Note on Combinatorial Engineering Frameworks for Hierarchical Modular Systems

The paper briefly describes a basic set of special combinatorial engineering frameworks for solving complex problems in the field of hierarchical modular systems. The frameworks consist of combinatorial problems (and corresponding models), which are interconnected/linked (e.g., by preference relation). Mainly, hierarchical morphological system model is used. The list of basic standard combinatorial engineering (technological) frameworks is the following: (1) design of system hierarchical model, (2) combinatorial synthesis ('bottom-up' process for system design), (3) system evaluation, (4) detection of system bottlenecks, (5) system improvement (re-design, upgrade), (6) multi-stage design (design of system trajectory), (7) combinatorial modeling of system evolution/development and system forecasting. The combinatorial engineering frameworks are targeted to maintenance of some system life cycle stages. The list of main underlaying combinatorial optimization problems involves the following: knapsack problem, multiple-choice problem, assignment problem, spanning trees, morphological clique problem.Comment: 11 pages, 7 figures, 3 table

Discrete Route/Trajectory Decision Making Problems

The paper focuses on composite multistage decision making problems which are targeted to design a route/trajectory from an initial decision situation (origin) to goal (destination) decision situation(s). Automobile routing problem is considered as a basic physical metaphor. The problems are based on a discrete (combinatorial) operations/states design/solving space (e.g., digraph). The described types of discrete decision making problems can be considered as intelligent design of a route (trajectory, strategy) and can be used in many domains: (a) education (planning of student educational trajectory), (b) medicine (medical treatment), (c) economics (trajectory of start-up development). Several types of the route decision making problems are described: (i) basic route decision making, (ii) multi-goal route decision making, (iii) multi-route decision making, (iv) multi-route decision making with route/trajectory change(s), (v) composite multi-route decision making (solution is a composition of several routes/trajectories at several corresponding domains), and (vi) composite multi-route decision making with coordinated routes/trajectories. In addition, problems of modeling and building the design spaces are considered. Numerical examples illustrate the suggested approach. Three applications are considered: educational trajectory (orienteering problem), plan of start-up company (modular three-stage design), and plan of medical treatment (planning over digraph with two-component vertices).Comment: 25 pages, 34 figures, 16 table

Towards Decision Support Technology Platform for Modular Systems

The survey methodological paper addresses a glance to a general decision support platform technology for modular systems (modular/composite alterantives/solutions) in various applied domains. The decision support platform consists of seven basic combinatorial engineering frameworks (system synthesis, system modeling, evaluation, detection of bottleneck, improvement/extension, multistage design, combinatorial evolution and forecasting). The decision support platform is based on decision support procedures (e.g., multicriteria selection/sorting, clustering), combinatorial optimization problems (e.g., knapsack, multiple choice problem, clique, assignment/allocation, covering, spanning trees), and their combinations. The following is described: (1) general scheme of the decision support platform technology; (2) brief descriptions of modular (composite) systems (or composite alternatives); (3) trends in moving from chocie/selection of alternatives to processing of composite alternatives which correspond to hierarchical modular products/systems; (4) scheme of resource requirements (i.e., human, information-computer); and (5) basic combinatorial engineering frameworks and their applications in various domains.Comment: 10 pages, 9 figures, 2 table

Towards balanced clustering - part 1 (preliminaries)

The article contains a preliminary glance at balanced clustering problems. Basic balanced structures and combinatorial balanced problems are briefly described. A special attention is targeted to various balance/unbalance indices (including some new versions of the indices): by cluster cardinality, by cluster weights, by inter-cluster edge/arc weights, by cluster element structure (for element multi-type clustering). Further, versions of optimization clustering problems are suggested (including multicriteria problem formulations). Illustrative numerical examples describe calculation of balance indices and element multi-type balance clustering problems (including example for design of student teams).Comment: 21 pages, 17 figures, 14 table

Composite Strategy for Multicriteria Ranking/Sorting (methodological issues, examples)

The paper addresses the modular design of composite solving strategies for multicriteria ranking (sorting). Here a 'scale of creativity' that is close to creative levels proposed by Altshuller is used as the reference viewpoint: (i) a basic object, (ii) a selected object, (iii) a modified object, and (iv) a designed object (e.g., composition of object components). These levels maybe used in various parts of decision support systems (DSS) (e.g., information, operations, user). The paper focuses on the more creative above-mentioned level (i.e., composition or combinatorial synthesis) for the operational part (i.e., composite solving strategy). This is important for a search/exploration mode of decision making process with usage of various procedures and techniques and analysis/integration of obtained results. The paper describes methodological issues of decision technology and synthesis of composite strategy for multicriteria ranking. The synthesis of composite strategies is based on 'hierarchical morphological multicriteria design' (HMMD) which is based on selection and combination of design alternatives (DAs) (here: local procedures or techniques) while taking into account their quality and quality of their interconnections (IC). A new version of HMMD with interval multiset estimates for DAs is used. The operational environment of DSS COMBI for multicriteria ranking, consisting of a morphology of local procedures or techniques (as design alternatives DAs), is examined as a basic one.Comment: 24 pages, 28 figures, 5 table

