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A two-stage, stochastic programming approach is proposed for incorporating demand uncertainty in multisite midterm supply-chain planning problems. In this bilevel decision-making framework, the production decisions are made "here-and-now" prior to the resolution of uncertainty, while the supply-chain decisions are postponed in a "wait-and-see" mode. The challenge associated with the expectation evaluation of the inner optimization problem is resolved by obtaining its closed-form solution using linear programming (LP) duality. At the expense of imposing the normality assumption for the stochastic product demands, the evaluation of the expected secondstage costs is achieved by analytical integration yielding an equivalent convex mixed-integer nonlinear problem (MINLP). Computational requirements for the proposed methodology are shown to be much smaller than those for Monte Carlo sampling. In addition, the cost savings achieved by modeling uncertainty at the planning stage are quantified on the basis of a rolling horizon simulation study.

This paper utilizes the framework of midterm, multisite supply chain planning under demand uncertainty (Gupta and Maranas, 2000) to safeguard against inventory depletion at the production sites and excessive shortage at the customer. A chance constraint programming approach in conjunction with a two-stage stochastic programming methodology is utilized for capturing the trade-off between customer demand satisfaction (CDS) and production costs. In the proposed model, the production decisions are made before demand realization while the supply chain decisions are delayed. The challenge associated with obtaining the second stage recourse function is resolved by first obtaining a closed-form solution of the inner optimization problem using linear programming duality followed by expectation evaluation by analytical integration. In addition, analytical expressions for the mean and standard deviation of the inventory are derived and used for setting the appropriate CDS level in the supply chain. A three-site example supply chain is studied within the proposed framework for providing quantitative guidelines for setting customer satisfaction levels and uncovering effective inventory management options. Results indicate that significant improvement in guaranteed service levels can be obtained for a small increase in the total cost.3

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Competition in the aircraft industry market and global warming has driven the industry to think along economic and environmental lines. This has resulted in the emergence of more electric aircraft (MEA). The increase in the power demand of aircraft, especially in the last two decades, coupled with advancement in battery materials and technology has led to the development of many high energy density batteries. This study presents an overview of the battery systems for MEA. In this paper, a study on the battery technologies used in aircraft in the last five decades is being done. A general background of the battery system is presented and the performance of the batteries based on energy densities and low temperature capabilities are evaluated and discussed. Evolution of MEA with its power system architecture and load profile is presented to understand the requirements of the battery system. Weight saving and cost analysis is done for the Li-ion and Ni-Cd batteries with respect to the requirement of an MEA 'Aircraft X'. Battery management system (BMS) for Li-ion batteries is also explored and discussed. Based on the analysis, Li-ion battery is selected and integrated with the power distribution DC network for future MEA.

An efficient decomposition procedure for solving midterm planning problems is developed based on Lagrangean relaxation. The basic idea of the proposed solution technique is the successive partitioning of the original problem into smaller, more computationally tractable subproblems by hierarchical relaxation of key complicating constraints. The systematic identification of these complicating constraints is accomplished by utilizing linear programming relaxation dual-multiplier information. This hierarchical Lagrangean relaxation procedure, along with an upper bound generating heuristic, is incorporated within a subgradient optimization framework. This solution strategy is found to be much more effective, in terms of both quality of solution and computational requirements, than commercial mixed-integer linear programming solvers in bracketing the optimal value, especially for larger problems.

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