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Two methods of decomposition of probabilistic relations are presented in this paper. They consist of splitting relations (blocks) into pairs of smaller blocks related to each other by new variables generated in such a way so as to minimize a cost function which depends on the size and structure of the result. The decomposition is repeated iteratively until a stopping criterion is met. Topology and contents of the resulting structure develop dynamically in the decomposition process and reflect relationships hidden in the data.

The study aims to optimize truck routes by minimizing social and economic costs. It introduces a strategy involving diverse drones and their potential for reusing at DNs based on flight range. In HTDRP-DC, trucks can select and transport various drones to LDs to reduce deprivation time. This study estimates the nonlinear deprivation cost function using a linear two-piece-wise function, leading to MILP formulations. A heuristic-based Benders Decomposition approach is implemented to address medium and large instances. Valid inequalities and a heuristic method enhance convergence boundaries, ensuring an efficient solution methodology.

Research has yet to address critical factors in disaster logistics: minimizing the social and economic costs simultaneously and using drones in relief distribution; deprivation as a social cost measures the human suffering from a shortage of relief supplies. The proposed hybrid truck-drone routing problem minimizing deprivation cost (HTDRP-DC) involves distributing relief supplies to dispersed demand nodes with undamaged (LDs) or damaged (DNs) access roads, utilizing multiple trucks and diverse drones. A Benders Decomposition approach is enhanced by accelerating techniques.

Incorporating deprivation and economic costs results in selecting optimal routes, effectively reducing the time required to assist affected areas. Additionally, employing various drone types and their reuse in damaged nodes reduces deprivation time and associated deprivation costs. The study employs valid inequalities and the heuristic method to solve the master problem, substantially reducing computational time and iterations compared to GAMS and classical Benders Decomposition Algorithm. The proposed heuristic-based Benders Decomposition approach is applied to a disaster in Tehran, demonstrating efficient solutions for the HTDRP-DC regarding computational time and convergence rate.

Current research introduces an HTDRP-DC problem that addresses minimizing deprivation costs considering the vehicle’s arrival time as the deprivation time, offering a unique solution to optimize route selection in relief distribution. Furthermore, integrating heuristic methods and valid inequalities into the Benders Decomposition approach enhances its effectiveness in solving complex routing challenges in disaster scenarios.

In this chapter we explore different ways to obtain decompositions of rank-dependent indices of socioeconomic inequality of health, such as the Concentration Index. Our focus is on the regression-based type of decomposition. Depending on whether the regression explains the health variable, or the socioeconomic variable, or both, a different decomposition formula is generated. We illustrate the differences using data from the Ethiopia 2011 Demographic and Health Survey (DHS).

To provide a new proof of convergence of the Adomian decomposition series for solving nonlinear ordinary and partial differential equations based upon a thorough examination of the historical milieu preceding the Adomian decomposition method.

Develops a theoretical background of the Adomian decomposition method under the auspices of the Cauchy‐Kovalevskaya theorem of existence and uniqueness for solution of differential equations. Beginning from the concepts of a parametrized Taylor expansion series as previously introduced in the Murray‐Miller theorem based on analytic parameters, and the Banach‐space analog of the Taylor expansion series about a function instead of a constant as briefly discussed by Cherruault et al., the Adomian decompositions series and the series of Adomian polynomials are found to be a uniformly convergent series of analytic functions for the solution u and the nonlinear composite function f(u). To derive the unifying formula for the family of classes of Adomian polynomials, the author develops the novel notion of a sequence of parametrized partial sums as defined by truncation operators, acting upon infinite series, which induce these parametrized sums for simple discard rules and appropriate decomposition parameters. Thus, the defining algorithm of the Adomian polynomials is the difference of these consecutive parametrized partial sums.

The four classes of Adomian polynomials are shown to belong to a common family of decomposition series, which admit solution by recursion, and are derived from one unifying formula. The series of Adomian polynomials and hence the solution as computed as an Adomian decomposition series are shown to be uniformly convergent. Furthermore, the limiting value of the mth Adomian polynomial approaches zero as the index m approaches infinity for the prerequisites of the Cauchy‐Kovalevskaya theorem. The novel truncation operators as governed by discard rules are analogous to an ideal low‐pass filter, where the decomposition parameters represent the cut‐off frequency for rearranging a uniformly convergent series so as to induce the parametrized partial sums.

This paper unifies the notion of the family of Adomian polynomials for solving nonlinear differential equations. Further it presents the new notion of parametrized partial sums as a tool for rearranging a uniformly convergent series. It offers a deeper understanding of the elegant and powerful Adomian decomposition method for solving nonlinear ordinary and partial differential equations, which are of paramount importance in modeling natural phenomena and man‐made device performance parameters.

