Our first main result says t.hat solvability of disullificatioll problems ill tile cOlllbi- nation of disjoint equational theories can be reduced to solvability of disl[r]

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Keywords: entropy fuzzy, F-PROMETHEE, multi criteria **decision** making, fuzzy set, **decision** maker, fuzzy environment
1. Introduction
Multiple criteria **decision** making (MCDM) is a powerful tool used widely **for** appraisement **and** ranking **problems** containing multiple, usually conflicting, criteria (Bilsel, Buyukozkan, & Ruan, 2006). Selecting a proper method requires an insight analysis among available MCDM **techniques**. Multi Criteria **Decision** Making (MDCM) methods have been implemented frequently in terms of solving different **problems** in both of certain **and** uncertain environments. Additionally, it is increasingly used in environmental policy evaluation as (a) it offers the possibility to deal with intricate issues, (b) it incorporates criteria that are difficult to monetize, (c) it represents a holistic view incorporating tangible as well as intangible (or ‘fuzzier’) aspects, often neglected by other evaluation **techniques** such as AHP (Munda, 2004).

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The considered engineering system is a portal frame under vertical **and** horizontal point loads, see Section 2. It is a simple mechanical system that helps avoiding false interpretation of results with respect to system behavior **and** failure states. Two competing failure modes, material failure **and** stability failure, make the limit state function strongly nonlinear **and** sensitive to uncertainties. Such a **combination** of failure modes is typical **for** many structural systems. Data **for** the structural **and** loading parameters are available more or less sparsely in Section 3. Using these information, three challenges formulated in Section 4 have to be solved. One of the crucial issues is an objective comparison of results caused by different underlying models **for** the uncertain input parameters. Obviously, this could be done on the basis of the decisions made. The second challenge of the benchmark deals with the question how data assimilation can help in the **decision** making. Finally, a design problem under polymorphic uncertainty is posed in which more demanding operation requirements have to be fulfilled. Reference solutions are given **for** comparison in Section 5 which have been determined by using the "true" structural **and** loading parameters which are only known to the authors. Final remarks can be found in Section 6.

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This chapter discusses the application of the model-reduction methods presented in the previous chapter to the specified TPM models derived in Chapter 2, as well as their numer- ical implementation. Therefore, it is necessary to provide further information on the corre- sponding problem settings such as the particularly chosen initial-boundary-value problem, the respective reduced system or the underlying processes to sample the snapshots. Starting with the relatively simple porous-soil model with linear-elastic material be- haviour, which results in a system of equations with (approximately) time-invariant sys- tem matrices (hereinafter referred to as linear equation systems), the suitability of a system reduction using the (modified) POD method is demonstrated **for** different prob- lem settings. Afterwards, the transition to time-efficient simulations of a nonlinear porous material undergoing large deformations, modelled with a Neo-Hookean approach, is dis- cussed. In this context, the DEIM is used as an additional method in **combination** with the POD method to deal with the occurring nonlinearities. Next, a reduction of dif- ferent biomechanical **problems** is further examined. Therefore, reduced simulations of drug-infusion processes in the human brain are initially performed. In particular, the POD method is first applied to the simplified brain-tissue model with a linear equation system, implying isotropic permeability conditions **and** the further assumptions of the simplified model derived in Subsection 3.3.2. Then, the POD-DEIM approach is used to reduce the system of equations of the general (nonlinear) brain-tissue model under anisotropic permeability conditions, investigating a variation of the material parameters. Subsequently, the POD-DEIM approach is applied to the intervertebral-disc model spec- ified in Subsection 3.3.3, studying the simulation of different deformation states. In this regard, the selection of specific snapshots is extensively investigated, since an appropriate choice strongly affects the quality of the reduced simulations. The main focus in all the numerical examples is on the investigation of the accuracy of the reduced models, as well as on the examination of the resulting time savings due to the system reduction. Finally, a generalised approach **for** a reduction of a coupled system of equations using the evolved modifications is presented in order to enable a systematic adaptation to other models. **For** all intents, the numerical implementation is realised with the coupled FE solver PANDAS . All computations were performed on a single core of an Intel i5-4590 with 32 GB of memory running at clock speed of 3.30 GHz. While the FE meshes **for** sim- ple geometrical **problems** are directly generated in the FE solver, the program package CUBIT 1 is used to define FE meshes **for** more complex geometries. Moreover, the deter-

