The single vendor single buyer integrated production inventory problem with lead time proportional to lot size and space restriction is studied. Demand is assumed to be stochastic and the continuous review inventory policy is used for the buyer. If the buyer places an order with lots of products, then the vendor will produce lots of products and the products will be transferred to the buyer with equal shipments many times. Mathematical model for this problem is defined and a Lagrangian relaxation approach is developed.

The modular assembly system can make it possible for the variety of products to be assembled in a short lead time. In this system, necessary components are assembled to optional components tailor to customers’ orders. Budget for inventory investments composed of inventory and purchasing costs are practically limited and the purchasing cost is often paid when an order is arrived. Service cost is assumed to be proportional to service level and it is included in budget constraint. We develop a heuristic procedure to find a good solution for a continuous review inventory system of the modular assembly system with a budget constraint. A regression analysis using a quadratic function based on the exponential function is applied to the cumulative density function of a normal distribution. With the regression result, an efficient heuristics is proposed by using an approximation for some complex functions that are composed of exponential functions only. A simple problem is introduced to illustrate the proposed heuristics.