An Inventory Model for Growing Items with Imperfect Quality When the Demand Is Price Sensitive under Carbon Emissions and Shortages

This research develops an optimization model for growing items in a supply chain with three stages: farmer, processor, and retailer while considering imperfect quality, mortality, shortages with full backordering, and carbon emissions. In the farmer stage, during the growing period, not all articles survive until the end of the period, so a density function of the probability of survival and death of the growing articles is taken into account. Moreover, it is considered imperfect quality in the retailer’s stage because as the supply chain goes down, there exists a greater probability of product defects. Here, the end customer (consumer) can detect poor-quality aspects such as poorly cut, poorly packed, expired products, etc. An inventory model that maximizes the expected total profit is formulated for a single type of growing items with price-dependent polynomial demand. An algorithm is developed to solve the optimization problem generating the optimal solution for order quantity, backordering quantity, selling price, and the number of shipments that maximizes the expected total profit per unit of time, and a numerical example is used to describe the applicability of the proposed inventory model. Finally, a sensitivity analysis has been carried out for all the input parameters of the inventory model, where the effect of each of the parameters on the decision variables is shown to extract some management knowledge. It was found that holding costs in the three stages of the supply chain have a substantial impact on the total profit per unit of time. In addition, as the demand scale parameter increases, the company must raise the selling price, which directly impacts the expected total profit per unit of time. This inventory model has the advantage that it can be applied to any growing item, including animals or plants, so it helps the owners of farms or crops to generate the most significant possible profit with their existing resources.

Inventory models have evolved to incorporate a wide range of realistic factors, including growing items, imperfect quality, deterioration, and sustainability concerns. While these areas have received significant individual attention, no model has yet integrated the complexities of growing items, imperfect quality, deterioration, and carbon emissions. This study addresses this gap by introducing an economic order quantity (EOQ) model for growing items that simultaneously accounts for imperfect quality, deterioration, carbon emissions, and a demand rate that is influenced by both stock levels and the freshness condition. The goal is to determine the replenishment cycle and the optimal order quantity that will maximise profit. A numerical example is presented to illustrate the model’s feasibility. A sensitivity analysis on key parameters is also conducted to provide critical managerial insights. The results reveal that the shelf life of items and the scaling parameter of demand are among the most influential factors of profit, causing up to 150% and 112% increase in profit, respectively. The findings also indicate that deterioration significantly impacts system profitability by up to −45%. Another critical insight is that profit decreases by up to 80% when the weight of the growing items increases. Furthermore, emissions can be most effectively reduced by focusing on the feeding process, which represents the most impactful factor for improving sustainability, whereas emissions from the screening process, purchasing, deterioration, and storage hold minimal financial consequence.

Inventory models that consider environmental and quality concerns have received some attention in the literature, yet no model developed to date has investigated these features in combination with growing items. Therefore, there is a need to incorporate these three relevant aspects together in a single inventory model to support decisions, compare results, and obtain new knowledge for the complexities of the real world. Moreover, current sustainable inventory management practices aim at mitigating the ecological consequences of an industry while preserving its profitability. The present study aligns with this perspective and introduces an economic order quantity (EOQ) model that considers imperfect quality while also accounting for sustainability principles. More specifically, the model addresses growing items, which have a demand dependent on selling price and the unique ability to grow while being stored in inventory. Additionally, the analysis acknowledges the possibility of classification errors during the inspection process, encompassing both Type-I and Type-II inspection errors. Furthermore, the model permits shortages and ensures that any shortage is completely fulfilled through backorders. The optimization model produces an optimal solution for the proposed model that is derived by optimizing three decision variables: order quantity of newborn items, backordering quantity, and the selling price of perfect items. A numerical example is presented, and the results are discussed. Finally, a sensitivity analysis on variations of parameters such as Type-I and Type-II errors shows that it is advantageous to reduce the percentage of good items that are misclassified as defective (i.e., Type-I error). As there is a direct impact of such errors on sales, it is imperative to address and mitigate this issue. When defective items are mistakenly classified as good Type-II errors, adverse consequences ensue, including a heightened rate of product returns. This, in turn, results in additional costs for the company, such as penalties and diminished customer confidence. Hence, the findings clearly suggest that the presence of Type-I and Type-II errors has a negative effect on the ordering policy and on the total expected profit. Moreover, this work provides a model that can be used with any growing item (including plants), so the decision-maker has the opportunity to analyze a wide variety of scenarios.