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The outline of the chapter is as follows. The construction of lower **and** upper bounds on one component of the optimal value vector is described in Section 3.1. In Section 3.2 we present our approximation theorem showing that an ε-approximation **for** the component can be obtained by taking into account only a local part of the state space whose size is independent of the total number of states in the MDP. Section 3.3 introduces the foundation of our approximation algorithm based on column generation, which is the main contribution of this thesis. In Section 3.4 we describe how the algorithm can be utilized to obtain also approximations **for** a concrete policy or a single action. Finally, Section 3.5 is devoted to several theoretical **and** practical is- sues related to the column generation algorithm. In particular, we investigate the structure of the encountered dual linear programs, derive a combinato- rial formula **for** the reduced profits, **and** propose various pricing strategies **and** approximation heuristics. Moreover, we develop an equivalent variant of the approximation algorithm that refrains from using linear programming **techniques**, but employs the policy iteration method instead.

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The main characteristics of all versions of ELECTRE methods were summarized by Roy (68), as shown in Table 2.4. Considering different problem statements, some guide- lines on how to choose among ELECTRE methods were also suggested. **For** instance, if it is truly essential to work with a very simple method **and** it is realistic to have no information on the indifference threshold **and** preference threshold, ELECTRE I should be selected in order to eliminate the non-dominated alternatives, while ELECTRE II should be used in order to build a partial pre-order of alternatives. ELECTRE VI would be convenient only if there exists a good reason to refusing the introduction of importance coefficients. In general, ELECTRE IS, II, III, IV, **and** TRI do provide powerful support **for** the classification of the alternatives. However, they require too many threshold definitions from DMs, thus, it is rather complex to implement these methods in real world **problems** (60).

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The considered engineering system is a portal frame under vertical **and** horizontal point loads. It is a simple mechanical system that helps avoiding false interpretation of results with respect to system behavior **and** failure states. On the other hand, it possesses two competing failure modes, material failure **and** stability failure, which make the limit state function strongly nonlinear **and** sensitive to uncertainties. Such a **combination** of failure modes is typical **for** many structural systems. The benchmark should demonstrate how real **decision** making can work in the case of polymorphic uncertainties. One of the crucial issues is an objective comparison of results with different origin, namely, probabilistic **and** non-probabilistic ones. Perhaps, this could be done on the basis of the decisions made. The second challenge of the benchmark deals with the question how data assimilation can help in the **decision** making. Finally, a design problem under polymorphic uncertainty is given in which more demanding operation requirements have to be fulfilled.

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Keywords: Classification; Data mining; Machine learning; **Decision** making; Asthma; Pulmonary sound signals; Discrete wavelet transformation
Background
As the **decision** situations become increasingly more complex, advanced analytical **techniques** are gaining popularity in addressing wide variety of problem types (descrip- tive, predictive **and** prescriptive) in many fields including healthcare **and** medicine (Delen et al. 2009). Because of the rapid increase in the collection **and** storage of large quantities of data (facilitated by improving software **and** hardware capabilities coupled with increasingly lower cost of acquiring **and** using them), data **and** model driven deci- sion making (a.k.a. analytics) is becoming a mainstream practice in every field imagin- able (from art to business, medicine to science). One area where faster **and** better decisions could make a significant difference is in healthcare/medicine. This data rich field can undoubtedly use what modern day **decision** analytics has to offer (Oztekin et al. 2009). In this study, we used analytics to address a classification type **decision** problem, namely prediction of asthma using only the chest sound signals obtained from actual patients using ordinary microphones.

