Covering location problem of emergency service facilities in an uncertain environment

In practical location problems on networks, the response time between any pair of vertices and the demands of vertices are usually indeterminate. This paper employs uncertainty theory to address the location problem of emergency service facilities under uncertainty. We first model the location set covering problem in an uncertain environment, which is called the uncertain location set covering model. Using the inverse uncertainty distribution, the uncertain location set covering model can be transformed into an equivalent deterministic location model. Based on this equivalence relation, the uncertain location set covering model can be solved. Second, the maximal covering location problem is investigated in an uncertain environment. This paper first studies the uncertainty distribution of the covered demand that is associated with the covering constraint confidence level α. In addition, we model the maximal covering location problem in an uncertain environment using different modelling ideas, namely, the (α, β)-maximal covering location model and the α-chance maximal covering location model. It is also proved that the (α, β)-maximal covering location model can be transformed into an equivalent deterministic location model, and then, it can be solved. We also point out that there exists an equivalence relation between the (α, β)-maximal covering location model and the α-chance maximal covering location model, which leads to a method for solving the α-chance maximal covering location model. Finally, the ideas of uncertain models are illustrated by a case study.     

Uncertain programming model for uncertain optimal assignment problem

This paper employs uncertain programming to deal with uncertain optimal assignment problem in which profit is uncertain. Within the framework of uncertain programming, it gives the uncertainty distribution of the optimal assignment profit, and the concept of α-optimal assignment for uncertain optimal assignment problem is proposed. Then α-optimal model is also constructed. Taking advantage of properties of uncertainty theory, α-optimal model can be transformed into a corresponding deterministic form, which can be solved by Kuhn–Munkres algorithm.     

Forward–Backward Stochastic Differential Games and Stochastic Control under Model Uncertainty

We study optimal stochastic control problems with jumps under model uncertainty. We rewrite such problems as stochastic differential games of forward–backward stochastic differential equations. We prove general stochastic maximum principles for such games, both in the zero-sum case (finding conditions for saddle points) and for the nonzero sum games (finding conditions for Nash equilibria). We then apply these results to study robust optimal portfolio-consumption problems with penalty. We establish a connection between market viability under model uncertainty and equivalent martingale measures. In the case with entropic penalty, we prove a general reduction theorem, stating that a optimal portfolio-consumption problem under model uncertainty can be reduced to a classical portfolio-consumption problem under model certainty, with a change in the utility function, and we relate this to risk sensitive control. In particular, this result shows that model uncertainty increases the Arrow–Pratt risk aversion index.     

The balanced minimum evolution problem under uncertain data

We investigate the Robust Deviation Balanced Minimum Evolution Problem (RDBMEP), a combinatorial optimization problem that arises in computational biology when the evolutionary distances from taxa are uncertain and varying inside intervals. By exploiting some fundamental properties of the objective function, we present a mixed integer programming model to exactly solve instances of the RDBMEP and discuss the biological impact of uncertainty on the solutions to the problem. Our results give perspective on the mathematics of the RDBMEP and suggest new directions to tackle phylogeny estimation problems affected by uncertainty.     

Choosing realistic values of indifference,preference and veto thresholds for use with environmental criteria within ELECTRE

The ELECTRE III outranking model is particularly suited to aiding the choice between project alternatives on the basis of mainly environmental criteria. The model requires values of three criterion thresholds, the indifference threshold (q), the preference threshold (p) and the veto threshold (v). These allow the uncertainties inherent in the criteria valuations to be incorporated into the decision process. There is, at present, a high degree of subjectivity involved in determining these thresholds, which are expressed in terms of the error/uncertainty associated with the valuations of each of the criteria under scrutiny. If, however, the ELECTRE III outranking model is to be used within a formal environmental appraisal system, the thresholds which govern the outranking relationship of one project option over another must take account of the effect on human beings of the difference between any two criterion scores. The authors suggest a new method for applying the standard ELECTRE III model to decision-aid problems within the formal mechanism of environmental impact assessment. This involves a new, more comprehensive approach for specifying realistic limits for p, q and v, within the context of an environmental appraisal, where both criterion error/uncertainty and human sensitivity to differing levels of the criterion are taken into account. Threshold valuations for noise impacts from a highway project are used to illustrate the proposed method.     

