This paper presents an analytical technique to determine the optimum number of cards to control material release in a CONWIP system. The work focuses on the card setting problem for a flow-shop system characterised by the presence of a batch processing machine (e.g. a kiln for long heat treatment). To control production, two different static approaches are developed: the first one is used when the bottleneck coincides with the batch processing machine and the second one is proposed when the bottleneck is another machine of the flow shop. In both contexts, by means of the appropriate model, one can optimize the performance of the flow-shop by maximizing the throughput and keeping the work in process at a minimum level. Numerical examples are also included in the paper to confirm the validity of the models and to demonstrate their practical utility.

This paper examines the optimization of production involving a tandem two-machine system producing a single part type, with each machine being subject to random breakdowns and repairs. An analytical model is formulated with a view to solving an optimal stochastic production problem of the system with machines having up-downtime non-exponential distributions. The model developed is obtained by using a dynamic programming approach and a semi-Markov process. The control problem aims to find the production rates needed by the machines to meet the demand rate, through a minimization of the inventory/shortage cost. Using the Bellman principle, the optimality conditions obtained satisfy the Hamilton-Jacobi-Bellman equation, which depends on time and system states, and ultimately, leads to a feedback control. Consequently, the new model enables us to improve the coefficient of variation (CVup/down) to be less than one while it is equal to one in Markov model. Heuristics methods are used to involve the problem because of the difficulty of the analytical model using several states, and to show what control law should be used in each system state (i.e., including Kanban, feedback and CONWIP control). Numerical methods are used to solve the optimality conditions and to show how a machine should produce.