Abraham Grosfeld-Nir

Lot-Sizing Two-Echelon Assembly Systems with Random Yields and Rigid Demand

We consider a two-echelon assembly system producing a single final product for which the demand is known. The first echelon consists of several parallel stages, whereas the second echelon consists of a single assembly stage. We assume that the yield at each stage is random and that demand needs to be satisfied in its entirety; thus, several production runs may be required. A production policy should specify, for each possible configuration of intermediate inventories, on which stage to produce next and the lot size to be processed. The objective is to minimize the expected total of setup and variable production costs. We prove that the expected cost of any production policy can be calculated by solving a finite set of linear equations whose solution is unique. The result is general in that it applies to any yield distribution. We also develop efficient algorithms leading to heuristic solutions with high precision and, as an example, provide numerical results for binomial yields.

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.

Control limits for two-state partially observable Markov decision processes

Abstract A production process can be in either a GOOD or a BAD state. The true state is unknown and can only be inferred from observations. If the state is good during one period it may deteriorate and become bad during the next period. Two actions are available: CONTINUE or REPLACE (for a fixed cost). The objective is to maximize the expected discounted value of the total future profits. We prove that “dominance in expectation” (the expected profit is larger in the good state than in the bad state) suffices for the optimal policy to be of a CONTROL LIMIT (CLT) type: continue if and only if the good state probability exceeds the CLT. This condition is weaker than “stochastic dominance”, which has been prevailing. We also show that the “expected profit function” is convex, strictly increasing.

Single bottleneck systems with proportional expected yields and rigid demand

Abstract This study considers single bottleneck systems: multistage production systems where all setup costs except one are zero. The stage with non-zero setup cost is defined to be the bottleneck (BN). Production is in lots and demand needs to be satisfied in its entirety. The expected yield (number of good units) at each stage is proportional to the lot size and only good items may proceed from one stage to another. We reveal several general features of the optimal policy and discuss some specific yields in detail. In particular it is shown that if units are processed one-by-one on the non-BNs, the optimal lotsizing problem can be reduced to the problem of optimally lotsizing a single-stage ‘equivalent machine’. Then, if the BN has Binomial yeild, the ‘equivalent machine’ itself has Binomial yields. Also, examples are given which demonstrate the non-intuitive structure of the optimal policy.

An Optimal Lot-Sizing and Offline Inspection Policy in the Case of Nonrigid Demand

A batch production process that is initially in the in-control state can fail with constant failure rate to the out-of-control state. The probability that a unit is conforming if produced while the process is in control is constant and higher than the respective constant conformance probability while the process is out of control. When production ends, the units are inspected in the order they have been produced. The objective is to design a production and inspection policy that guarantees a zero defective delivery in minimum expected total cost. The inspection problem is formulated as a partially observable Markov decision process (POMDP): Given the observations about the quality of the items that have already been inspected, the inspector should determine whether to inspect the next unit or stop inspection and possibly pay shortage costs. We show that the optimal policy is of the control limit threshold (CLT) type: The observations are used to update the probability that the production process was still in control while producing the candidate unit for inspection. The optimal policy is to continue inspection if and only if this probability exceeds a CLT value that may depend on the outstanding demand and the number of uninspected items. Structural properties satisfied by the various CLT values are presented.

Serial production systems with random yield and rigid demand: A heuristic

We consider a heuristic for serial production systems with random yields and rigid demand: all usable units exiting a stage move forward. We calculate optimal lots and corresponding expected costs for binomial, interrupted-geometric, and all-or-nothing yields. Our method is that it makes it easy to analyze large systems.

Production to order on a two machine line with random yields and rigid demand

Abstract Random yields may necessitate multiple production runs whenever demand is ‘rigid’; i.e., whenever shortages are not permitted. In this paper, we extend the analysis of problems with rigid demand to a two machine system which has an intermediate inventory of inspected units. For such systems, the production policy must specify, as a function of the intermediate inventory level and the remaining demand, which machine to run next and its lot size. After showing that the optimal policy can be calculated through a series of linear programs, we use the dual formulation to demonstrate that a straightforward policy improvement algorithm will yield the optimal policy. Finally, some empirically observed properties of both the optimal policy and its cost function are used to develop a simpler heuristic algorithm which performs within 1/2% of the optimal over a test bed of 180 problems.

Partially observed Markov decision processes with binomial observations

We consider partially observed Markov decision processes with control limits. We analytically show how the finite-horizon control limits are non-monotonic in (a) the time remaining and (b) the probability of obtaining a conforming unit. We also prove that the infinite-horizon control limit can be calculated by solving a finite set of linear equations.

Multistage production systems with random yields and rigid demand

The increased complexity of contemporary manufacturing makes it hard to maintain a predictable level of output. We consider several types of multistage systems in an environment where yield is random, production is in lots and demand needs to be satisfied in its entirety. For each possible configuration of intermediate inventories a production policy must specify the stage at which to produce next and the lot size to be processed. The objective is to minimise the expected total of setup and variable production costs. We show that the expected cost associated with any arbitrary production policy can be calculated by solving a finite set of linear equations. The result is general in that it applies to any yield structure. We also develop efficient heuristics to solve various multistage systems. Selected numerical results are presented for binomial yields.