Towards combinatorial clustering: preliminary research survey

The paper describes clustering problems from the combinatorial viewpoint. A brief systemic survey is presented including the following: (i) basic clustering problems (e.g., classification, clustering, sorting, clustering with an order over cluster), (ii) basic approaches to assessment of objects and object proximities (i.e., scales, comparison, aggregation issues), (iii) basic approaches to evaluation of local quality characteristics for clusters and total quality characteristics for clustering solutions, (iv) clustering as multicriteria optimization problem, (v) generalized modular clustering framework, (vi) basic clustering models/methods (e.g., hierarchical clustering, k-means clustering, minimum spanning tree based clustering, clustering as assignment, detection of clisue/quasi-clique based clustering, correlation clustering, network communities based clustering), Special attention is targeted to formulation of clustering as multicriteria optimization models. Combinatorial optimization models are used as auxiliary problems (e.g., assignment, partitioning, knapsack problem, multiple choice problem, morphological clique problem, searching for consensus/median for structures). Numerical examples illustrate problem formulations, solving methods, and applications. The material can be used as follows: (a) a research survey, (b) a fundamental for designing the structure/architecture of composite modular clustering software, (c) a bibliography reference collection, and (d) a tutorial.Comment: 102 pages, 66 figures, 67 table

Multiset Estimates and Combinatorial Synthesis

The paper addresses an approach to ordinal assessment of alternatives based on assignment of elements into an ordinal scale. Basic versions of the assessment problems are formulated while taking into account the number of levels at a basic ordinal scale [1,2,...,l] and the number of assigned elements (e.g., 1,2,3). The obtained estimates are multisets (or bags) (cardinality of the multiset equals a constant). Scale-posets for the examined assessment problems are presented. 'Interval multiset estimates' are suggested. Further, operations over multiset estimates are examined: (a) integration of multiset estimates, (b) proximity for multiset estimates, (c) comparison of multiset estimates, (d) aggregation of multiset estimates, and (e) alignment of multiset estimates. Combinatorial synthesis based on morphological approach is examined including the modified version of the approach with multiset estimates of design alternatives. Knapsack-like problems with multiset estimates are briefly described as well. The assessment approach, multiset-estimates, and corresponding combinatorial problems are illustrated by numerical examples.Comment: 30 pages, 24 figures, 10 table

Course on System Design (structural approach)

The article describes a course on system design (structural approach) which involves the following: issues of systems engineering; structural models; basic technological problems (structural system modeling, modular design, evaluation/comparison, revelation of bottlenecks, improvement/upgrade, multistage design, modeling of system evolution); solving methods (optimization, combinatorial optimization, multicriteria decision making); design frameworks; and applications. The course contains lectures and a set of special laboratory works. The laboratory works consist in designing and implementing a set of programs to solve multicriteria problems (ranking/selection, multiple choice problem, clustering, assignment). The programs above are used to solve some standard problems (e.g., hierarchical design of a student plan, design of a marketing strategy). Concurrently, each student can examine a unique applied problem from his/her applied domain(s) (e.g., telemetric system, GSM network, integrated security system, testing of microprocessor systems, wireless sensor, corporative communication network, network topology). Mainly, the course is targeted to developing the student skills in modular analysis and design of various multidisciplinary composite systems (e.g., software, electronic devices, information, computers, communications). The course was implemented in Moscow Institute of Physics and Technology (State University).Comment: 22 pages, 14 figure

TauRieL: Targeting Traveling Salesman Problem with a deep reinforcement learning inspired architecture

In this paper, we propose TauRieL and target Traveling Salesman Problem (TSP) since it has broad applicability in theoretical and applied sciences. TauRieL utilizes an actor-critic inspired architecture that adopts ordinary feedforward nets to obtain a policy update vector $v$. Then, we use $v$ to improve the state transition matrix from which we generate the policy. Also, the state transition matrix allows the solver to initialize from precomputed solutions such as nearest neighbors. In an online learning setting, TauRieL unifies the training and the search where it can generate near-optimal results in seconds. The input to the neural nets in the actor-critic architecture are raw 2-D inputs, and the design idea behind this decision is to keep neural nets relatively smaller than the architectures with wide embeddings with the tradeoff of omitting any distributed representations of the embeddings. Consequently, TauRieL generates TSP solutions two orders of magnitude faster per TSP instance as compared to state-of-the-art offline techniques with a performance impact of 6.1\% in the worst case.Comment: 10 pages, 5 figures, 1 Algorithm, 4 Table

Continuous Toolpath Planning in Additive Manufacturing

We develop a framework that creates a new polygonal mesh representation of the sparse infill domain of a layer-by-layer 3D printing job. We guarantee the existence of a single, continuous tool path covering each connected piece of the domain in every layer. We present a tool path algorithm that traverses each such continuous tool path with no crossovers. The key construction at the heart of our framework is an Euler transformation which converts a 2-dimensional cell complex K into a new 2-complex K^ such that every vertex in the 1-skeleton G^ of K^ has even degree. Hence G^ is Eulerian, and a Eulerian tour can be followed to print all edges in a continuous fashion. We start with a mesh K of the union of polygons obtained by projecting all layers to the plane. We compute its Euler transformation K^. In the slicing step, we clip K^ at each layer using its polygon to obtain a complex that may not necessarily be Euler. We then patch this complex by adding edges such that any odd-degree nodes created by slicing are transformed to have even degrees again. We print extra support edges in place of any segments left out to ensure there are no edges without support in the next layer. These support edges maintain the Euler nature of the complex. Finally we describe a tree-based search algorithm that builds the continuous tool path by traversing "concentric" cycles in the Euler complex. Our algorithm produces a tool path that avoids material collisions and crossovers, and can be printed in a continuous fashion irrespective of complex geometry or topology of the domain (e.g., holes). We implement our test our framework on several 3D objects. Apart from standard geometric shapes, we demonstrate the framework on the Stanford bunny.Comment: Accepted in SPM2020; implementation details expanded, manuscript revised after revie