### 数学代写|组合优化代写Combinatorial optimization代考|New Facets

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Several families of valid inequalities were introduced for the problem, in the following we will give the basic valid inequalities that define facets.
The NLP with two cables types is formulated as follows.
Let $G=(V, E)$ be an undirected graph, a set of $M$ commodities $\left{\left(s_{1}, t_{1}\right), \ldots,\left(s_{|M|}, t_{|M|}\right)\right}$, a commodity $q \in{1, \ldots, M}$ having a demand $d_{q}$.
\begin{aligned} &\min \sum_{u \in E E} l_{u v} \gamma^{1} x_{u E}^{1}+l_{w v} \gamma^{2} x_{u E}^{2} \ &\sum_{v \in V} f_{(u, v)}^{q}-\sum_{v \in V} f_{(v, w)}^{q}=\left{\begin{array}{ll} d_{q}, \text { if } u=s_{q} \ -d_{q}, \text { if } u=t_{q}, \ 0, & \text { otherwise. } \end{array} \quad \text { for all } u \in V, q \in{1, \ldots, M}(17)\right. \end{aligned}
\begin{aligned} &\sum_{q=1}^{M}\left(f_{(u, v)}^{k}+f_{(v, u)}^{k}\right) \leq x_{u v}^{1}+C x_{u v}^{2} \ &x_{e}^{1}, x_{e}^{2} \geq 0, \quad \text { for all } e \in E, \ &f_{(u, v)}^{k}, f_{(v, u)}^{k} \geq 0 \text { for all } u v \in E, k \in K . \end{aligned}
Rounded Cut-Set Inequalities. The rounded cut-set inequalities are obtained by partitioning the node set into two subsets, namely $S$ and $\bar{S}$. Magnanti et al. [14] introduced the following form for the rounded cut-set inequalities:
$$x^{1}(\delta(S))+r x^{2}(\delta(S)) \geq r\left\lceil\frac{D_{S, \bar{S}}}{C}\right\rceil$$

where $D_{S, \bar{S}}$ is the total demand whose origin lays in $S$ and destination in $\bar{S}$ and vice-versa, and
$$r=\left{\begin{array}{l} C, \text { if } \quad D_{S, \bar{S}} \quad(\bmod C)=0 \ D_{S, \bar{S}} \quad(\bmod C), \text { otherwise. } \end{array}\right.$$
Theorem 3 [14]. A rounded cut set inequality defines a facet for the $N L P$ if and only if

1. the subgraphs induced by $S$ and by $\bar{S}$ are connected,
2. $D_{S, \bar{S}}>0$.
Partition Inequalities. Several version of partition inequalities have been developed in the literature, in particular, Magnanti et al. [14] gave a family of three-partition inequalities of the following form:
$$x_{12}^{1}+x_{13}^{1}+x_{23}^{1}+r\left(x_{12}^{2}+x_{13}^{2}+x_{23}^{2}\right) \geq\left\lceil\left[\frac{r\left(\left\lceil\frac{d_{12}+d_{13}}{C}\right\rceil+\left\lceil\frac{d_{12}+d_{23}}{C}\right\rceil+\left\lceil\frac{d_{13}+d_{23}}{C}\right\rceil\right)}{2}\right\rceil\right.$$
The following metric inequalities are valid for NLP with one cable type:
Metric Inequalities. Metric inequalities can be introduced after projecting out the flow variables from the polyhedron and obtain a “capacity formulation” in the space of capacity variables. In particular Avella et al. [2] gave a family of metric inequalities that completely describe the polyhedron, they take the form
$$\mu x \geq \rho(\mu, G, d), \mu \in \operatorname{Met}(G)$$
In the next section we present a lifting procedure that extends the Rounded cut-set inequalities defined for NLP to be valid for BBFLP.

## 数学代写|组合优化代写Combinatorial optimization代考|Lifted Rounded Cut-Set Inequalities

Now, we give the main result of the paper which is a new family of valid inequalities for the BBFLP. These latter inequalities are obtained by lifting the rounded cut-set inequalities.

Consider the situation where all the facilities are opened and all the clients are assigned to a facility. For every $j \in D$, let $h_{j}$ be the facility to which it is assigned (Fig. 1).
Let $Q_{0}=\left{(x, y, t) \in Q\right.$ such that $t_{j}^{h}=0$, for all $\left.h \in H \backslash h_{j}, j \in D\right}$.
Lemma 1. The solutions of $Q_{0}$ are exactly solutions of a NLP where the demand set is composed of the pairs $\left(j, h_{j}\right)$, with $j$ the origin, $h_{j}$ the destination and $d_{j}$ is the commodity.