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provided in literature. This dissertation contributes a pixel-based algorithm to detect increased backscattering in SAR images by analyzing the SAR pixel values according to simulated layers. To detect demolished buildings, simulated images are generated using LiDAR data. Two comparison operators (normalized mutual information **and** joint histogram slope) are used to compare image patches related to same buildings. An experiment using Munich data has shown that both of them provide an overall accuracy of more than 90%. A **combination** of these two comparison operators using **decision** trees improves the result. The fourth objective is to detect changes between SAR images acquired with different incidence angles. **For** this purpose, three algorithms are presented in this dissertation. The first algorithm is a building-level algorithm based on layer fill. Image patches related to the same buildings in the two SAR images are extracted using simulation methods. **For** each extracted image patch pair, the change ratio based on the fill ratio of building layers is estimated. The change ratio values of all buildings are then classified into two classes using the EM-algorithm. This algorithm works well **for** buildings with different size **and** shape in complex urban scenarios. Since the whole building is analyzed as one object, buildings with partly demolished walls may not be detected. Under the same idea, a wall-level change detection algorithm was developed. Image patches related to the same walls in the two SAR images were extracted **and** converted to have the same geometry. These converted patch pairs are then compared using change ratios based on fill ratio or fill position. Lastly, the wall change results are fused to provide building change result. Compared to the building-level change detection algorithm, this method is more time consuming, but yields better results **for** partly demolished buildings. A **combination** of these two algorithms is therefore suggested, whereby the building-level method is used **for** all buildings **and** wall-level method additionally **for** selected large buildings. The third developed algorithm is a wall-level change detection algorithm based on point-feature location. To this end, local maximum points in two SAR images corresponding to the same building façade are compared. This method provides promising result **for** the present data. It may work better **for** future data with increased resolution to detect changes of detailed façade structures.

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Received: 14 October 2016 / Accepted: 14 March 2017 / Published online: 23 March 2017 The Author(s) 2017. This article is an open access publication
Abstract Many hierarchical **techniques** to solve large Markov **decision** processes (MDPs) are based on the par- tition of the state space into strongly connected compo- nents (SCCs) that can be classified into some levels. In each level, smaller **problems** named restricted MDPs are solved, **and** then these partial solutions are combined to obtain the global solution. In this paper, we first propose a novel algorithm, which is a variant of Tarjan’s algorithm that simultaneously finds the SCCs **and** their belonging levels. Second, a new definition of the restricted MDPs is presented to ameliorate some hierarchical solutions in discounted MDPs using value iteration (VI) algorithm based on a list of state-action successors. Finally, a robotic motion-planning example **and** the experiment results are presented to illustrate the benefit of the proposed decom- position algorithms.

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The considered engineering system is a portal frame under vertical **and** horizontal point loads. It is a simple mechanical system that helps avoiding false interpretation of results with respect to system behavior **and** failure states. On the other hand, it possesses two competing failure modes, material failure **and** stability failure, which make the limit state function strongly nonlinear **and** sensitive to uncertainties. Such a **combination** of failure modes is typical **for** many structural systems. The benchmark should demonstrate how real **decision** making can work in the case of polymorphic uncertainties. One of the crucial issues is an objective comparison of results with different origin, namely, probabilistic **and** non-probabilistic ones. Perhaps, this could be done on the basis of the decisions made. The second challenge of the benchmark deals with the question how data assimilation can help in the **decision** making. Finally, a design problem under polymorphic uncertainty is given in which more demanding operation requirements have to be fulfilled.

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the systems of ODEs arising in the high-order context is that constraints, such as memory requirements **and** stability limits (**for** explicit relaxation), become stricter. Furthermore, experience shows that more effort is needed in order to have both algo- rithmic **and** storage efficiency [122]. Some advances have already been made: just as an example, one of the 5 cores of the European ADIGMA research project [121, 122] was dedicated to solution strategies. Nonetheless, many questions have still to be considered in order to make high-order methods competitive. The role played by re- laxation is essential to efficiency, with respect to both computational cost **and** storage requirements. In general, the larger the number of ‘difficulties’ in the problem to be solved (stability issues, stiffness of the matrix, memory limitations, ill-conditioning, slow convergence), the larger the number of choices (which **combination** of methods leads to the solution with minimum effort) **and** the number of parameters to be tuned. Sections 3.2 **and** 3.3 are devoted to explicit **and** implicit **techniques**, respectively. Therein we turn our attention also to stability issues, multigrid methods, storage issues **and** preconditioning. In particular, we review those existing time-relaxation **techniques** which play a role in our best-practice relaxation strategy (which we will present in Chapter 4), focusing on those aspects which are relevant to our work. In this chapter efficiency is addressed both from a computational **and** implementation point of view, in the latter case involving the support of libraries **and** automatic differentiation (see Section 3.4).

