A robustness approach to uncapacitated network design problems

In this paper, we address uncapacitated network design problems characterised by uncertainty in the input data. Network design choices have a determinant impact on the effectiveness of the system. Design decisions are frequently made with a great degree of uncertainty about the conditions under which the system will be required to operate. Instead of finding optimal designs for a given future scenario, designers often search for network configurations that are “good” for a variety of likely future scenarios. This approach is referred to as the “robustness” approach to system design. We present a formal definition of “robustness” for the uncapacitated network design problem, and develop algorithms aimed at finding robust network designs. These algorithms are adaptations of the Benders decomposition methodology that are tailored so they can efficiently identify robust network designs. We tested the proposed algorithms on a set of randomly generated problems. Our computational experiments showed two important properties. First, robust solutions are abundant in uncapacitated network design problems, and second, the proposed algorithms performance is satisfactory in terms of cost and number of robust network designs obtained.     

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.     

On one approach to the problem of robustness of density estimators

In this paper, we characterize the robustness of a density estimator by the square error increment caused by the “contamination” of a sample by a preassigned value ε. The dependence of this characteristic on ε is presented. A model example is considered. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 28–33, Perm, 1993.     

An efficient approach for dual sourcing inventory systems with Tailored Base-Surge policies

Nowadays most companies use dual sourcing strategies to manage supply chains. Recently, a practical policy called the Tailored Base-Surge (TBS) policy has been proposed to manage the system, and shown to perform well. We study a dual sourcing inventory system with i.i.d. demands and TBS policies. Using the techniques of MacLaurin series analysis, we develop an efficient method to calculate the moments of the inventory position, the bullwhip effect, and performance measures. Numerical experiments show that our method works well.     

Modeling sourcing strategies to mitigate part obsolescence

Part obsolescence is a common problem across industries, from avionics and military sectors to most original equipment manufacturers serving industrial markets. When a part supplier announces that a part will become obsolete, the OEM can choose from a number of sourcing options. In practice, the three most commonly adopted mitigation strategies are: (1) a lifetime, or life-of-type (LOT), buy from the original supplier; (2) part substitution, which finds a suitable alternative; and (3) line redesign, which modifies the production line to accommodate a new part. We first develop a framework incorporating fixed cost, variable cost, leadtime, demand uncertainty and the discount rate to directly compare and characterize these three sourcing strategies in a static context. We next formulate an integrated sourcing approach that starts with a bridge buy and may continue with part substitution or line redesign when the originals parts are depleted. Through numerical studies, we identify the joint impact of the problem parameters on the static and integrated sourcing strategies and the optimal choice among them. While the integrated sourcing approach outperforms the static ones in many cases it is not a dominant strategy.     

A framework for robustness to ambiguity of higher-order beliefs

Standard type spaces induce belief structures defined by precise beliefs. This paper proposes and analyzes simple procedures for constructing perturbations of such belief structures in which beliefs have a degree of ambiguity. Specifically, we construct ambiguous type spaces whose induced (ambiguous) belief hierarchies approximate the standard, precise, belief hierarchies corresponding to the initial type space. Based on a metric that captures the resulting approximation, two alternative procedures to construct such perturbations are introduced, and are shown to yield a simple and intuitive characterization of convergence to the initial unperturbed environment. As a special case, one of these procedures is shown to characterize the set of all finite perturbations. The introduced perturbations and their convergence properties provide conceptual foundations for the analysis of robustness to ambiguity of various solutions concepts, and for various decision rules under ambiguity.     

A unified approach for different concepts of robustness and stochastic programming via non-linear scalarizing functionals

Abstract

We show that many different concepts of robustness and of stochastic programming can be described as special cases of a general non-linear scalarization method by choosing the involved parameters and sets appropriately. This leads to a unifying concept which can be used to handle robust and stochastic optimization problems. Furthermore, we introduce multiple objective (deterministic) counterparts for uncertain optimization problems and discuss their relations to well-known scalar robust optimization problems by using the non-linear scalarization concept. Finally, we mention some relations between robustness and coherent risk measures.     

A condition for null robustness

Sufficient conditions are given that certain statistics have a common distribution under a wide class of underlying distributions. Invariance methods are the primary technical tool in establishing the theoretical results. These results are applied to MANOVA problems, problems involving canonical correlations, and certain statistics associated with the complex normal distribution.     

A systematic approach to pricing and hedging international derivatives with interest rate risk: analysis of international derivatives under stochastic interest rates

This paper deals with the valuation and the hedging of non-path-dependent European options on one or several underlying assets in a model of an international economy allowing for both, interest rate risk and exchange rate risk. Using martingale theory and, in particular, the change of numeraire technique we provide a unified and easily applicable approach to pricing and hedging exchange options on stocks, bonds, futures, interest rates and exchange rates. We also cover the pricing and hedging of compound exchange options.     

