Finite valued feedback laws and piecewise classical solutions

Often, in engineering literature, we find control systems in which the open loop inputs are piecewise constant and take values in a finite set. Such open loop inputs cause the system to have fairly regular solutions. On the other hand, when acting in closed loop, feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type. In fact, pathological behaviours such as the accumulation of discontinuities may appear (Zeno phenomenon). We give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours.     

Partial Data for the Neumann-to-Dirichlet Map

We show that measurements of a Neumann-to-Dirichlet map, with either inputs or outputs restricted to part of the boundary, can determine an electric potential on that domain. Given a convexity condition on the domain, either the set on which measurements are taken, or the set on which input functions are supported, can be made to be arbitrarily small. The result is analogous to the result by Kenig, Sjöstrand, and Uhlmann for the Dirichlet-to-Neumann map. The main new ingredient in the proof is an improved Carleman estimate for the Schrödinger operator with appropriate boundary conditions. This is proved by Fourier analysis of a conjugated operator along the boundary of the domain.     

Unobservable or omitted production variables in data envelopment analysis through unit-specific production trade-offs

The paper focuses upon situations in which decision-making units carry out their production activities with some inputs or outputs unobservable (or possibly omitted), and when there are a priori known constraints on the relative significance of otherwise observable (or explicitly considered) inputs and outputs. For such settings, the paper proposes a modification that alters traditional construction of the production possibility set and isolates the role of the unobservable (or omitted) variables in production by means of restrictions on virtual inputs and outputs being converted into production trade-offs. In effect, the proposed procedure induces unit-specific production possibility sets that derive from production trade-offs framed for units assessed separately to reflect their specific observed production conditions. The modification is implemented within a weighted slacks-based measure with restricted direction of slacks in order to make technical efficiency measurement more informative and consistent with the operating conditions under which production activities are accomplished. These ideas are illustrated and models implemented in a case study of bank branch performance measurement.     

Estimating output gains by means of Luenberger efficiency measures

In this paper we propose a new measure of input allocative efficiency that we estimate using directional distance functions. Our new measure compares the gain in output if a firm reduces technical inefficiency for the direct production possibility set and the gain in output if the firm reduces technical inefficiency for the indirect production possibility set. Because the directional distance function uses a translated origin, the gain in output from an optimal reallocation of inputs can be estimated for non-radial expansions in output. We estimate efficiency for Japanese banks during 1992–1999. The gains in outputs from reducing allocative inefficiency by reallocating inputs are greater than the gains in outputs that can be attained by reducing technical inefficiency.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

The LeChatelier principle in a DEA model

This paper aims to relate the LeChatelier principle, first introduced into economics by Samuelson (1947), with the DEA approach through two propositions. These propositions allow for bridging the principle over a DEA model with and without the presence of non-discretionary inputs and enable one to make comparisons for the various efficiency measures under different conditions. The quasi-fixity of some inputs hinders a firm’s capacity from instantly and freely adjusting its input combination in order to minimize its production costs. The assumption that all inputs are discretionary tends to exaggerate managers’ ability to dispense resources and renders invalid information on the adjustment of the current input mix.     

Common set of weights in data envelopment analysis: a linear programming problem

Data Envelopment Analysis is used to determine the relative efficiency of Decision Making Units as the ratio of weighted sum of outputs by weighted sum of inputs. To accomplish the purpose, a DEA model calculates the weights of inputs and outputs of each DMU individually so that the highest efficiency can be estimated. Thus, the present study suggests an innovative method using a common set of weights leading to solving a linear programming problem. The method determines the efficiency score of all DMUs and rank them too.     