Robust and reliable forward–reverse logistics network design under demand uncertainty and facility disruptions

There are two broad categories of risk, which influence the supply chain design and management. The first category is concerned with uncertainty embedded in the model parameters, which affects the problem of balancing supply and demand. The second category of risks may arise from natural disasters, strikes and economic disruptions, terroristic acts, and etc. Most of the existing studies surveyed these types of risk, separately. This paper proposes a robust and reliable model for an integrated forward–reverse logistics network design, which simultaneously takes uncertain parameters and facility disruptions into account. The proposed model is formulated based on a recent robust optimization approach to protect the network against uncertainty. Furthermore, a mixed integer linear programing model with augmented p-robust constraints is proposed to control the reliability of the network among disruption scenarios. The objective function of the proposed model is minimizing the nominal cost, while reducing disruption risk using the p-robustness criterion. To study the behavior of the robustness and reliability of the concerned network, several numerical examples are considered. Finally, a comparative analysis is carried out to study the performance of the augmented p-robust criterion and other conventional robust criteria.     

A Midpoint-Radius approach to regression with interval data

In this paper, a revisited interval approach for linear regression is proposed. In this context, according to the Midpoint-Radius (MR) representation, the uncertainty attached to the set-valued model can be decoupled from its trend. The estimated interval model is built from interval input-output data with the objective of covering all available data. The constrained optimization problem is addressed using a linear programming approach in which a new criterion is proposed for representing the global uncertainty of the interval model. The potential of the proposed method is illustrated by simulation examples.     

A two-person game of timing with random termination

We consider a marksmanship contest in which the first contestant to hit his target wins and the contest is to be terminated at a random timeT with cdfH(t). The model is evidently an extension of the classical discrete fire duel to the timing problem under an uncertain environment. It is shown that the uncertainty on the termination of the contest has influence on the equilibrium strategies and the equilibrium values.     

Sensory evaluation based on linguistic decision analysis

The evaluation processes are widely used for quality inspection, design, marketing exploitation and other fields in industrial companies. In many of these fields the items, products, designs, etc., are evaluated according to the knowledge acquired via human senses (sight, taste, touch, smell and hearing), in such cases, we talk about sensory evaluation, in it an important problem arises as it is the modelling and management of uncertain knowledge in the evaluation process, because the information acquired by our senses throughout human perceptions always involves uncertainty, vagueness and imprecision.The decision analysis techniques have been utilized in many evaluation processes, hence this paper proposes and shows the application of the linguistic decision analysis to sensory evaluation and its advantages, particularly based on the linguistic 2-tuple representation model, in order to model and manage consistently the uncertainty and vagueness of the information in this type of problems.     

Portfolio selection under model uncertainty: a penalized moment-based optimization approach

We present a new approach that enables investors to seek a reasonably robust policy for portfolio selection in the presence of rare but high-impact realization of moment uncertainty. In practice, portfolio managers face difficulty in seeking a balance between relying on their knowledge of a reference financial model and taking into account possible ambiguity of the model. Based on the concept of Distributionally Robust Optimization (DRO), we introduce a new penalty framework that provides investors flexibility to define prior reference models using the distributional information of the first two moments and accounts for model ambiguity in terms of extreme moment uncertainty. We show that in our approach a globally-optimal portfolio can in general be obtained in a computationally tractable manner. We also show that for a wide range of specifications our proposed model can be recast as semidefinite programs. Computational experiments show that our penalized moment-based approach outperforms classical DRO approaches in terms of both average and downside-risk performance using historical data.     

