Solving the Two-Stage Robust FTTH network design Problem under Demand Uncertainty

For the past few years, the increase in high bandwidth requiring services forced telecommunication operators like France Telecom - Orange to engage the deployment of optical networks, the Fiber To The Home Gigabit Passive Optical Network (FTTH GPON) technology, leading to new design problems. Such problems have already been studied. However, to the best of our knowledge, without taking into account the future demand uncertainty. In this paper, we propose a model for a two-stage robust optimization FTTH network design problem tackling the demand uncertainty. We propose an exact algorithm, based on column and constraint generation algorithms, and we show some preliminary results.     

The complexity of computing a robust flow

The Maximum Robust Flow problem asks for a flow on the paths of a network maximizing the guaranteed amount of flow surviving the removal of any $k$ arcs. We point out a flaw in a previous publication that claimed $NP$-hardness for this problem when $k=2$. For the case that $k$ is part of the input, we present a new hardness proof. We also discuss the complexity of the integral version of the problem.     

Revisiting a game theoretic framework for the robust railway network design against intentional attacks

This paper discusses and extends some competitive aspects of the games proposed in an earlier work, where a robust railway network design problem was proposed as a non-cooperative zero-sum game in normal form between a designer/operator and an attacker. Due to the importance of the order of play and the information available to the players at the moment of their decisions, we here extend those previous models by proposing a formulation of this situation as a dynamic game. Besides, we propose a new mathematical programming model that optimizes both the network design and the allocation of security resources over the network. The paper also proposes a model to distribute security resources over an already existing railway network in order to minimize the negative effects of an intentional attack. For the sake of readability, all concepts are introduced with the help of an illustrative example.     

A robust optimization approach to closed-loop supply chain network design under uncertainty

The concern about significant changes in the business environment (such as customer demands and transportation costs) has spurred an interest in designing scalable and robust supply chains. This paper proposes a robust optimization model for handling the inherent uncertainty of input data in a closed-loop supply chain network design problem. First, a deterministic mixed-integer linear programming model is developed for designing a closed-loop supply chain network. Then, the robust counterpart of the proposed mixed-integer linear programming model is presented by using the recent extensions in robust optimization theory. Finally, to assess the robustness of the solutions obtained by the novel robust optimization model, they are compared to those generated by the deterministic mixed-integer linear programming model in a number of realizations under different test problems.     

A new methodology for evaluating sustainable product design performance with two-stage network data envelopment analysis

Sustainable product design has been considered as one of the most important practices for achieving sustainability. To improve the environmental performances of a product through product design, however, a firm often needs to deal with some difficult technical trade-offs between traditional and environmental attributes which require new design concepts and engineering specifications. In this paper, we propose a novel use of the two-stage network Data Envelopment Analysis (DEA) to evaluate sustainable product design performances. We conceptualize “design efficiency” as a key measurement of design performance in terms of how well multiple product specifications and attributes are combined in a product design that leads to lower environmental impacts or better environmental performances. A two-stage network DEA model is developed for sustainable design performance evaluation with an “industrial design module” and a “bio design module.” To demonstrate the applications of our DEA-based methodology, we use data of key engineering specifications, product attributes, and emissions performances in the vehicle emissions testing database published by the US EPA to evaluate the sustainable design performances of different automobile manufacturers. Our test results show that sustainable design does not need to mean compromise between traditional and environmental attributes. Through addressing the interrelatedness of subsystems in product design, a firm can find the most efficient way to combine product specifications and attributes which leads to lower environmental impacts or better environmental performances. This paper contributes to the existing literature by developing a new research framework for evaluating sustainable design performances as well as by proposing an innovative application of the two-stage network DEA for finding the most eco-efficient way to achieve better environmental performances through product design.     

Complexity of Source-Sink Monotone 2-parameter min cut

There are many applications of max flow with capacities that depend on one or more parameters. Many of these applications fall into the “Source-Sink Monotone” framework, a special case of Topkis's monotonic optimization framework, which implies that the parametric min cuts are nested. When there is a single parameter, this property implies that the number of distinct min cuts is linear in the number of nodes, which is quite useful for constructing algorithms to identify all possible min cuts.When there are multiple Source-Sink Monotone parameters, and vectors of parameters are ordered in the usual vector sense, the resulting min cuts are still nested. However, the number of distinct min cuts was an open question. We show that even with only two parameters, the number of distinct min cuts can be exponential in the number of nodes.     

Polynomial-time identification of robust network flows under uncertain arc failures

We propose Linear Programming (LP)-based solution methods for network flow problems subject to multiple uncertain arc failures, which allow finding robust optimal solutions in polynomial time under certain conditions. We justify this fact by proving that for the considered class of problems under uncertainty with linear loss functions, the number of entities in the corresponding LP formulations is polynomial with respect to the number of arcs in the network. The proposed formulation is efficient for sparse networks, as well as for time-critical networked systems, where quick and robust decisions play a crucial role.     

