یک مدل برنامه ریزی تولید یکپارچه با " زمان انجام تولید" وابسته به بار و ذخایر ایمنی

The divergence over the years of research paradigms addressing the production planning problem has led to the development of an extensive set of techniques, each of which can address a particular aspect of the practical problem and none of which provides a complete solution. In particular, most approaches fail to address the circular, non-linear dependency between resource utilization, lead-times and safety stocks. We present a non-linear programming formulation of the integrated problem using clearing functions that determines a work release schedule guaranteeing a specified service level in the face of stochastic demand. We introduce an iterative heuristic solution procedure that solves a relaxed LP approximation of the original NLP at each iteration to determine the lead-time profile to set safety-stock levels. Computational experiments suggest that our proposed iterative procedure performs well relative to conventional LP models that assume fixed, workload-independent lead-times.

Since the late 1950s, a wide variety of optimization models have been applied to problems of supply chain planning and control. As the years have gone by, three distinct paradigms have emerged which have become largely independent of each other, presumably due to the quite different mathematical tools they require. Mathematical programming models generally seek a minimum cost or maximum profit allocation of production resources to products over time. One approach, widely applied in the process industries, generates aggregate production plans from optimal short-term schedules constrained by highly detailed resource and process limitations. A second approach, more common in the discrete manufacturing industries, takes an aggregate view of the problem, dividing the planning horizon into discrete time buckets and determining a production rate for each product at each resource in each period ( Johnson and Montgomery, 1974 and Voss and Woodruff, 2003). Although both approaches are based on discrete time buckets, they use them in different ways. In the process systems engineering formulations, the time bucket is defined to be much smaller than the smallest processing time (raw processing time), in a manner analogous to numerical integration, justifying the assumption of linearity within a time bucket, as in system dynamics approaches (Sterman, 2000). In the industrial engineering literature, where discrete manufacturing systems dominate, the time bucket is much larger than the raw processing time. Thus many events such as job completions occur within a time bucket, and average values provide a good approximation for the behavior of the system being studied. Both these approaches have their disadvantages. While the industrial engineering models with large time buckets often fail to capture the dynamics of the production system, the process engineering models with small time buckets suffer from the curse of dimensionality due to the size of the resulting formulations. In both approaches, the models are primarily deterministic in nature, and do not explicitly consider the stochastic nature of either the production system or the demand it faces. Queueing models (e.g., Buzacott and Shanthikumar, 1993 and Hopp and Spearman, 2001), on the other hand, explicitly model the stochastic nature of the production process and the material flow between stages of the system. These models capture the non-linear relationships between key system parameters and performance measures, such as utilization and cycle time, effectively. However, they are primarily descriptive in nature, and their emphasis on steady-state solutions limits their usefulness for situations where the demand is a non-time-stationary stochastic process, which is generally the case in industrial practice. A third stream of research, inventory theory (Hadley and Whitin, 1963 and Zipkin, 1997), has developed a rich body of work. The dominant focus here has been on the uncertainty of demand, with very simple models of capacity being used to represent the production process. This work has resulted in the development of robust inventory management policies such as the base stock policy that can provide useful insights into how to address the consequences of uncertain demand. However, this work has also concentrated primarily on time-stationary systems, and almost exclusively addresses long-run expected behavior. The result of this segregation of research paradigms has been the development of an extensive set of techniques, each of which can address a particular aspect of the problem faced in practice and none of which can provide a complete solution. In particular, almost all prior approaches fail to address the circular, non-linear dependence between plant loading/utilization, lead-time and safety stocks. Jobs are released into the plant to meet demand, increasing resource utilization and hence the lead-time (the time elapsing between the release of a job into the facility and its completion) in a highly non-linear manner. This, together with demand uncertainty, introduces the need to hold safety stock to satisfy a desired customer service level, requiring additional releases that further increase utilization. In this paper we present a non-linear programming formulation of the complete problem using clearing functions and introduce an iterative heuristic that solves a relaxed LP approximation of the original NLP at each iteration to determine a load-dependent lead-time profile to set the safety-stock limits. We present computational experiments that evaluate the performance of the proposed heuristic, and show that it performs favorably compared to conventional linear programming models with a fixed, exogenous lead-time. Thus the contribution of this paper is to extend previous work using clearing functions for production planning in a deterministic environment (Asmundsson et al., 2009 and Asmundsson et al., 2006) to an environment with stochastic demand, where modelling the relationship between safety stocks, lead-times and resource utilization is critical.

In this work, we present an integrated non-linear programming formulation of the problem of planning production releases in the face of stochastic demand to maintain a specified service level. We capture the non-linear dependence between resource utilization and lead-times using clearing functions, and explicitly model the dependency between workload, lead-time and safety stocks. We propose an iterative heuristic procedure that solves a relaxed LP approximation of the original NLP at each iteration to determine the lead-time profile that is used to set safety-stock levels. The heuristic is thus capable of making the tradeoff between increasing the lead-time, and hence the safety-stock level, by increasing the utilization, and producing ahead of time to build finished goods inventory based on the costs. We present preliminary experimental results on the performance of the proposed heuristic, which are encouraging. Our future work constitutes a formal investigation of the convergence and optimality/quality of the iteration procedure, and its extension to multi-product and multi-supplier, multi-retailer and multi-market supply chain networks.