We consider an EOQ inventory model for growing items, wherein the value and size of items increase during time, some instances of these items are livestock, fish, and poultry. The main difference between this inventory system and older ones is weight increment of products during stocking without buying more. This paper studies an inventory system of poultries that new-born items are fed to reach the ideal weight for consumers. In this study, based on the consumers’ preference of fresh foods over frozen items, we assume that shortage is permitted and consumers wait for fresh items when company pays some additional penalties, i.e., the shortage is fully backordered. On the other hand, for each cycle, the producer must prepare the place in terms of hygiene conditions; thus, a setup time per cycle is considered. The aim of this study is to obtain optimum system solution, such that total costs, including setup, purchasing, holding, feeding, and shortage, are minimized. To do so, we employ mathematical measures to approximate growing rates and model the system as a non-linear programming. To solve the obtained optimization model, we employ hessian matrix to obtain optimal solution for this inventory system. The proposed EOQ inventory model helps poultry industries in Iran to optimize their system considering costs and permissible shortage, and it can be employed in other countries. Finally, we provide a numerical example and its sensitivity analysis, plus some potential future directions.
Trying to optimize organization costs by managing inventories goes back to more than a century ago when the first economic order quantity (EOQ) inventory model was proposed by Harris . Harris’s inventory model minimizes total costs, including holding and ordering costs, such that inventory system faces no shortages. An important modification of EOQ inventory model is the economic production quantity (EPQ) model proposed by Taft , where instead of receiving products in orders at once; they are produced at a known rate.
Recently, Rezaei  investigated an EOQ inventory model for products that are growing during storage, for industries such as livestock, fish farming, and poultry. In these inventory systems, the weight of products increases during the period of stocking without ordering additional items. The aforementioned study is the first systematic research that considers this class of inventory. To do so, it develops a general inventory model then extends that inventory model for poultry. After this study, Zhang et al.  developed the aforementioned inventory model for cases that legislator imposes a constraint carbon emission or put penalty for it.
Before these studies, the EOQ inventory model has been modified by several researchers and academicians to relax some particular boundaries of products specifications. One of the first attempts to model inventory systems for unconventional items dates back to Whitin , who addressed goods that became old-fashioned after a specified period. Afterwards, Ghare and Schrader  studied an inventory system, wherein items decayed exponentially. Then, Covert and Philip  extended Ghare and Schrader  study for cases that deterioration rate could vary during time. Other studies that dealt with relaxing infinite life cycle and modifying the model for perishable products such as vegetable, dairies, batteries, and drugs are: Muriana ; Dobson et al. ; Yan and Wang ; and Boxma et al. . For comprehensive reviews about inventory models of perishable items, see Goyal and Giri  and Bakker et al. .
On the other hand, some other studies extended the EOQ/EPQ inventory models for imperfect products, for example, Rosenblatt and Lee  considered a production system that after some time the manufacturing system alters to out of control and starts to produce imperfect items. Afterwards, Salameh and Jaber  studied an EOQ/EPQ inventory model with imperfect quality items. These items were either suitable for other processes or could be sold in a batch after inspection process. Hayek and Salameh  investigated the rework of produced imperfect quality items in an imperfect production system for an EPQ inventory model. Moreover, some research addressed EOQ/EPQ model when the received batch should be examined to identify defective items. Examples are Manna et al. , Mukhopadhyay and Goswami , Nobil et al.  and Pasandideh et al. .
Furthermore, some researchers extended classical EOQ inventory model considering trade credits policies to overcome capital boundaries of system. First kind of these studies is Goyal , wherein purchasing costs could be paid back with a permissible delay. In this model, the supplier permits the vendor to pay part of the costs during the period. Then, Rajan and Uthayakumar  developed an EOQ inventory model with permissible delays in payment for cases that demand and holding costs change as a function of time. In another study, Pasandideh et al.  studied trade credits for an EOQ inventory model considering several items, permissible shortage, and warehouse constraint using genetic algorithm.
Another way to overcome system costs in inventory models is by decreasing inventory level considering permissible shortage. Using this, system managers avoid investing too much on building warehouses, especially for items such as food that requires temperature regulation, hygiene, and other special conditions. One of the first kinds of these studies was performed by Hadley and Whitin . They revised Harris  model for cases that system faces some shortages. San-José et al.  developed an EOQ inventory model considering partial backordering, i.e., some of the customers do not wait for new items, considering non-linear holding costs. Pervin et al.  addressed an inventory system with permissible shortage, random deterioration, and time-dependent demand and holding cost to obtain optimum replenishment policy.
This paper considers an inventory system for a single product. By suitable nutrition, new-born items grow and reach the ideal weight for satisfying customers demand. It is assumed that shortage is permitted. On the other hand, a setup time is considered to prepare nurturing environment before a new cycle. The aim of this study is to determine the optimum values of shortage and cycle-length subject to minimizing inventory system total costs including setup, purchasing, holding, feeding, and shortage. Moreover, the mathematical model of this inventory system has a non-linear programming (NLP) form. In “Appendix A”, it is proven that this NLP is a convex problem. Furthermore, an exact solution algorithm for this problem is proposed by employing convexity property of the model. The proposed EOQ inventory model with growing items is applicable in instances that the growing function estimation with a linear function does not introduce unacceptable errors to the system. The proposed EOQ inventory model helps livestock, poultry, and aquaculture industries to optimize their inventory system considering feeding costs, growing rate, and improvement of revenue management. Using these analytical approaches, industry managers can make the optimal decisions about ordering time and purchasing new-born items, growing and shortage period length. This model developed based on historical data so is more beneficial than decision making by a conjecture or trial and error.
Considering permissible shortage in the proposed model makes it closer to real-world scenarios and helps managers to calculate optimum shortage period when face flexible customers, i.e. their customers do not supply their demand from other suppliers in case of shortage. Moreover, in most poultry and aquaculture cases, a setup time is required for system inspection and preparation for another run. Finally, proposing a straightforward solution procedure and a linear estimation of growth function helps managers to obtain a near optimum solution. A comparison between this work and former studies is proposed in Table 1.
The rest of this paper is organized as follows. Section 3 provides the notation, assumptions, and problem definition. Section 4 presents an exact algorithm for solving the EOQ inventory model with growing items. Section 5 solves a numerical example with proposed algorithm and also presents a sensitivity analysis of the inventory model. Finally, Sect. 6 presents some conclusions and potential future research directions.