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

Authors: Cynthia Griselle De-la-Cruz-Márquez; Leopoldo Eduardo Cárdenas-Barrón; Buddhadev Mandal
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

Nowadays, it is well known that global warming is a great hazard to the planet, and the carbon emissions are a principal source of global warming. For this reason, the customers have become more environment and quality conscious than before, and as a result, they request the firms to be ecofriendly. In this context, it is desirable that companies develop and implement inventory models which consider sustainability issues. Furthermore, the companies face problems of shortages and setting prices in order to persist in a competitive and challenging business. Besides, there exists a kind of items different than the traditional products that it is necessary to feed them until a target weight is reached in order to slaughter and sell to customers. These are named as growing items. In this sense, this research work proposes an inventory model for growing items with imperfect quality when the demand is price sensitive under carbon emissions and shortages. The shortages are fully backordered. The demand is price sensitive according to a polynomial function. The proposed inventory model determines jointly the optimal policy for the selling price of perfect-quality growing items, the order quantity, and the backordering quantity which maximize the expected total profit per unit of time. Some numerical examples are resolved in order to illustrate the use and the applicability of the inventory model. Finally, a sensitivity analysis is conducted and some managerial insights are given.


---

## Body

## 1. Introduction

Inventory management is a valued function for all firms. The main purpose of the inventory management department is to control the materials from the acquisition to sale, taking decisions of how much and when to buy the items avoiding excess or unplanned stockout.Since the introduction of the classical economic order quantity (EOQ) inventory model proposed by Harris [1], industries have shown great interest in the development of inventory control models with new features. For example, Ghosh et al. [2] built an inventory model for a deteriorating production system when there are random machine breakdowns considering that there exists a selling price discount. Manna et al. [3] studied the impacts of inspection errors on an imperfect production-inventory model when the demand rate depends on both selling-price discount and warranty time. Khan et al. [4] and Shaikh et al. [5] introduced a two-warehouse inventory model for instantaneously deteriorating goods considering an advance payment agreement. In the same research direction, Khan et al. [6] investigated the effects of noninstantaneous deterioration on the ordering decisions for a two-warehouse inventory system taking into consideration an advance payment scheme. Shaikh et al. [7] developed an EOQ inventory model for deteriorating goods with stock-dependent demand when there exists a price discount facility.It is well known that the EOQ determines the optimal order quantity to minimize total inventory costs. Although this inventory model has the groundwork for inventory systems, it has unrealistic assumptions. For that reason, many researchers have incorporated different characteristics in order to model real world applications such as imperfect quality, carbon emissions reduction, growing items, shortages, and pricing, among others.Harris’s [1] inventory model assumes that all items are of perfect quality. However, this assumption not always occurs because the majority of the manufacturing processes generate a portion of imperfect quality items. So, Salameh and Jaber [8] formulated an EOQ inventory model considering that the ordered lot contains imperfect-quality items and these items are sold as a single batch at a lower price. Cárdenas-Barrón [9] identified and corrected an error in the inventory model presented by Salameh and Jaber [8]. Wee et al. [10] extended Salameh and Jaber’s [8] inventory model by incorporating shortages with full backordering. Afterwards, Maddah and Jaber [11] detected and rectified one flaw in Salameh and Jaber’s [8] inventory model. The flaw is in the manner of determining the expected total profit per unit of time ETPU. Salameh and Jaber [8] utilized the equation ETPU=ETP/T which is not exact. The research work of Wee et al. [10] contains the same flaw. It is important to remark that the process of producing the profit is a renewal process with renewal points at order placement periods. Therefore, in order to calculate the expected total profit per unit of time, it is needed to apply the well-known elementary renewal reward theorem given by Ross [12]. For this reason, the expected total profit per unit of time must be computed with the following equation: ETPU=ETP/ET. Chang and Ho [13] revisited and rescued Wee et al.’s [10] inventory model by applying the well-known elementary renewal reward theorem to get the expected total profit per unit time. Cárdenas-Barrón [14] derived closed-form expressions to determining the optimal solution to an EOQ inventory model considering items with imperfect quality. Ghiami and Beullens [15] developed a production-inventory model for a deteriorating product with partial backordering using a cash-flow net present value analysis. They generated managerial insights related if it is convenient financially to have planned shortages with partial backordering and lost sales. Zhou et al. [16] developed an economic-order quantity model considering shortages, imperfect quality, and inspection errors. Taleizadeh et al. [17] studied an inventory model when the lot contains imperfect-quality items, and these cannot be replaced with perfect-quality ones immediately, but these are repaired and then they are sold at lower price. In this line of research, Rezaei [18] introduced an inventory model considering imperfect-quality items. Sampling inspection plans are designed so the buyer would be able to decide regarding the next step.At the present time, as we know, global warming is a big threat to the world and carbon emissions are a leading cause of global warming. For this reason, the sustainability is an extremely significant concern for all people that desire to maintain the planet healthy. It is well known that industries generate large amounts of carbon emissions. Furthermore, the transportation of goods through the supply chains produces carbon emissions too. Consequently, these damage the environment and cause global warming at the same time. In this context, the governments of the countries have imposed strict policies that limit the amounts of carbon emissions with the aim of having a sustainable environment free of pollution. The regulations related to avoid pollution impose additional costs to firms; thus, it is convenient for these to adapt and obey the regulations. In this direction, several companies have adopted inventory models that consider reducing carbon emissions. For example, Hua et al. [19] investigated how companies manage carbon emissions in an inventory management. Arslan and Turkay [20] integrated sustainability features into the design of an inventory model and proposed different policies that include environmental and social criteria. Battini et al. [21] introduced a sustainable EOQ inventory model integrating factors that have a great impact on the environment. Lin and Sarker [22] presented an inventory model that considers carbon tax policy and imperfect-quality items avoiding shortages and incorporated and evaluated some carbon tax systems. Tiwari et al. [23] investigated an integrated vendor-buyer inventory model for deteriorating goods with imperfect quality, taking into account that all processes in the supply chain cause carbon emissions. The target of their inventory model is to decrease the ecological effects. Kazemi et al. [24] proposed an inventory model with imperfect quality from a sustainable point of view, and the aim is to know the impact of emission costs into the total profit. Modak et al. [25] addressed a manufacturer-retailer supply chain inventory model by considering that there is a cost of GHG emission of the production system and determined the optimal pricing policies. Afterwards, Sinha and Modak [26] presented an economic production quantity (EPQ) inventory model taking into account aspects of carbon emissions and carbon trading. Li and Hai [27] studied an inventory system for a warehouse with multiple retailers considering carbon emissions. Manna et al. [28] constructed a production-inventory model for controlling the GHG emissions when the pollution parameters are fuzzy. Later, Manna et al. [29] studied the effects of carbon emissions on an imperfect production-inventory model. Huang et al. [30] proposed an inventory model for a two-echelon supply chain considering that all operations within the chain produce carbon emissions. Their inventory model helps companies to determine the optimal production quantity and delivery quantity which minimize the total costs under different carbon emission policies. Recently, Mishra et al. [31] revised the standard EOQ inventory model to integrate sustainability matters and developed three inventory models with and without shortage situations. Medina-Santana and Cárdenas-Barrón [32] formulated an inventory model considering a discontinuous transportation cost function and carbon emissions function. The most recent investigation about carbon emissions belongs to the authors Modak and Kelle [33]. They proposed to apply the social work donation as a scheme of corporate social responsibility into a closed-loop supply chain, taking into consideration the carbon emission