# Stochastic Scheduling for a Network of MEMS Job Shops

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This work is motivated by the pressing need for operational control in the fabrication of Microelectromechanical systems or MEMS. MEMS are miniature three-dimensional integrated electromechanical systems with the ability to absorb information from the environment, process this information and suitably react to it. These devices offer tremendous advantages owing to their small size, low power consumption, low mass and high functionality, which makes them very attractive in applications with stringent demands on weight, functionality and cost. While the system''s "brain" (device electronics) is fabricated using traditional IC technology, the micromechanical components necessitate very intricate and sophisticated processing of silicon or other suitable substrates. A dearth of fabrication facilities with micromachining capabilities and a lengthy gestation period from design to mass fabrication and commercial acceptance of the product in the market are factors most often implicated in hampering the growth of MEMS. These devices are highly application specific with low production volumes and the few fabs that do possess micromachining capabilities are unable to offer a complete array of fabrication processes in order to be able to cater to the needs of the MEMS R&D community. A distributed fabrication network has, therefore, emerged to serve the evolving needs of this high investment, low volume MEMS industry. Under this environment, a central facility coordinates between a network of fabrication centers (Network of MEMS job shops -- NMJS) containing micromachining capabilities. These fabrication centers include commercial, academic and government fabs, which make their services available to the ordinary customer. Wafers are shipped from one facility to another until all processing requirements are met. The lengthy and intricate process sequences that need to be performed over a network of capital intensive facilities are complicated by dynamic job arrivals, stochastic processing times, sequence-dependent set ups and travel between fabs. Unless the production of these novel devices is carefully optimized, the benefits of distributed fabrication could be completely overshadowed by lengthy lead times, chaotic routings and costly processing. Our goal, therefore, is to develop and validate an approach for optimal routing (assignment) and sequencing of MEMS devices in a network of stochastic job shops with the objective of minimizing the sum of completion times and the cost incurred, given a set of fabs, machines and an expected product mix.

In view of our goal, we begin by modeling the stochastic NMJS problem as a two-stage stochastic program with recourse where the first-stage variables are binary and the second-stage variables are continuous. The key decision variables are binary and pertain to the assignment of jobs to machines and their sequencing for processing on the machines. The assignment variables essentially fix the route of a job as it travels through the network because these variables specify the machine on which each job-operation must be performed out of several candidate machines. Once the assignment is decided upon, sequencing of job-operations on each machine follows. The assignment and sequencing must be such that they offer the best solution (in terms of the objective) possible in light of all the processing time scenarios that can be realized. We present two approaches for solving the stochastic NMJS problem. The first approach is based on the L-shaped method (credited to van Slyke and Wets, 1969). Since the NMJS problem lacks relatively complete recourse, the first-stage solution can be infeasible to the second-stage problem in that the first stage solution may either violate the reentrant flow conditions or it may create a deadlock. In order to alleviate these infeasibilities, we develop feasibility cuts which when appended to the master problem eliminate the infeasible solution. Alternatively, we also develop constraints to explicitly address these infeasibilities directly within the master problem. We show how a deadlock involving 2 or 3 machines arises if and only if a certain relationship between operations and a certain sequence amongst them exists. We generalize this argument to the case of m machines, which forms the basis for our deadlock prevention constraints. Computational results at the end of Chapter 3 compare the relative merits of a model which relies solely on feasibility cuts with models that incorporate reentrant flow and deadlock prevention constraints within the master problem. Experimental evidence reveals that the latter offers appreciable time savings over the former. Moreover, in a majority of instances we see that models that carry deadlock prevention constraints in addition to the reentrant flow constraints provide at par or better performance than those that solely carry reentrant flow constraints.