Conservative independence-based causal structure learning in absence of adjacency faithfulness

This paper presents an extension to the Conservative PC algorithm which is able to detect violations of adjacency faithfulness under causal sufficiency and triangle faithfulness. Violations can be characterized by pseudo-independent relations and equivalent edges, both generating a pattern of conditional independencies that cannot be modeled faithfully. Both cases lead to uncertainty about specific parts of the skeleton of the causal graph. These ambiguities are modeled by an f-pattern. We prove that our Adjacency Conservative PC algorithm is able to correctly learn the f-pattern. We argue that the solution also applies for the finite sample case if we accept that only strong edges can be identified. Experiments based on simulations and the ALARM benchmark model show that the rate of false edge removals is significantly reduced, at the expense of uncertainty on the skeleton and a higher sensitivity for accidental correlations.     

Forecasting using belief functions: An application to marketing econometrics

A method is proposed to quantify uncertainty on statistical forecasts using the formalism of belief functions. The approach is based on two steps. In the estimation step, a belief function on the parameter space is constructed from the normalized likelihood given the observed data. In the prediction step, the variable Y to be forecasted is written as a function of the parameter θ and an auxiliary random variable Z with known distribution not depending on the parameter, a model initially proposed by Dempster for statistical inference. Propagating beliefs about θ and Z through this model yields a predictive belief function on Y. The method is demonstrated on the problem of forecasting innovation diffusion using the Bass model, yielding a belief function on the number of adopters of an innovation in some future time period, based on past adoption data.     

Generalized light robustness and the trade-off between robustness and nominal quality

Robust optimization considers optimization problems with uncertainty in the data. The common data model assumes that the uncertainty can be represented by an uncertainty set. Classic robust optimization considers the solution under the worst case scenario. The resulting solutions are often too conservative, e.g. they have high costs compared to non-robust solutions. This is a reason for the development of less conservative robust models. In this paper we extract the basic idea of the concept of light robustness originally developed in Fischetti and Monaci (Robust and online large-scale optimization, volume 5868 of lecture note on computer science. Springer, Berlin, pp 61–84, 2009) for interval-based uncertainty sets and linear programs: fix a quality standard for the nominal solution and among all solutions satisfying this standard choose the most reliable one. We then use this idea in order to formulate the concept of light robustness for arbitrary optimization problems and arbitrary uncertainty sets. We call the resulting concept generalized light robustness. We analyze the concept and discuss its relation to other well-known robustness concepts such as strict robustness (Ben-Tal et al. in Robust optimization. Princeton University Press, Princeton, 2009), reliability (Ben-Tal and Nemirovski in Math Program A 88:411–424, 2000) or the approach of Bertsimas and Sim (Oper Res 52(1):35–53, 2004). We show that the light robust counterpart is computationally tractable for many different types of uncertainty sets, among them polyhedral or ellipsoidal uncertainty sets. We furthermore discuss the trade-off between robustness and nominal quality and show that non-dominated solutions with respect to nominal quality and robustness can be computed by the generalized light robustness approach.     

A jump-diffusion model for option pricing under fuzzy environments

Owing to fluctuations in the financial markets from time to time, the rate λ of Poisson process and jump sequence {V_i} in the Merton’s normal jump-diffusion model cannot be expected in a precise sense. Therefore, the fuzzy set theory proposed by Zadeh [Zadeh, L.A., 1965. Fuzzy sets. Inform. Control 8, 338-353] and the fuzzy random variable introduced by Kwakernaak [Kwakernaak, H., 1978. Fuzzy random variables I: Definitions and theorems. Inform. Sci. 15, 1-29] and Puri and Ralescu [Puri, M.L., Ralescu, D.A., 1986. Fuzzy random variables. J. Math. Anal. Appl. 114, 409-422] may be useful for modeling this kind of imprecise problem. In this paper, probability is applied to characterize the uncertainty as to whether jumps occur or not, and what the amplitudes are, while fuzziness is applied to characterize the uncertainty related to the exact number of jump times and the jump amplitudes, due to a lack of knowledge regarding financial markets. This paper presents a fuzzy normal jump-diffusion model for European option pricing, with uncertainty of both randomness and fuzziness in the jumps, which is a reasonable and a natural extension of the Merton [Merton, R.C., 1976. Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125-144] normal jump-diffusion model. Based on the crisp weighted possibilistic mean values of the fuzzy variables in fuzzy normal jump-diffusion model, we also obtain the crisp weighted possibilistic mean normal jump-diffusion model. Numerical analysis shows that the fuzzy normal jump-diffusion model and the crisp weighted possibilistic mean normal jump-diffusion model proposed in this paper are reasonable, and can be taken as reference pricing tools for financial investors.     

