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## NP-completeness of cell formation problem with grouping efficacy objective

In the current paper we provide a proof of NP-completeness for the Cell Formation Problem (CFP) with the fractional grouping efficacy objective function. First the CFP with a linear objective function is considered. Following the ideas of Pinheiro et al. (2016) we show that it is equivalent to the Bicluster Graph Editing Problem (BGEP), which is known to be NP-complete due to the reduction from the 3-Exact 3-Cover Problem – 3E3CP (Amit, 2004). Then we suggest a polynomial reduction of the CFP with the linear objective to the CFP with the grouping efficacy objective. It proves the NP-completeness of this fractional CFP formulation. Along with the NP-status our paper presents important connections of the CFP with the BGEP and 3E3CP. Such connections could be used for ”transferring” of known theoretical properties, efficient algorithms, polynomial cases, and other features of well-studied graph editing and exact covering problems to the CFP.

The Cell Formation Problem (CFP) consists in an optimal grouping of the given machines and parts into cells, so that machines in every cell process as much as possible parts from this cell (intra-cell operations) and as less as possible parts from another cells (inter-cell operations). The grouping efficacy is the objective function for the CFP which simultaneously maximizes the number of intra-cell operations and minimizes the number of inter-cell operations. Currently there are no exact approaches (known to the authors) suggested for solving the CFP with the grouping efficacy objective. The only exact model which solves the CFP in a restricted formulation is due to Elbenani & Ferland [10]. The restriction consists in fixing the number of production cells. The main difficulty of the CFP is the fractional objective function - the grouping efficacy. In this paper we address this issue for the CFP in its common formulation with a variable number of cells. Our computational experiments are made for the most popular set of 35 benchmark instances. For the 14 of these instances using CPLEX software we prove that the best known solutions are exact global optimums.

In this chapter, we introduce a new heuristic for Cell Formation Problem in its most general formulation with grouping efficiency as an objective function. Suggested approach applies an improvement procedure to obtain solutions with high grouping efficiency. This procedure is repeated until efficiency can be increased for randomly generated configurations of cells. We consider our preliminary results for 10 popular benchmark instances taken from the literature. Also source instances with the solutions we got can be found in the Appendix.

In this paper we introduce a new pattern-based approach within the Linear Assignment Model with the purpose to design heuristics for a combinatorial optimization problem (COP). We assume that the COP has an additive (separable) objective function and the structure of a feasible (optimal) solution to the COP is predefined by a collection of cells (positions) in an input file. We define a pattern as a collection of positions in an instance problem represented by its input file (matrix). We illustrate the notion of pattern by means of some well known problems in COP among them the Linear Ordering Problem, Cell Formation Problem (CFP) just to mention a couple. The CFP is defined on a Boolean input matrix which rows represent machines and columns - parts. The CFP consists in finding three optimal objects: a block-diagonal collection of rectangles, a rows (machines) permutation, and a columns (parts) permutation such that the grouping efficacy is maximized. The suggested heuristic combines two procedures: the pattern-based procedure to build an initial solution and an improvement procedure to obtain a final solution with high grouping efficacy for the CFP. Our computational experiments with the most popular set of 35 benchmark instances show that our heuristic outperforms all well known heuristics and returns either the best known or improved solutions to the CFP.

The Cell Formation Problem (CFP) is an NP-hard optimization problem considered for cellular man- ufacturing systems. Because of its high computational complexity there have been developed a lot of heuristics and almost no exact algorithms for solving this problem. In this paper we suggest a branch- and-bound algorithm which provides exact solutions for the CFP with the grouping efficacy objective function. To linearize this fractional objective function we apply the Dinkelbach approach. Our algorithm finds optimal solutions for 24 of the 35 popular benchmark instances from literature and for the remaining instances it finds good solutions close to the best known. The difference in the grouping efficacy with the best known solutions is always less than 1.5%.

The earliest approaches to the cell formation problem in group technology, dealing with a binary machine-part incidence matrix, were aimed only at minimizing the number of intercell moves (exceptional elements in the block-diagonalized matrix). Later on this goal was extended to simultaneous minimization of the numbers of exceptions and voids, and minimization of intercell moves and within-cell load variation, respectively. In this paper we design the first exact branch-and-bound algorithm to create a Pareto-optimal front for the bi-criterion cell formation problem.

The cell formation problem (CFP) is an NP-hard optimization problem considered for cell manufacturing systems. Because of its high computational complexity several heuristics have been developed for solving this problem. In this paper we present a branch and bound algorithm which provides exact solutions of the CFP. This algorithm finds optimal solutions for 13 problems of the 35 popular benchmark instances from the literature.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

Existing approaches suggest that IT strategy should be a reflection of business strategy. However, actually organisations do not often follow business strategy even if it is formally declared. In these conditions, IT strategy can be viewed not as a plan, but as an organisational shared view on the role of information systems. This approach generally reflects only a top-down perspective of IT strategy. So, it can be supplemented by a strategic behaviour pattern (i.e., more or less standard response to a changes that is formed as result of previous experience) to implement bottom-up approach. Two components that can help to establish effective reaction regarding new initiatives in IT are proposed here: model of IT-related decision making, and efficiency measurement metric to estimate maturity of business processes and appropriate IT. Usage of proposed tools is demonstrated in practical cases.