## 4 Model Selection and Development
### Model Development  
In the study, a DLP model was used to examine a case study, which needs to reduce operational costs related to fuel consumption, maintenance and tax expenses by optimizing the number of service vehicles required for employee transportation. Also, the examined shuttle service consists of vehicles with different capacities (17 + 1 and 27 + 1), and DLP helps allocate the right number of each type of vehicle to meet demand while minimizing waste. On the other hand, the study implements dynamic linear programming (DLP) to adjust to changing workforce transportation needs, optimizing the routes and fleet size weekly. Lastly, the study proposes fleet scheduling under variable demand, and DLP is the most suitable method to ensure that the shuttle fleet is used efficiently, considering variations in shift patterns, overtime and demand fluctuations. Linear programming is a mathematical modeling technique that helps achieve the optimum outcome through efficient use of limited resources. The method has three basic components: objective function, constraint function and decision variables [53]. The objective function of a general linear programming structure is given in Equations (1)-(6), and the constraint function is given in Equations (7)-(10) [54]. \begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{**Criteria**} & \multicolumn{2}{c}{**Method**} \\  & \
\begin{tabular}{c} **Dynamic Linear** \\ **Programming** \\ \end{tabular} &
\begin{tabular}{c} **Optimization Methods** \\ \end{tabular} \\ \hline Optimality & Excellent & Approximate \\ \hline Sequential decision-making suitability & Excellent & Moderate \\ \hline Handling linear/structured constraints & Excellent & Limited \\ \hline Robustness in fleet allocation problems & High & Moderate \\ \hline Computational predictability & High & Medium/low \\ \hline Ease of parameter tuning & Simple & Moderate/complex \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The performance comparison of DLP and other optimization methods [65; 66; 67; 68]. \[Z_{max/min}=\sum_{j=1}^{n}c_{j}x_{j}=c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+\ldots \ldots\ldots\ldots+c_{n}x_{n} \tag{1}\]  
\[a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}+\ldots\ldots\ldots+a_{1n}x_{n}\leq,=;\geq b_ {1} \tag{2}\]  
\[a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}+\ldots\ldots\ldots+a_{1n}x_{n}\leq,=;\geq b_ {2} \tag{3}\]  
\[a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}+\ldots\ldots\ldots+a_{3n}x_{n}\leq,=;\geq b_ {3} \tag{4}\]  
\[\ldots\]  
\[a_{m1}x_{1}+a_{m2}x_{2}+a_{m3}x_{3}+\ldots\ldots\ldots+a_{mn}x_{n}\leq,=;\geq b _{m} \tag{5}\]  
\[x_{1},x_{2},\ldots\ldots\ldots,x_{n}\geq 0\]  
\[\sum_{j=1}^{n}a_{j}x_{n}\leq,=;\geq b_{i}(i=1,2,3,\ldots,m) \tag{6}\]  
where  
\(Z_{max/min}\): The objective function (either maximization or minimization). \(c_{j}\): The coefficients of decision variables \(x_{j}\). \(a_{ij}\): The coefficients in the constraints. \(b_{i}\): The right-hand-side values of constraints. \(x_{j}\geq 0\): The non-negativity constraints. The matrices for the objective function (Equation (7)) and the constraints (Equations (8)-(10)) of a general linear programming model are shown as follows [54; 69]:  
\[Z_{max/min}=(c_{1}c_{2}c_{3}\ldots\ldots\ldots c_{n})_{1\times n} \tag{7}\]  
\[\left[ \begin{array}{c}x_{1}\\ x_{2}\\.\\.\\ x_{n}\\ x_{n+1}\\.\\.\\ x_{n+m}\end{array} \right]_{n\times 1} \tag{8}\]  
\[\left[ \begin{array}{cccc}a_{11}&a_{12}&\ldots&a_{1n}\\ a_{21}&a_{22}&\ldots&a_{2n}\\.&.&.&.\\.&.&.&.\\.&.&.&.\\ a_{m1}&a_{m2}&\ldots&a_{mn}\end{array} \right]_{m\times n} \tag{9}\]  
\[\left[ \begin{array}{c}x_{1}\\ x_{2}\\.\\.\\.\\ x_{n}\end{array} \right]_{n\times 1} \tag{10}\]Dynamic linear programming (DLP) is an optimization framework that extends classical linear programming into a multi-period setting, where decisions at one stage affect future stages. The general form of dynamic linear programming, as formulated in Equation (11), includes the objective function, which maximizes the total profit (or minimizes the total cost) over \(T\) time periods. \[