Probability,fuzziness and borderline cases

An integrated approach to truth-gaps and epistemic uncertainty is described, based on probability distributions defined over a set of three-valued truth models. This combines the explicit representation of borderline cases with both semantic and stochastic uncertainty, in order to define measures of subjective belief in vague propositions. Within this framework we investigate bridges between probability theory and fuzziness in a propositional logic setting. In particular, when the underlying truth model is from Kleene's three-valued logic then we provide a complete characterisation of compositional min–max fuzzy truth degrees. For classical and supervaluationist truth models we find partial bridges, with min and max combination rules only recoverable on a fragment of the language. Across all of these different types of truth valuations, min–max operators are resultant in those cases in which there is only uncertainty about the relative sharpness or vagueness of the interpretation of the language.     

Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ? ⁿ . Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform $\tilde f$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ? ⁿ , the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.     

Multicriteria estimation of probabilities on basis of expert non-numeric,non-exact and non-complete knowledge

A new method of alternatives’ probabilities estimation under deficiency of expert numeric information (obtained from different sources) is proposed. The method is based on the Bayesian model of uncertainty randomization. Additional non-numeric, non-exact, and non-complete expert knowledge (NNN-knowledge, NNN-information) is used for final estimation of the alternatives’ probabilities. An illustrative example demonstrates the proposed method application to forecasting of oil shares price with the use of NNN-information obtained from different experts (investment firms).     

On the confidence preferences model

In this paper we study the model of decision under uncertainty consistent with confidence preferences. In that model, a decision maker held beliefs represented by a fuzzy set of priors and tastes captured by a standard affine utility index on consequences. First, we find some interesting properties concerning the well-known maxmin expected utility model, taking into account the point of view of the confidence preferences model. Further, we provide new examples of preferences that capture ambiguity-averse attitudes weaker than ambiguity attitudes featured by maxmin expected utility theory. Finally, we discuss the axiomatic foundations for the confidence preferences model with optimistic behavior.     

MRP-controlled batch-manufacturing environment under uncertainty

The overall aim of this research is to model the effects of uncertainty on delivery performance in an MRP-controlled batch-manufacturing environment with multi-product and multi-level dependent demand. To this end, MRP planning and batch-manufacturing system control architectures were modelled using simulation to implement the MRP release logic. Simulation and experimental design were carried out based on a real case enterprise. ANOVA showed that four uncertainty factors—namely late delivery from suppliers, machine breakdowns, process batch size increments and customer design changes—have significant effects on delivery performance. This ANOVA further showed that uncertainties create knock-on and compound effects; the latter are difficult to predict in practice. Significant two-way and three-way interactions among some uncertainty factors were also found, making it more difficult to characterise the precise factor effects. It was found that the more uncertain the environment is, the later the deliveries are. It can be concluded that MRP-controlled batch-manufacturing enterprises should diagnose uncertainties that are significantly affecting delivery performance, and tackle these uncertainties most urgently to prevent diffusion of knock-on and compound effects and improve delivery performance. This conclusion was validated through the case enterprise, for which significant delivery improvement has been achieved.     

Uncertainty Inequalities on Laguerre Hypergroup

In this paper, we give analogues of local uncertainty inequality on ${\mathbb{R}^n}$ for stratified Laguerre hypergroup, connected with the spectral analysis of a given homogeneous sublaplacian L, also indicate how local uncertainty inequalities imply global uncertainty inequalities. It would be interesting to note that we deduce the local uncertainty inequalities for the radial functions on the Heisenberg group. Finally, we extend Heisenberg-Pauli-Weyl uncertainty inequality by ultracontractive properties of the semigroups generated by the differential operator and on the estimate on the heat kernel.