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provided in literature. This dissertation contributes a pixel-based algorithm to detect increased backscattering in SAR images by analyzing the SAR pixel values according to simulated layers. To detect demolished buildings, simulated images are generated using LiDAR data. Two comparison operators (normalized mutual information **and** joint histogram slope) are used to compare image patches related to same buildings. An experiment using Munich data has shown that both of them provide an overall accuracy of more than 90%. A **combination** of these two comparison operators using **decision** trees improves the result. The fourth objective is to detect changes between SAR images acquired with different incidence angles. **For** this purpose, three algorithms are presented in this dissertation. The first algorithm is a building-level algorithm based on layer fill. Image patches related to the same buildings in the two SAR images are extracted using simulation methods. **For** each extracted image patch pair, the change ratio based on the fill ratio of building layers is estimated. The change ratio values of all buildings are then classified into two classes using the EM-algorithm. This algorithm works well **for** buildings with different size **and** shape in complex urban scenarios. Since the whole building is analyzed as one object, buildings with partly demolished walls may not be detected. Under the same idea, a wall-level change detection algorithm was developed. Image patches related to the same walls in the two SAR images were extracted **and** converted to have the same geometry. These converted patch pairs are then compared using change ratios based on fill ratio or fill position. Lastly, the wall change results are fused to provide building change result. Compared to the building-level change detection algorithm, this method is more time consuming, but yields better results **for** partly demolished buildings. A **combination** of these two algorithms is therefore suggested, whereby the building-level method is used **for** all buildings **and** wall-level method additionally **for** selected large buildings. The third developed algorithm is a wall-level change detection algorithm based on point-feature location. To this end, local maximum points in two SAR images corresponding to the same building façade are compared. This method provides promising result **for** the present data. It may work better **for** future data with increased resolution to detect changes of detailed façade structures.

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The search process in SAT works as follows: Initially all variables are unassigned. In each node of the search tree, unit domain propagation is performed, i.e., constraint propagation over each boolean formula is applied. Thereby, variables of one constraint (one formula) are checked whether they can be fixed to 1 (true) or 0 (false). E.g., if all but one literal of a clause are false, the remaining variable can be set such that the clause is satisfied. Local search heuristics are used in order to find a feasible assignment. As soon as the first feasible assignment has been found, the algorithm stops. When constraint propagation cannot detect any further fixings, some variable is fixed to true or false (the branching step). The branching variable is chosen according to its importance in former propagations **and** its involvement in conflicts. If all literals of at least one clause are fixed to zero, the subproblem is infeasible. This is called a conflict. Conflicts are analyzed in order to produce conflict clauses [180]. During this analysis, a conflict graph is created. Such a graph is a logical expression tree where vertices correspond to variable fixings (zero or one). There is a directed edge in that graph between two vertices u **and** v if the fixing of the variable belonging to u deduced the fixing of the other variable in node v. Cuts in that graph that separate the branching decisions from the infeasibility yield new clauses. Furthermore, the analysis may show that some branching **decision** (variable fixing) in the last subtree at depth level ` yielded the infeasibility together with other fixings at some depth level ` 0 < `. Then, the negation of the branching **decision** can be already applied at depth level ` 0 . This is called non-chronological backtracking. The concept of conflict analysis in our framework with integer variables is explained in more detail in Section 2.2.1.

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2.4 Search **and** classification procedure
The selection of materials (the documents) was performed in two stages. Firstly, we carried out a search in the Scopus database, including a comprehensive set of keywords related to both the field of corporate finance (capital budgeting, working capital, financial planning, financial performance evaluation, etc.) **and** the field of MCDM (multi-attribute utility theory, multi-objective programming, goal programming, preference disaggregation, etc.). The keywords were combined using the logical operators “OR”, indicating that at least one word from each field had to appear in the search output **and** “**AND**”, in order to obtain the intersection of the keywords of the two knowledge fields. In this first stage 1,417 papers were obtained. In the second stage, we read the Abstracts **and** eliminated those not related to the field of corporate finance **and** those papers that did not really use MCDM **techniques**. Thus, the sample was reduced to 339 papers **and** 8 books.