tax and that demand is uncertain.In recent years, a new topic, that is, inventory management for growing items, has also been included in the derivation of inventory models. A growing item is a kind of product which is capable of growth during time like farmyard animals, unlike inert products that do not increase in weight during storage. Growing items are incorporated for first time in an EOQ inventory model by Rezaei [34]. Here, the weight increment of items is the main difference between the inventory models proposed by Rezaei [34] and Harris [1]. Rezaei [34] developed an EOQ for growing items for different types of animals, specifically for a kind of poultry. Zhang et al. [35] revisited and extended the research work of Rezaei [34] by creating an inventory model for growing items reducing the carbon emissions in operations. Sebatjane [36] presented three inventory models for growing items. Afterwards, Khalilpourazari and Pasandideh [37] worked with growing items too. They formulated a mathematical model for multi-item economic order quantity subjected to some operational constrains. Nobil et al. [38] derived an EOQ inventory model for growing items when the shortages are allowed, and these are fully backordered. They argued that their inventory model helps poultry industries. Under an environment of a two-echelon supply chain composed by one supplier and one farmer, Malekitabar et al. [39] developed an inventory model for growing mortal items considering both the growth function and the mortality rate for the items. Recently, Sebatjane and Adetunji [40] presented an EOQ inventory model for growing items with imperfect quality. It is assumed that a certain fraction of the growing items has lower quality than desired. They mentioned that it is necessary to define the growth function of the items in order to calculate the feeding cost. In this context, Sebatjane and Adetunji [40] considered three growth functions: logistic, linear, and split linear. In the same year, Sebatjane and Adetunji [41] introduced an economic-order quantity inventory model for growing items with incremental quantity discounts. They proposed an optimal inventory policy to minimize the total inventory cost in both the owned and rented facilities. Sebatjane and Adetunji [42] continued working with growing items, but in this case, they formulated an inventory control model more realistically, considering that the items need to be transformed and packaged into a consumable form before customer demand is met. The next year, Sebatjane and Adetunji [43] built a model for managing inventory in a perishable food products supply chain that begins with farming operations where growing items live and finishes with the consumption of processed inventory. Sebatjane and Adetunji [44] derived an inventory model for a four-echelon supply chain with farming, processing, screening, and retail operations. Gharaei and Almehdawe [45] provided an economic growing quantity (EGQ) inventory model to determine the optimal economic growth and slaughter period and the economic growing quantity to minimize the total cost of the inventory system. Hidayat et al. [46] presented an EOQ inventory model with a capacitated warehouse facility and limited budget for growing products when the seller provides an incremental quantity discounts scheme. Mokhtari et al. [47] addressed a production-inventory model for growing goods which deteriorate through time. Nishandhi [48] studied an EOQ inventory model with budget-capacity constraint for growing items when a portion of the items are of imperfect quality. Pourmohammad-Zia and Karimi [49] determined the optimal replenishment and breeding policies for growing products. Afzal and Alfares [50] and Alfares and Afzal [51] developed EOQ inventory models for growing items considering shortages with full backordering. Recently, Sebatjane and Adetunji [52] created an inventory model for growing items under a three-echelon supply chain environment taking into account farming, processing, and retail echelons. Table 1 shows the inventory models related with growing items.Table 1
Inventory models with growing items.
AuthorsPrice-dependent demandType of price-dependent demandAllowed shortagesType of backorderingImperfect qualityCarbon taxType of objective functionOptimizeRezaei [34]No—No—NoNoMax. profitOrder quantity and slaughter timeZhang et al. [35]No—No—NoYesMin. costOrder quantity and slaughter timeSebatjane [36]No—No—YesNoMax. profitOrder quantity and cycle timeNo—No—NoNoMin. costOrder quantity and cycle timeNo—No—NoNoMin. costOrder quantity and cycle timeKhalilpourazari and Pasandideh [37]No—No—NoNoMax. profitOrder quantity and slaughter timeNobil et al. [38]No—YesFullNoNoMin. costOrder quantity, backordering quantity, and cycle timeMalekitabar et al. [39]YesLinearNo—NoNoMax. profitSelling price and cycle timeSebatjane and Adetunji [40]No—No—YesNoMax. profitOrder quantity and cycle timeSebatjane and Adetunji [41]No—No—NoNoMin. costOrder quantity and cycle timeSebatjane and Adetunji [42]No—No—NoNoMin. costOrder quantity, cycle time, and number of shipmentsSebatjane and Adetunji [43]YesExponentialNo—NoNoMax. profitSelling price, order quantity, cycle time, and number of shipmentsSebatjane and Adetunji [44]No—No—YesNoMax. profitOrder quantity, cycle time, and number of shipmentsGharaei and Almehdawe [45]No—No—NoNoMin. costOrder quantity and cycle timeHidayat et al. [46]No—No—NoNoMin. costOrder quantity and cycle timeMokhtari et al. [47]No—No—NoNoMax. profitOrder quantity and slaughter timeNishandhi [48]No—YesFullYesNoMin. costOrder quantity, backordering quantity, and cycle timePourmohammad-Zia and Karimi [49]No—No—YesNoMin. costOrder quantity and cycle timeAfzal and Alfares [50]No—YesFullYesNoMin. costOrder quantity, backordering quantity, and cycle timeAlfares and Afzal [51]No—YesFullYesNoMin. costOrder quantity, backordering quantity, and cycle timeSebatjane and Adetunji [52]No—No—NoNoMax. profitOrder quantity, cycle time, and number of shipmentsThis paperYesPolynomialYesFullYesYesMax. profitSelling price, order quantity, backordering quantity, and cycle timeThe organizations need to determine the optimal selling prices for the products in order to encourage that the customers buy more and more products, and therefore, these organizations can survive in the competitive business environment. In this context, the academicians and researchers are modelling the demand rate as dependent of price. For example, Khan et al. [53] analyzed two supply chain inventory systems when the demand rate depends on price. Khan et al. [54] proposed an inventory model for deteriorating merchandises when the demand is price sensitive, and there exists a discount policy according all-units arrangement. Khan et al. [55] developed inventory models for perishable products when the demand rate is dependent on both price and advertisement. On the one hand, Khan et al. [56] examined the effects of an advance payment with discount facility on ordering decisions for perishable items taking into account that the demand rate is both price and stock dependent. On the other hand, Panda et al. [57] dealt with a two-warehouse system for deteriorating items when the demand rate is dependent on both price and stock; however, the price is not optimized. Sinha et al. [58] formulated an entropic-order quantity inventory model when the demand of the product is dependent on selling price and there is an inspection process to split the imperfect-quality products from the perfect ones. Modak [59] and Modak and Kelle [60] introduced omni- and dual-channel supply chain models, respectively, by considering that there exists a price- and delivery-time-sensitive stochastic demand.Nowadays, the firms face problems of shortages and setting the selling price in order to survive in a competitive business, which is becoming more challenging day by day. Moreover, the customers are more quality and environmental conscious than before, and therefore, they demand products of perfect quality and request firms to minimize the carbon emissions to the environment. However, the firms always face problems of process quality and contaminate the planet. Besides, there exists a type of goods different than the conventional products which requires feeding until a target weight is reached in order to slaughter and sell to customers. These are named as growing items (chickens, cows, pigs, goats, fish, shrimps, etc.). In this direction, this research work develops an inventory model for growing items with imperfect quality when the demand is price sensitive under carbon emissions and shortages. The shortages are fully backordered. The demand is price sensitive according to a polynomial function. The proposed inventory model determines jointly the optimal policy for the selling price of perfect-quality growing items, the order quantity, and backordering quantity which maximize the total profit per unit of time.The remaining parts of this research work are described as follows: Section2 introduces the notation defining symbols for the parameters, decision variables, decision-dependent variables, and functions. Section 3 states the assumptions under which the inventory model is built. Section 4 formulates the inventory model with shortages for growing items with imperfect quality when the demand is price sensitive considering carbon emissions. Section 5 determines some theoretical results, develops the solution procedure to determine the optimal solution to the inventory problem, and identifies some inventory models as special cases. Section 6 solves some numerical examples. Section 7 provides a sensitivity analysis and some managerial insights. Section 8 gives some conclusions and summaries several research points that can be addressed in the near future.


