<p>Economies of scale refer to the cost advantages that companies gain when production becomes more efficient. This concept is crucial for any business in any industry, representing the cost savings and competitive advantages that larger businesses have over smaller ones. Companies can achieve economies of scale by increasing production and lowering costs, as the costs are spread over a larger number of goods. Combining the replenishment of multiple products can further reduce costs, forming an effective strategy for cost savings by consolidating expenditures. Traditional inventory models, such as the Economic Order Quantity (EOQ) model, assume that demand is exogenous and seek to minimize ordering and inventory holding costs. However, when inventories stimulate consumer purchases, demand becomes an endogenous variable, influenced by the retailer’s stocking decisions. Academic literature often suggests that higher inventory levels can lead to higher sales, emphasizing the positive relationship between inventory decisions and sales realization, typically modeled with an increasing concave curve. In practical situations, the inventory-sales relationship can be complex. There are at least three reasons explaining why higher inventory levels could potentially increase sales. First, when inventory levels are zero, potential customers may leave empty-handed, reducing sales. Second, when inventory levels are positive but close to zero, the products in the store may not fully meet the tastes of potential customers. Third, in certain contexts, higher inventory levels may still drive sales growth even when the levels are already relatively high. In this paper, our focus is on an inventory system where the demand rate of the product is a function of the inventory level (quantity on hand). For such products, the probability of making a sale increases as the inventory quantity of the product grows. We plan to present a formulation for Jointly Replenishment Problems (JRP) of multiple products. Unlike traditional JRP, the demand rate of all products is dependent on inventory levels. In addition to modeling, we provide an algorithm to solve the problem, viewing it as a generalization of the algorithm proposed by [<CitationRef CitationID="CR13">13</CitationRef>]. Based on this algorithm, we discuss how to generate junction points to provide a suitable solution for our problem. Furthermore, we provide some complexity analysis and implement it on some numerical examples.</p>

An algorithm for jointly replenishment problem under stock dependent demand and power-of-two policy

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发表日期：

2025年11月13日

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Jen-Yen Lin

摘要

Economies of scale refer to the cost advantages that companies gain when production becomes more efficient. This concept is crucial for any business in any industry, representing the cost savings and competitive advantages that larger businesses have over smaller ones. Companies can achieve economies of scale by increasing production and lowering costs, as the costs are spread over a larger number of goods. Combining the replenishment of multiple products can further reduce costs, forming an effective strategy for cost savings by consolidating expenditures. Traditional inventory models, such as the Economic Order Quantity (EOQ) model, assume that demand is exogenous and seek to minimize ordering and inventory holding costs. However, when inventories stimulate consumer purchases, demand becomes an endogenous variable, influenced by the retailer’s stocking decisions. Academic literature often suggests that higher inventory levels can lead to higher sales, emphasizing the positive relationship between inventory decisions and sales realization, typically modeled with an increasing concave curve. In practical situations, the inventory-sales relationship can be complex. There are at least three reasons explaining why higher inventory levels could potentially increase sales. First, when inventory levels are zero, potential customers may leave empty-handed, reducing sales. Second, when inventory levels are positive but close to zero, the products in the store may not fully meet the tastes of potential customers. Third, in certain contexts, higher inventory levels may still drive sales growth even when the levels are already relatively high. In this paper, our focus is on an inventory system where the demand rate of the product is a function of the inventory level (quantity on hand). For such products, the probability of making a sale increases as the inventory quantity of the product grows. We plan to present a formulation for Jointly Replenishment Problems (JRP) of multiple products. Unlike traditional JRP, the demand rate of all products is dependent on inventory levels. In addition to modeling, we provide an algorithm to solve the problem, viewing it as a generalization of the algorithm proposed by [13]. Based on this algorithm, we discuss how to generate junction points to provide a suitable solution for our problem. Furthermore, we provide some complexity analysis and implement it on some numerical examples.