یک روش حساس به خطر نگهداری و تعمیرات بهره ور

While risk-sensitive (RS) approaches for designing plans of total productive maintenance are critical in manufacturing systems, there is little in the literature by way of theoretical modeling. Developing such plans often requires the solution of a discrete-time stochastic control-optimization problem. Renewal theory and Markov decision processes (MDPs) are commonly employed tools for solving the underlying problem. The literature on preventive maintenance, for the most part, focuses on minimizing the expected net cost, and disregards issues related to minimizing risks. RS maintenance managers employ safety factors to modify the risk-neutral solution in an attempt to heuristically accommodate elements of risk in their decision making. In this paper, our efforts are directed toward developing a formal theory for developing RS preventive-maintenance plans. We employ the Markowitz paradigm in which one seeks to optimize a function of the expected cost and its variance. In particular, we present (i) a result for an RS approach in the setting of renewal processes and (ii) a result for solving an RS MDP. We also provide computational results to demonstrate the efficacy of these results. Finally, the theory developed here is of sufficiently general nature that can be applied to problems in other relevant domains.

Total productive maintenance (TPM) is a management initiative that has been widely embraced in the industry. A positive strategic outcome of such implementations is the reduced occurrence of unexpected machine breakdowns that disrupt production and lead to losses which can exceed millions of dollars annually. Additionally, frequent machine breakdowns indirectly can lead to a host of other problems, e.g., difficulties in meeting customer deadlines, which makes the transition from make-to-stock to make-to-order difficult ( Suri, 1998) and magnifies the need to keep extra safety stocks, increasing inventory-holding costs ( Askin & Goldberg, 2002). An important tool of a TPM program is the stochastic model used to determine the optimal time for preventive maintenance (PM) ( Askin & Goldberg, 2002). PM can help reduce the frequency of unexpected repairs when the failure rate is of an increasing nature ( Das & Sarkar, 1999; Lewis, 1994). Renewal processes (Kao, 1997; Ross, 1992) and Markov decision processes (MDPs) (Bertsekas, 1995; Puterman, 1994) are frequently used as the underlying stochastic models in a TPM program. A critical drawback of a traditional approach in TPM is to use the expected value of the long-run cost as the objective function. Such an approach overlooks the risk associated with the occasional high cost that can occur in system optimized with respect to the expected cost. As a result, risk-sensitive (RS) managers, whom we interacted with in a local automobile industry, modify the predicted optimal (with respect to the expected cost) time for maintenance, τ*τ*, by using a factor of safety, ηη, where η>1η>1, such that the time for PM is then: τ*/ητ*/η. While this certainly results in a more conservative time for maintenance, it is a heuristic approach. What managers really need is a more sophisticated approach that would help them (i) quantify their risk sensitivity on a scale from 0 to 1 and (ii) determine the optimal maintenance time using a model that incorporates this factor. This clearly motivates the need for embedding the well-known Markowitz criterion ( Markowitz, 1952) within the stochastic model. Another significant demand of managers from the model is the ability to quantify risks in terms of dollars (or Euros) and hours—units that they are comfortable with. In particular, senior managers involved in developing long-term plans for an enterprise are familiar with the idea of using variance per unit time as a measure of risk in strategic decision making (see Ruefli, Collins, & Lacugna, 1999 for an extensive survey). Since TPM has a significant strategic impact on the organization, the units of risk in these calculations should ideally match those used in strategic management. One of the goals of this research is to develop models that can be conveniently used by managers. A general cost function (objective function) using the Markowitz criterion is equation(1) g(τ)=μC+θσ2withθ>0, Turn MathJax on where μCμC and σ2σ2 denote the long-run mean and the long-run variance, respectively, of the net cost per unit time incurred from following a preventive maintenance plan that prescribes ττ as the time for PM. An alternative formulation in terms of rewards , in which the objective function is maximized , is gR(τ)=μR-θσ2, with θ>0gR(τ)=μR-θσ2,withθ>0, where μRμR and σ2σ2 denote the long-run mean and variance of the net reward per unit time, respectively. Since μR=-μCμR=-μC, both formulations are equivalent. Risk-neutral (RN) statistical models for PM use θ=0θ=0. Typically, θθ is selected by experimentation by the manager and is a function of the variability in the system. A very large value for θθ is undesirable, since that could produce a solution with a very low variability but also with a very high cost. This is because a very large value for θθ amplifies the importance of the variance and diminishes that of the mean. A very low value for θθ, on the other hand, is indicative of a manager who is neutral to risks. Clearly, the smaller the value of θθ, the closer the model gets to becoming RN. The time for PM, it must be understood, is the time since the last repair or PM. A common assumption is that the unit or the line is as good as new when it is repaired or preventively maintained. A typically made second assumption is that when the machine is not working, it is assumed not to age. We will stick to these two assumptions here. The main focus of this paper is to develop a theory when θ>0θ>0. The work of Chen and Jin (2003) also employs the Markowitz criterion, but their approach is quite different than ours; this will be clarified via our discussions below. TPM plans for the production line in its entirety tend to be distinct from those for individual units that operate independently of the line. Most factories are full of such units, e.g., fork-lift trucks, electrical pumps, etc. We will develop separate models for the individual-unit scenario and the production-line scenario. For the case of the individual unit, we will present a renewal-theory model and for the case of the production line, we will present a more involved model based on MDPs. The analysis will involve presentation of some key results that could be applied to a large number of other management-science problems involving control theory. Thereafter, we will present results from computational experiments with both models. The remainder of this paper is organized as follows. Section 2 presents the renewal-theory model, and Section 3 presents the Markov decision model. Section 4 describes empirical work done using these two models, and Section 5 concludes the paper.

The literature on RS PM is limited.We developed two mathematically sound models, based on renewal theory and MDPs, for RS TPM. We need to point out that while our models were developed for optimizing a combination of mean and variance, they could be adapted easily for combining mean and the standard deviation, which have the same units. Our renewal-theory model was motivated by an industrial need for a model that quantifies risk in tractable units, e.g., dollar2 per hour. For the MDP model, we developed a surrogate objective function that closely mimicked the exact objective function, computationally. What is interesting is that for the surrogate, we were able to develop a computationally attractive DP approach, whose convergence we were able to show. Both models developed above are of a sufficiently general nature, and can be applied to other problems in management science. Other problem domains for our models that we will pursue in future work are supply chain management and airline revenue management.