## 2. Notations

With the aim of having a standard notation for the inventory models with growing items, the nomenclature of Sebatjane and Adetunji [40] is used and extra symbols are defined too. Therefore, the following notations are adopted in order to develop the inventory model for growing items with imperfect quality, carbon emissions, and planned shortages.


### 2.1. Parameters

π: Scale parameter for the price-dependent demandρ: Sensitivity parameter for the price-dependent demandn: Demand power indexv: Selling price of imperfect items (currency symbol/unit of weight)K: Setup cost (currency symbol/cycle)h: Holding cost (currency symbol/unit of weight/unit of time)b: Shortage cost (currency symbol/unit of weight/unit of time)c: Feeding cost (currency symbol/unit of weight)p: Purchasing cost (currency symbol/unit of weight)z: Inspection cost (currency symbol/unit of weight)θ: Carbon tax rate (currency symbol/amount of carbon emissions)Ec: Carbon emissions cost (currency symbol)K⌢: Amount of carbon emissions produced during the setup process (unit of weight/unit of time)h⌢: Amount of carbon emissions caused by holding items in the warehouse (unit of weight/unit of time)c⌢: Amount of carbon emissions generated during the feeding period (unit of weight/unit of time)p⌢: Amount of carbon emissions made during the purchasing activity (unit of weight/unit of time)z⌢: Amount of carbon emissions created during the inspection process (unit of weight/unit of time)r: Inspection rate (unit of weight/unit of time)α: Asymptotic weight of each item (unit of weight)β: Integration constant (numeric value)λ: Growth rate (numeric value/unit of time)x: Percentage of slaughtered items that are of imperfect quality 0≤x≤1Ex: Expected value of the percentage of imperfect items 0≤Ex≤11−Ex: Expected value of the percentage of perfect items 0≤1−Ex≤1w0: Weight of a newborn item (unit of weight)w1: Target weight of a grown item (unit of weight)wt: Weight of an item at time t (unit of weight)t1: Growing period (unit of time)t2: Inspection period for the backordering quantity B (unit of time)t3: Inspection period for yw1−B units of weight (unit of time)t4: Consumption period of perfect items after inspection time (unit of time)t5: Shortages accumulation period (unit of time)Decision variables:y: Order quantity of newborn items (units)B: Backordering quantity (unit of weight)s: Selling price of perfect items (currency symbol/unit of weight)Decision-dependent variables:T: Cycle time (unit of time)Q0: Total weight at the beginning of the growing period, Q0=yw0 (unit of weight)Qt1: Total weight at the end of the growing period t1, Qt1=yw1 (unit of weight)Functions:Ds: Price-dependent demand function (unit of weight/unit of time)wtt: Growth functiongx: Probability density function of the percentage of imperfect itemsTPUy,B,s: Total profit (currency symbol/unit of time)