The multi-layered network design problem

We address the problem of designing a network built on several layers. This problem occurs in practical applications but has not been studied extensively from the point of view of global optimisation, since the problem of designing a single-layered network is complex. An example of an application is the design of a virtual network (Internet Protocol) built on a sparse optical transport network.     

Single-commodity robust network design problem: Complexity,instances and heuristic solutions

We study a single-commodity Robust Network Design problem (RND) in which an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. In each scenario, a subset of the nodes is exchanging flow. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. Previously conducted computational investigations on the problem motivated the study of the complexity of some special cases and we present complexity results on them, including hypercubes. In turn, these results lead to the definition of new instances (random graphs with {−1, 0, 1} balances) that are computationally hard for the natural flow formulation. These instances are then solved by means of a new heuristic algorithm for RND, which consists of three phases. In the first phase the graph representing the network is reduced by heuristically deleting a subset of the arcs, and a feasible solution is built. The second phase consists of a neighborhood search on the reduced graph based on a Mixed-Integer (Linear) Programming (MIP) flow model. Finally, the third phase applies a proximity search approach to further improve the solution, taking into account the original graph. The heuristic is tested on the new instances, and the comparison with the solutions obtained by Cplex on a natural flow formulation shows the effectiveness of the proposed method.     

General network design: A unified view of combined location and network design problems

This paper presents a unified framework for the general network design problem which encompasses several classical problems involving combined location and network design decisions. In some of these problems the service demand relates users and facilities, whereas in other cases the service demand relates pairs of users between them, and facilities are used to consolidate and re-route flows between users. Problems of this type arise in the design of transportation and telecommunication systems and include well-known problems such as location-network design problems, hub location problems, extensive facility location problems, tree-star location problems and cycle-star location problems, among others. Relevant modeling aspects, alternative formulations and possible algorithmic strategies are presented and analyzed.     

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.     

Necessary and sufficient condition in Lyapunov robust control: Multi-input case

We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A ^T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB ^T PB>0 and that the determinant ofB ^T PW be Hurwitz, whereW=block diag[w ₁,...,w _m ], with elementw _i =[s ^k _i –1,s ^k _i –2,...,s, 1] ^T ; here, the symbolsk _i ,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems.     

GRASP algorithms for the robust railway network design problem

This paper analyzes the solvability of a railway network design problem and its robust version. These problems are modeled as integer linear programming problems with binary variables, and their solutions provide topological railway networks maximizing the trip coverage in the presence of a competing mode, both assuming that the network works fine and that links can fail, respectively. Since these problems are computationally intractable for realistic sizes, GRASP heuristics are proposed for finding good feasible solutions. The results obtained in a computational experience indicate that our GRASP algorithms are suitable for railway network design problems.     

A comparison of routing sets for robust network design

Designing a network able to route a set of non-simultaneous demand vectors is an important problem arising in telecommunications. In this paper, we compare the optimal capacity allocation costs for six routing sets: affine routing, volume routing and its two simplifications, the routing based on an unrestricted 2-cover of the uncertainty set, and the routing based on a cover delimited by a hyperplane.     

An integrated model for logistics network design

In this paper we introduce a new formulation of the logistics network design problem encountered in deterministic, single-country, single-period contexts. Our formulation is flexible and integrates location and capacity choices for plants and warehouses with supplier and transportation mode selection, product range assignment and product flows. We next describe two approaches for solving the problem---a simplex-based branch-and-bound and a Benders decomposition approach. We then propose valid inequalities to strengthen the LP relaxation of the model and improve both algorithms. The computational experiments we conducted on realistic randomly generated data sets show that Benders decomposition is somewhat more advantageous on the more difficult problems. They also highlight the considerable performance improvement that the valid inequalities produce in both solution methods. Furthermore, when these constraints are incorporated in the Benders decomposition algorithm, this offers outstanding reoptimization capabilities.     

Survivable network design with demand uncertainty

The objective in designing a communications network is to find the most cost efficient network design that specifies hardware devices to be installed, the type of transmission links to be installed, and the routing strategy to be followed. In this paper algorithmic ideas are presented for improving tractability in solving the survivable network design problem by taking into account uncertainty in the traffic requirements. Strategies for improving separation of metric inequalities are presented and an iterative approach for obtaining solutions, that significantly reduces computing times, is introduced. Computational results are provided based on data collected from an operational network.     

Geometric partitioning and robust ad-hoc network design

We consider bounds for the price of a European-style call option under regime switching. Stochastic semidefinite programming models are developed that incorporate a lattice generated by a finite-state Markov chain regime-switching model as a representation of scenarios (uncertainty) to compute bounds. The optimal first-stage bound value is equivalent to a Value at Risk quantity, and the optimal solution can be obtained via simple sorting. The upper (lower) bounds from the stochastic model are bounded below (above) by the corresponding deterministic bounds and are always less conservative than their robust optimization (min-max) counterparts. In addition, penalty parameters in the model allow controllability in the degree to which the regime switching dynamics are incorporated into the bounds. We demonstrate the value of the stochastic solution (bound) and computational experiments using the S&P 500 index are performed that illustrate the advantages of the stochastic programming approach over the deterministic strategy.