Optimizing a partial vertically integrated supply chain with hierarchical internal coordination: Bilevel model and solution algorithms

Partial vertically integrated supply chains (PVI-SCs) involve multiple stakeholders, with some controlling specific stages of the supply chain and others managing the remaining ones, all working collaboratively toward a common objective. While stakeholders are often modeled as single entities, certain contexts require recognizing their internal structure, where distinct departments operate under separate decision makers. This paper analyzes a PVI-SC in which a company owns the distribution centers and assembly plants, has access to customer demand information and outsources the procurement of raw materials to a set of independent suppliers. Coordination is assumed within the company, from which two departments are considered: sales and production. A hierarchical relationship exists between them, with the sales department holding a higher decision-making level than the production department. Consequently, the sales manager's decisions serve as inputs for designing the production plan. The production manager's decisions affect total service time, as they must account for the lead time of components provided by suppliers and the duration of the assembly process. To address this problem, we propose a mixed-integer nonlinear bilevel programming model and its equivalent linear formulation. To solve the linearized model optimally, we develop a customized Branch & Bound algorithm, employing two bounding strategies: one based on the high point relaxation and the other on an infeasibility criterion. To further reduce computational time for solving the bilevel problem, we design an effective and efficient nested iterated local search algorithm. A realistic case study validates our modeling approach, while numerical experiments on synthetic instances with realistic parameters confirm the strong performance of the proposed algorithms. The results highlight the importance of analyzing potential decision-making conflicts between departments as a critical step in effective supply chain management, ultimately reflected in the fulfillment of customer requirements.