Abstract

We consider a two-stage production systemfaced by semiconductor manufacturing which produces a hierarchy of multiplegrades of outputs. In the first stage, a single type of input (wafer) is usedto produce multiple types of semi-finished parts with dependent yield rates,and in the second stage, each type of semi-finished parts can be transformedinto a corresponding type of final products, or downgraded to a type of lowergrade final products. Random customer demands are faced on the final products,and demands of different types of final products are not allowed to besubstituted. The advantage of this production system is that it can preventunhealthy ordering from customers who intentionally send out false demandsignals for high grade products and revise the orders to lower grade productswhen the delivery time is close, which was observed in semiconductormanufacturing. The objective of the study is to plan the quantity of the inputat the first stage and the respective downgrade quantities at the second stageso as to meet the required service level at the minimum cost. With some commonassumptions, we propose a modified base-stock policy for this two-stageproduction system and show that the occurrence of nil excess inventory abovethe base-stock level follows a renewal process. We further extend the modifiedbase-stock policy to a better policy that invokes risk pooling over multiplegrade products. The performance of these two polices are evaluated viasimulation to provide managerial insights. 

Keywords: Semiconductor production system; Modified base-stockpolicy; Yield rates  

1. Introduction

Weconsider a two-stage production system that produces a hierarchy of multiplegrades of outputs. In the first stage, a single type of input material is usedto produce multiple grades of semi-finished parts according to different yieldrates (or split ratios). In the second stage, a semi-finished part can betransformed into the corresponding grade product or downgraded to a lower gradeproduct to satisfy the customer’s demand which is stochastic. This problem isan abstracted sub-problem faced by semiconductor manufacturing that produceschips from wafers where input of one type of wafer results in multiple gradesof parts. To produce the final products or chips, these parts need to gothrough another operation where they may be processed retaining the same speedor downgrading to a lower speed. An illustration of this production system isgiven in Figure 1. The downgrade production at the last operation is practicedby manufacturers in order to protect their profit margins and market prices. Ithas been instilled because it was discovered that direct downgrade substitutionon demands induced unhealthy ordering from certain customers who intentionallysent out false demand signals for high grade products and revised the orders tolower grade products when the delivery time was close. In our study, thedemands of the final products are not allowed to be substituted. Even thoughthe production system with downgrade substitution on demands are widelystudied, little study has been done on the downgrade production system. A basicproblem arising from the system is how best to plan the input quantity at stageone and the respective downgrade quantities at stage two at each time period soas to meet the service constraints at the minimum cost. For a single product,it is well known that the base-stock policy is the optimal policy to use forperiodic review models when no setup cost is involved and identical independentdistribution is assumed for demands over time. For multiple products, however,when production is governed by the yield rates, the dependence relationshipamong the products makes it impossible to replenish to the base-stock levelsseparately for each of the products at every time period. In this paper, basedon the notion of the critical or bottleneck product, we propose a modifiedbase-stock policy which keeps the excess inventory over the base-stock levelonly at the critical product. We show that the occurrences of nil excessinventory of the critical product above the base-stock level follow a renewalprocess. We further extend the modified base-stock policy to a better policythat invokes risk pooling over the multiple grade products. After that, weconduct a numerical experiment to evaluate the performance of these two policesand provide some managerial insights.