Simultaneous reductions in inventory of raw materials, work-in-process, and finished items have recently become a major focus in supply chain management. Vendor-managed inventory is a well-known practice in supply chain collaborations, in which manufacturer manages inventory at the retailer and decides about the time and replenishment. In this paper, an integrated vendor-managed inventory model is presented for a two-level supply chain structured as a single capacitated manufacturer at the first level and multiple retailers at the second level. Manufacturer produces different products where demands are assumed decreasing functions of retail prices. In this chain, both the manufacturer and retailers contribute to determine their own decision variables in order to maximize their benefits. While previous research on this topic mainly included a single objective optimization model where the objective was either to minimize total supply chain costs or to maximize total supply chain benefits, in this research a fair profit contract is designed for the manufacturer and the retailers. The problem is first formulated into a bi-objective non-linear mathematical model and then the lexicographic max–min approach is utilized to obtain a fair non-dominated solution. Finally, different test problems are investigated in order to demonstrate the applicability of the proposed methodology and to evaluate the solution obtained. 
Lexicographic max–min approach for an integrated vendor-managed inventory problem