## 4 Mathematical Formulations
### Simultaneous Pickup and Delivery Vehicle Routing Problem: MILP 1
\right\}\)  
The \(MILP1\) formulation is as follows:  
\[Min\sum_{k=1}^{m}\sum_{j=0}^{n}\sum_{i=0,\tau j}^{n}c_{ij}x_{ij}^{k} \tag{1a}\] \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\] (1b) \[\sum_{i=0,\tau j}^{n}x_{ji}^{k}=\sum_{i=0,\tau j}^{n}x_{ij}^{k}\;\;(\forall j =1,...,n;\;\forall k=1,...,m)\] (1c) \[\sum_{j=0,\tau j}^{n}\sum_{i=0,\tau j}^{n}(t_{ij}+T{\rm s}_{j}+T{\rm c}_{j})x_{ ij}^{k}\leq T_{(max)}\;\;(\forall k=1,...,m)\] (1d) \[W_{j}\leq\sum_{k=1}^{m}\left[Q_{K}^{(w)}\sum_{i=0,\tau j}^{n}x_{ij}^{k}\right] (\forall j=1,...,n)\] (1e) \[W_{(0,k)}\leq Q_{k}^{(w)}(\forall k=1,...,m)\] (1f) \[C_{j}\leq\sum_{k=1}^{m}\left[Q_{k}^{(c)}\sum_{i=0,\tau j}^{n}x_{ji}^{k}\right] (\forall j=1,...,n)\] (1g) \[C_{(0,k)}\leq Q_{k}^{(c)}(\forall k=1,...,m)\] (1h) \[W_{j}-W_{i}\geq-d_{j}^{(w)}+(Sc_{j}^{ini}+d_{j}^{(c)})*\beta+( \sum_{k=1}^{m}x_{ij}^{k}-1)M_{9}\;\;(\forall i,j=1,...,n,i\
eq j)\] (1i) \[W_{j}-W_{i}\geq-d_{j}^{(w)}+(Sc_{j}^{ini}+d_{j}^{(c)})*\beta+(1- \sum_{k=1}^{m}x_{ij}^{k})M_{10}(\forall i,j=1,...,n,i\
eq j)\] (1j) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]0 (1k) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]1 (1l) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]2\[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]3 (1n) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]4 (1o) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]5 (1p) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]6 (1q) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]7 (1r) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]8 (1s) \[\sum_{k=1}^{m}\sum_{i=0,\tau j}^{n}x_{ij}^{k}=1\;\;(\forall j=1,...,n)\]9 (1t) Knowing that: \[\sum_{i=0,\tau j}^{n}x_{ji}^{k}=\sum_{i=0,\tau j}^{n}x_{ij}^{k}\;\;(\forall j =1,...,n;\;\forall k=1,...,m)\]0 \[\sum_{i=0,\tau j}^{n}x_{ji}^{k}=\sum_{i=0,\tau j}^{n}x_{ij}^{k}\;\;(\forall j =1,...,n;\;\forall k=1,...,m)\]1 \[\sum_{i=0,\tau j}^{n}x_{ji}^{k}=\sum_{i=0,\tau j}^{n}x_{ij}^{k}\;\;(\forall j =1,...,n;\;\forall k=1,...,m)\]2 \[\sum_{i=0,\tau j}^{n}x_{ji}^{k}=\sum_{i=0,\tau j}^{n}x_{ij}^{k}\;\;(\forall j =1,...,n;\;\forall k=1,...,m)\]3 According to the \(MILP1\) formulation, the objective function (1a) seeks to minimize the total transportation cost. Constraints (1b) and (1c) ensure that the first vertex (node) associated with each customer is visited once for a simultaneous pickup and delivery. Constraint (1d) is related to the time service. The latter includes the travel time between two customers and the time necessary for loading goods into the vehicle. It prevent exceeding the predefined time capacity. Constraint (1e) ensures that the weight of the vehicle load before leaving a customer \(j\) does not exceed the vehicle capacity if there is a path between customers \(i\) and \(j\). Constraint (1f) ensures that the weight of goods required by all customers does not exceed the vehicle capacity in terms of weight. Constraint (1g) guarantees that the number of boxes loaded into the vehicle before leaving a customer \(j\) does not exceed the vehicle volume whenever it travels from \(j\) to another customer \(i\). Constraint (1h) ensures that the number of boxes required by all customers does not exceed the vehicle volume capacity. Constraints (1i) and (1j) express the relation between two successive customers in terms of weight. We considered here two types of weight; one is associated with the delivery boxes, while the other one is associated with the initial boxes existing at each customer store. Constraint (1k) defines the vehicle load before leaving the depot. Constraints (1l) and (1m) present the weight of goods loaded into the vehicle before traveling from the depot to a customer \(j\). Constraints (1n) and (1o) represent the relation between two successive customers in terms of volume (number of boxes).