Robustness Optimization of Fms under Production Plan Variations: the Case of Cyclic Production

This paper discusses an extension of our previous work (Saitou, 1998) on robustness optimization of flexible manufacturing systems (FMS) that undergo forecasted production plan variations. The extension is made to a more general class of FMS performing “non-linear” or “cyclic” production that allows multiple operation types per one machine type. As in our previous work (Saitou, 1998), a configuration of an FMS is modeled as a colored Petri net and the associated transition firing sequence, and the robustness of FMS is defined as the insensitivity of production performances against variations in production plan. The optimization of the robustness of the colored Petri net model is formulated as a multi-objective optimization problem which minimizes production costs under multiple production plans (batch sizes for all jobs), and reconfiguration cost due to production plan changes. A genetic algorithm, coupled with a dispatching rule based on shortest imminent operation time (SIO), 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. The simulation results suggest that the proposed robustness optimization scheme should be considered when the products are moderately different in their job specifications so that optimizing for a particular production plan creates inevitably bottlenecks in product flow and/or deadlock under other production plans. INTRODUCTION Flexible manufacturing systems (FMS) are a class of manufacturing system which can be quickly configured to produce multiple types of products (jobs). Recent increase in the use of FMS is driven by the need of agile manufacturing that can quickly adopt changes in production plans (batch sizes for all jobs) 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 operations 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 (Saitou, 1998). 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 an 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 similar in their job specifications, then, it is conjectured that one would 1 Copyright  1999 by ASME not need to consider 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 moderately different, slight change in the production plan will heavily impact production efficiency, possibly due to the creation of bottlenecks in product flow. This would necessitate the system reconfiguration in order to achieve efficient production under the new production plan. The above conjecture motivated our previous work on Petrinet based robustness optimization of FMS under production plan variation (Saitou, 1998). The simple production scenarios discussed in the work validated this conjecture for a class of FMS performing “linear” production that only allows one operation type per one machine type. This paper presents an extension of this previous work to a more general class of FMS performing “non-linear” or “cyclic” production that allows multiple operation types per one machine type. As in our previous work (Saitou, 1998), a configuration of an FMS is modeled as a colored Petri net and the associated transition firing sequence, and the robustness of FMS is defined as the insensitivity of production performances against variations in production plan. The optimization of the robustness of the colored Petri net model is formulated as a multi-objective optimization problem which minimizes production costs under multiple production plans (batch sizes for all jobs), and reconfiguration cost due to production plan changes. A genetic algorithm, coupled with a dispatching rule based on shortest imminent operation time (SIO), is used to simultaneously find an semi-optimal resource (machine) 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 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). In addition to such use as an analysis tool, Petri net models are often used for FMS scheduling problems. Given a job specification (operation sequences needed for each job, the machine types and processing time for each operation), and the corresponding resource allocation (the number of machines in each type), 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, 1997), coupled with discrete-event simulation of the operation of the Petri net model. In general, the quality of the optimal schedule is influenced by the quality of resource allocation (i.e., the topology of the Petri net model) for a given job specification. 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 discrete programming problems and solved by heuristic search algorithms (Sprecher, 1994). The solution provides an optimal allocation of a given resources (i.e., machines) and timedriven 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. PROBLEM FORMULATION Colored Petri net model of manufacturing systems Colored Petri nets (Alla, 1985; David, 1992) are an extension of ordinary Petri nets where a place can contain multiple tokens distinguished by a “color” associated with each token. This extension allows colored Petri nets to model manufacturing systems capable of simultaneous production of multiple products in a graphically elegant manner by associating types of products with colors of tokens. As an ordinary Petri net, a colored Petri net is a directed graph consisting of two types of node, places and transitions. Two nodes are connected by an directed edge which connects either a place to a transition or a transition to a place (see Figures 1–3). In a basic form, a colored Petri net R is defined as a six-tuple: 2 Copyright  1999 by ASME R = 〈P,T, pre, post,m0,C〉 (1) where P is a set of places, T is a set of transitions and C is a set of colors. pre and post are functions of the type P× T × C 7→ Z|C| and m0 : P 7→ Z|C| is the initial marking, where Z is a set of integers. A place p ∈ P is graphically represented by a circle, and a transition t ∈ T is represented by a bar. A place can contain one or more tokens (with possibly different colors). The number and colors of tokens at a place p∈ P is called marking of the place denoted as m(p), where m : P 7→ Z|C|, and represented graphically as colored dots in a circle1. Let places p, q and a transition t are connected by edges (p,t) and (t,q). The place p is called an input place of the transition t, and the place q is called an output place of the transition t. Marking of places change according to the following rules: 1. For each input place p of a transition t, if m(p)≥ pre(p,t,c) for a color, t is called enabled with respect to the color c. 2. If a transition t is enabled with respect to a color c, it can fire. 3. If a transition t enabled with respect to a color c fires, m(p) changes to m(p)− pre(p,t,c), and for each output place q of t, m(q) changes to m(q)+ post(q,t,c). In addition to the above basic definition, capacities to places and time associated with places are often defined in FMS modeling (timed places with capacities). In this case, an enabled transition t can fire only if an enabling token has been in the input place p longer than or equal to a specified time2, and the total number of tokens does not exceed the capacity of the output place q as a result of marking change. A sequence of marking changes in all places of a colored Petri net is called evolution of marking. The evolution of marking in a colored Petri net from the initial marking represents the sequence of event occurrences in the modeled discrete-event system. Figures 1–3 illustrate the evolution of marking in a simple colored Petri net that models a production facility consisting of one “start” buffer ps, and one machine pm1 of type M1, and one machine pm2 of type M2. The production facility is to produce two types of products and which both need just one operation to finish. The machines of type M1 is capable of performing this operation on both product types and with an unit time, while the machines of type M2 can only perform the operation on product with an unit time. This job specification is summarized in Table 1, where columns indicate jobs (product types) and rows indicate operations (only 1In most literature, however, a token is represented as < c >, where c is a symbol representing the color of the token, as they are not normally printed in color. 2This assumes a “clock” keeping track of the marking changes. Table 1. example job specifications.