The optimization models devised by the PSE community to assist in the operation and design of chemical supply chains have usually concentrated on maximizing the economic benefit of the company. However, there has been recently a growing awareness of the importance of including environmental concerns along with traditional economic criteria in the optimization procedure. Unfortunately, despite the effort made in the area, there are still some important aspects which merit further attention. Specifically, one of the main drawbacks of the existing models in the area of sustainable process design is their rather limited scope. Thus, they usually reduce the emissions of a plant locally at the expense of increasing burdens elsewhere in the life cycle, in such a way that the overall environmental impact is increased. Furthermore, almost all of the methods available in the literature are deterministic, i.e., they assume nominal values for the parameters involved in the impact assessment model. Optimizing the environmental impact in the mean scenario may lead to optimistic solutions with high probabilities of exceeding the nominal performance.
This paper proposes a holistic framework for the design of sustainable chemical supply chains that takes into account the uncertainty associated with the impact assessment model. The environmental impact is measured through the Eco-Indicator 991, which reflects the advances in the damaged oriented method recently developed for Life Cycle Impact Assessment2. This strategy covers the entire life cycle of the product, process or activity, including extracting and processing of raw materials; manufacturing, transportation and distribution; reuse and maintenance; recycling and final disposal. This type of analysis helps to reduce the impact globally, as all the activities carried out by the supply chain entities are considered when assessing the environmental performance. Furthermore, the strategy proposed explicitly considers the uncertainty associated with the damage model based on the Life Cycle Assessment principles. Specifically, the variability of the Eco-Indicator 99 under the uncertain environment is controlled by reducing the probability of exceeding a specified target level.
The problem is mathematically formulated as a multi-objective mixed integer linear program (moMILP). Two objectives are considered in the model, the net present value (NPV) and the probability of an Eco-Indicator 99 value below a given target. The inclusion of the latter criteria gives rise to a chance-constraint whose deterministic equivalent is obtained by applying concepts from chance-constrained programming3. The resulting multi-objective model is reformulated as a parametric MILP. The probability associated with the Eco-Indicator 99 is pursued as main objective whereas the NPV is constrained to be greater than an epsilon value. By parametrically changing this epsilon value, the entire set of Pareto optimal solutions of the problem can be generated.
The parametric MILP is solved by decomposing it into two sub-problems and iterating between them. The first sub-problem is a parametric LP obtained by fixing all the binary variables of the original formulation. This problem is solved by performing a sensitivity analysis with respect to the target imposed to the NPV. This yields a parametric profile that is intersected with the current best one providing an approximation to the Pareto curve of the original problem. The second sub-problem is a MILP where the epsilon parameter is relaxed as a variable. This MILP includes parametric cuts and logic cuts that are derived from previous explored solutions. The parametric cuts force the model to seek solutions improving the best current parametric profile in at least one point. The logic cuts are added to exclude the solutions found in previous iterations.
The capabilities of the proposed model and solution procedure are illustrated through several case studies. Numerical results show how environmentally friendlier solutions in the face of uncertainty in the damage model can be attained by systematically trading-off the economic benefit of the process. These robust solutions are achieved by structural modifications in the SC and also by properly adjusting the production rates and transportation flows between the SC nodes. Furthermore, the proposed decomposition strategy is able to provide the whole set of Pareto optimal solutions in a fraction of the CPU time required in the standard epsilon-constraint method.
 The Eco-indicator 99, "A damage oriented method for life cycle impact assessment. Methodology Report and Manual for Designers", Technical Report, PRé Consultants, Amersfoort, The Netherlands, 2000.
 ISO 14040. Environmental Management - Life Cycle Assessment - Part 1: Principles and Framework. ISO, 1997.
 Charnes, A.; Cooper, W. W. Normal Deviates and Chance Constraints. J. Am. Stat. Assoc. 1962, 52, 134.