### 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

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

The Buy-at-Bulk facility location problem (BBFLP for short) is defined by an undirected graph $G=(V, E)$, where $V$ denotes the set of nodes and $E$ denotes the set of edges. We are given a set of clients $D \subseteq V$ and a set of facilities $H \subseteq V$. Each client $j \in D$ has a positive demand $d_{j}$. Each facility $h \in H$ is associated with an opening cost $\mu_{h}$. We also have a set $K$ of different cable types. Each cable type $k \in K$ has a capacity $u_{k}$ and a set-up cost per unit length denoted by $\gamma_{k}$. Finally, for each edge $e \in E$ we consider a length $l_{e} \in \mathbb{Z}^{+}$. We assume that the cable types satisfy the so-called economies of scales in that if we let $K={1, \ldots,|K|}$ such that $u_{1} \leq u_{2} \leq \ldots \leq u_{K}, \gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{K}$, and $\gamma_{1} / u_{1} \geq \gamma_{2} / u_{2} \geq \ldots \geq \gamma_{K} / u_{K}$. A solution of the BBFLP consists in choosing

• a set of facilities to open.
• an assignment of each client to exactly one opened facility,
• a number of cables of each type to be installed on each edge of the graph in order to route the demands from each client to the facility to which it is assigned.

The BBFLP is closely related to the facility location problem and network loading problem. It has many applications in telecommunications and transportation network design. It is not hard to see that the BBFLP contains the Facility Location problem and hence it is NP-hard.

In the literature, the BBFLP was mainly studied in the perspective of approximation algorithms. It was first studied by Meyrson et al. [17] who considered a cost-distance problem and present the BBFLP as a special case of this latter problem. In their work, they provided an $O(\log |D|)$ approximation algorithm.
Ravi et al. [19] gave an $O(k)$ approximation algorithm for the BBFLP where $k$ is the number of cable types. Later, Friggst et al. [13] considered an integer programming formulation for the BBFLP, and showed that this formulation has an integrality gap of $O(k)$. They also considered the variant of the BBFLP where the opened facilities must be connected and gave an integrality gap of $O(1)$. Recently, Bley et al. [7] presented the first exact algorithm for the problem. They introduced a path-based formulation for the problem and compare it with a compact flow-based formulation. They also design an exact branch-and-priceand-cut algorithm for solving the path-based formulation.

As mentioned before, the BBFLP is related to the Network Loading Problem (NLP) and the Facility Location Problem (FLP). Both problems have received a lot of attention. Concerning, the NLP, Magnanti et al. [14] studied the NLP from a polyhedral point of view. They introduced some classes of valid inequalities and devised a Branch-and-Cut algorithm. In [15], Magnanti et al. considered the NLP with two cable types and some particular graphs and gave a complete description of the associated polyhedron in these cases. Bienstock et al. [6] studied the NLP with two cable types with possible extension to more than three cable types. Barahona [9] addressed the same problem, he used a relaxation without flow variables, this relaxation is based on cut condition for multicommodity flows. Gülnük [11] gave a branch and cut algorithm using spanning tree inequalities and mixed integer rounding inequalities. Agarwal [3] has introduced 4-partition based facets. Agarwal [4] extended his previous work and get a complete description of the 4-node network design problem. Raacker et al. [18] have extended the polyhedral results for cut-based inequalities for network design problem with directed, bidirected and undirected link-capacity models. Agarwal [5] developed the total-capacity inequalities, one-two inequalities and spanning trees inequalities based on a p-partition of the graph and discuss conditions under which these inequalities are facet-defining.

