Solid waste management in European countries: a review of systems analysis techniques.

Robust possibilistic programming for socially responsible supply chain network design: A new approach

Multistage stochastic integer programming (MSIP) combines the difficulty of uncertainty, dynamics, and non-convexity, and constitutes a class of extremely challenging problems. A common formulation for these problems is a dynamic programming formulation involving nested cost-to-go functions. In the linear setting, the cost-to-go functions are convex polyhedral, and decomposition algorithms, such as nested Benders’ decomposition and its stochastic variant, stochastic dual dynamic programming (SDDP), which proceed by iteratively approximating these functions by cuts or linear inequalities, have been established as effective approaches. However, it is difficult to directly adapt these algorithms to MSIP due to the nonconvexity of integer programming value functions. In this paper we propose an extension to SDDP—called stochastic dual dynamic integer programming (SDDiP)—for solving MSIP problems with binary state variables. The crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts, termed Lagrangian cuts, derived from a Lagrangian relaxation of a specific reformulation of the subproblems in each stage, where local copies of state variables are introduced. We show that the Lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of SDDiP with probability one. We show that, under fairly reasonable assumptions, an MSIP problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size. Thus our proposed SDDiP approach is applicable to very general classes of MSIP problems. Extensive computational experiments on three classes of real-world problems, namely electric generation expansion, financial portfolio management, and network revenue management, show that the proposed methodology is very effective in solving large-scale multistage stochastic integer optimization problems.

Stochastic dual dynamic integer programming

In this study, a robust two-step method (RTSM) is developed to solve the interval linear programming (ILP) problem. It improved upon the two-step method (TSM) proposed by Huang et al. (1992) through incorporating additional constraints into solution procedures to avoid absolute violation. RTSM was applied to a simple case related to environmental management. The results demonstrated applicability of the developed methodology. Compare with the modified interval linear programming (MILP) method proposed by Zhou et al., (2008) and the three-step method (ThSM) developed by Cao and Huang (2011), RTSM can generate a relatively larger solution space and thus avoid significant loss of decision-related information. Besides, RTSM has simpler solution procedures than ThSM, and will not lead to great computational requirement.

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Y. P. Li

Papers

Inexact multistage stochastic integer programming for water resources management under uncertainty.

A two-stage fuzzy robust integer programming approach for capacity planning of environmental management systems