We, next, develop an optimality cut which when appended to the master problem helps in eliminating the suboptimal master solution. We also present alternative optimality and feasibility cuts obtained by modifying the disjunctive constraints in the subproblem so as to eliminate the big H terms in it. Although any large positive number can be used as the value of H, a conservative estimate may improve computational performance. In light of this, we develop a conservative upper bound for operation completion times and use it as the value of H. Test instances have been generated using a problem generator written in JAVA. We present computational results to evaluate the impact of a conservative estimate for big H on run time, analyze the effect of the different optimality cuts and demonstrate the performance of the multicut method (Wets, 1981) which differs from the L-shaped method in that the number of optimality cuts it appends is equal to the number of scenarios in each iteration. Experimentation indicates that Model 2, which uses the standard optimality cut in conjunction with the conservative estimate for big H, almost always outperforms Model 1, which also uses the standard optimality cut but uses a fixed value of 1000 for big H. Model 3, which employs the alternative optimality cut with the conservative estimate for big H, requires the fewest number of iterations to converge to the optimum but it also incurs the maximum premium in terms of computational time. This is because the alternative optimality cut adds to the complexity of the problem in that it appends additional variables and constraints to the master as well as the subproblems. In the case of Model 4 (multicut method), the segregated optimality cuts accurately reflect the shape of the recourse function resulting in fewer overall iterations but the large number of these cuts accumulate over the iterations making the master problem sluggish and so this model exhibits a variable performance for the various datasets. These experiments reveal that a compact master problem and a conservative estimate for big H positively impact the run time performance of a model. Finally, we develop a framework for a branch-and-bound scheme within which the L-shaped method, as applied to the NMJS problem, can be incorporated so as to further enhance its performance.

Our second approach for solving the stochastic NMJS problem relies on the tight LP relaxation observed for the deterministic equivalent of the model. We, first, solve the LP relaxation of the deterministic equivalent problem, and then, fix certain binary assignment variables that take on a value of either a 0 or a 1 in the relaxation. Based on this fixing of certain assignment variables, additional logical constraints have been developed that lead to the fixing of some of the sequencing variables too. Experimental results, comparing the performance of the above LP heuristic procedure with CPLEX over the generated test instances, illustrate the effectiveness of the heuristic procedure. For the largest problems (5 jobs, 10 operations/job, 12 machines, 7 workcenters, 7 scenarios) solved in this experiment, an average savings of as much as 4154 seconds and 1188 seconds was recorded in a comparison with Models 1 and 2, respectively. Both of these models solve the deterministic equivalent of the stochastic NMJS problem but differ in that Model 1 uses a big H value of 1000 whereas Model 2 uses the conservative upper bound for big H developed in this work. The maximum optimality gap observed for the LP heuristic over all the data instances solved was 1.35%. The LP heuristic, therefore, offers a powerful alternative to solving these problems to near-optimality with a very low computational burden. We also present results pertaining to the value of the stochastic solution for various data instances. The observed savings of up to 8.8% over the mean value approach underscores the importance of using a solution that is robust over all scenarios versus a solution that approximates the randomness through expected values.

We, next, present a dynamic stochastic scheduling approach (DSSP) for the NMJS problem. The premise behind this undertaking is that in a real-life implementation that is faithful to the two-stage procedure, assignment (routing) and sequencing decisions will be made for all the operations of all the jobs at the outset and these will be followed through regardless of the actual processing times realized for individual operations. However, it may be possible to refine this procedure if information on actual processing time realizations for completed operations could be utilized so that assignment and sequencing decisions for impending operations are adjusted based on the evolving scenario (which may be very different from the scenarios modeled) while still hedging against future uncertainty. In the DSSP approach, the stochastic programming model for the NMJS problem is solved at each decision point using the LP heuristic in a rolling horizon fashion while incorporating constraints that model existing conditions in the shop floor and the actual processing times realized for the operations that have been completed.

The implementation of the DSSP algorithm is illustrated through an example problem. The results of the DSSP approach as applied to two large problem instances are presented. The performance of the DSSP approach is evaluated on three fronts; first, by using the LP heuristic at each decision point, second, by using an optimal algorithm at each decision point, and third, against the two-stage stochastic programming approach. Results from the experimentation indicate that the DSSP approach using the LP heuristic at each decision point generates superior assignment and sequencing decisions than the two-stage stochastic programming approach and provides solutions that are near-optimal with a very low computational burden. For the first instance involving 40 operations, 12 machines and 3 processing time scenarios, the DSSP approach using the LP heuristic yields the same solution as the optimal algorithm with a total time savings of 71.4% and also improves upon the two-stage stochastic programming solution by 1.7%. In the second instance, the DSSP approach using the LP heuristic yields a solution with an optimality gap of 1.77% and a total time savings of 98% over the optimal algorithm. In this case, the DSSP approach with the LP heuristic improves upon the two-stage stochastic programming solution by 6.38%. We conclude by presenting a framework for the DSSP approach that extends the basic DSSP algorithm to accommodate jobs whose arrival times may not be known in advance.