## 数学代写|组合优化代写Combinatorial optimization代考|Integer Programming Formulation and Polyhedron

Now, we give the so-called cut formulation for the BBFLP. This formulation can be obtained by slightly modifying the flow-based formulation introduced by [7] and projecting out the flow variables. The cut formulation is given below. Variable $t_{j}^{h}$ equals 1 if the client $j$ is assigned to facility $h$, for all $j \in D$ and $h \in H$, and $x_{e}^{k}$ is the number of cable of type $k$ installed on edge $e$, for all $e \in E$ and $k \in K$.
$\min \sum_{h \in H} \mu_{h} y_{h}+\sum_{e \in E} \sum_{k \in K} \gamma^{k} l_{e}^{k} x_{e}^{k}$
$\sum_{h \in H} t_{j}^{h}=1$,
for all $j \in D$
$t_{j}^{h} \leq y_{h}$,
for all $h \in H, j \in D$
$h \in H, j \in D$
$\sum_{\varepsilon \in \delta(W)} \sum_{k \in K} u_{k} x_{e}^{k} \geq \sum_{j \in W \cap D_{h}} \sum_{h \cap H \bar{S}} t_{j}^{h} d_{j}+\sum_{j \in W \cap D} \sum_{h \in H \cap S} t_{j}^{h} d_{j}$, for all $W \subseteq D$,
$S \subseteq V, \bar{S} \subseteq V \backslash S \quad$ (3)
$t_{j}^{h} \geq 0, \quad$ for all $h \in H, j \in D$, (4)
$y_{h} \leq 1, \quad$ for all $h \in H$,
$x_{e}^{k} \geq 0, \quad$ for all $k \in K, e \in E$, (6)
$t_{j}^{h} \in{0,1}, \quad$ for all $h \in H, j \in D$, (7)
$y_{h} \in{0,1}, \quad$ for all $h \in H$,
$x_{e}^{k} \in \mathbb{Z}^{+}, \quad$ for all $k \in K, e \in E$.
The constraints (1) impose that each client must be assigned to exactly one facility. Constraints (2) are the linking constraints and state that the clients can not be assigned to not open facilities. Constraints (3) are the so-called cut-set inequalities ensuring that the capacity on the edges of the graph is enough for routing all the demands. Constraints (4), (5) and (6) are trivial constraints. Constraints ( 7$),(8)$ and $(9)$ are the integrality constraints.
In the remain of this paper we focus on the cut formulation.
Let $Q=\left{(x, y, t) \in \mathbb{R}^{|E||K|} \times \mathbb{R}^{|H|} \times \mathbb{R}^{|H||D|}\right.$ such that $(x, y, t)$ satisfying (1)-(9) $}$. In the following, we give the dimension of the polyhedron and show that all the trivial inequalities define facets.

## 数学代写|组合优化代写Combinatorial optimization代考|Facets from Facility Location Problem

One can easily see that the projection of $Q$ on variables $y$ and $t$ corresponds to the solutions of a FLP. Thus every valid inequality (resp. facet) for the FLP polytope, in the space of $y$ and $t$ variables is also valid (resp. facet) for $Q$. Let $l_{j h}$ be the cost of assigning client $j$ to facility $h$, recall that the FLP is formulated as follows.
$$\begin{gathered} \min \sum_{j \in D} \sum_{h \in H} l_{j h} t_{j}^{h}+\sum_{h \in H} \mu_{h} y_{h} \ \sum_{h \in H} t_{j}^{h}=1, \quad j \in D \ t_{j}^{h} \leq y_{h}, \quad h \in H, j \in D \ t_{j}^{h} \in{0,1}, \quad h \in H, j \in D \ y_{i} \in{0,1} \quad h \in H . \end{gathered}$$
In what follows, we give some valid inequalities for the FLP which, from the above remarks, are also valid for the BBFLP. Note that under some conditions, these inequalities define facets of $Q$. For more details on valid inequalities and facets associated with FLP, the reader can refer to [10]. In particularly, the following inequalities are valid for facility location problem.

