Purpose The purpose of this paper is to illustrate the importance of redesigning, reusing, remanufacturing, recovering, recycling and reducing (6R) to sustainable manufacturing and discuss the general procedure to reconfigure robots. Two critical challenges in adopting industrial robots in small and medium-sized enterprise (SMEs) are flexibility and cost, as the number of tasks of the same type can be limited because of the size of an SME. The challenges can be alleviated by 6R. The 6R processes allow a robot to adopt new tasks, increase its utilization rate and reduce unit costs of products. Design/methodology/approach There is no shortcut to implement sustainable manufacturing. All of the manufacturing resources in a system should be planned optimally to reduce waste and maximize the utilization rates of resources. In this paper, modularization and reconfiguration are emphasized to implement 6R processes in sustainable manufacturing; robots are especially taken into consideration as core functional modules in the system. Modular architecture makes it feasible to integrate robots with low-cost customized modules for various tasks for the high utilization rates. A case study is provided to show the feasibility. Findings Finding the ways to reuse manufacturing resources could bring significant competitiveness to an SME, in the sense that sophisticated machines and tools, such as robots, can be highly utilized even in a manufacturing environment with low or medium product volumes. The concepts of modularization and 6R processes can be synergized to achieve this goal. Research limitations/implications The authors propose the strategy to enhance the utilization rates of core manufacturing resources using modular architecture and 6R practice. The axiomatic design theory can be applied as the theoretical fundamental to guide the 6R processes; however, a universal solution in the implementation is not available. The solutions have to be tailored to specific SMEs, and the solutions should vary with respect to time. Practical implications To operate a sustainable manufacturing system, a continuous design effort is required to reconfigure existing resources and enhance their capabilities to fulfill new tasks in the dynamic environment. Social implications The authors focus on the importance of sustainable manufacturing to modern society, and they achieve this goal by reusing robots as system components in different applications. Originality/value Sustainable manufacturing has attracted a great deal of attention, although the operable guidance for system implementation is scarce. The presented work has thrown some light in this research area. The 6R concept has been introduced in a modular system to maximize the utilizations of critical manufacturing resources. It is particularly advantageous for SMEs to adopt sophisticated robots cost-effectively.

Given a finite simple graph G, an odd cover of G is a collection of complete bipartite graphs, or bicliques, in which each edge of G appears in an odd number of bicliques and each non-edge of G appears in an even number of bicliques. We denote the minimum cardinality of an odd cover of G by b2(G) and prove that b2(G) is bounded below by half of the rank over F2 of the adjacency matrix of G. We show that this lower bound is tight in the case when G is a bipartite graph and almost tight when G is an odd cycle. However, we also present an infinite family of graphs which shows that this lower bound can be arbitrarily far away from b2(G).Babai and Frankl (1992) proposed the "odd cover problem," which in our language is equivalent to determining b2(Kn). Radhakrishnan, Sen, and Vishwanathan (2000) determined b2(Kn) for an infinite but density zero subset of positive integers n. In this paper, we determine b2(Kn) for a density 3/8 subset of the positive integers.

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We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the A-and Z-sequences of these sums of Riordan arrays, and also identify an analog for A-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations 'Der' and 'Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.