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Introduction 7 cases which have the same algebraic behavior, but lead to different degrees of complexity. Hence we need to go beyond the classification provided by the algebraic properties, **and** perform a finer analysis of the cases. It turns out that the problem still is dichotomic in nature, revealing that each of these **problems** is equivalent to the standard “complete” **problems** of standard complexity classes inside P. Finally, in Chapter 5, we consider quantified constraint formulas. These are generalizations of the usual constraint formu- las, where additionally the quantifiers ∃ **and** ∀ are allowed to occur. As hinted above, such formulas can be used to describe settings where two opponents are working against each other. It is well-known that adding these quantifiers to the formulas raises the complexity of the involved **decision** **problems** significantly: the **problems** we consider in this chapter are prototypical **for** the classes of the polynomial hierarchy, **and** **for** the class PSPACE, containing all computational **problems** which can be solved in polyno- mial space. We study various **problems** **for** these formulas: first, we consider the formula evaluation problem in this context, **and** the closely related model checking problem. An- other **decision** problem which is very interesting is the equivalence problem, where we ask if two formulas have the same set of satisfying assignments. This question is very important in practice, since it can be used to decide whether two given database queries are equivalent, if a program behaves as its specification demands, or if two games have the same winning strategies.

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First **and** foremost, my utmost gratitude to Dr. Dhish Kumar Saxena not only as a supervisor but also as a friend. As his student, I have been extremely lucky to benefit from his very rigorous expertise, constant encouragement, **and** from his vision. He is a truly remarkable person, **and** this thesis would not have been possible without his constant support. I am also truly indebted **and** thankful **for** all the time that he spent during innumerous technical discussions **and** also as a father-figure in teaching me how to become a better human-being through: sincerity, hard-work, honesty, ambition, **and** most important, attitude. I would like also to thank my two co-supervisors, Professor Ashutosh Tiwari **and** Dr. Evan Hughes, **for** their constant support, inputs **and** encouragement during my PhD. I am also grateful to Professor Qingfu Zhang **for** his expertise **and** also to Professor Rajkumar Roy **for** his support. Moreover, I want to thank Dr. Keshav Dahal **and** Professor Mark Savill **for** having read **and** marked this thesis **and** **for** their useful detailed comments, positive feedback, **and** challenging discussion.

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Our experiments show that the ANS heuristic is capable of nding a larger number of feasible delivery slots than the Simple Insertion heuristic, requiring run times that are suited for A[r]

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The lean combustion concept is considered the most promising approach to achieve an engine which is both more fuel efficient **and** at the same time has low emissions. Here the air fuel mixture injected by the burners is already lean **and** in conjunction with partially pre-mixing the existence of near stoichiometric regions can be reduced significantly. By this means the peak temperature **and** thus the NOx production rate is substantially reduced allowing a further increase of the overall temperature level to improve the fuel efficiency. In comparison to conventional combustors a complete redistribution of the air flow into the combustor is required which also affects the amount of air available **for** cooling of the combustor walls. **For** a combustor with lean burning primary zone the cooling air consumption has to be reduced by roughly 50% leading to a sparse wall cooling film. Combustor cooling becomes even more demanding as the overall temperature (including cooling air temperature) **and** pressure level will increase. This requires not only the development of cooling concepts with increased efficiency (see e.g. [3]). The key **for** the understanding of lean combustion in aero engines is the characterization **and** understanding of the interaction between the highly turbulent, swirling **and** reacting burner flow field with the wall cooling film. To the knowledge of the authors, aside from experiments performed at the laboratory scale [18][7][13], there is very little experimental data obtained at realistic operating conditions using non-intrusive laser-optical diagnostic methods.

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In the regime of small σ (or γ), however, the TS-CGS iteration cannot provide robust smoothing because the coupling in the space direction becomes weak **and** therefore, pointwise relaxation in space is not eﬀective in reducing the high-frequency components of the error. To overcome this problem, block-relaxation of the variables that are strongly connected must be performed. In our case, this means solving **for** the pairs of state **and** adjoint variables along the time-direction **for** each space coordinate. To describe this block Gauss–Seidel procedure, consider the discrete optimality system (11) at any i, j **and** **for** all time steps. **For** simplicity, we use the optimality condition to eliminate the control variable. Thus **for** each spatial grid point i, j, a block-tridiagonal system is obtained, where each block is a 2 ×2 matrix corresponding to the pair (y, p). This block-tridiagonal system has the following form:

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