The inverse 1-median location problem on uncertain tree networks with tail value at risk criterion

Highlights

• A necessary and sufficient condition for the α-1-median is obtained.

• The uncertain inverse 1-median problem is transformed into a programming model with deterministic constraints.

• A new model for the problem is developed.

• An O(nlog n) time algorithm for the problem under weighted l₁ norm is proposed.

Abstract

In an inverse 1-median location problem on a tree network, the goal is to modify the vertex weights of the underlying tree network at the minimum total cost such that a predetermined vertex becomes the 1-median. This paper investigates the case that the vertex weights and modification costs are considered as uncertain variables. We first obtain a necessary and sufficient condition for the α-1-median on uncertain trees. Based on this condition, we transform the problem into a linear programming model with deterministic constraints. Finally, we consider the proposed model with tail value at risk objective under the weighted l₁ norm and present a solution algorithm for the problem with time complexity of O(nlog n).

Introduction

One of the important aspects of the inverse optimization which has recently been studied by many researchers is the inverse location problem, in which the task is to change the parameters of the original problem at the minimum total cost with respect to modification bounds such that the given facility location under the new values of the parameters becomes optimal. One of the well-known models is the inverse 1-median location problem (I1-MLP) which is stated on a tree network as follows:

Consider $T=(V,E)$ as a tree network with vertex set $V=\{v_1,v_2,\dots ,v_n\}$ and edge set E. Let d(v_i, v_j) be the distance between two vertices v_i and v_j in T. Moreover, let w_i be a nonnegative weight associated to v_i ∈ V, and each edge e ∈ E has a positive length. We call a vertex v* as a 1-median of the tree network T if and only if for all v_j ∈ V the inequality

$$\sum _{i=1}^n{w}_{i}d(v^{*},v_i)\le \sum _{i=1}^n{w}_{i}d(v_j,v_i).$$

is satisfied.

Assume that v_i, $i=1,\dots ,p,$ are the immediate neighbors of an arbitrary vertex v*. Denote by $T_1,T_2,\dots ,T_p$ the subtrees rooted at the vertices $v_1,v_2,\dots ,v_p,$ respectively. Define

$$V\left(T_j\right)=\left\{v_k\right|v_k\text{is}\text{a}\text{vertex}\text{of}{T}_j\},j=1,2,\dots ,p.$$

Goldman [10] proved that a vertex v* ∈ V is a 1-median of the tree network T, if and only if

$$\sum _{v_i\in V\left(T_j\right)}{w}_i-\frac{1}{2}\sum _{v_i\in V}{w}_i\le 0,j=1,2,\dots ,p.$$

In an I1-MLP on a tree network, the task is to modify the vertex weights at the minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to new vertex weights. Let the vertex weights obey the upper and lower bounds ${\overline{w}}_i$ and w_i, respectively. Suppose that v* is not a 1-median with respect to the original vertex weights. In an I1-MLP model, the task is to modify w_i to $w_i^{*},$ for all v_i ∈ V such that

• The vertex v* ∈ V becomes a 1-median of the tree with respect to $w_i^{*}$.

• The following bound constraints are satisfied:

  $${\underline{w}}_i\le w_i^{*}\le {\overline{w}}_i\text{for}\text{all}{v}_i\in V.$$

• The objective cost function $\parallel w-w^{*}\parallel$ becomes minimum under a specific norm.

Assume that we incur the nonnegative costs $C_i^{+}$ and $C_i^{-},$ if the vertex weight of v_i is increased and decreased by one unit, respectively. Define $C_i=C_i^{+},$ if w_i is increased and $C_i=C_i^{-},$ if w_i is decreased. The total modification costs can be measured by the weighted l₁ norm (Wl₁N), weighted l_∞ norm (Wl_∞N) and weighted bottleneck-Hamming distance (WBHD) which are stated as follows, respectively:

$$\sum _{v_i\in V}{C}_i\mid w_i-w_i^{*}\mid ,$$

$$\underset{v_i\in V}{\max }{C}_i\mid w_i-w_i^{*}\mid ,$$

$$\underset{v_i\in V}{\max }{C}_{i}H(w_i,w_i^{*}),$$

where $H(w_i,w_i^{*})=1$ if $w_i=w_i^{*},$ and 0 otherwise.

The literature review shows that the inverse location problems have recently been studied by many researchers. For a survey on inverse 1-median location problems, see e.g., [1], [4], [5], [9], [11], [12], [13], [20], [24].

In the real life, we are usually faced with various types of uncertainty. For example, some parameters of the location problems like, the vertex weights, the travel times between vertices, the establishing costs of facilities and the network modification costs may not be certainly known. The uncertainty theory introduced by Liu [16] is a suitable tool to deal with such parameters. Gao [7] modeled the single facility location problems with uncertain demands. Wen et al. [29] investigated the capacitated facility location-allocation problem with uncertain demands. Nguyen and Chi [21] studied I1-MLP on a tree with uncertain costs and showed that the inverse distribution function of the minimum cost can be obtained at O(n²log n) time. Zhang et al. [32] investigated the sustainable multi-depot emergency facilities location-routing problem with uncertain information. For a survey on uncertain location problems, we refer the interested reader to [8], [14], [15], [19], [23], [25], [26], [31]. Forthermore, see [6] as a related work.

Note that the uncertainty leads to risk. Liu [18] introduced the risk concept in the uncertain environment. Measuring the risk is one of the important steps in the decision making process. The risk metrics contain techniques and data sets used to calculate the risk value of the problem under investigation. Tail value at risk (TVaR) metric [22] is one of the measures of the risk that is widely acceptable among industry segments and market participants. For a survey on the risk management in the location problems with random and fuzzy variables, see e.g., [2], [3], [27], [28], [30].

In this paper, we concentrate on the I1-MLP model in a tree network with uncertain vertex weights and weight modification costs under Wl₁N. The article is organized as follows: In the next section, we first present some basic definitions and theorems in the uncertainty theory. Then, the TVaR metric and its property in an uncertain environment are introduced. In Section 3, we derive a necessary and sufficient condition for the α-1-median on an uncertain tree and apply this condition to propose an equivalent model with deterministic constraints for the uncertain inverse 1-median location problem (UI1-MLP). Then, we consider the vertex weights as linear uncertain variables and develop a new model for the problem. Since the costs are considered as uncertain variables, we solve the obtained model with TVaR objective. Finally, we suppose that the uncertain costs have linear or zigzag distributions and then we present a combinatorial algorithm with time complexity of O(nlog n) for the problem under Wl₁N. The conclusion of the paper is given in Section 4.