Global climate change requires stakeholders to consider energy elements in their decision-making. Electricity costs, in particular, constitute a significant portion of operational costs in most manufacturing systems. The electricity bills can be lowered if electricity-consuming operations are correctly scheduled. We consider a manufacturing operations control problem with known time-varying electricity prices in a finite planning horizon. Each operation is unique and has its own concave electricity consumption function. Pre-emptions of operations are allowed, yet postponing an operation incurs a cumulative penalty for each time period. In addition, each pre-emption is considered a new operation. The electricity cost in each time period is exogenous and there exists a capacity constraint on the total electricity amount consumed in each period due to infrastructure and provider's limitations. There is a fixed start-up cost incurred for switching on the machine and a fixed reservation cost incurred for keeping the machine 'On'. The system also includes a rechargeable battery. The customer has to determine when to process each operation within the time horizon so as to minimise total electricity consumption and operations postponement penalty costs. A dynamic programming solution is proposed and the complexity of the models is analysed. After examining several special cases of the model, the optimum times to charge and discharge the rechargeable battery are determined. A polynomial time algorithm for a special case of a single operation with uniform capacity is proposed.
Microgel particles with pore structure and ligands distributed evenly throughout their matrices overcome the major limitations of protein purification systems: low ligand density on the immobilized matrix and protein access to those ligands. A straightforward synthetic scheme for a highly efficient microgel matrix is reported.
Single machine scheduling problems have been extensively studied in the literature under the assumption that all jobs have to be processed. However, in many practical cases, one may wish to reject the processing of some jobs in the shop, which results in a rejection cost. A solution for a scheduling problem with rejection is given by partitioning the jobs into a set of accepted and a set of rejected jobs, and by scheduling the set of accepted jobs among the machines. The quality of a solution is measured by two criteria: a scheduling criterion, F1, which is dependent on the completion times of the accepted jobs, and the total rejection cost, F2. Problems of scheduling with rejection have been previously studied, but usually within a narrow framework-focusing on one scheduling criterion at a time. This paper provides a robust unified bicriteria analysis of a large set of single machine problems sharing a common property, namely, all problems can be represented by or reduced to a scheduling problem with a scheduling criterion which includes positional penalties. Among these problems are the minimization of the makespan, the sum of completion times, the sum and variation of completion times, and the total earliness plus tardiness costs where the due dates are assignable. Four different problem variations for deal- 396 J Comb Optim (2012) 23:395-424 ing with the two criteria are studied. The variation of minimizing F1 + F2 is shown to be solvable in polynomial time, while all other three variations are shown to be N P-hard. For those hard problems we provide a pseudo polynomial time algorithm. An FPTAS for obtaining an approximate efficient schedule is provided as well. In addition, we present some interesting special cases which are solvable in polynomial time.
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