## 2.1. Parameters

π: Scale parameter for the price-dependent demandρ: Sensitivity parameter for the price-dependent demandn: Demand power indexv: Selling price of imperfect items (currency symbol/unit of weight)K: Setup cost (currency symbol/cycle)h: Holding cost (currency symbol/unit of weight/unit of time)b: Shortage cost (currency symbol/unit of weight/unit of time)c: Feeding cost (currency symbol/unit of weight)p: Purchasing cost (currency symbol/unit of weight)z: Inspection cost (currency symbol/unit of weight)θ: Carbon tax rate (currency symbol/amount of carbon emissions)Ec: Carbon emissions cost (currency symbol)K⌢: Amount of carbon emissions produced during the setup process (unit of weight/unit of time)h⌢: Amount of carbon emissions caused by holding items in the warehouse (unit of weight/unit of time)c⌢: Amount of carbon emissions generated during the feeding period (unit of weight/unit of time)p⌢: Amount of carbon emissions made during the purchasing activity (unit of weight/unit of time)z⌢: Amount of carbon emissions created during the inspection process (unit of weight/unit of time)r: Inspection rate (unit of weight/unit of time)α: Asymptotic weight of each item (unit of weight)β: Integration constant (numeric value)λ: Growth rate (numeric value/unit of time)x: Percentage of slaughtered items that are of imperfect quality 0≤x≤1Ex: Expected value of the percentage of imperfect items 0≤Ex≤11−Ex: Expected value of the percentage of perfect items 0≤1−Ex≤1w0: Weight of a newborn item (unit of weight)w1: Target weight of a grown item (unit of weight)wt: Weight of an item at time t (unit of weight)t1: Growing period (unit of time)t2: Inspection period for the backordering quantity B (unit of time)t3: Inspection period for yw1−B units of weight (unit of time)t4: Consumption period of perfect items after inspection time (unit of time)t5: Shortages accumulation period (unit of time)Decision variables:y: Order quantity of newborn items (units)B: Backordering quantity (unit of weight)s: Selling price of perfect items (currency symbol/unit of weight)Decision-dependent variables:T: Cycle time (unit of time)Q0: Total weight at the beginning of the growing period, Q0=yw0 (unit of weight)Qt1: Total weight at the end of the growing period t1, Qt1=yw1 (unit of weight)Functions:Ds: Price-dependent demand function (unit of weight/unit of time)wtt: Growth functiongx: Probability density function of the percentage of imperfect itemsTPUy,B,s: Total profit (currency symbol/unit of time)


## 3. Assumptions

The inventory model is based on the following assumptions:(1)
The planning horizon is infinite and a single kind of items is purchased, and these are capable of growing before the slaughter process.(2)
The shortages are permitted, and these are completely backordered.(3)
The items are slaughtered and are immediately inspected in order to sell them to consumers. Firstly, the backordering quantity is inspected in order to cover the shortages of the previous cycle.(4)
There exists an inspection process that is 100% effective.(5)
A random percentage of the slaughtered items is of imperfect quality.(6)
Imperfect-quality items are not reworked or replaced.(7)
All imperfect-quality items are salvaged and sold as a single lot at the end of the inspection process.(8)
The feeding cost for growing the items is directly related to weight gained by these.(9)
The holding cost to keep a weight unit of the slaughtered item in storage is incurred during both the inspection process and the consumption period.(10)
The demand rateDs is a polynomial function of selling price of the perfect-quality items. It is as follows: Ds=π−ρsn.(11)
The selling price of perfect-quality items is optimized, and it must be greater than that of the imperfect-quality items.(12)
Carbon emissions are taken into account, and these occur in all operations of the inventory system, except in the shortage period.


