## Appendix A  
The list of notations to represent the production quantities of manufacturer and suppliers are represented as follows. Indices  
\(i\): supplier index, \(i\) = 1, 2,... \(I\)  
\(j\): item index, \(j\) = 1, 2,... \(J\)  
\(t\): cycle time index, \(t\) = 1, 2,... \(T\)  
decision variables  
\(CQjt\): units of _j_th item crashed in _t_th time cycle (units)  
\(MQ_{jt}\): quantity of _j_th item manufactured in _t_th time cycle (units)  
\(QQijt\): number of _j_th item outsourced to _i_th supplier in _t_th time cycle (units)  
Parameters  
\(AS_{jtt}\): minimum available _j_th item from _i_th supplier in _t_th time cycle (units)  
\(BS_{jtt}\): maximum available _j_th item from _i_th supplier in _t_th time cycle (units)  
\(CC_{jt}\): cost of crashing _j_th item in time cycle _t_th ($/unit)  
\(CE\): carbon emission produced from production system (kg)  
\(D_{jt}\): demand for _j_th item in _t_th time cycle (units) \begin{tabular}{l l} \(DO_{ij}\) & diesel consumed to manufacture \(j\)th item from \(i\)th supplier (L/unit) \\ \(DP_{j}\) & consumption of diesel to manufacture \(j\)th item (L/unit) \\ \(DR_{j}\) & usage of diesel for reworking \(j\)th item (L/unit) \\ \(EO_{ij}\) & required electricity to manufacture \(j\)th item from \(i\)th supplier (kWh/unit) \\ \(EP_{j}\) & usage of electricity to manufacture \(j\)th item (kWh/unit) \\ \(ER_{j}\) & electricity consumed to rework \(j\)th item (kWh/unit) \\ \(FC_{jt}\) & fixed cost for production of \(j\)th item in \(t\)th time cycle (\$) \\ \(GO_{ij}\) & gasoline consumed to manufacture \(j\)th item from \(i\)th supplier (m\({}^{3}\)/unit) \\ \(GP_{j}\) & gasoline for manufacturing of \(j\)th item (m\({}^{3}\)/unit) \\ \(RG_{j}\) & required gasoline to rework \(j\)th item (m\({}^{3}\)/unit) \\ \(HC_{jt}\) & holding cost for \(j\)th item in \(t\)th time cycle (\$/unit) \\ \(mC_{jt}\) & cost of maintenance for \(j\)th item in \(t\)th time cycle (\$) \\ \(MC_{jt}\) & manufacturing cost of \(j\)th item in \(t\)th time cycle (\$/unit) \\ \(ML_{jt}\) & minimum number of \(j\)th items in \(t\)th time cycle (units) \\ \(MQL_{jt}\) & cost of MQL for \(j\)th item in \(t\)th time cycle (\$/unit) \\ \(MU_{jt}\) & maximum number of \(j\)th items in \(t\)th time cycle (units) \\ \(OC_{ijt}\) & outsourcing cost for \(j\)th item in \(t\)th time cycle to \(i\)th supplier (\$/unit) \\ \(S_{j}\) & per unit scrap cost (\$/unit) \\ \(SO_{ij}\) & solar energy consumed to manufacture \(j\)th item from \(i\)th supplier (kWh/unit) \\ \(SP_{i}\) & solar energy used to manufacture \(j\)th item (kWh/unit) \\ \(SR_{j}\) & utilization of solar energy for reworking \(j\)th item (kWh/unit) \\ \(TC\) & total cost of production (\$/unit) \\ \(TEC\) & total cost of energy utilized (\$/unit) \\ \(VC_{ijt}\) & cost of reworking for \(i\)th supplier and \(j\)th item in \(t\)th time cycle (\$/unit) \\ \(WC_{jt}\) & warehousing cost (\$/unit) \\ \(\alpha_{1}\) & electricity coefficient equivalent to carbon emission \\ \(\alpha_{2}\) & diesel fuel coefficient equivalent to carbon emission \\ \(\alpha_{3}\) & gasoline coefficient equivalent to carbon emission \\ \(\alpha_{4}\) & solar energy coefficient equivalent to carbon emission \\ \(\alpha_{ijt}\) & proportion of scrap produced in defective \(j\)th item, manufactured by \(i\)th supplier in \(t\)th time cycle (\%) \\ \(\beta_{jit}\) & defective rate for \(j\)th item, manufactured by \(i\)th supplier in \(t\)th time cycle, percentage (\%) \\ \(\gamma_{1}\) & unit cost of electricity (\$/kWh) \\ \(\gamma_{2}\) & per unit cost of diesel oil (\$/kWh) \\ \(\gamma_{3}\) & cost of gasoline per unit (\$/m\({}^{3}\)) \\ \(\gamma_{4}\) & unit cost of solar energy (\$/kWh) \\ \(\gamma_{5}\) & carbon emission cost (\$/kg) \\ \(NT_{jt}\) & additional cost limit \(j\)th item in \(t\)th time cycle (units) \\ \end{tabular}   
**deviational variables**