Integrated maintenance and production planning with endogenous uncertain yield
Introduction
In many production environments, diminishing machine conditions and high production rates can adversely affect the product quality. The machine yield can be jointly impacted by both the production quantity and maintenance decisions. This impact is especially important where the maintenance cycles are short compared to production cycles. Increasing wear in machine shop tool bits may cause defects over time which would result with faulty products. For instance, as part of the etch operation in semiconductor water fabrication, chemicals are used to strip materials from the surface of the silicon wafers. The contamination of the inner chambers of the etch equipment rapidly increases with higher production levels. This results in defective products and in lower machine yield. It creates unique challenges in multi-stage maintenance and production planning, and can affect the inventory and outsourcing costs. Proper understanding of such relationships has the potential to result in significant savings in operational costs and improved efficiency for the overall production system. Therefore, production and maintenance decisions as well as their impact on uncertain yield need to be considered simultaneously. The complexity of these relationships makes the integrated system-based approaches crucial. This paper introduces a novel integrated decision making framework to investigate the trade-offs among production planning and maintenance decisions under random demand and endogenous (decision dependent) yield.
Decision makers need to adapt to dynamic behavior of the production environments because of the uncertainty involved with the markets. This motivates models that allow integrated decision making under uncertainty while addressing a high number of decision alternatives. Integrated and simultaneous production and maintenance models are shown to outperform traditional sequential approaches [4], [41]. Within such integrated decision making frameworks, the machine yield can be a function of maintenance and/or production. However, most of the existing models can deal with only a relatively small number of decision alternatives. The main challenge is the computational complexity; especially for high levels of decision alternatives and when uncertainty is considered. Stochastic programming methods can be employed to deal with many alternatives for the decision variables under long term uncertainty [6]. Mula et al. [31] point out the frequent use of stochastic programming models in their survey of production planning models. However, the dependence of a random quantity on previous decisions require models that consider endogenous randomness. Particularly, the probability distribution of the random variable may depend on the previous decisions. The solution of such models is not straightforward, therefore mostly models with discrete random variables and small number of scenarios are considered. Recent computational and methodogical advances in simulation based optimization algorithms such as those in [11] can be utilized to address problems with continuous uncertainty which may depend on decisions with higher number of alternatives.
This paper contributes to the literature in two main ways. First, the integrated model lets the decision maker evaluate the trade-offs involved among production quantity, maintenance, outsourcing, salvaging decisions and uncertain endogenous machine yield in a two-stage setting with random demand. We utilize a stochastically proportional yield based approach in which yield rate is modeled via a Truncated Normal distribution. This lets the decision maker to explicitly model variance independently from the batch size. Our approach is general enough to accommodate any discrete or continuous probability distribution as well as any form of objective function. In addition, our stochastic program is flexible enough to accommodate more decision alternatives than the existing literature such as [40] that can deal with relatively small number (up to 20) of maximum production quantities. Second, this is the first application of augmented probability simulation (APS) based stochastic programming solution approach of Ekin et al. [11] to solve a production planning and maintenance problem. Finding the solution for the proposed model with endogenous uncertainty is not straightforward and had not been considered previously. We solve the proposed two-stage nonlinear stochastic program using an augmented probability simulation based optimization method. This is one of the first applications of augmented probability simulation based optimization method, the first to solve a stochastic optimization model that includes direct endogenous randomness or nonlinearity.
The paper is organized as follows. Section 2 presents the relevant literature review. Section 3 presents the modeling framework and details about modeling yield. Section 4 describes the stochastic optimization method used to solve the proposed model. Section 5 provides a numerical illustration with a large numerical study and sensitivity analysis, and presents a discussion of the results. Section 6 concludes with a summary of findings and directions for future work.
Section snippets
Literature review
This section provides a literature review that is relevant to the proposed model and solution approach from three perspectives. First, the literature of modeling yield and integrated production models is presented. This is followed by a brief overview of endogenous stochastic models and simulation based stochastic programming approaches.
Modeling framework
The production environment for a single machine and a single product is modeled in a two stage setting. In the following, the notation for decision variables and parameters is presented. It is followed by a discussion of the optimization model and modeling endogenous random yield.
Decision variables
x1: production quantity, continuous first stage
x2: maintenance decision, integer first stage; x2 ∈ {1, 2, 3} that correspond to zero maintenance, preventive or corrective maintenance respectively
x2a, x
Methodology
A closed form solution for the proposed model is not available, which requires the use of simulation based methods. SAA is computationally inefficient for endogenous problems, since the scenario trees become very large for continuous uncertainty and/or many decision alternatives. We use the APS approach of Ekin et al. [11], which provides a natural way to deal with conditional dependence between the state and the decision spaces. In what follows, we present the implementation of APS for the
Numerical illustration
This section presents the computational results for various production environments. We investigate the trade-offs among the uncertain yield, optimal production and maintenance decisions and objective function. First, we report and analyze the results for 7,776 test problems that consider different specifications of the parameters. Then, we use particular cases to investigate the sensitivity of the results with respect to changes in some parameters and discuss the potential use of other
Conclusion and directions for future work
In this study, we present a novel integrated decision making framework for production planning and maintenance under endogenous uncertain yield. This is the first such approach that treats the uncertain yield using a stochastically proportional model with endogenous mean. This lets the decision maker measure the effects of previous decisions on production environment and machine condition. In addition, the variance of the yield rate can be specified independently from the endogenous mean and
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