## 数学代写|组合优化代写Combinatorial optimization代考|Computational Results

The polyhedral theory behind valid inequalities and how they are capable of sculpting the convex hull of integer solutions is quite elegant. From the optimization point of view however, our goal is clear: finding an optimal solution. Consequently, it is not relevant whether or not an inequality is valid, as long as we guarantee that there is at least one optimal solution in the feasible region.
Many assignment problems hide a natural symmetry issue that slows down typical branch and bound applications. Symmetric solutions can be seen as different solutions with the same objective function value. In [7], this issue is addressed by introducing symmetry-breaking constraints. Such constraints are intentionally not valid inequalities in the sense that they attempt to remove some integer feasible solutions from the feasible region, while keeping a symmetric solution for each solution removed.

We have adapted these constraints to fit USNMP. Constraints (11) reinforces constraints (2) in a way that the $i$-th demand must be assigned to one of the first $i$ vehicles. Constraints (12) assures that demand $e$ can be assigned to vehicle $i$ only if vehicle $i-1$ serves at least one of the first $e-1$ demands.
$$\begin{array}{lr} \sum_{i=1}^e x_e^i=1 & \forall e \in E \ \sum_{j=i}^e x_e^j \leq \sum_{u=i-1}^{e-1} x_u^{i-1} & \forall e \in E, i \in K: e \geq j \end{array}$$
Table 1 provides a comparison of performances between the formulation (1)(7) given in [18] solved by CPLEX $12.7$ (column CPLEX) and the reinforced formulation (1)-(9) with the addition of symmetry-breaking constraints (11)(12) and $k$-tree inequalities (10) embedded on the branch-and-cut procedure (column $\mathrm{B} \& \mathrm{C}$ ). The instances were generated randomly with the number of demands $|E|$ ranging from 30 to 50 , the number of stations $|V|$ from 10 to 30 and the capacity was fixed at 5 . For each set of parameters, 3 instances were tested (column $i$ ). The time of resolution $(C P U)$ is displayed in seconds. Time limit was set to one hour and the instances that were not solved within 1 hour are marked with an asterisks. The number of nodes on the branch-and-bound tree is displayed under column Nodes. The gap percentage between the best integer solution found and the lower bound provided by the linear relaxation is displayed under column GAP. Finally, Time on Cuts shows the relative amount of time spent on solving the separation problem.

## 数学代写|组合优化代写Combinatorial optimization代考|The Maximum Concurrent Flow

Our motivation for studying this problem stems from transportation networks, be they road or rail-oriented. As a critical increase of the load can induce a decrease of the quality of services, an hypothesis consists of assuming that the passengers traffic tends naturally to balance itself to an equilibrium [18]. One can model this problem by minimizing the maximum capacity utilization, and the latter can be reformulated as a MCFP [20]. In our context, we have historical traffic data including a partial observation of the arc loads for a certain subset of arcs. For some networks, we are also given a subset $A^{\prime} \subset A$ of arcs with known capacities.

In general, however, we do not know the arc capacities. The problem we are interested in is the MCFP with incomplete arc capacities. The MCFP in LP formulation (2) without the capacity constraints is clearly an unbounded LP. To avoid this situation, we employ a given set $S$ of scenarios from our historical arc load database. Each scenario $s=\left(A^s, \ell^s, K^s\right) \in S$ consists of a subset $A^s \subset A$ of arcs, a partial are load function $\ell^s: A^s \rightarrow \mathbb{R}_{+}$, and a set of commodities $K^s$. We require that: (i) missing capacities should be estimated so as to allow the maximum known arc loads over all scenarios, (ii) arc loads from computed flows should be as close as possible to the loads given in the scenarios, and (iii) each flow solution for a scenario should describe an optimal solution of the MCFP w.r.t. capacity and commodity values. We therefore define the following problem, which is new as far as we could ascertain.

Although the IMCFP is natively cast in a multi-objective fashion (see condition (ii)), in practice we minimize a max norm over all arcs and all scenarios. We remark that condition (iii) is only apparently recursive: we want to decide $f, c$ at the same time and also require that every $f^s$ should be optimal flows w.r.t. a putative MCFP instance defined over the values of the $c$ variables and the $K^s$ parameters. We shall see below that the IMCFP can be formulated by means of a Mixed-Integer Linear Programming formulation that combines both primal and dual variables.

# 组合优化代考

## 数学代写|组合优化代写Combinatorial optimization代考|Computational Results

$$\sum_{i=1}^e x_e^i=1 \quad \forall e \in E \sum_{j=i}^e x_e^j \leq \sum_{u=i-1}^{e-1} x_u^{i-1} \quad \forall e \in E, i \in K: e \geq j$$

## 数学代写|组合优化代写Combinatorial optimization代考|The Maximum Concurrent Flow

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