Neural network as a simulation metamodel in economic analysis of risky projects

An artificial neural network (ANN) model for economic analysis of risky projects is presented in this paper. Outputs of conventional simulation models are used as neural network training inputs. The neural network model is then used to predict the potential returns from an investment project having stochastic parameters. The nondeterministic aspects of the project include the initial investment, the magnitude of the rate of return, and the investment period. Backpropagation method is used in the neural network modeling. Sigmoid and hyperbolic tangent functions are used in the learning aspect of the system. Analysis of the outputs of the neural network model indicates that more predictive capability can be achieved by coupling conventional simulation with neural network approaches. The trained network was able to predict simulation output based on the input values with very good accuracy for conditions not in its training set. This allowed an analysis of the future performance of the investment project without having to run additional expensive and time-consuming simulation experiments.     

On the control of a generalized storage model

A finite-capacity storage model is considered. The random inputs (negative inputs represent demands) are of various types, determined by a Markov chain, and occur at discrete times. Under suitable assumptions on the costs involved, including a penalty cost for unmet demand, an optimal control policy is determined for the releases from the storage facility, when operated over a finite horizon. Stationary control policies for the unbounded horizon are also determined and conditions for their optimality are discussed. Finally, a few simple examples are considered.The author would like to acknowledge the constructive comments of the referee, which led to an improved exposition of the present paper.     

Sensitivity analysis of efficient units in the presence of non-discretionary inputs

Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs can be varied at the discretion of management or other users. In any realistic situation, however, there may exist “exogenously fixed” or non-discretionary factors that are beyond the control of a DMU’s management, which also need to be considered. This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses when some inputs are exogenously fixed. Super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. In this paper by means of modified Banker and Morey’s (BM hereafter) model [R.D. Banker, R. Morey, Efficiency analysis for exogenously fixed inputs and outputs, Operations Research 34 (1986) 513–521], in which the test DMU is excluded from the reference set, we are able to determine what perturbations of discretionary data can be tolerated before frontier DMUs become nonfrontier.     

Mittag-Leffler stability of fractional-order Hopfield neural networks

Fractional-order Hopfield neural networks are often used to model how interacting neurons process information. To show reliability of the processed information, it is needed to perform stability analysis of these systems. Here, we perform Mittag-Leffler stability analysis for them. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality that can be effectively used to this analysis. Importantly, these general results can help construct Lyapunov functions used to Mittag-Leffler stability analysis of fractional-order Hopfield neural networks. As a result, a set of sufficient conditions is derived to guarantee this stability. In addition, the general results can be easily used to the establishment of stability conditions for achieving complete and quasi synchronization in the coupling case of these networks with constant or time-dependent external inputs. Finally, two numerical examples are presented to show the effectiveness of our theoretical results.     

Evaluating returns to scale and congestion by production possibility set in intersection form

This research proposes a new method to estimate returns to scale(RTS) of decision making units(DM Us) with multiple inputs and outputs.The state of return to scale includes increasing RTS,constant RTS,decreasing RTS and evidence of congestion.The method is based on the production possibility set in the intersection form given by a set of linear inequalities.We propose and prove the necessary and sufficient conditions for the RTS estimation.With the new procedure,to estimate the RTS of a DM U is simply to ch...     

Triple integral approach to reachable set bounding for linear singular systems with time‐varying delay

This paper is concerned with the reachable set estimation problem of singular systems with time‐varying delay and bounded disturbance inputs. Based on a novel Lyapunov–Krasovskii functional that contains four triple integral terms, reciprocally convex approach and free‐weighting matrix method, two sufficient conditions are derived in terms of linear matrix inequalities to guarantee that the reachable set of singular systems with time‐varying delay is bounded by the intersection of ellipsoid. Finally, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Fluid-Structure Port-Hamiltonian Model for Incompressible Flows in Tubes with Time Varying Geometries

ABSTRACT

A simple and scalable finite-dimensional model based on the port-Hamiltonian framework is proposed to describe the fluid–structure interaction in tubes with time-varying geometries. For this purpose, the moving tube wall is described by a set of mass-spring-damper systems while the fluid is considered as a one-dimensional incompressible flow described by the average momentum dynamics in a set of incompressible flow sections. To couple these flow sections small compressible volumes are defined to describe the pressure between two adjacent fluid sections. The fluid-structure coupling is done through a power-preserving interconnection between velocities and forces. The resultant model includes external inputs for the fluid and inputs for external forces over the mechanical part that can be used for control or interconnection purposes. Numerical examples show the accordance of this simplified model with finite-element models reported in the literature.     

