method. A project from the literature review is adopted and solved by the presented methodology. The results demonstrate the accuracy and efficiency of the two proposed approaches for the introduced practical problem. timec-cost trade-off (TCT)fuzzy credibility theory; multi-objective optimization; augmented \(\varepsilon\)-constraint method; goal attainment method +
Footnote †: journal:  
==WARNING: Truncated because of repetitions==
}Pr_{jt}\cdot\sum_{r=1}^{m} I_{jt}\cdot u_{jt}\] (27)  
\[\text{min}z_{3}=\sum_{j=1}^{n}We_{j}\sum_{t=1}^{M\!R_{j}}Q_{t}\cdot u_{jt}\] (28)  
\[f_{L}^{l}=ft\] (29)  
Subject to:  
\[\sum_{t=1}^{M\!R_{j}}u_{jt}\leq 1\quad\forall j\] (30)  
\[f_{j}^{l}-s_{j}^{E}\geq(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})-\sum_{t=1}^{M\!R _{j}}tu_{jt}),\forall j\] (31)  
\[s_{j}^{E}-s_{i}^{E}\geq(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})-\sum_{t=1}^{M\!R _{j}}tu_{it})+FS_{ij}^{\text{min}}\quad\forall i,j\in FS\text{ relationships}\] (32)  
\[f_{k}^{l}-f_{j}^{l}\geq(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})-\sum_{t=1}^{M\!R _{k}}tu_{kt})+FS_{jk}^{\text{min}}\quad\forall j,k\in FS\text{ relationships}\] (33)  
\[s_{j}^{E}-s_{i}^{E}\geq SS_{ij}^{\text{min}}\quad\forall i,j\in SS\text{ relationships}\] (34)
+++  
\[ \begin{split}& f_{k}^{l}-f_{j}^{l}\geq(((2-2\alpha)d_{3}+(2\alpha-1)d_ {4})-\sum\limits_{t=1}^{MR_{k}}tu_{kt})-(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})-\\ &\sum\limits_{t=1}^{MR_{j}}tu_{it})+SS_{jk}^{\min}\forall j,k\in SS \text{ relationships}\\ & s_{j}^{E}-s_{i}^{E}\geq(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})- \sum\limits_{t=1}^{MR_{i}}tu_{it})-(((2-2\alpha)d_{3}+(2\alpha-1)d_{4})-\\ &\sum\limits_{t=1}^{MR_{j}}tu_{it})+FF_{ij}^{\min}\forall i,j\in FF \text{ relationships}\\ & f_{k}^{l}-f_{j}^{l}\geq F_{jk}^{\min}\quad\forall j,k\in FF \text{ relationships}\\ & s_{j}^{E}-s_{i}^{E}\geq SF_{ij}^{\min}-(((2-2\alpha)d_{3}+(2 \alpha-1)d_{4})-\sum\limits_{t=1}^{MR_{j}}tu_{it})\forall i,j\in SF\text{ relationships}\\ & f_{k}^{l}-f_{j}^{l}\geq SF_{jk}^{\min}-(((2-2\alpha)d_{3}+(2 \alpha-1)d_{4})-\sum\limits_{t=1}^{MR_{j}}tu_{it})\forall j,k\in SF\text{ relationships}\\ & s_{j}^{E},f_{j}^{l}\geq 0\quad\forall j\\ & u_{it}\in\{0,1\}\quad\forall j,t\end{split}  \tag{35}\]  
The goal attainment method is considered a very efficient method for dealing with optimization problems with multiple objective functions like the model presented in this paper. \(Wo_{i}\) and \(B_{i}\) are the inputs of the goal attainment method, and they represent the weight and goal values determined for each goal, respectively.