There are many research topics that are related to inventory management. For example, Sarkar et al. [
1] and Quezada-Téllez et al. [
2] studied inventory management in society. Lee et al. [
3] and Zimpel-Leal, and Lettice [
4] considered industry. Rubio-Valdehita et al. [
5] and Górska-Warsewicz et al. [
6] examined retailing companies. Gładysz et al. [
7] and Orynycz et al. [
8] presented lean management. De Giovanni [
9] and Malik, and Sarkar [
10] developed supply chain philosophies. Inventory management depends on the type of product that a firm retails. Stable products have no expiration date, while perishable products feature a limited time range, gradually losing their value over time. Whereas perishable products, such as fresh fish, lose their value steadily, others, such as milk, have a fixed shelf life. Before the expiry date is reached, these products do not change in appearance, taste, texture, or flavor. There exists a large body of research on inventory models for products with a fixed shelf life. Most inventory models dealing with deteriorated items have a constant decay rate. For example, Lin et al. [
11] developed a production inventory model with the constant deteriorated date and the production rate dependent on demand and inventory level, and Yang et al. [
12] studied an inventory model with a constant decay rate proportional to the inventory level. In these models, a dichotomous shelf-life effect for perishable items is assumed. The item is either as good as brand-new or the item is deteriorated and cannot be used. However, this assumption is not reasonable in some situations. An obvious case is an agricultural product. The quality of products perceived by consumers may be reduced within the shelf-life duration. They may purchase substitute products or brands offering a later expiry date. The demand reaches the maximum when the product is fresh and declines as time goes on. Just a few inventory models considered perishable items with a fixed shelf-life. The current paper is based on the work of Avinadav and Arponen [
13], who considered the effect of perceived quality, using a declining demand rate, which is dependent on the time until the expiry date. When an item is fresh, its perceived quality is considered perfect. They constructed an inventory model with polynomial-type demand which is proportional to the remaining time to full shelf-life duration. The research gap is that Avinadav and Arponen [
13] provided those numerical examples with negative maximum profit. It seems trivial that for an example of a maximum profit inventory model obtains a negative profit then this example or the computation of this numerical example must contain errors. However, Avinadav and Arponen [
13] informed researchers that they examined 729 numerical examples to find 255 examples with negative maximum profits, and then they discard those 255 examples. For the remaining 474 examples with positive maximum profits, Avinadav, and Arponen [
13] execute sensitivity analysis. In this paper, we will provide theoretical evidenceto show that under two conditions of Theorem 2, obtaining a negative maximum profit is reasonable.
Up to now, 26 papers have cited Avinadav and Arponen [
13] in their references. We will provide a brief discussion of those 26. Avinadav et al. [
14] expanded this inventory system to derive the optimal order quantity, price, and replenishment cycles for a demand that is linearly dependent on price and time. We recall that Avinadav and Arponen [
13] considered the polynomial type demand
and then Avinadav et al. [
14] extended to
where
is the selling price such that
is the multiplication of a polynomial type function in
and a linearly decreasing function in
. Leśniewski and Bartoszewicz [
15] study inventory models with defect items, warehouse capacity, and deterioration during storage in a warehouse under the bullwhip effect and supply chain. Aiello et al. [
16] constructed a mathematical model to optimize the food supply chain consist of retailers and potential recipients that operate the food recovery to include the benefits of donors and the operational charges for the food recovery. Avinadav [
17] studied periodic inventory models to revise a classical newsvendor system such that the holding cost is related to the stock levels within the selling period to optimize the expected profit. Avinadav et al. [
18] further generalized this approach so that generalized demand is the multiplication of a decreasing function and a linearly decreasing function. However, Lemma 1 of Avinadav et al. [
18] is questionable, as its proof is based on the existence of an interior optimal solution. In
Section 3, this paper will provide a reasonable explanation for the restriction. In Avinadav et al. [
14] and Avinadav et al. [
18], they cannot prove the uniqueness of the optimal solution and they were not aware that sometimes the optimal solution occurs at the boundary. Herbon [
19] examined an inventory model with a perishable product with a fixed shelf life, and a dynamic pricing policy to study consumer sensitivity to price and freshness. Aiello et al. [
20] developed a food supply chain for food recovery policies to find the optimal time to remove perishable items from the shelves and then decided to the livestock market or donated organizations. Chuang and Lin [
21] studied a two-echelon inventory model with a supplier and one retailer to decide the optimal solutions for selling price under a fixed shelf-life and a ramp type demand with shortages. Ma [
22] analyzed the maximization of profit for fresh products with two quality levels by a consumer utility function to solve the optimal solutions for ordering quantity and selling price. Muriana [
23] considered a food supply chain product with an uncertain shelf life that could be withdrawn and shipped to alternative destinations. Avinadav [
24] considered a stochastic periodic-review inventory system to investigate its relation to the classical newsvendor problems and then derived several approximated solutions by a Brownian motion demand process. Muriana [
25] developed a deterioration model with a stochastic demand following a normal distribution for open-dating foods with a shelf life. Yamazaki et al. [
26] analyzed safety stock and cycle stock to study the relationship between fluctuation stock and safety stock under various cases of differences between supply and demand. Avinadav et al. [
27] constructed a new inventory system with deteriorated items to determine the optimal promotion expenditures, ordering quantity, selling price, and cycle length. Demirag et al. [
28] built an economic order quantity model with a decreasing demand, expired date, and a linear expression that is closely related to Avinadav and Arponen [
13]. Demirag et al. [
28] obtained a cubic polynomial for the optimal cycle length and then they derived three formulated candidates for the optimal solution to replace the numerical approximated approach proposed by Avinadav and Arponen [
13]. Muriana [
29] tried to diminish the influence of production losses along the supply chain when demand is restricted by various economic, political, climatic, and legal factors. Sharma et al. [
30] studied partial backlogging policy when demand is dependent on selling price and expiry date. Chernonog and Avinadav [
31] considered a two-echelon supply chain consisting of a leader (manufacturer) and a follower (retailer) where the demand is related to investment in advertising, product age, and price. Hanukov et al. [
32] developed a combined queueing and inventory system to consider inventory management techniques and time management policies simultaneously for the fast-food industry. They obtained the steady-state probabilities by matrix geometric methods to decide the optimal level of investment in preservation technologies and the optimal preliminary services capacity. Lin et al. [
33] examined a system to the reserves and procurement management for humanitarian logistics by sensitivity analyses and numerical examples to reveal managerial insights and then provided practical suggestions for the government. Avinadav [
34] formulated a two-echelon supply chain with a manufacturer and a retailer to examine conditions where the demand is related to the age of the product on a shelf-life, sales effort, and selling price. Chernonog [
35] developed a two-echelon supply chain consisting of a manufacturer and a retailer under a Stackelberg game to decide the investment in advertising, cycle length, and the selling price. He studied two cases: retailer-leader and manufacturer-leader to derive relations between advertising investment and cycle length. Hanukov et al. [
36] constructed service models for the fast-food market for fastidious or strategic customers for fresh items or pre-prepared items under different prices to optimize the expected profit. Krommyda et al. [
37] developed a maximum profit inventory model in which products that are near their expiry date are reduced in price, donated to charity, or sold to the livestock market. Muriana [
38] examined inventory models for deteriorated items with Weibull perishable rate to consider the relationship between the characteristic life and the perishable rate to locate the optimal solution. Hanukov et al. [
39] developed a multi-server model to short sojourn time that servers will use their idle time to make partially prepared items. They applied a Markovian queueing method to derive closed-form solutions for the entries of the rate matrix.