## 4. Inventory Model Development

The inventory model for growing items with imperfect quality, carbon emissions, and shortages is depicted in Figure1. Consider a situation where a company orders y newborn growing items from an outside supplier at the beginning of the growing period t1. Each newborn growing item has an initial weight of w0. In this moment, the total initial weight of the inventory is Q0=yw0. The growing items are fed, and eventually, they grow through time until an objective weight of w1 is attained. Then, the growing items are slaughtered at the end of the growing time t1. At this point, the final weight of the inventory is Qt1=yw1, and this total weight contains a percentage x of imperfect items. The portion of imperfect items is a random variable with a known probability density distribution fx, and its expected value is Ex. The shortages are permitted, and these are fully backordered. Therefore, immediately, the inspection process starts to screen the items to complete the backordering quantity B at a rate of r units of weight per unit of time during the period t2 in order to satisfy immediate shortages from the previous cycle. So, at the end of the inspection period t2, the inventory model diminishes vertically by B units of weight. It is important to remark that the items continue to be inspected at the same rate r during the period t3 till the total weight is screened. The length of the inspection time is t2+t3. It is worth mentioning that, for the duration of t3, the on-hand inventory declines by both removing the imperfect items and current demand rate. At the end of period t3, the imperfect items are salvaged and sold as a single lot with a less price. Consequently, the on-hand inventory drops vertically by xyw1 units of weight. On the other hand, in the course of t4, the on-hand inventory decreases due to the current demand rate. The inventory level continues gradually consuming until it reaches zero at the end of period t4 and the shortages period starts. As a final point, during t5, the shortages are accumulated at the current demand rate which are eventually satisfied in the next cycle. Without loss of generality and with the purpose to make the mathematical expressions more tractable, the cycle time is determined as follows: T=t3+t4+t5.Figure 1
Inventory behavior for growing items with imperfect quality and shortages.Throughoutt1, the items are growing according to a logistic growth function which relates the weight of items with time using three input parameters. These input parameters are the asymptotic weight of the items, the integration constant, and the growth rate, which are represented by α, β, and λ, respectively. Thus, the logistic growth function of the items is mathematically expressed by(1)wtt=α1+βe−λt.As it was mentioned above, the growing items are slaughtered when their weight attains the objective weight ofw1 which occurs at the end of the growth period t1. Hence,(2)w1=wtt=t1=α1+βe−λt1.The duration of the growth periodt1 is calculated by solving equation (2) for t1. Thus,(3)t1=−ln1/βα/w1−1λ.The inventory must be inspected before being sold in order to avoid to vend imperfect quality as good ones; therefore, firstly, the backordering quantityB of the previous cycle must be screened at an inspection rate r. Therefore, the duration of the inspection period t2 is computed as follows:(4)t2=Br.After the inspection of the backordering quantity is conducted, the screening process continues until the total weight is screened due to the fact that there are pendingyw1−B units of weight to be inspected. This is performed during the inspection time t3 which is given as follows:(5)t3=yw1−Br.After the inspection timet3, the imperfect-quality items are withdrawn from the storage and sold. Now, the on-hand inventory contains only perfect-quality items, and these are consumed during t4. The time t4 is determined as follows:(6)Et4=yw1−B−Dt3−Exyw1Ds.The shortages accumulation periodt5 is obtained with(7)t5=BDs.With the aim to have more manageable equations and without loss of generality, it is defined that the expected cycle timeT is computed by the sum of t3,t4, and t5. From equations (5)–(7), the expected cycle time ET is calculated as(8)ET=t3+Et4+t5=yw11−ExDs.The main goal of the inventory model for growing items with imperfect quality, carbon emissions, and shortages is to determine the optimal values for the selling price, order quantity, and backordering quantity that maximize the total profit of the company. The total profit is obtained by the difference between the total revenue and the total cost of the inventory system. The details of how to calculate each component of the total revenue and the total cost are given below.


### 4.1. Expected Revenue per Period

Due to the fact that the company vends both perfect- and imperfect-quality items, for that reason, the expected total revenueETR is calculated by the sum of total sales of imperfect and perfect items. The perfect-quality items are sold at a price of s per unit of weight, while at the end of the inspection period t3, the imperfect-quality items are vended as a single batch with a less price of v per unit of weight. As a result, the expected total revenue ETR per period is given by(9)ETR=syw11−Ex+vyw1Ex.


### 4.2. Total Cost per Period

The expected total costETC per period includes the purchasing cost, setup cost, feeding cost, expected holding cost, inspection cost, backordering cost, and expected carbon emissions cost. One has(10)ETC=Pc+Sc+Fc+EHc+Zc+Bc+EEc.The objective of this inventory model is to maximize the expected total profitETP which is obtained by subtracting the expected total cost per period ETC from the expected total revenue ETR per period. Then,(11)ETP=ETR−Pc−Sc−Fc−EHc−Zc−Bc−EEc.The following section provides a detailed discussion of the calculations of the aforesaid costs.


### 4.3. Purchasing Cost per Period

At the beginning of each period, the company buysy newborn items at a cost of p per unit of weight; each one with a weight of w0. Thus, the purchasing cost per period is obtained with(12)Pc=pyw0.


### 4.4. Setup Cost per Period

At the commencement of each period, a setup cost ofK is carried out. So, the setup cost per period is determined as(13)Sc=K.


### 4.5. Feeding Cost per Period

The growing items are fed duringt1, and a feeding cost is incurred by the company at c per unit of weight. As the growing items become older and bigger, they need more food. For that reason, the quantity of food consumed by the items depends on the age (weight) of the items according to the growth function wtt. The feeding cost per period is calculated with(14)Fc=cy∫0t1wt1tdt=cy∫0t1α1+βe−λtdt=cyαt1+αλln1+βe−λt1−ln1+β.


### 4.6. Expected Holding Cost per Period

The company incurs in a holding cost for maintaining in good conditions the items in storage during the cycle timeT. The expected holding cost is determined by multiplying the holding cost h per unit of weight per unit of time by the inventory carried out during t2+t3+t4. The whole inventory held is calculated as the sum of the areas A1+A2+A3+A4+A5 (see Figure 5 in Appendix A). So, the expected holding cost is given by(15)EHc=hy2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.The detailed derivation of the expected holding costEHc is given in Appendix A.


### 4.7. Inspection Cost per Period

Throughoutt2+t3, a 100% inspection process is performed at a rate of r with the aim of splitting the perfect-quality items from imperfect-quality items. The company incurs in an inspection cost of z per unit of weight. In consequence, the inspection cost per period is expressed as(16)Zc=zyw1.


### 4.8. Shortage Cost per Period

During the course oft5, the shortages are accumulated till these reach the backordering quantity of B units of weight. The company has a backordering cost for the management of the accumulation of shortages. The backordering cost is computed by multiplying shortage cost b per unit of weight per unit of time by area A6 (see Figure 5 in Appendix A). Therefore, the backordering cost is calculated as(17)Bc=bB22Ds.The detailed derivation of the backordering costBc is specified in Appendix B.There are several research works that suppose that the carbon emissions generated by the companies are due to the transportation and warehousing activities. But, in fact, there are a lot of operations that emit carbon emissions. For example, the operations involved in the following process also cause carbon emissions: purchasing, setup, feeding, holding inventory, and inspection, among others. It is important to mention that some growing items per se generate carbon emissions (cows, goats, pigs, etc.). The carbon emissions caused by these processes are determined as follows:


### 4.9. Carbon Emissions Caused by the Purchasing Action

(18)P⌢c=p⌢yw0.


