Joint replenishment model of both-ways and one-way substitution among products in fixed time horizon

A general joint replenishment model (JRM) has been developed for two substitutable products with different demand structures depending on their prices and availabilities over the time cycle. When both products are available, the demand of a substitute product deceases against its own price and increases with the other's price. When one product is out of stock, a portion of demand of the stock-out product goes to the available product. The model is formulated to determine the optimal order quantities and total profits for different scenarios are determined using the classical optimisation method and parametric study. The optimal criteria for maximum profit is outlined and the model is illustrated numerically. Some sensitivity analyses with different substitution ratios are performed and some interesting conclusions have been derived.