Joint Replenishment of Partially Substitutable Imperfect Items with Cost of Substitution

In inventory management, some items can be substituted for others. This means that in case of a shortage, some of the demand can be met by the substitute item, leading to reduced shortage costs. This paper examines two partially substitutable products with imperfect items in the order       quantity. Imperfect items are removed after inspection. The two items have the same cycle time, and in the case of a shortage, part of the demand is substituted. The model is presented and solved symbolically for some special cases, and a numerical example is provided for the general case. The findings show that increasing substitution costs, decreasing holding costs of the substitutable item, and decreasing the percentage of substitution could reduce the time with a shortage.     Moreover, increasing substitution and holding costs increases the total inventory costs, while the increase in the percentage of substitution could either increase or decrease the total inventory costs in different intervals.