Document Type : Research Studies
Author
Department of Mechanical Power Engineering, Faculty of Engineering, Tanta University, Tanta, Egypt,
Abstract
The production of energy exists in different shapes and forms, e.g. thermal, mechanical, electrical or other forms of energy. Each of these forms exists in different types and arts. The production and scheduling of energy types are decision processes which are basically concerned with the adoption or more precisely the allocation of the natural and industrial resources in order to best satisfy marketing and customer requircinents at minimum best economic conditions. There are many reasons that make energy production planning a challenging problem. The variable structure of both the demand and costs, the difficulty to precisely forecast the demand at the very detailed level, the less flexibility to modify the operating conditions, and that there are usually long and uncertain delays in obtaining the industrial resources (new machines, workers training, subcontracting capacities, raw materials item components, etc.)
Researchers and seientists have paid a great part of attention for developing scheduling and planning systems to support such decision making processes. Two different approaches for production planning and scheduling are known. The first, called the monolithic approach, formulates the problems as a large scale mixed-integer linear programming problems and is usually solved approximately using Lagrangean relaxation to the mixed integer linear program. The second approach, called the hierarchical approach, partitions the planning and scheduling problem into a hierarchy of smaller subproblems. The upper hierarchy deals with strategic decisions for the planning horizon, while the lower hierarchy deals with the more detailed short-termed scheduling.
The computational effort required for the monolithic approach is generally greater than that required for the hierarchical approach. The hierarchical approach may require less detailed demand input data. Also the uncertainty is better treated by the hierarchical approach.
In this study, a comparison is presented between two different hybrid procedures for solving in hierarchical energy production planning and scheduling problem. A simplification in one of them is made. In the first procedure, hierarchical subproblems are included in an overall mixed integer linear programming formulation. Accordingly, the problem is partitioned into two hierarchics; families in the lower hierarchy, which, in turn, are aggregated into types in the upper hierarchy. In this procedure, the set of inventory constraints are priced out using Lagrangeau multipliers in the main objective function, which is then partitioned into two subproblems. The first subproblem is an optimal control problem on the family level (lower hierarchy) while the other subproblem is a linear program on the type level (upper hierarchy). The two subproblems are linked together by an inventory aggregation constraint. By this approach, a feedback is included automatically in the solution procedure, but it does not pass directly from the lower level to the upper level since both subproblems are solved in parallel. In the second procedure, instead of using Lagrangean relaxation with respect to a group of constraints in the original model, the model is first partitioned, ill which cuts are priced out by means of a set of Lagrangean multipliers into the subproblem generated on the family level. Accordingly, one subproblem is trivial and the other subproblem is an incapacitated lot-sizing oplinal control problem. The size of this subproblem is exactly the same as that of the equivalent subproblem in the previous procedure. In both procedures, the optimal control subproblem is solved using dynamic programming and the set of Lagrangean multipliers are updated using subgradient optimization algorithm.
Results show that both algorithms are quite acceptable and efficient. A comparison between the results of both algorithms shows considerable improvement compared to the result found in the literature. The enhancement of these results may be returned to the fact that step sizes are more suitably chosen and to the simplification done in the solution procedure.
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