Line Balancing in Parallel M/M/1 Lines and Loss Systems as Cooperative Games

We consider production and service systems that consist of parallel lines of two types: (i) M/M/1 lines and (ii) lines that have no buffers (loss systems). Each line is assumed to be controlled by a dedicated supervisor. The management measures the effectiveness of the supervisors by the long run expected cost of their line. Unbalanced lines cause congestion and bottlenecks, large variation in output, unnecessary wastes and, ultimately, high operating costs. Thus, the supervisors are expected to join forces and reduce the cost of the whole system by applying line-balancing techniques, possibly combined with either strategic outsourcing or capacity reduction practices. By solving appropriate mathematical programming formulations, the policy that minimizes the long run expected cost of each of the parallel-lines system, is identified. The next question to be asked is how to allocate the new total cost of each system among the lines' supervisors so that the cooperation's stability is preserved. For that sake, we associate a cooperative game to each system and we investigate its core. We show that the cooperative games are reducible to market games and therefore they are totally balanced, that is, their core and the core of their subgames are non-empty. For each game a core cost allocation based on competitive equilibrium prices is identified.