# Optimal Deployment of Container Weighing Equipment: Models and Properties
## 3 Mathematical Analyses  
In this section, we present several propositions derived from the established model and provide detailed proofs for them. These propositions can help verify the effectiveness and practicality of the model. Proposition 1: _If \(Q\) is large enough, then any optimal solution shows that all containers are weighed at their origin ports._  
Proof: We first show that an upper bound of the optimal objective function value is \(\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus\{i\}}\left(\overline{u }_{ij}+\sum_{k\in\mathcal{P}}n_{ij}^{k}\right)=\sum_{i\in\mathcal{P}}\sum_{j \in\mathcal{P}\setminus\{i\}}N_{ij}\). To this end, we have  
\[ \begin{array}{l}\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus\{i\}} \sum_{k\in\mathcal{P}\setminus\{i\}}\left(1\!\cdot\!u_{ij}^{k}+\alpha\!\cdot\!w_ {ij}^{k}+\lambda\!\cdot\!v_{ij}^{k}\right)+\sum_{i\in\mathcal{P}}\sum_{j\in \mathcal{P}\setminus\{i\}}\left(1\!\cdot\!\overline{u}_{ij}+\lambda\!\cdot\! \overline{v}_{ij}\right)\\ \leq\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus\{i\}}\sum_{k\in \mathcal{P}\setminus\{i,j\}}\left(1\!\cdot\!u_{ij}^{k}+1\!\cdot\!w_{ij}^{k}+1 \!\cdot\!v_{ij}^{k}\right)+\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus \{i\}}\left(1\!\cdot\!\overline{u}_{ij}+1\!\cdot\!\overline{v}_{ij}\right)\\ \leq\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus\{i\}}\left(\sum_{k\in \mathcal{P}\setminus\{i,j\}}n_{ij}^{k}+\overline{n}_{ij}\right)\\ =\sum_{i\in\mathcal{P}}\sum_{j\in\mathcal{P}\setminus\{i\}}N_{ij}\end{array} \]where the first inequality holds because \(\alpha<1\) and \(\lambda<1\), and the second inequality holds because of Constraints (3)-(4). Next, if \(Q\) is large enough, we can construct the following solution: \(x_{i}^{\mathbf{x}}=\left[\frac{\sum_{j\in\mathcal{P}\setminus\{i\}}N_{ij}}{M}\right], i\in\mathcal{P};u_{ij}^{\mathbf{k}\mathbf{q}}=n_{ij}^{\mathbf{k}},w_{ij}^{\mathbf{k}\bm  
Obviously, increasing the term \(\sum_{k\in\mathcal{P}\setminus\{j,j\}}1\cdot u_{\hat{ij}}^{k*}+1\cdot\overline{u}_{ ij}^{*}\) by one and simultaneously decreasing \(w_{ij}^{\widehat{\imath}}\) by one ensures the feasibility of the solution. This adjustment reflects an additional container originating and weighed at port \(\hat{i}\), offset by a reduction in containers that start from port \(i\) and are transshipped at port \(\hat{i}\) being weighed at the transshipped port \(\hat{i}\); the solution is still feasible. The inequality holds because \(\alpha<1\); therefore, we see that the value of the original objective function is less than that of the new one, which contradicts with the assumption. This concludes the proof of the proposition. **Proposition 4**. : _If an optimal solution \((x_{i}^{*},\,i\in\mathcal{P};u_{ij}^{k*},w_{ij}^{k*},v_{ij}^{k*},i,j,k\in \mathcal{P},i\
eq j\
eq k;\overline{u}_{ij}^{*},\overline{v}_{ij}^{*},i,j\in \mathcal{P},i\
eq j)\) has an intermediate port \(\hat{i}\) weighing more than 0 containers, then it is possible that not all containers starting from port \(\hat{i}\) are weighed at port \(\hat{i}\). Mathematically, it is possible that there exist ports \(\hat{i}\), \(\hat{k}\), and \(\hat{j}\) such that \(w_{\hat{ij}}^{i*}>0\) and \(\sum_{j\in\mathcal{P}\setminus\{\hat{i}\}}\sum_{k\in\mathcal{P}\setminus\{ \hat{i}\}}u_{\hat{ij}}^{k*}+\sum_{j\in\mathcal{P}\setminus\{\hat{i}\}} \overline{u}_{ij}^{*}<\sum_{j\in\mathcal{P}\setminus\{\hat{i}\}}N_{\hat{ij}}\)._  
Proof. : We prove this proposition by finding an example that satisfies it. Assume that \(\mathcal{P}=\{1,2,3,4\}\), \(Q=2\), and \(M=10\). The MS needs to transfer 10 containers from port 2 to port 4 without transshipment, 1 container from port 1 to port 3 via 2, and 5 containers from port 4 to port 1 without transshipment, which can be described as  
\[\left\{ \begin{array}{c}Q=2\\ M=10\\ \alpha=0.8\\ \lambda=0.4\end{array} \right\}N_{13}=1\\ N_{31}=N_{32}=N_{34}=0\\ N_{42}=N_{43}=0\]  
\[\left\{ \begin{array}{c}n_{13}^{2}=1\\ n_{ij}^{k}=0\,,(i,j,k)\
eq(1,3,2)\\ i\
eq j\
eq k,i,j,k\in P\end{array} \right\}\left\{