Control limits in use at metrology stations are traditionally set by yield requirements. Since deviations from these limits usually trigger machine stoppage, the monitor design has a direct impact on the availability of the station, and thus on the product cycle time (CT). We developed a bi-criteria tradeoff formulation between the expected and the die yield based on the impact of the inspection control limits on both performance measures. In this research we extend our previous result by formulating the Pareto optima set for an entire flow-shop network rather than just a single station.
We consider a multi-step flow-shop network of m=1,...,M stations, each of which is immediately followed by an imperfect monitor operation. Production stations are afflicted by a two-state particle deposition process whose rate can be either low, or high. If the monitor indicates a high deposition rate, the process is stopped for repair which guarantees machine recalibration. These stoppages are referred to as vacations.
First we formulate CT and yield for each station in the network. Then, we integrate the stations and refer to the whole network to compute the overall line CT and yield. We explore the impact of the upper control limit of station m, (UCL(m)), on the expected yield and CT of station m, denoted yield(m) and CT(m) respectively. We model the process as a Markov Decision Process to obtain the stationary distribution function. This allows us to compute the expected run length and expected contamination rate and to explicitly formulate yield(m) and CT(m) as functions of UCL(m). While the expectation of yield(m) is calculated using a close form equation, CT(m) is obtained by using G/G/1 queuing model.
Once CT(m) and yield(m) are well defined, we generate the tradeoff curve between these two criteria, where each point on the curve corresponds to a unique UCL(m) value for given station parameters. Constructing the tradeoff curve allows us to recommend the decision makers a UCL(m) that results in the optimal CT(m) given a constraint on yield(m) (or vice versa). However, in order to ensure that this recommendation is, in fact, Pareto optimal, the tradeoff curve has to be convex. Since using a combination of different UCL(m) values is feasible, we develop a simple algorithm that constructs a convex Pareto optimal set for each station.
To extended our result to a flow-shop network, we present an optimal greedy algorithm that recommends a set of UCL(m) values for each point on the CT to yield Pareto optima curve.Since we model the CT(m) for each station using a G/G/1 queuing model, the CT of the entire flow-shop is modeled as a G/G/1 queuing network. However, there is no exact solution for the CT of such a model. Therefore, we approximate CT using a G/G/1 queuing network approximation technique.
This technique allows us to construct the CT and yield Pareto optimal set for an entire network in polynomial time.
