In this paper, we discuss a single-stage, single-product, multi-period production planning problem under the condition of uncertain demand. Our primal assumption is that the forecasted product demand is given via a set of discrete scenarios, i.e., demand quantity and its realization probability. We also assume that the planner must determine the production plan before he or she knows any realization of the product demand. In the traditional approach, the problem is modeled as a stochastic linear program based on the realization order of product demand, and thus the model often produces a wide difference between the minimum and maximum production quantities for two or later periods. The preceding activities such as ordering materials may have difficulty in planning operations because many possible production quantities with wide variations are provided from the production plan. In this paper, we consider the situation such that a limited quantity of additional production can be conducted if necessary after receiving the realized demand quantity provided that the required production capacity is allocated in advance. In this mechanism, the planner decides the basic production quantity and the slack capacity for additional production. Using numerical experiments, we have found that by keeping a slack capacity, the level of inventory at the end of the first period is lower than that of the traditional approach under some conditions, and the proposed mechanism can control the difference between the maximum and minimum production quantities.