An input relaxation measure of efficiency in stochastic data envelopment analysis

We introduce stochastic version of an input relaxation model in data envelopment analysis (DEA). The input relaxation model, recently developed in DEA, is useful to resource management [e.g. G.R. Jahanshahloo, M. Khodabakhshi, Suitable combination of inputs for improving outputs in DEA with determining input congestion, Appl. Math. Comput. 151(1) (2004) 263–273]. This model allows more changes in the input combinations of decision making units than those in the observed inputs of evaluating decision making units. Using this extra flexibility in input combinations we can find better outputs. We obtain a non-linear deterministic equivalent to this stochastic model. It is shown that under fairly general conditions this non-linear model can be replaced by an ordinary deterministic DEA model. The model is illustrated using a real data set.     

Ranking DMUs by l1-norm with fuzzy data in DEA

The relative efficiency of a DMU is the result of comparing the inputs and outputs of the DMU and those of other DMUs in the PPS (production possibility set). If the inputs and outputs are fuzzy, the DMUs cannot be easily evaluated and ranked using the obtained efficiency scores. In this paper, presenting a new idea for ranking of DMUs with fuzzy data. And finally, we introduce a numerical example.     

Data envelopment analysis with nonlinear virtual inputs and outputs

An underlying assumption in DEA is that the weights coupled with the ratio scales of the inputs and outputs imply linear value functions. In this paper, we present a general modeling approach to deal with outputs and/or inputs that are characterized by nonlinear value functions. To this end, we represent the nonlinear virtual outputs and/or inputs in a piece-wise linear fashion. We give the CCR model that can assess the efficiency of the units in the presence of nonlinear virtual inputs and outputs. Further, we extend the models with the assurance region approach to deal with concave output and convex input value functions. Actually, our formulations indicate a transformation of the original data set to an augmented data set where standard DEA models can then be applied, remaining thus in the grounds of the standard DEA methodology. To underline the usefulness of such a new development, we revisit a previous work of one of the authors dealing with the assessment of the human development index on the light of DEA.     

Using a group support system to aid input–output identification in DEA

In data envelopment analysis (DEA), operating units are compared on their outputs relative to their inputs. The identification of an appropriate input–output set is of decisive significance if assessment of the relative performance of the units is not to be biased. This paper reports on a novel approach used for identifying a suitable input–output set for assessing central administrative services at universities. A computer-supported group support system was used with an advisory board to enable the analysts to extract information pertaining to the boundaries of the unit of assessment and the corresponding input–output variables. The approach provides for a more comprehensive and less inhibited discussion of input–output variables to inform the DEA model.     

Perturbation analysis of the analytic hierarchy decision method

The input set for the Analytic Hierarchy Process consists of the integers less than ten and their reciprocals. A perturbation in the input may cause significant change in the output vector, particularly when the dimension of the input matrix is small (n ≤ 4) and the inputs are close together. We examine that change using numerical computational schemes. The cardinal values of the output vector are seen to suffer significant perturbation due to small integral changes on all the inputs. Ordinal perturbations are less pronounced and can be better quantified by a numerical scheme. Known indices of consistency provide an indicator of truly poor input data. For an unskilled modeler, they can provide a false sense of assurance, but these consistency measures do not protect the modeler from consistently inaccurate inputs.     

Stabilization of a stochastic nonlinear economy

Methodology for dynamical systems described by nonlinear stochastic differential equations is applied to economic systems, particularly the Phillips-Turnovsky model, generalizing the model further to nonlinear terms and stochastic coefficients and inputs, to determine expectations and correlation matrices for the variables which represent solution processes of a set of coupled equations. The theory is quite general and will apply if the model is changed to use different nonlinear forms, e.g., for taxes as a function of income, or different statistics for fluctuations (which need not be stationary or Gaussian or small), or, to include retarded effects.