### 4.10. Carbon Emissions Produced by the Setup Activity

(19)S⌢c=K⌢.


### 4.11. Carbon Emissions Generated during the Feeding Process

(20)F⌢c=c⌢y∫0t1wt1tdt=c⌢y∫0t1α1+βe−λtdt=c⌢yαt1+αλln1+βe−λt1−ln1+β.


### 4.12. Carbon Emissions Created in Holding Inventory’s Operations

(21)EH⌢c=h⌢y2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.


### 4.13. Carbon Emissions Made by the Inspection Process

(22)Z⌢c=z⌢yw1.Carbon taxθ is one of the well-known mechanisms imposed by government regulations as a penalty. This means that the companies need to pay a tax on the amount of carbon emissions. As a result, the carbon emissions cost per period that the company must pay is(23)Ec=θp⌢yw0+K⌢+z⌢yw1+c⌢yαt1+αλln1+βe−λt1−ln1+β+h⌢y2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.


### 4.14. Expected Total Profit Function

The expected total profitETP per period is calculated in the following manner: the expected total revenue ETR per period minus the expected total cost ETC per period. Basically, the expected total profit ETP is formulated by substituting equations (9) and (12) to (17) and equation (23) into equation (11). Then, the expected total profit ETPU per unit time per period is determined as ETPU=ETP/ET. Thus,(24)ETPUy,B,s=sDs+vDsEx1−Ex−pDsw0w11−Ex−KDsyw11−Ex−zDs1−Ex−cDsw11−Exαt1+αλln1+βe−λt1−ln1+β−hyw1E1−x221−Ex−B+B22yw11−Ex+Dsyw1Exr1−Ex+DsBr−bB22yw11−Ex−θp⌢Dsw0w11−Ex+K⌢Dsyw11−Ex+z⌢Ds1−Ex+c⌢Dsw11−Exαt1+αλln1+βe−λt1−ln1+β+h⌢yw1E1−x221−Ex−B+B22yw11−Ex+Dsyw1Exr1−Ex+DsBr.Considering that the demand rate depends on selling price with a polynomial functionDs=π−ρsn, the expected total profit ETPUy,B,s per unit of time becomes(25)ETPUy,B,s=sπ−ρsn+vπ−ρsnEx1−Ex−pπ−ρsnw0w11−Ex−Kπ−ρsnyw11−Ex−zπ−ρsn1−Ex−cπ−ρsnw11−Exαt1+αλln1+βe−λt1−ln1+β−hyw1E1−x221−Ex−B+B22yw11−Ex+π−ρsnyw1Exr1−Ex+π−ρsnBr−bB22yw11−Ex−θp⌢π−ρsnw0w11−Ex+K⌢π−ρsnyw11−Ex+z⌢π−ρsn1−Ex+c⌢π−ρsnw11−Exαt1+αλln1+βe−λt1−ln1+β+h⌢yw1E1−x221−Ex−B+B22yw11−Ex+π−ρsnyw1Exr1−Ex+π−ρsnBr.Then, the objective is to maximize the expected total profitETPUy,B,s per unit of time. Therefore, the optimization problem is formulated as follows:(26)Maxy,B,s∈ΩETPUy,B,swhereΩ=y,B,s:y>0,0≤B≤yw1andp≤s≤πρ1/n.It is important to remark that the abovementioned maximization formulation is a nonlinear optimization problem.


## 4.1. Expected Revenue per Period

Due to the fact that the company vends both perfect- and imperfect-quality items, for that reason, the expected total revenueETR is calculated by the sum of total sales of imperfect and perfect items. The perfect-quality items are sold at a price of s per unit of weight, while at the end of the inspection period t3, the imperfect-quality items are vended as a single batch with a less price of v per unit of weight. As a result, the expected total revenue ETR per period is given by(9)ETR=syw11−Ex+vyw1Ex.


## 4.2. Total Cost per Period

The expected total costETC per period includes the purchasing cost, setup cost, feeding cost, expected holding cost, inspection cost, backordering cost, and expected carbon emissions cost. One has(10)ETC=Pc+Sc+Fc+EHc+Zc+Bc+EEc.The objective of this inventory model is to maximize the expected total profitETP which is obtained by subtracting the expected total cost per period ETC from the expected total revenue ETR per period. Then,(11)ETP=ETR−Pc−Sc−Fc−EHc−Zc−Bc−EEc.The following section provides a detailed discussion of the calculations of the aforesaid costs.


## 4.3. Purchasing Cost per Period

At the beginning of each period, the company buysy newborn items at a cost of p per unit of weight; each one with a weight of w0. Thus, the purchasing cost per period is obtained with(12)Pc=pyw0.


## 4.4. Setup Cost per Period

At the commencement of each period, a setup cost ofK is carried out. So, the setup cost per period is determined as(13)Sc=K.


## 4.5. Feeding Cost per Period

The growing items are fed duringt1, and a feeding cost is incurred by the company at c per unit of weight. As the growing items become older and bigger, they need more food. For that reason, the quantity of food consumed by the items depends on the age (weight) of the items according to the growth function wtt. The feeding cost per period is calculated with(14)Fc=cy∫0t1wt1tdt=cy∫0t1α1+βe−λtdt=cyαt1+αλln1+βe−λt1−ln1+β.


## 4.6. Expected Holding Cost per Period

The company incurs in a holding cost for maintaining in good conditions the items in storage during the cycle timeT. The expected holding cost is determined by multiplying the holding cost h per unit of weight per unit of time by the inventory carried out during t2+t3+t4. The whole inventory held is calculated as the sum of the areas A1+A2+A3+A4+A5 (see Figure 5 in Appendix A). So, the expected holding cost is given by(15)EHc=hy2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.The detailed derivation of the expected holding costEHc is given in Appendix A.


