Infeasible schedules

If a schedule is computed based on a model that does not consider uncertainties, this schedule can become suboptimal or even infeasible when the situation has changed. For example, a schedule can become suboptimal if a batch is unexpectedly of inferior quality and the revenues are a function of its quality. A schedule can become infeasible if there is an unexpected plant failure that reduces the plant capacity a batch has to be immediately transferred to another unit, but no unit is available. Then it is impossible to modify the infeasible schedule to a feasible one. 

When these uncertainties are not considered in the computation of a schedule, the uncertainties in the capacity may lead to infeasible schedules, e.g., a schedule requires more capacity than available, whereas the uncertain demands have an effect on the value of the profit, e.g., when a schedule results in more or less product than demanded. 

The paper is structured as follows. The next section describes the inter-dependent decision variables that have to be determined during the operation of a pipeless plant. Section 3 motivates and describes the use of an EA for the scheduling of the steps of the recipes. Special attention is paid to the handling of infeasibilities by repair algorithms... 

It is assumed that an unexpected event unveils along the time horizon. If the proactive schedule results infeasible, the right-shifting policy is applied to it. This schedule is compared with the reactive one for the nominal scenario. Both schedules are compared in terms of the proposed evaluation criterion, Eq.(20) excluding operational costs for the sake of simplicity. 

For this problem, we use exactly the same algorithm as in Section 1.1, except that a different sequence rearrangement strategy is used and the initial values of kT are determined differently. We specify appropriate values for initial kT, the final kT, the number of schedules (NS) actually evaluated at each kT and the reduction factor a by which kT is reduced at each iteration. As mentioned in section 3.2, we select a job randomly and insert it at randomly selected positions in each group sequence to create a new production sequence from the current one. Then we check its feasibility as discussed in section 3.3. Only if the sequence is feasible, we determine its completion times using the simulation algorithm and we do not count the infeasible sequences in the number of schedules NS at each kT. 

Second period constraint 100 < 105 is not satisfied. We don t need to check remaining constraints since the problem became infeasible. We caimot satisfy the demands of the first two periods with our available resources for the first two periods. However, all of the constraints were satisfied, then the next step would be to find an initial feasible solution. For example, as we increase the capacities for each period to 60, the problem becomes feasible. We can shift back demands to find initial solution. Fifth period net requirements is more than our capacity, so five units are shifted to third period. Then our new production/ordering schedule becomes D = (45, 60, 50, 60, 60, 44). Now we can improve the initial solution. There may be different approaches to improve the solution, we adopt one mentioned by Nahmias [3]. The idea is to shift production orders back as long as the holding costs is less than the set-up costs starting from the last period. In our example, we don t have enough capacity in previous periods to shift 44 back. 

With classical structure, the proposed GA is implemented in Matlab to search the optimal/good solutions for the problem. This study considers the problem associated with soft precedence constraints, which will incur a penalty if violated rather rendering the sequence and schedule infeasible. A penalty implies that the respective chromosome is less likely to pass in the next generation, but still may have very valuable characteristics to pass on through the evolution process. 

It is evident that uncertainties in batch operations may arise from different sources (i.e., external demand, prices of raw and final products, processing times, and equipment availability) causing that programmed schedules become nonoptimal and in some cases infeasible. Despite the uncertain nature of scheduling problems, research efforts over last decades have primarily focused on deterministic formulations, which assume that all parameters are precisely known in advance. 

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