By Lemma 1, every valid inequality (respectively facet defining) for the corresponding NLP polyhedron are valid for $Q_{0}$ (respectively facet defining). This means that rounded cut-set inequalities are valid for $Q_{0}$.

Our objective is to extend the rounded cut-set inequalities by a lifting procedure to obtain facet defining inequalities for BBFLP. Let $(x, y, t) \in Q_{0}$ and $S \subseteq V$ be a node set such that $S \cap V \neq \emptyset \neq \bar{S} \cap V$. Let $A$ (resp. $A^{\prime}$ ) be the node subset of $D \cap S$ (resp. $D \cap \bar{S}$ ) which are assigned to a facility of $H \cap \bar{S}$ (resp. $H \cap S$ ). Also, let $B$ (resp. $B^{\prime}$ ) be the node subset of $D \cap S$ (resp. $D \cap \bar{S}$ ) which are assigned to a facility of $H \cap S$ (resp. $H \cap \bar{S}$ ).
The rounded cut-set inequality induced by $S$ is
$$x^{1}(\delta(S))+r x^{2}(\delta(S)) \geq r\left\lceil\frac{d\left(A \cup A^{\prime}\right)}{C}\right\rceil,$$
where $r=d\left(A \cup A^{\prime}\right)(\bmod C)$. Clearly (24) is valid for $Q_{0}$.
For convenience, we let,
\begin{aligned} &A_{1}=\left{j \in A \cup A^{\prime} \text { such that } d_{j} \leq r\right} \ &A_{2}=\left{j \in A \cup A^{\prime} \text { such that } d_{j}>r\right} \ &B_{1}=\left{j \in B \cup B^{\prime} \text { such that } d_{j} \leq C-r\right}, \ &B_{2}=\left{j \in B \cup B^{\prime} \text { such that } d_{j}>C-r\right} . \end{aligned}
Now we give our main result.
Theorem 4. Let $S \subseteq V$ with $S \cap V \neq \emptyset \neq V \cap \bar{S}$. The inequality
$$x^{1}(\delta(S))+r x^{2}(\delta(S))+\sum_{j \in D} \sum_{\left.h \in H \backslash h_{j}\right}} C_{j}^{h} t_{j}^{h} \geq r\left\lceil\frac{d\left(A \cup A^{\prime}\right)}{C}\right\rceil$$
where
$$C_{j}^{h}=\left{\begin{array}{l} d_{j}, \text { for } j \in A_{1} \cap A, h \in H \cap S \text { and for } \in A_{1} \cap A^{\prime}, h \in H \cap \bar{S}, \ r, \text { for } j \in A_{2} \cap A, h \in H \cap S \text { and for } j \in A_{2} \cap A^{\prime}, h \in H \cap \bar{S} \ C-r-d_{j}, \text { for } j \in B_{1} \cap B, h \in H \cap \bar{S} \text { and for } j \in B_{1} \cap B^{\prime}, h \in H \cap S, \ 0, \text { otherwise. } \end{array}\right.$$
is valid for BBFLP. Moreover (25) is facet for $Q$ if (24) is facet for $Q_{0}$.

## 数学代写|组合优化代写Combinatorial optimization代考|A Polyhedral Study for the Buy-at-Bulk Facility Location Problem

Lemma 3. If $D^{\prime}=\emptyset$ then, for every $j_{0} \in A \cup A^{\prime}$ and $h_{0} \in H \backslash\left{h_{j_{0}}\right}$,
$$\xi_{j_{0}}^{h_{0}}=r\left\lceil\frac{d(W)}{C}\right\rceil+\varepsilon_{W}$$
where $W=\left(A \cup A^{\prime}\right) \backslash\left{j_{0}\right}$.
Proof. First, let $\beta$ be the number of cables of type 1 loaded on the edges of $\delta(S)$. Also, w.l.o.g., we assume that the number of cables of type 2 loaded on the edges of $\delta(S)$ is $\left\lceil\frac{d(W)}{C}\right\rceil-\lambda$, for some $\lambda \geq 0$. In any feasible solution of the BBFLP, we have that $\beta+C\left(\left\lceil\frac{d(W)}{C}\right\rceil-\lambda\right) \geq d(W)$.

