## 3 Resolution Approaches
### Exact Approach
\[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\] \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\] (3) \[C_{i^{\prime}j^{\prime}}+trtr_{r}\leqslant C_{ij}-t_{ij}+M(4-Y_{ ij}-Y_{i^{\prime}j^{\prime}}-G_{j\eta}-G_{j^{\prime}\eta})+M\text{U}_{i\vec{i}r \eta}\] \[\qquad\forall i,i^{\prime}\in 1\dots m,i<i^{\prime},\forall j^{ \prime}\in 1\dots n,j<j^{\prime},\forall r\in 1\dots k,\forall q\in 1\dots l_{r}\] (4) \[C_{ij}+trtr_{r}\leqslant C_{i^{\prime}j^{\prime}}-t_{i^{\prime} j^{\prime}}+M(4-Y_{ij}-Y_{i^{\prime}j^{\prime}}-G_{j\eta}-G_{i^{\prime}\eta})+M(1- U_{i^{\prime}r\eta})\] \[\qquad\forall i,i^{\prime}\in 1\dots m,i<i^{\prime},\forall j,i^{ \prime}\in 1\dots n,j<j^{\prime},\forall r\in 1\dots k,\forall q\in 1\dots l_{r}\] (5) \[STR_{\eta}\leqslant\sum_{i=1}^{m}C_{ij}-\sum_{i=1}^{m}t_{ij}Y_{ ij}+M(1-G_{j\eta})\] \[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r}\] (6) \[FTR_{\eta}\geqslant\sum_{i=1}^{m}C_{ij}-M(1-G_{j\eta})\qquad \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r}\] (7) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]0 \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]1 (8) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]2 \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]3 (9) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]4 (10) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]5 (11) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]6 (12) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]7 (13) \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]8 \[Y_{ij}\leqslant Z_{i\eta}+(1-G_{j\eta})\quad\forall i\in 1\dots m, \forall j\in 1\dots n,\forall r\in 1\dots k,\forall q\in 1\dots l_{r} \tag{2}\]9 (14) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]0 \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]1 (15) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]2 (16) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]3 (17) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]4 (18) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]5 (19) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]6 (20) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]7 (21) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]8 (22) \[NTR_{\eta}\geqslant\sum_{i=1}^{n}Z_{i\eta}\ \text{-}\ 1 \forall r\in 1\dots k,\forall q\in 1\dots l_{r}\]9 (23) \[C_{i^{\prime}j^{\prime}}+trtr_{r}\leqslant C_{ij}-t_{ij}+M(4-Y_{ ij}-Y_{i^{\prime}j^{\prime}}-G_{j\eta}-G_{j^{\prime}\eta})+M\text{U}_{i\vec{i}r \eta}\]0 (24) \[C_{i^{\prime}j^{\prime}}+trtr_{r}\leqslant C_{ij}-t_{ij}+M(4-Y_{ ij}-Y_{i^{\prime}j^{\prime}}-G_{j\eta}-G_{j^{\prime}\eta})+M\text{U}_{i\vec{i}r \eta}\]1The objective function (1) is set to maximize the net profit, which consists of the total revenue minus the total costs, including job fixed costs, tardiness costs, job rejection costs, renting cost of machines, renting cost of resources, and transportation cost of jobs and resources.