## 4.7. Inspection Cost per Period

Throughoutt2+t3, a 100% inspection process is performed at a rate of r with the aim of splitting the perfect-quality items from imperfect-quality items. The company incurs in an inspection cost of z per unit of weight. In consequence, the inspection cost per period is expressed as(16)Zc=zyw1.


## 4.8. Shortage Cost per Period

During the course oft5, the shortages are accumulated till these reach the backordering quantity of B units of weight. The company has a backordering cost for the management of the accumulation of shortages. The backordering cost is computed by multiplying shortage cost b per unit of weight per unit of time by area A6 (see Figure 5 in Appendix A). Therefore, the backordering cost is calculated as(17)Bc=bB22Ds.The detailed derivation of the backordering costBc is specified in Appendix B.There are several research works that suppose that the carbon emissions generated by the companies are due to the transportation and warehousing activities. But, in fact, there are a lot of operations that emit carbon emissions. For example, the operations involved in the following process also cause carbon emissions: purchasing, setup, feeding, holding inventory, and inspection, among others. It is important to mention that some growing items per se generate carbon emissions (cows, goats, pigs, etc.). The carbon emissions caused by these processes are determined as follows:


## 4.9. Carbon Emissions Caused by the Purchasing Action

(18)P⌢c=p⌢yw0.


## 4.10. Carbon Emissions Produced by the Setup Activity

(19)S⌢c=K⌢.


## 4.11. Carbon Emissions Generated during the Feeding Process

(20)F⌢c=c⌢y∫0t1wt1tdt=c⌢y∫0t1α1+βe−λtdt=c⌢yαt1+αλln1+βe−λt1−ln1+β.


## 4.12. Carbon Emissions Created in Holding Inventory’s Operations

(21)EH⌢c=h⌢y2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.


## 4.13. Carbon Emissions Made by the Inspection Process

(22)Z⌢c=z⌢yw1.Carbon taxθ is one of the well-known mechanisms imposed by government regulations as a penalty. This means that the companies need to pay a tax on the amount of carbon emissions. As a result, the carbon emissions cost per period that the company must pay is(23)Ec=θp⌢yw0+K⌢+z⌢yw1+c⌢yαt1+αλln1+βe−λt1−ln1+β+h⌢y2w12E1−x22Ds−yw11−ExBDs+B22Ds+y2w12Exr−yw1ExBr+yw1Br.


## 4.14. Expected Total Profit Function

The expected total profitETP per period is calculated in the following manner: the expected total revenue ETR per period minus the expected total cost ETC per period. Basically, the expected total profit ETP is formulated by substituting equations (9) and (12) to (17) and equation (23) into equation (11). Then, the expected total profit ETPU per unit time per period is determined as ETPU=ETP/ET. Thus,(24)ETPUy,B,s=sDs+vDsEx1−Ex−pDsw0w11−Ex−KDsyw11−Ex−zDs1−Ex−cDsw11−Exαt1+αλln1+βe−λt1−ln1+β−hyw1E1−x221−Ex−B+B22yw11−Ex+Dsyw1Exr1−Ex+DsBr−bB22yw11−Ex−θp⌢Dsw0w11−Ex+K⌢Dsyw11−Ex+z⌢Ds1−Ex+c⌢Dsw11−Exαt1+αλln1+βe−λt1−ln1+β+h⌢yw1E1−x221−Ex−B+B22yw11−Ex+Dsyw1Exr1−Ex+DsBr.Considering that the demand rate depends on selling price with a polynomial functionDs=π−ρsn, the expected total profit ETPUy,B,s per unit of time becomes(25)ETPUy,B,s=sπ−ρsn+vπ−ρsnEx1−Ex−pπ−ρsnw0w11−Ex−Kπ−ρsnyw11−Ex−zπ−ρsn1−Ex−cπ−ρsnw11−Exαt1+αλln1+βe−λt1−ln1+β−hyw1E1−x221−Ex−B+B22yw11−Ex+π−ρsnyw1Exr1−Ex+π−ρsnBr−bB22yw11−Ex−θp⌢π−ρsnw0w11−Ex+K⌢π−ρsnyw11−Ex+z⌢π−ρsn1−Ex+c⌢π−ρsnw11−Exαt1+αλln1+βe−λt1−ln1+β+h⌢yw1E1−x221−Ex−B+B22yw11−Ex+π−ρsnyw1Exr1−Ex+π−ρsnBr.Then, the objective is to maximize the expected total profitETPUy,B,s per unit of time. Therefore, the optimization problem is formulated as follows:(26)Maxy,B,s∈ΩETPUy,B,swhereΩ=y,B,s:y>0,0≤B≤yw1andp≤s≤πρ1/n.It is important to remark that the abovementioned maximization formulation is a nonlinear optimization problem.


## 5. Solution Procedure

Firstly, the unconstrained optimization problem is considered in order to obtain some theoretical results. Secondly, an algorithm for finding the optimal solution to the constrained problem given by equation (26) is developed.