Circulant and Odd Cycle Inequalities. Cornuejols et al. [8] introduced the following. Let $p, q$ be integers satisfying $2 \leq q<p \leq m$ and $p \leq n, p$ is not multiple of $q, s_{1}, \ldots, s_{p}$ be distinct facilities, $m_{1}, \ldots, m_{p}$ be distinct clients, all the indices are modulo $p$ the following inequality is valid for facility location problem.
$$\sum_{i=1}^{p} \sum_{j=i}^{i+q-1} t\left(s_{i}, m_{j}\right) \leq \sum_{h=1}^{p} y\left(s_{i}\right)+p-\lceil p / q\rceil$$
Where $t\left(s_{i}, m_{j}\right)=\sum_{i \in s_{i}} \sum_{i \in m_{j}} t_{j}^{i}, y(S s i)=\sum_{i \in s_{i}} y_{i}$, they called the inequality above circulant inequality. Guignard [12] showed that this inequality defines facet when $p=q+1$, and it is called simple, and when $q=2$ it is called odd cycle inequality.

(p,q) Inequalities. Ardal et al. [1] addressed a family of valid inequalities $(p, q)$ inequalities. Let $p, q$ be integers, $2 \leq q \leq p \leq n, p$ is not multiple of $q, H^{\prime} \subseteq H$, $\left|H^{\prime}\right| \geq\lceil p / q\rceil, D^{\prime} \subseteq D,\left|D^{\prime}\right|=p, \tilde{G}$ is a bipartie graph having $H^{\prime}$ and $J^{\prime}$ as a node set, and $\forall h \in H^{\prime}$ degree(h) $=q$, and the set of edges of $G$ is $\tilde{E}$. The $(p, q)$ inequalities are defined as follows.
$$\text { where } t(\tilde{E})=\sum_{{i, j} \in E} t_{j}^{h} .$$

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

Buy-at-Bulk 设施选址问题（简称 BBFLP）由无向图定义G=(在,和)， 在哪里在表示节点集和和表示边的集合。我们有一组客户D⊆在和一套设施H⊆在. 每个客户j∈D有积极的需求dj. 各设施H∈H与开盘成本相关μH. 我们还有一套ķ不同的电缆类型。每种电缆类型ķ∈ķ有能力在ķ每单位长度的设置成本表示为Cķ. 最后，对于每条边和∈和我们考虑一个长度l和∈从+. 我们假设电缆类型满足所谓的规模经济，如果我们让ķ=1,…,|ķ|这样在1≤在2≤…≤在ķ,C1≤C2≤…≤Cķ， 和C1/在1≥C2/在2≥…≥Cķ/在ķ. BBFLP 的解决方案在于选择

• 一套设施开放。
• 将每个客户分配到一个完全开放的设施，
• 将在图表的每条边上安装许多每种类型的电缆，以便将每个客户的需求路由到分配给它的设施。

BBFLP 与设施位置问题和网络负载问题密切相关。它在电信和交通网络设计中有许多应用。不难看出 BBFLP 包含设施位置问题，因此它是 NP 难的。

## 数学代写|组合优化代写Combinatorial optimization代考|Integer Programming Formulation and Polyhedron

∑H∈H吨jH=1,

H∈H,j∈D
∑e∈d(在)∑ķ∈ķ在ķX和ķ≥∑j∈在∩DH∑H∩H小号¯吨jHdj+∑j∈在∩D∑H∈H∩小号吨jHdj， 对全部在⊆D,

X和ķ≥0,对全部ķ∈ķ,和∈和, (6)

X和ķ∈从+,对全部ķ∈ķ,和∈和.

## 数学代写|组合优化代写Combinatorial optimization代考|Facets from Facility Location Problem

∑一世=1p∑j=一世一世+q−1吨(s一世,米j)≤∑H=1p是(s一世)+p−⌈p/q⌉

(p,q) 不等式。阿达尔等人。[1] 解决了一系列有效的不等式(p,q)不平等。让p,q为整数，2≤q≤p≤n,p不是的倍数q,H′⊆H, |H′|≥⌈p/q⌉,D′⊆D,|D′|=p,G~是一个具有H′和Ĵ′作为一个节点集，并且∀H∈H′学位（h）=q, 和边的集合G是和~. 这(p,q)不等式定义如下。
在哪里 吨(和~)=∑一世,j∈和吨jH.

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