Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

A heuristic approach for big bucket multi-level production planning problems

Multi-level production planning problems in which multiple items compete for the same resources frequently occur in practice, yet remain daunting in their difficulty to solve. In this paper, we propose a heuristic framework that can generate high quality feasible solutions quickly for various kinds of lot-sizing problems. In addition, unlike many other heuristics, it generates high quality lower bounds using strong formulations, and its simple scheme allows it to be easily implemented in the Xpress-Mosel modeling language. Extensive computational results from widely used test sets that include a variety of problems demonstrate the efficiency of the heuristic, particularly for challenging problems

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times

Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures

Stochastic Programming with Economic and Operational Risk Management in Petroleum Refinery Planning under Uncertainty

Rising crude oil price and global energy concerns have revived great interests in the oil and gas industry, including the optimization of oil refinery operations. However, the economic environment of the refining industry is typically one of low margins with intense competition. This state of the industry calls for a continuous improvement in operating efficiency by reducing costs through business-driven engineering strategies. These strategies are derived based on an acute understanding of the world energy market and business processes, with the incorporation of advanced financial modeling and computational tools. With regards to this present situation, this work proposes the application of the two-stage stochastic programming approach with fixed recourse to effectively account for both economic and operational risk management in the planning of oil refineries under uncertainty. The scenario analysis approach is adopted to consider uncertainty in three parameters: prices of crude oil and commercial products, market demand for products, and production yields. However, a large number of scenarios are required to capture the probabilistic nature of the problem. Therefore, to circumvent the problem posed by the resulting large-scale model, a Monte Carlo simulation approach is implemented based on the sample average approximation (SAA) technique. The SAA technique enables the determination of the minimum number of scenarios required yet still able to compute the true optimal solution of the problem for a desired level of accuracy within the specified confidence intervals. We consider Conditional Value-at-Risk (CVaR) as the risk metric to hedge against the three parameters of uncertainty, which affords a framework that also involves the use of the Value-at-Risk (VaR) measure. We adopt two approaches in formulating appropriate two-stage stochastic programs with mean–CVaR objective function. The first approach is by using the conventional definition of CVaR that leads to a linear optimization model approximation coupled with a graphical-based solution strategy to determine the value of VaR using SAA in order to arrive at the optimal solution. The second approach is to utilize auxiliary variables to formulate a suite of stochastic linear programs with CVaR-based constraints. We conduct computational studies on a representative refinery planning problem to investigate the various model formulations using GAMS/CPLEX and offer some remarks about the merits of these formulations