We have that $x^{1}(\delta(S))+r x^{2}(\delta(S))=\beta+r\left(\left\lceil\frac{d(W)}{C}\right\rceil-\lambda\right)$. If $\lambda=0$, we can show that the minimum value of $x^{1}(\delta(S))+r x^{2}(\delta(S))$ is obtained for $\beta=0$. Thus, $\xi_{j_{0}}^{h_{0}}=r\left\lceil\frac{d(W)}{C}\right\rceil$ in this case. Now, if $\lambda \geq 1$, we have that
$$\beta+r\left(\left\lceil\frac{d(W)}{C}\right\rceil-\lambda\right) \geq(C-r) \lambda-C+r_{W}+r\left\lceil\frac{d(W)}{C}\right\rceil .$$
Here also, we can show that the minimum of $x^{1}(\delta(S))+r x^{2}(\delta(S))$ is obtained for $\lambda=1$ and $\beta=C(\lambda-1)+r_{W}=r_{W}$, that is $\xi_{j_{0}}^{h_{0}}=r_{W}-r+r\left\lceil\frac{d(W)}{C}\right\rceil$.
Thus, $\xi_{j_{0}}^{h_{0}}=r\left\lceil\frac{d(W)}{C}\right\rceil+\min \left{0, r_{W}-r\right}=r\left\lceil\frac{d(W)}{C}\right\rceil+\varepsilon_{W}$.
Lemma 4. If $D^{\prime}=\emptyset$ then, for every $j_{0} \in B \cup B^{\prime}$ and $h_{0} \in H \backslash\left{h_{j_{0}}\right}$,
$$\xi_{j_{0}}^{h_{0}}=r\left\lceil\left[\frac{d(W)}{C}\right\rceil+\varepsilon_{W}\right.$$
where $W=A \cup A^{\prime} \cup\left{j_{0}\right}$.
Now, for a node set $U \subseteq D$, we let
$$\alpha_{U}=\left\lceil\frac{d\left(A \cup A^{\prime}\right)}{C}\right\rceil-\left\lceil\frac{d(U)}{C}\right\rceil .$$
From Lemmas 3 and 4 , when $D^{\prime}=\emptyset$, the lifting coefficient of a variable $t_{j_{0}}^{h_{0}}$ is of the form
$$C_{j_{0}}^{h_{0}}=r \alpha_{W}-\varepsilon_{W}$$
where
$$W=\left{\begin{array}{l} \left(A \cup A^{\prime}\right) \backslash\left{j_{0}\right}, \text { for } j_{0} \in A \cup A^{\prime}, \ A \cup A^{\prime} \cup\left{j_{0}\right}, \text { for } j_{0} \in B \cup B^{\prime} . \end{array}\right.$$
From this, we can see that the lifting coefficient of variable $t_{j}^{h}$, when $D^{\prime}=\emptyset$ ({i.e. $}$ when the variable is lifted at first), is
$-C_{j}^{h}=r \alpha_{W}-\varepsilon_{W}=d_{j}$, for $j \in A_{1}$,
$-C_{j}^{h}=r \alpha_{W}-\varepsilon_{W}=r$, for $j \in A_{2}$,
$-C_{j}^{h}=r \alpha_{W}-\varepsilon_{W}=0$, for $j \in B_{1}$,
$-C_{j}^{h}=r \alpha_{W}-\varepsilon_{W}=C-r-d_{j}$, for $j \in B_{2} .$

## 组合优化代写

\begin{aligned} &\min \sum_{u \in EE} l_{uv} \gamma^{1} x_{u E}^{1}+l_{wv} \gamma^{2} x_{ u E}^{2} \ &\sum_{v \in V} f_{(u, v)}^{q}-\sum_{v \in V} f_{(v, w)}^{q} =\左{dq, if u=sq −dq, if u=tq, 0, otherwise. \quad \text { for all } u \in V, q \in{1, \ldots, M}(17)\right. \end{对齐} ∑q=1M(f(u,v)k+f(v,u)k)≤xuv1+Cxuv2 xe1,xe2≥0, for all e∈E, f(u,v)k,f(v,u)k≥0 for all uv∈E,k∈K. RoundedCut−SetInequalities.Theroundedcut−setinequalitiesareobtainedbypartitioningthenodesetintotwosubsets,namelySandS¯.Magnantietal.[14]introducedthefollowingformfortheroundedcut−setinequalities: x^{1}(\delta(S))+rx^{2}(\delta(S)) \geq r\left\lceil\frac{D_{S, \bar{S}}}{C}\对\rceil

