## 2 Materials and Methods
### Multi-Objective Optimization
#### 2.2.2 Binary Integer Programming Model and Definition of Parameters  
In this section, we present the suggested multi-objective optimization model, detailing its parameters, objective functions, constraints, and decision variables along with their respective definitions. * Parameters; \(P_{e}\): Production target for excavator e. This represents the desired amount of material (e.g., tons of ore or soil) that excavator e should process or load onto trucks within a given time period. It is a goal the model aims to meet as closely as possible. \(RT_{e}\): Ready time for excavator e at a given timestamp. This is the time at which excavator e is available to start working (e.g., after maintenance or previous tasks). It affects scheduling and assignment decisions. \(TT_{de}\): Travel time from dump d to excavator e. This measures the time it takes for a truck to travel from a specific dump site d to excavator e. It is critical for calculating total time in the system and ensuring efficient assignments. * Decision Variables: \(x_{tde}\in\ \{0,\ 1\}\): Binary decision variable \(x_{tde}=\left\{ \begin{array}{cl}1\,\ if\ truck\ t\ on\ dump\ d\ is\ assigned\ to\ excavator\ e\\ 0,\ \ otherwise.\end{array} \right.\) This variable indicates whether truck t at dump d is assigned to excavator e (1 = yes, 0 = no). It is binary because each truck can only be assigned or not assigned, with no partial assignments allowed. \(\Delta_{e}\): Deviation from the production target for excavator e. This variable captures how much the actual production (based on truck assignments) deviates from the target \(P_{e}\) for excavator e. It is split into positive \(\Delta_{e}^{+}\) and negative \(\Delta_{e}^{-}\) components in the model to measure over- or under-achievement. * Objective Functions: Minimization of total deviation from production targets. This objective aims to ensure that the production achieved by each excavator is as close as possible to its target \(P_{e}\). The positive deviation (\(\Delta_{e}^{+}\), exceeding the target) and negative deviation (\(\Delta_{e}^{-}\), falling short) are summed across all excavators e. \[Minimize:\ Z_{1}=\sum_{e\in\ E}\Delta_{e}^{+}+\Delta_{e}^{-}\] Minimization of the total ready time of excavators. This objective seeks to reduce the combined ready time and travel time associated with truck assignments, making the operation more efficient. \[Minimize:\ Z_{2}=\sum_{e\in E}\sum_{t\in T}\sum_{d\in D}x_{tde}\cdot(RT_{e}+TT _{de})\]
* Constraints: Each truck can only be assigned to one excavator. This ensures that a truck t is assigned to exactly one combination of dump d and excavator e, preventing double-booking or unassigned trucks. \[\sum_{e\in E}\sum_{d\in D}x_{tde}=1\hskip 56.905512 pt\forall t\in T\] Each dump site can have one truck requiring an excavator assignment at the same time. This limits each dump site d to having at most one truck assigned to it at any given time, reflecting capacity or operational constraints at the dump. \[\sum_{e\in E}\sum_{t\in T}x_{tde}\leq 1\hskip 56.905512 pt\forall d\in D\] Decision variables of the trucks on the road are set to zero at the given time. This constraint forces \(x_{tde}\) to be 0 for trucks that are currently on the road (unavailable), ensuring that only available trucks are assigned. \[x_{tde}=0\hskip 56.905512 pt\forall t\in T.\forall d\in D,\forall e\in E\]  
Deviation constraints: These define the positive and negative deviations from the production target for each excavator. The positive deviation \(\Delta_{e}^{+}\) measures how much production exceeds \(P_{e}\), while the negative deviation \(\Delta_{e}^{-}\) measures how much it falls short. These are calculated based on the number of trucks assigned.