PurposeCarbon emission is a significant issue for the current business market and global warming. Nowadays, most countries have focused to reduce the environmental impact of business with durable financial benefits. The purpose of this study is to optimize the entire cost functions with carbon emission and to find the sustainable optimal ordering quantity for retailers.Design/methodology/approachThis paper illustrates a sustainable inventory model having a set of two non-instantaneous substitutable deteriorating items under joint replenishment with carbon emission. In this model demand and deterioration rate are considered as deterministic, constant and triangular fuzzy numbers. The objective is to find the optimal ordering quantity for retailers and to minimize the total cost function per unit time with carbon emission. The model is then solved with the help of Maple software.FindingsThis paper presents a solution method and also develop an algorithm to determine the order quantities which optimize the total cost function. A numerical experiment illustrates the improvement in optimal total cost of the inventory model with substitution over without substitution. The graphical results show the convexity of the cost function. Finally, sensitivity analysis is given to get the impact of parameters and validity of the model.Originality/valueThis study considers a set of two non-instantaneous substitutable deteriorating items under joint replenishment with carbon emission. From the literature review, in the authors’ knowledge no researcher has undergone this kind of study.

The present article establishes both crisp and fuzzy economic order quantity (EOQ) models with proportionate discount for items with imperfect quality under learning effect in a finite time horizon. The objective of the crisp model is to determine the optimal order lot size to maximise the total profit where the demand rate is constant and the defective items follow a learning curve. Then, we discussed on two different cases of fuzzy inventory models. Case-1: Defective rate follows a learning curve and the demand rate assumed to be a triangular fuzzy number. Case-2: Defective rate, fixed ordering cost and holding cost follow the learning curve and the demand rate taken to be a triangular fuzzy number. The objective of the fuzzy models is to estimate the maximum total profit per unit time in fuzzy sense and then to derive formula of the optimal lot size for each case and the corresponding total profit functions are defuzzified by using signed distance method. Numerical examples are provided to demonstrate the developed models. Sensitivity analysis is conducted both for crisp and fuzzy models to examine the effect of number of shipments on the order quantity and total profits under various conditions.

An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit

Regularly, manufacturing systems produce perfect and imperfect quality items. The perfect items start deteriorating as soon as these enter to the inventory. On the other hand, the suppliers provide a delay in payment in order to motivate their buyers to purchase more products. This paper develops a two-warehouse inventory model that considers jointly the imperfect quality items, deterioration and one level of trade credit. The proposed inventory model optimizes the order quantity to maximize the total profit per unit time. Finally, the proposed inventory model and its solution procedure are validated with numerical examples and a sensitivity analysis is done to show how inventory model reacts to changes in parameters.

Traditionally, the inventory models available in the literature assume that all articles in the purchased lot are perfect and the demand is constant. However, there are many causes that provoke the presence of defective goods and the demand is dependent on some factors. In this direction, this paper develops an economic order quantity (EOQ) inventory model for imperfect and perfect quality items, taking into account that the imperfect ones are sent as a single lot to a repair shop for reworking. After reparation, the items return to the inventory system and are inspected again. Depending on the moment at which the reworked lot arrives to the inventory system, two scenarios can occur: Case 1: The reworked lot enters when there still exists inventory; and Case 2: The reworked lot comes into when the inventory level is zero. Furthermore, it is considered that the holding costs of perfect and imperfect items are distinct. The demand of the products is nonlinear and dependent on price, which follows a polynomial function. The main goal is to optimize jointly the lot size and the selling price such that the expected total profit per unit of time is maximized. Some theoretic results are derived and algorithms are developed for determining the optimal solution for each modeled case. It is worth mentioning that the proposed inventory model is a general model due to the fact that this contains some published inventory models as particular cases. With the aim to illustrate the use of the proposed inventory model, some numerical examples are solved.

