(440d) Stochastic Bilevel Optimization of Agrochemical Supply Chains with Mean-Variance Risk Management

The global agrochemical market is highly consolidated, with large multinational companies accounting for a major share of the market. Thus, even for a single agrochemical product, its global supply chain typically involves numerous paths connecting the raw material sources to the final customers. Besides the structural complexity, agrochemical supply chains are also subject to seasonal variations and other risks and uncertainties unique to the industry, thereby posing a need for risk management tools and strategies. Furthermore, most multinational agrochemical companies have outsourced a significant portion of their intermediate production to contract manufacturing organizations (CMOs) located in developing countries, while they focus on the manufacturing of the active ingredient molecule from intermediates internally. This unique characteristic further complicates the supply chain network, increases uncertainties, and poses decision-making challenges as the agrochemical company needs to coordinate with one or more CMOs through a contract signed well before the start of intermediate production. In this work, we present a scenario-based stochastic mixed-integer quadratic constrained bilevel optimization framework for an agrochemical supply chain, in which we use the mean-variance method to manage extra costs associated with demand loss and exceeded demand. In this bilevel model, the leader is a multinational agrochemical company whose goal is to minimize its overall costs while keeping the extra costs at a low level, whereas the follower is a CMO whose goal is to maximize its own profit.

We adopt a two-step reformulation strategy introduced in [1] to solve the bilevel problem, in which the objective function of the lower-level problem contains continuous and integer variables. In the first step, the integer variables appearing in the objective function are transformed into a continuous form, making the lower-level problem an MINLP. In the second step, we convert the bilevel problem into a single-level problem by formulating the optimality conditions of the lower-level MINLP using the Karush-Kuhn-Tucker (KKT) conditions. As for the upper-level problem, we introduce perspective reformulation [2] to linearize the non-convex quadratic constraint that calculates and bounds the variance for the first time. Overall, this leads to a single-level MINLP, which can be solved using a global solver such as BARON. In a series of illustrative case studies of different sizes, we show that iteratively introducing perspective cuts to the reformulated single-level MINLP continuously improves the dual bound and solution time. And coupling the two-step reformulation strategy and perspective reformulation can be a powerful tool to solve large-scale bilevel supply chain optimization problems.

References

[1] S. Medina-González, L.G. Papageorgiou, V. Dua, 2021, A reformulation strategy for mixed-integer linear bi-level programming problems, Computers & Chemical Engineering, 153, 107409.

[2] O. Günlük, J. Linderoth, 2012, Perspective Reformulation and Applications. In: Lee, J., Leyffer, S. (eds) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol 154. Springer, New York, NY.