Robustness Optimization of Fms under Production Plan Variations: Preliminary Results

A method for robustness optimization of flexible manufacturing systems is presented which undergo forecasted product plan variations. A configuration of an FMS is modeled by a colored Petri net and the associated transition firing sequence. The robustness optimization of the colored Petri net model is formulated as a multi-objective optimization problem which minimizes production costs under multiple production plans, and reconfiguration cost due to production plan changes. As a first attempt, machines with limited flexibility are considered, and a genetic algorithm, coupled with a simple FIFO dispatching rule, is used to simultaneously find an semi-optimal resource allocation and event-driven schedule of a colored Petri net. The resulting Petri nets are then compared with the Petri nets optimized for a particular production plan in order to address the effectiveness of the robustness optimization for simultaneous production of products with different similarities. The simulation results suggest that the robustness optimization should be considered when the products are moderately different in their manufacturing processes. INTRODUCTION Flexible manufacturing systems (FMS’s) are a class of manufacturing system which can be rapidly configured to produce ∗Corresponding author †Formerly a visiting student, Department of Mechanical Engineering and Applied Mechanics, University of Michigan. multiple types of products. Recent increase in the use of FMS’s has been driven by the need of agile manufacturing systems which can quickly adopt changes in production plans due to market demand fluctuation. While the increased flexibility of an FMS provides greater productivity under various production scenario, it imposes increased complexity in allocation of given resources to different processes required in making each product, and the scheduling of the sequence of activities to accomplish the best production efficiency (Lee, 1994). In order to quickly adapt fluctuating market demand, the resource allocation and scheduling, or configuration in short, of an FMS should not simply be optimized for the current production plan. Rather, it should ideally be optimized for robustness against the variation in production plans, so that the system can deal with the variation with minimal reconfiguration (i.e. reallocation and rescheduling) while achieving consistently efficient production under all production plans of interest. For this, a reliable forecast on the future change in production plan must be provided, which may or may not be available at a given time. Assuming such forecasts are available, let us consider the scenario where the FMS simultaneously produces two kinds of products A and B, and the total number of production (sum of the numbers of A’s and B’s to be produced) per unit time (eg. a day) is kept constant with production plan variation (i.e. only a fraction of the two products varies). When A and B are very similar1, then, it is conjectured that one would not need to con1In which sense is left undefined here. This issue will be revisited in the 1 Copyright  1998 by ASME sider robustness optimization since the configuration optimized for the current production plan is robust enough such that little system reconfigurations are necessary to deal with production plan change (imagine the extreme of this case where A and B are identical). On the other hand, when products under simultaneous production are different (but not too different to impair the justification for simultaneous production), slight change in the production plan will heavily impact production efficiency, hence necessitating the system reconfiguration in order to achieve efficient production under the new production plan. The above conjecture motivated us to develop a methodology for robustness optimization of FMS configuration which undergo given product plan variations, and to study the effectiveness of the methodology for simultaneous production of products with different similarities. A configuration of an FMS is modeled by a colored Petri net and the associated transition firing sequence. The robustness optimization of the colored Petri net model is formulated as a multi-objective optimization problem which minimizes production costs under multiple production plans, and reconfiguration cost due to production plan changes. As an initial attempt, machines with limited flexibility are considered, and a genetic algorithm, coupled with a simple FIFO dispatching rule, is used to simultaneously find an semi-optimal resource (machine) allocation and event-driven schedule on a colored Petri net. The resulting Petri nets are then compared with the Petri nets optimized for a particular production plan in order to validate the above conjecture. RELATED WORK Petri nets (Petri, 1962) have been widely used for analysis and simulation of FMS due to their capability of modeling concurrency, synchronization and sequencing in discrete-event systems (Dubois, 1983; Narahari, 1985). Among the most recent is the work by Dhumal, Dhawan, Kona and Soni (Dhumal, 1996), where a Petri net model of a flexible forging cell was used to analyze the production performances under different production scenarios. In addition to such use as an analysis tool, Petri net models are often used for FMS scheduling problems. Given a job specification (the amount of production and the sequence of operations needed for each job), and the corresponding resource allocation (the type and number of machines for each operation, and the processing time), one can construct a Petri net model of an FMS, where event-driven operation schedules of the modeled FMS are represented as the transition firing sequences of the Petri net. Due to the NP-completeness of the underlying job-shop scheduling problem (JSSP) (Garey, 1979), an optimal schedule is often found via heuristic search algorithms such as beam search (Shih, 1991), A* algorithm (Lee, 1994) and genetic algorithms (Chiu, discussion section. 1997), coupled with discrete-event simulation of the operation of the Petri net model. In general, the quality of the optimal schedule is influenced the quality of resource allocation (i.e. the topology of the Petri net model) for a given job specifications. This motivates the simultaneous optimization of resource allocation and scheduling, a generalization of JSSP known as generalized resourceconstrained project scheduling problems (GRCPSP), which is also NP-complete (Garey, 1979). GRCPSP is typically formulated as mathematical programming problems and solved by heuristic search algorithms (Sprecher, 1994). The solution provides an optimal allocation of a given resources and time-driven operation schedules. Although event-driven schedules are often preferred for FMS scheduling due to their robustness (Lee, 1994), discrete-event based models such as Petri nets are rarely used for GRCPSP due to the computational time for the model simulation. In the above work, the search is directed towards the discovery of the schedule (and the resource allocation in the case of GRCPSP) optimized for a fixed production plan, which could potentially be sensitive to a small perturbation in the current production plan. In continuous mathematical programming, this issue is addressed as sensitivity analyses, where the sensitivity of the optimum to small parameter perturbation is computed, in most cases, in terms of Lagrange multipliers. Several method has been proposed to find an optimal (or suboptimal) solution of nonlinear programming problems which is less sensitive to parameter perturbations (d’Entremont, 1988; Parkinson, 1990; Sundaresan, 1993). Since these methods are essentially an application of Taguchi’s robust parameter design (Taguchi, 1978; Taguchi, 1987) to nonlinear programming, they are designed for continuous optimization problems, and hence do not directly apply to problems involving discrete design parameters, such as the FMS scheduling problems using Petri nets discussed above.