Joint pricing and lot-sizing policy for deteriorating items with price and stock dependent demand rate and shortage

This paper considers a joint pricing and production problem for deteriorating items with price and stock dependent demand rate, constant production rate, exponential deterioration distribution and allowing shortage. The main purpose is to determine the optimal selling price and production scheme simultaneously such that the average profit per unit time is maximised. In order to get the optimal solution, a two-stage method is proposed. In stage 1, we study the optimal solution for fixed selling price. Two cases that building inventory is profit or not are addressed. The existence and uniqueness of optimal solutions are also proved, respectively. In stage 2, we study the optimal solution for fixed production scheme and adopt Newton's method to numerically show that the solution get from the first-order necessary condition is the optimum of the problem. Based on the two-stage method, a specific iteration algorithm is presented to search the optimal solution. In the next, we conduct a numerical example to illustrate the effectiveness and practicability of model and algorithm, and verify that the result searched by the algorithm is close to the optimal solution. At last, some sensitive analyses on the main parameters are carried out and some managerial implications are put forward.