In this paper, we consider the production planning problem for a single manufacturer with the investment to improve the market impact under demand uncertainty. In the mathematical model, the average demand increases if the investment of market impact is increased for each product. The objective is to maximize the total profit with a piecewise linear investment cost and a budget constraint. The problem is formulated as a mixed integer nonlinear programming problem. A solution procedure based on Lagrangian relaxation is developed to solve the problem efficiently. In the proposed method, an analytical solution of the newsboy problem is effectively used to derive the lower and upper bounds. The condition of the concavity of the profit function is derived. The effectiveness of the proposed method is confirmed through computational experiments.

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