A comparison of two sourcing tactics for a new component

We compare two sourcing tactics for a manufacturer to purchase a new component to be used in a one-time production run of a new product with uncertain and price-elastic demand. One alternative is to issue a request-for-quote (RFQ), which is where the manufacturer requests a price-quantity schedule from suppliers. The manufacturer uses this information to determine a production quantity and the number of components to purchase from each supplier. The other alternative is to post a bid specifying how the manufacturer’s purchase quantity will depend on the supplier’s component price. The suppliers use this information to compete on quantity.We find that relative to RFQ, which is more challenging for the manufacturer to characterize the supplier response due to the possibility of supplier interaction, the benefit to the manufacturer from posting a bid increases with the number of suppliers due to increased intensity of competition. If the new component is from an emerging industry where there is little mutual awareness among candidate suppliers, then regardless of number of suppliers, expected manufacturer profit is higher under RFQ. Posting a bid is more likely to benefit the manufacturer when the new component is from a more established industry with a high degree of awareness among candidate suppliers.     

Tariff concessions in production sourcing

In this paper, we study a multi-stage production sourcing problem where tariff concessions can be exploited at the firm level using free trade agreements between countries. To solve the problem, an algorithm which embeds a very large-scale neighborhood (VSLN) search into a simulated annealing framework is developed. A numerical study is conducted to verify the effectiveness of the solution approach.     

A capacitated facility location and inventory management problem with single sourcing

The current paper addresses the integrated location and inventory problem with capacity constraints. Adopting more realistic assumptions comes at the cost of increased complexity and inability to solve the model with existing methods, mainly due to the non-linear terms that arise. We attempt to render the extended formulation solvable by linearizing its non-linear terms. Certain terms are replaced by exact reformulations while for the rest a piecewise linearization is implemented. The contribution of this work is not only the development of a formulation that is more practical, but also the reformulation that enables its solving with commercial software. We test our proposed approach on a benchmark dataset from the literature, including both small and large instances of the problem. Results clearly demonstrate the superiority of this approach in terms of both solution quality and computational time.     

A notion of robustness and stability of manifolds

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C¹ with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C² is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic.     

A robustness study in sampling on successive occasions

Summary In this paper, we have undertaken an investigation covering three occasions in sampling on successive occasions with a view to examining efficiency robustness of the best linear unbiased estimator (BLUE) visa-vis certain other potentially conceivable estimators when the usual correlation model breaks down. We have inferred that the BLUE, is by and large, an efficiency robust estimate in the face of unforeseen deviations from the usual correlation model.     

On structural stability and robustness to bounded rationality

In this paper, we study the model M$M$, a parameterized class of “general games” together with an associated abstract rationality function. We prove that model M$M$ is structurally stable and robust to ?$?$-equilibria for “almost all” parameter values.     

Single versus multiple supplier sourcing strategies

Successful supply chain management necessitates an effective sourcing strategy to combat uncertainties in both supply and demand. In particular, supply disruption results in excessive downtime of production resources, upstream and downstream supply chain repercussions, and eventually a loss in the market value of the firm. In this paper we analyze single period, single product sourcing decisions under demand uncertainty. Our approach integrates product prices, supplier costs, supplier capacities, historical supplier reliabilities and firm specific inventory costs. A unique feature of our approach is the integration of a firm specific supplier diversification function. We also extend our analysis to examine the impact of minimum supplier order quantities. Our results indicate that single sourcing is a dominant strategy only when supplier capacities are large relative to the product demand and when the firm does not obtain diversification benefits. In other cases, we find that multiple sourcing is an optimal sourcing strategy. We also characterize a non-intuitive trade-off between supplier minimum order quantities, costs, and supplier reliabilities. Finally, we examine the robustness of our results through an extensive numerical analysis of the key parameters of our model.     

Improving robustness of solutions to arc routing problems

This paper considers the stochastic capacitated arc routing problem (SCARP), obtained by taking random demands in the CARP. For real-world problems, it is important to create solutions that are insensitive to changes in demand, because these quantities are not deterministic but randomly distributed. This paper provides the basic concept of a new technique to compute such solutions, based upon the best method published for CARP: a hybrid genetic algorithm (HGA). The simulation analysis was achieved with the well-known DeArmon's, Eglese's and Belenguer's instances. This intensive evaluation process was carried out with 1000 replications providing high-quality statistical data. The results obtained prove that there is a great interest to optimize not only the solution cost but also the robustness of solutions. This work is a step forward to treat more realistic problems including industrial goals and constraints linked to demand variations.     

Ranking robustness and its application to evacuation planning

We present a new approach to handle uncertain combinatorial optimization problems that uses solution ranking procedures to determine the degree of robustness of a solution. Unlike classic concepts for robust optimization, our approach is not purely based on absolute quantitative performance, but also includes qualitative aspects that are of major importance for the decision maker.We discuss the two variants, solution ranking and objective ranking robustness, in more detail, presenting problem complexities and solution approaches. Using an uncertain shortest path problem as a computational example, the potential of our approach is demonstrated in the context of evacuation planning due to river flooding.