The purpose of this paper is to specify the decomposition conditions of Wand and Weber for the Business Process Model and Notation (BPMN). Therefore, an interpretation of the conditions for BPMN is derived and compared to a specification of the conditions for enhanced Event-Driven Process Chains (eEPCs). Based on these results, guidelines for a conformance check of BPMN and eEPC models with the decomposition conditions are shown. Further, guidelines for decomposition are formulated for BPMN models. The usability of the decomposition guidelines is tested with modelling experts.

An approach building on a representational mapping is used for specifying the decomposition conditions. Therefore, ontological constructs of the Bunge-Wand-Weber ontology are mapped to corresponding modelling constructs and an interpretation of the decomposition conditions for BPMN is derived. Guidelines for a conformance check are then defined. Based on these results, decomposition guidelines are formulated. Their usability is tested in interviews.

The research shows that the decomposition conditions stemming from the information systems discipline can be transferred to business process modelling. However, the interpretation of the decomposition conditions depends on specific characteristics of a modelling language. Based on a thorough specification of the conditions, it is possible to derive guidelines for a conformance check of process models with the conditions. In addition, guidelines for decomposition are developed and tested. In the study, these are perceived as understandable and helpful by experts.

Research approaches based on representational mappings are subjected to subjectivity. However, by having three researchers performing the approach independently, subjectivity can be mitigated. Further, only ten experts participated in the usability test, which is therefore to be considered as a first step in a more comprising evaluation.

This paper provides the process modeller with guidelines enabling a conformance check of BPMN and eEPC process models with the decomposition conditions. Further, guidelines for decomposing BPMN models are introduced.

This paper is the first to specify Wand and Weber's decomposition conditions for process modelling with BPMN. A comparison to eEPCs shows, that the ontological expressiveness influences the interpretation of the conditions. Further, guidelines for decomposing BPMN models as well as for checking their adherence to the decomposition conditions are presented.

Applying “modularity” principles in services is gaining in popularity. The purpose of this paper is to enrich existing service modularity theory and practice by exploring how services are being decomposed and how the modularization aim and the routineness of the service(s) involved may link to different decomposition logics. The authors argue that these are fundamental questions that have barely been addressed.

The authors first built a theoretical framework of decomposition steps and the design choices involved that distinguished six decomposition logics. The authors conducted a systematic literature search that generated 18 empirical articles describing 16 service modularity cases. The authors analysed these cases in terms of decomposition logic and two main contingencies: modularization aim and service routineness.

Only three of the 18 articles explicitly addressed the service decomposition by reflecting on the underlying design choices. By unravelling the decomposition in each case, the authors were able to identify the decomposition logic and found four of the six theoretically derived logics: single-level process oriented; single-level outcome oriented; multilevel outcome oriented; and multilevel combined orientation. Although the authors did not find a direct relationship between the modularization aim and the decomposition logic, the authors did find that single-level decomposition logics seem to be mainly applied in non-routine service offerings whereas the multilevel ones are mainly applied in routine service offerings.

By contributing to a common understanding of modular service decomposition and proposing a framework that explicates the design choices involved, the authors enable an enhanced application of the modularity concept in services.

This paper reports on the work done to decompose a large sized solid model into smaller solid components for rapid prototyping technology. The target geometric domain of the solid model includes quadrics and free form surfaces.

The decomposition criteria are based on the manufacturability of the model against a user‐defined manufacturing chamber size and the maintenance of geometrical information of the model. In the proposed algorithm, two types of manufacturing chamber are considered: cylindrical shape and rectangular shape. These two types of chamber shape are commonly implemented in rapid prototyping machines.

The proposed method uses a combination of the regular decomposition (RD)‐method and irregular decomposition (ID)‐method to split a non‐producible solid model into smaller producible subparts. In the ID‐method, the producible feature group decomposition (PFGD)‐method focuses on the decomposition by recognising producible feature groups. In the decomposition process, less additional geometrical and topological information are created. The RD‐method focuses on the splitting of non‐producible sub‐parts, which cannot be further decomposed by the PFGD‐method. Different types of regular split tool surface are studied.

Combination of the RD‐method and the ID‐method makes up the proposed volume decomposition process. The user can also define the sequence and priority of using these methods manually to achieve different decomposition patterns. The proposed idea is also applicable to other decomposition algorithm. Some implementation details and the corresponding problems of the proposed methods are also discussed.