In this paper, a multi-product newsvendor problem is formulated as a random nonlinear integrated optimization model by taking into consideration the selling price, the producing and outsourcing quantities, and the nonlinear budget constraint. Different from the existing models, the demands of products depend on the prices, as well as being time-varying due to random market fluctuation. In addition, outsourcing strategy is adopted to deal with possible shortage caused by the limited capacity. Consequently, the constructed model is involved with joint optimization of the producing and outsourcing quantities, and the selling prices of all the products. For this model with continuous random demands, we first transform it into a nonlinear programming problem by expectation method. Then, an efficient algorithm, called the feasible-direction-based spectral conjugate gradient algorithm, is developed to find a robust solution of the model. By case study and sensitivity analysis, some interesting conclusions are drawn as follows: (a) Budget is a critical constraint for optimizing the decision-making of the retailer, and there exist different threshold values of the budget for the substitute and complementarity scenarios. (b) The price sensitivity matrix seriously affects the maximal expected profit mainly through affecting the optimal outsourcing quantity.
Feasibility and competitiveness of intensive smallholder dairy farming in Brazil in comparison with soya and sugarcane: Case study of the Balde Cheio Programme