The paper develops the integrated inventory models with imperfect quality and environmental impact. Specially, the capital investments strategies in process quality improvement are adopted. In addition, we further consider the operations decisions in ordering, production, and transportation involving carbon emission costs. An efficient algorithm is presented to determine the optimal order quantity, process quality, shipment size and the number of shipments that minimizes the total expected cost of the integrated vendor-buyer inventory model in a coordinated supply chain. This integrated inventory model is useful particularly for JIT inventory systems where the vendor and the buyer form a strategic alliance for profit sharing. Finally, a numerical example is provided to illustrate the theoretical results. Sensitivity analysis is carried out to investigate the model parameters effects on the optimal solution. Managerial implications are also included.

ABSTRACTIncorporation of quality and environmental concerns in production and inventory models has received considerable attention in the inventory management literature; however, researchers studied these topics mostly independently. Thus, it is required to jointly incorporate those two relevant aspects in a single research to support decisions, compare the results and obtain new insights for complexities in practice. This paper takes a step in this line of thought and revisits some economic order quantity (EOQ) models with imperfect quality from a sustainable point of view. The objective is to investigate the impact of emission costs on the replenishment order sizes and the total profit of a buyer (retailer) in an imperfect supply process, where the buyer receives the batches containing a percentage of imperfect quality items. First, an EOQ model with imperfect quality items and emission costs, which are the result of warehousing and waste disposal activities, is formulated. Next, the model is extended to account for the situations where the buyer considers different areas for stocking the imperfect and good quality items, learning occurs in imperfect quality and the inspection process at the buyer's end contains error. The developed models are tested numerically and compared to investigate the optimal policies considering emission costs.

An economic order quantity with imperfect quality and quantity discounts

This paper corrects an improper expression in the economic order quantity (EOQ) model with two-warehouses and imperfect quality developed by Chung et al. [2009. A twowarehouse inventory model with imperfect quality production processes. Computers & Industrial Engineering 56, 193–197]. The modified model yields a simple and corrected expression for the optimal order quantity and expected profit per unit time. This paper then extends the above idea to a case involving a 100% inspection process with screening errors that may occur under imperfect quality and two warehouses. A model, in which it includes two scenarios, with imperfect quality and penalty costs under screening errors (Type I and Type II error) is developed. The effects of percentage of defective items and screening errors on the optimal solution are investigated. A numerical example is provided for the developed model and its result reveals that the order lot size and expected total profit considering in this paper is less than Chung et al. [2009]. Managerial insights are also draw.

This article investigates the inventory problem for item received with imperfect quality and inspection errors in an uncertain environment. Two inventory models are discussed with fuzzy parameters for crisp order quantity, or for fuzzy order quantity. Function principle is proposed as an arithmetic operation of fuzzy trapezoidal number to obtain fuzzy economic order quantity and fuzzy annual profit. Graded mean integration method is used for defuzzification of the annual profit. Extension of Lagrangian method is used to find optimal order quantity. Numerical examples are provided to illustrate the results of proposed models.

Optimizing price, order quantity, and backordering level using a nonlinear holding cost and a power demand pattern

Nowadays, consumers are more health conscious than before, and their demand of fresh items has intensely increased. In this context, an effective and efficient inventory management of the perishable items is needed in order to avoid the relevant losses due to their deterioration. Furthermore, the demand of products is influenced by several factors such as price, stock, and freshness state, among others. Hence, this research work develops an inventory model for perishable items, constrained by both physical and freshness condition degradations. The demand for perishable items is a multivariate function of price, current stock quantity, and freshness condition. Specific to price, six different price-dependent demand functions are used: linear, isoelastic, exponential, logit, logarithmic, and polynomial. By working with perishable items that eventually deteriorate, this inventory model also takes into consideration the expiration date, a salvage value, and the cost of deterioration. In addition, the holding cost is modelled as a quadratic function of time. The proposed inventory model jointly determines the optimal price, the replenishment cycle time, and the order quantity, which together result in maximum total profit per unit of time. The inventory model has a wide application since it can be implemented in several fields such as food goods (milk, vegetables, and meat), organisms, and ornamental flowers, among others. Some numerical examples are presented to illustrate the use of the inventory model. The results show that increasing the value of the shelf-life results in an increment in price, inventory cycle time, quantity ordered, and profits that are generated for all price demand functions. Finally, a sensitivity analysis is performed, and several managerial insights are provided.