Mixed-integer second order conic programming formulation for joint lot-sizing and pricing problem

Lot-sizing and pricing problems have been addressed quite extensively. Most literature study the problems and develop models from the perspective of operations management. Fewer works can give an operational research model as well as exact solutions methods. In this paper, we investigate a joint lot-sizing and pricing problem with the consideration of capacity constraints and backlogging. We make the decision on production planning over a finite multi-period horizon. The demand of each period is related to the price. Therefore, we first propose a nonlinear mixed integer model with the nonlinear objective function to maximize the total profit. The model cannot be solved by off-the-shelf solver directly. Second-order conic programming (SOCP) technique is applied to transform the model into a specific nonlinear model, which can be solved by CPLEX. We develop a mixed integer second-order program (MI-SOCP) model and finally, we conduct the experiment and give an exact solution. Moreover, we make sensitivity analyses on the two parameters.