$$r=\left{C, if DS,S¯(modC)=0 DS,S¯(modC), otherwise. \对。$$

1. 子图由S并通过S¯连接，
2. DS,S¯>0.
分区不等式。文献中已经发展了几个版本的分区不等式，特别是 Magnanti 等人。[14] 给出了以下形式的三分区不等式族：
x121+x131+x231+r(x122+x132+x232)≥⌈[r(⌈d12+d13C⌉+⌈d12+d23C⌉+⌈d13+d23C⌉)2⌉
以下度量不等式对具有一种电缆类型的 NLP 有效：
度量不等式。将多面体中的流量变量投影出来后，可以引入度量不等式，得到容量变量空间中的“容量公式”。特别是 Avella 等人。[2] 给出了一系列完全描述多面体的度量不等式，它们采用以下形式
μx≥ρ(μ,G,d),μ∈Met⁡(G)
在下一节中，我们将介绍一个提升程序，它将为 NLP 定义的圆角割集不等式扩展为对 BBFLP 有效。

## 数学代写|组合优化代写Combinatorial optimization代考|Lifted Rounded Cut-Set Inequalities

x1(δ(S))+rx2(δ(S))≥r⌈d(A∪A′)C⌉,

\begin{对齐} &A_{1}=\left{j \in A \cup A^{\prime} \text { 这样 } d_{j} \leq r\right} \ &A_{2}=\left{ j \in A \cup A^{\prime} \text { 这样 } d_{j}>r\right} \ &B_{1}=\left{j \in B \cup B^{\prime} \text { 这样 } d_{j} \leq Cr\right}, \ &B_{2}=\left{j \in B \cup B^{\prime} \text { 这样 } d_{j}>Cr\right } 。\end{对齐}\begin{aligned} &A_{1}=\left{j \in A \cup A^{\prime} \text { such that } d_{j} \leq r\right} \ &A_{2}=\left{j \in A \cup A^{\prime} \text { such that } d_{j}>r\right} \ &B_{1}=\left{j \in B \cup B^{\prime} \text { such that } d_{j} \leq C-r\right}, \ &B_{2}=\left{j \in B \cup B^{\prime} \text { such that } d_{j}>C-r\right} . \end{aligned}

x^{1}(\delta(S))+r x^{2}(\delta(S))+\sum_{j \in D} \sum_{\left.h \in H \反斜杠 h_{j} \right}} C_{j}^{h} t_{j}^{h} \geq r\left\lceil\frac{d\left(A \cup A^{\prime}\right)}{C} \对\rceilx^{1}(\delta(S))+r x^{2}(\delta(S))+\sum_{j \in D} \sum_{\left.h \in H \backslash h_{j}\right}} C_{j}^{h} t_{j}^{h} \geq r\left\lceil\frac{d\left(A \cup A^{\prime}\right)}{C}\right\rceil

$$C_{j}^{h}=\left{dj, for j∈A1∩A,h∈H∩S and for ∈A1∩A′,h∈H∩S¯, r, for j∈A2∩A,h∈H∩S and for j∈A2∩A′,h∈H∩S¯ C−r−dj, for j∈B1∩B,h∈H∩S¯ and for j∈B1∩B′,h∈H∩S, 0, otherwise. \对。$$

## 数学代写|组合优化代写Combinatorial optimization代考|A Polyhedral Study for the Buy-at-Bulk Facility Location Problem

ξj0h0=r⌈d(W)C⌉+εW

β+r(⌈d(W)C⌉−λ)≥(C−r)λ−C+rW+r⌈d(W)C⌉.

ξj0h0=r⌈[d(W)C⌉+εW

αU=⌈d(A∪A′)C⌉−⌈d(U)C⌉.

Cj0h0=rαW−εW

$$W=\left{\begin{array}{l} \left(A \cup A^{\prime}\right) \backslash\left{j_{0}\right}, \text { for } j_{0} \in A \cup A^{\prime}, \ A \cup A^{\prime} \cup\left{j_{0}\right}, \text { for } j_{0} \in B \cup B^{\prime} . \结束{数组}\begin{array}{l} \left(A \cup A^{\prime}\right) \backslash\left{j_{0}\right}, \text { for } j_{0} \in A \cup A^{\prime}, \ A \cup A^{\prime} \cup\left{j_{0}\right}, \text { for } j_{0} \in B \cup B^{\prime} . \end{array}\对。$$

−Cjh=rαW−εW=dj， 为了j∈A1,
−Cjh=rαW−εW=r， 为了j∈A2,
−Cjh=rαW−εW=0， 为了j∈B1,
−Cjh=rαW−εW=C−r−dj， 为了j∈B2.

## 有限元方法代写

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