text: Given a family @math of graphs and a positive integer @math , a graph @math is called vertex @math -fault-tolerant with respect to @math , denoted by @math -FT @math , if @math contains some @math as a subgraph, for every @math with @math . Vertex-fault-tolerance has been introduced by Hayes [A graph model for fault-tolerant computing systems, IEEE Transactions on Computers, C-25 (1976), pp. 875-884.], and has been studied in view of potential applications in the design of interconnection networks operating correctly in the presence of faults. We define the Fault-Tolerant Complete Matching (FTCM) Problem in bipartite graphs of order @math : to design a bipartite @math , with @math , @math , @math , that has a FTCM, and the tuple @math , where @math and @math are the maximum degree in @math and @math , respectively, is lexicographically minimum. @math has a FTCM if deleting at most @math vertices from @math creates @math that has a complete matching, i.e., a matching of size @math . We show that if @math is integer, solutions of the FTCM Problem can be found among @math -regular bipartite graphs of order @math , with @math , and @math . If @math then all @math -regular bipartite graphs of order @math have a FTCM, and for @math , it is not the case. We characterize the values of @math , @math , @math , and @math that admit an @math -regular bipartite graph of order @math , with @math , and give a simple construction that creates such a graph with a FTCM whenever possible. Our techniques are based on Hall's marriage theorem, elementary number theory, linear Diophantine equations, properties of integer functions and congruences, and equations involving them.
  @cite_21: Several design guidelines and flexibility indices have been developed in the literature to inform the design of flexible production networks. In this paper, we propose additional flexibility design guidelines for unbalanced networks, where the numbers of plants and products are not equal, by refining the well-known Chaining Guidelines. We study symmetric networks, where all plants have the same capacity and product demands are independent and identically distributed, and focus mainly on the case where each product is built at two plants. We also briefly discuss cases where (1) each product is built at three plants and (2) some products are built at only one plant. An extensive computational study suggests that our refinements work very well for finding flexible configurations with minimum shortfall in unbalanced networks.
  @cite_4: Abstract A new methodology is presented for structuring the multiskilling characteristics of a set of initially single-skilled employees in the context of a multi-department service sector business, more specifically in the retail industry. Assuming stochastic demand under continuous modeling of the workforce, the methodology decomposes the multiskilling problem into three stages that are then solved sequentially. The first stage delivers a novel analytic expression for estimating the number of multiskilled employees required by each store department, the second stage applies constructive heuristics to these estimates to generate a feasible set of closed chaining structures, and the third stage uses Monte Carlo simulation and a linear programming model to evaluate the chains so generated. The results show that in unbalanced systems, the solutions involve closed long chains and closed short chains of different lengths, which are robust to demand variability, minimize the expected total cost of staff shortages, and exhibit the best cost-effective performance. The methodology provides a tool for company decision makers to address the three fundamental multiskilling issues in unbalanced systems, namely, where to add multiskilling, how much to add and how it should be added.
summary: Nevertheless, the relation of our model with the process flexibility literature is still remote, since the process flexibility community has so far focused on systems where a server can process different kinds of compatible requests within the same time period. Moreover, only recently unbalanced systems, i.e., with @math , have started to be considered (see, @cite_21 and @cite_4 for examples).