A two-bottleneck system with binomial yields and rigid demand

Abstract

This study considers multistage production systems where production is in lots and only two stages have non-zero setup costs. Yields are binomial and demand, needing to be satisfied in its entirety, is “rigid”. We refer to a stage with non-zero setup cost as a “bottleneck” (BN) and thus to the system as “a two-bottleneck system” (2-BNS). A close examination of the simplest 2-BNS reveals that costs corresponding to a particular level of work in process (WIP) depend upon costs for higher levels of WIP, making it impossible to formulate a recursive solution.

For each possible configuration of intermediate inventories a production policy must specify at which stage to produce next and the number of units to be processed. We prove that any arbitrarily “fixed” production policy gives rise to a finite set of linear equations, and develop algorithms to solve the two-stage problem. We also show how the general 2-BNS can be reduced to a three-stage problem, where the middle stage is a non-BN, and that the algorithms developed can be modified to solve this problem.

Introduction

This study considers a serial multistage manufacturing system where production at each stage (machine, work-center) is in lots or runs and only two stages have non-zero setup costs. The cost of processing a lot of size N at stage k is α_k+β_kN, where the parameters α_k and β_k are the “setup” and “variable” (per-unit) processing costs, respectively. Using the terminology in Grosfeld-Nir and Ronen (1993) and Grosfeld-Nir (1995), we refer to a stage with non-zero setup cost as a “bottleneck” (BN) and to the system as a “two-bottleneck system” (2-BNS). Similarly, we refer to a system with no BNs (all setup costs are 0) as a “zero-bottleneck system” (0-BNS) and to a system with only one BN as a “single-bottleneck system” (SBNS). Typically, unless a BN is at either end or the two BNs are next to each other, a 2-BNS consists of three 0-BNSs and two BNs (Fig. 1).

We assume binomial yield, i.e., each of the units within a lot has the same probability of success and its quality is independent of the other units. We refer to a manufacturing facility with binomial yield as a “binomial machine”. As a batch exits a machine its quality is examined; defective units are worthless and must be scrapped, and only good units may proceed to the next stage. We assume that the demand must be satisfied in its entirety. That is, as long as the number of usable units exiting the last stage is short of the outstanding demand, further production runs, aimed at satisfying the remaining demand, must be initiated, as necessary. We refer to units that have been successfully processed by only some of the machines as “work in process” (WIP). If at the end of production, the number of final usable products exceeds the demand, or there is WIP remaining, the excess (or the WIP remaining) has no value. This situation often arises in environments where orders are for small quantities and products are custom made.

Given the system parameters and the demand size, the objective is to find the “optimal policy” which minimizes the expected total of setup and variable production costs. Such a policy must specify, for each possible configuration of intermediate inventories (WIP), on which machine to produce next and the number of units to process. The most elementary 2-BNS consists of two BNs without any 0-BNs. We will refer to such a system simply as a two-stage system (see Fig. 4).

The modeling of manufacturing with random yields has attracted the attention of many researchers; see Yano and Lee (1995) for a literature review. As far as demand is concerned, two options have been addressed for the most part: (i) “rigid demand”, where an order must be satisfied in its entirety, possibly necessitating multiple production runs (see Grosfeld-Nir and Gerchak (2004), for a review of such models); and (ii) “non-rigid demand”, where there is only one production run and a penalty for shortage (see Section 9.4.8 of Zipkin (2000) for a description of such single-attempt scenarios).

The single stage with binomial yields and rigid demand has been analyzed since the mid-1950s, often under the label of “reject allowance” (Sepheri et al., 1986). For binomial machines, Beja (1977) proved that the optimal lot strictly increases in the demand. Anily (1993) proved similar results for a machine with discrete uniform yield. It is intuitive to assume that the optimal lot always exceeds the demand, which for binomial and discrete uniform yields is implied from the fact that the optimal lot strictly increases in the demand. However, in their study of some fundamental questions concerning the single stage, Grosfeld-Nir and Gerchak (1996) provide examples where the optimal lot sometimes decreases in the demand and in the setup cost, and show that for some yield patterns the optimal lot may be smaller than the demand.

Only a few results are known concerning multistage systems with random yields and rigid demand. Grosfeld-Nir and Ronen (1993) studied a SBNS with binomial machines. They showed that the SBNS can be replaced by a single stage with truncated negative-binomial yields. Grosfeld-Nir (1995) studied SBNSs where the expected yield is proportional to the lot size. He proved that if the BN is a binomial machine, the SBNS can be reduced to a single-stage binomial machine. Wein (1992) assumed “perfect rework”: if the number of usable units exiting a certain stage is not sufficient, bad units can be perfectly reworked, thus necessitating at most two production runs at each stage. Grosfeld-Nir and Gerchak (2002) expanded on Wein's idea, allowing unlimited reworks at each stage and recognizing the possible difference between work and rework. Pentico (1994) analyzed a heuristic where, if the number of good units ready to enter a stage is sufficiently large, all units are processed by the next stage. Grosfeld-Nir and Robinson (1995) studied a binomial two-stage system. They used LP to obtain the optimal policy for demand of 1 or 2 units and a heuristic for larger demand.

In 2 The single stage, 3 A single-bottleneck system with binomial yields we summarize relevant known results concerning the single stage and the SBNS. In Section 4 we formulate the two-stage problem and explain why a direct recursive solution is impossible. In Section 5 we introduce algorithms used to solve the two-stage problem. The results are compared to optimal results obtained by LP and to special test cases where the optimal solution is known, namely SBNSs. In Section 6 we address the “basic 2-BNS” (three machines: BN, 0-BN, BN) and show how the “general 2-BNS” can be reduced to a basic 2-BNS. Section 7 provides concluding remarks.