## 3 Design of the Fitness Function  
The metaheuristic algorithms to be developed based on \"Two heads are better than one\" follow the paradigm of evolutionary algorithms. Therefore, a fitness function that plays the role of an indicator to provide information on the solution quality in the search processes is required. The fitness function must take into account the objective function as well as the constraints to characterize the quality of a solution quantitatively. In the literature, a lot of approaches based on penalty methods have been proposed to improve the solution quality and reduce the violation of constraints. The fitness function used in this study is biased against infeasible solutions [39]. The fitness function plays a pivotal role in guiding the evolutionary processes through considering the factors that are able to improve the value of the objective function and feasibility of the solution found. Our approach to the design of the fitness function takes into account these factors by adopting the proven technology from [39] that works for a wide variety of optimization problems to effectively move toward the feasible region through biasing feasible solutions against infeasible ones by penalizing the violation of constraints and replacing the current solution with a better solution. The effectiveness of this design approach has been verified in our previous works, e.g., [5,10,31,32]. To describe the fitness function, let \(S_{f}\) be the set of feasible solutions in the current population. Then the objective function value of the worst feasible solution in \(S_{f}\) is \(S_{f\text{min}}=\min\limits_{(x,y)\in S_{f}}F(x,y)\). To penalize violation of Constraints (2) and (3), the terms \(\left|\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}q_{djk}^{1}-y_{p}s_{ pk}^{1}\right|\) and \(\left|\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}q_{djk}^{2}-y_{p}s_{ pk}^{2}\right|\) are introduced, respectively. To penalize violation of Constraint (4), the term \(\min(\sum\limits_{p=1}^{p}y_{p}f_{p}-\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_ {d}}x_{dj}(c_{dj}-o_{dj}),0.0)\) is introduced. To penalize violation of Constraint (5), the term \(\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}\min(1-\sum\limits_{j=1}^{J_{d} }x_{dj},0.0)\) is introduced. To penalize violation of Constraints (6) and (7), the terms \(\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{J_{d}}x_{dj}\text{min}\)\(((\frac{F_{dj}(x,y)}{\sum\limits_{p\in\Gamma_{dj}}y_{p}cf_{pdj}+x_{dj}c_{dj}})-r_{P},0.0)\) and \(\sum\limits_{p=1}^{p}y_{p}\text{min}((\frac{F_{dj}(x,y)}{\sum\limits_{p\in \Gamma_{dj}}y_{p}cf_{pdj}+x_{dj}c_{dj}})-r_{P},0.0)\) are introduced, respectively. The fitness function \(F_{1}(x,y)\) used in this paper is defined in (9):  
\[F_{1}(x,y)=\left\{ \begin{array}{l}F(x,y)\ if\ (x,y)\ is\ feasible\\ U(x,y)\ otherwise\end{array} \right., \tag{9}\]  
where \(U(x,y)\) is defined in (10). \[ \begin{array}{l}U(x,y)=S_{f\min}-\sum\limits_{p=1}^{P}\sum\limits_{k=1}^{K} (\left|\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{d}}x_{dj}q_{djk}^{1}-y_{p}s_{ pk}^{1}\right|+\left|\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{d}}x_{dj}q_{djk}^{2}-y_{p}s_{ pk}^{2}\right|)\\ +\min(\sum\limits_{p=1}^{P}y_{p}f_{p}-\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{ d}}x_{dj}(c_{dj}-o_{dj}),0.0)\\ +\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{d}}\min(1-\sum\limits_{j=1}^{I_{ d}}x_{dj},0.0)\\ +\sum\limits_{d=1}^{D}\sum\limits_{j=1}^{I_{d}}x_{dj}\min(((\frac{F_{dj}(x,y)}{ \sum\limits_{p\in\Gamma_{dj}}y_{p}cf_{pdj}+x_{dj}c_{dj}})-r_{D},0.0)\\ +\sum\limits_{p=1}^{P}y_{p}\min((\frac{F_{dj}(x,y)}{\sum\limits_{p\in\Gamma_{ dj}}y_{p}cf_{pdj}+x_{dj}c_{dj}})-r_{P},0.0)\end{array}  \tag{10}\]  
The original DE approach was proposed for problems with continuous solution space. As the decision variables of the DGRP are discrete, the solution space is discrete. The approach adopted in this study is to use a function to convert a solution in continuous solution space to a discrete solution systematically.