### 5.1. Theoretical Results

The aim is to find the optimal values for order quantityy, backordering quantity B, and selling price s that maximize the expected total profit per unit of time. As the expected total profit per unit of time function ETPUy,B,s is continuous and twice differentiable with respect to the three decision variables y,B,s on the interval 0,∞, there exists a global maximum on that interval.The necessary conditions which must be satisfied for the optimality of solution that maximizes the expected total profit per unit of time function are∂ETPUy,B,s/∂y=0, ∂ETPUy,B,s/∂B=0, and ∂ETPUy,B,s/∂s=0.The first partial derivative ofETPUy,B,s given in equation (25) with respect to y is expressed as follows:(27)∂ETPUy,B,s∂y=Kπ−ρsny2w11−Ex−hw1E1−x221−Ex−B22y2w11−Ex+π−ρsnw1Exr1−Ex+bB22y2w11−Ex−θ−K⌢π−ρsny2w11−Ex+h⌢w1E1−x221−Ex−B22y2w11−Ex+π−ρsnw1Exr1−Ex=0.The first partial derivative ofETPUy,B,s with respect to B is given by(28)∂TPUy,B,s∂B=−h−1+Byw11−Ex+π−ρsnr−bByw11−Ex−θh⌢−1+Byw11−Ex+π−ρsnr=0.The first partial derivative ofETPUy,B,s with respect to s is(29)∂ETPUy,B,s∂s=π−n+1ρsn−ρnsn−1w11−ExvExw1−p+θp⌢w0−K+θK⌢y−z+θz⌢w1−c+θc⌢αt1+αλln1+βe−λt1−ln1+β−h+θh⌢ryw12Ex+Bw11−Ex=0.By solving equations (27) and (28) simultaneously for the decision variables y and B, one has(30)y=2π−ρsnrK+θK⌢w12h+θh⌢rE1−x2+2π−ρsnEx−h+θh⌢w11−Exr−π−ρsn2/h+b+θh⌢r,(31)B=h+θh⌢yw11−Exr−π−ρsnh+b+θh⌢r.The sufficient conditions for the optimality of the solution that maximizeETPUy,B,s are given in Appendix C.


### 5.2. Algorithm for Finding the Optimal Solution

The following Algorithm1 is proposed taking into account the theoretical results and constrains.Algorithm 1:Algorithm to find the optimal solution.
Step 1. Provide the input parameters of the inventory system.Step 2. Compute the selling prices′, the order quantity y′, and the backordering quantity B′ by solving simultaneously equations (29)–(31).Step 3. If the optimality conditions are satisfied, then go to Step 4. Else, go to Step 8.Step 4. Ifp≤s≤π/ρ1/n, then go to Step 7. Else, go to Step 5.Step 5. Ifs>π/ρ1/n, then set s′=π/ρ1/n, determine the order quantity y′ with equation (30) and the backordering quantity B′ with equation (31), and go to Step 6. Else, set s′=p and calculate the order quantity y′ with equation (30) and the backordering quantity B′ with equation (31) and go to Step 6.Step 6. Calculate the expected total profit per unit of timeETPUy′,B′,s′ with equation (25).Step 7. Report the solution:y∗,B∗,s∗=y′,B′,s′ and ETPU∗y∗,B∗,s∗=ETPUy′,B′,s′.Step 8. Stop.


### 5.3. Special Cases

The proposed inventory model developed in this research work is a general inventory model because it contains the following inventory models as special cases: Sebatjane and Adetunji’s [40] improved inventory model, Maddah and Jaber [11], Chang and Ho [13], Shih [61], Silver [62], and the traditional EOQ inventory models with and without shortages.(i)
Whenn=1, it means that the product has a linear price-dependent demand π−ρs. It is expressed as follows:(32)y=2π−ρsrK+θK⌢w12h+θh⌢rE1−x2+2π−ρsEx−h+θh⌢w11−Exr−π−ρs2/h+b+θh⌢r,(33)B=h+θh⌢yw11−Exr−π−ρsh+b+θh⌢r,(34)π−2ρs−ρw11−ExvExw1−p+θp⌢w0−K+θK⌢y−z+θz⌢w1−c+θc⌢αt1+αλln1+βe−λt1−ln1+β−h+θh⌢ryw12Ex+Bw11−Ex=0.(ii)
When shortages and carbon tax rate are not considered and the demand is not dependent on price (i.e.,b⟶∞,θ⟶0 and ρ⟶0), then an improved version of Sebatjane and Adetunji’s [40] inventory model is obtained. It is shown as follows:(35)y=2πKhw12E1−x2+2πEx/r.(iii)
When shortages are permitted, carbon tax rate is not involved, the demand is not dependent on price, and the products are not of the type of growing items, and thus, the feeding process is not incorporated (i.e.,b>0,θ⟶0,ρ⟶0, and c⟶0), and then, the inventory model of Chang and Ho [13] is determined. It is given by(36)y=2πKhE1−x2+2πEx/r−h1−Ex2/h+b,(37)B=hy1−Exh+b.(iv)
When shortages are not allowed, carbon tax rate is not taken account, the demand is not dependent on price, and the products are not of the type of growing items (b⟶∞,θ⟶0,ρ⟶0, and c⟶0); then, Maddah and Jaber’s [11] inventory model is derived. It is presented as follows:(38)y=2πKhE1−x2+2πEx/r.(v)
When shortages are not tolerable, carbon tax rate is not considered, the demand does not depend on price, the goods are not of the type of growing items, and inspection rate is sufficiently large (i.e.,b⟶∞,θ⟶0,ρ⟶0,c⟶0 and r⟶∞); then, it is converted to the inventory model of Shih [61] and Silver [62]. It is expressed as(39)y=2πKhE1−x2.(vi)
When shortages are permissible, carbon tax rate is not incorporated, the demand is not dependent on price, the products are not of the type of growing items, and the products are of perfect quality (i.e.,b>0,θ⟶0,ρ⟶0,c⟶0, and x=0); then, the traditional inventory model with shortages and with full backordering is found. It is written as follows:(40)y=2πKhh+bb,(41)B=hyh+b.(vii)
When shortages are not allowable, carbon tax rate is not taken into consideration, the demand does not depend on price, the goods are not of type of growing items, and these are of good quality (i.e.,b⟶∞,θ⟶0,ρ⟶0,c⟶0, and x=0); then, it is transformed to the traditional inventory model without shortages proposed by Harris [1]. It is(42)y=2πKh.It is important to remark that the inventory model developed in this research does not reduce to Salameh and Jaber’s [8] and Wee et al.’s [10] inventory models because those inventory models contain a flaw that was identified and corrected by Maddah and Jaber [11] and Chang and Ho [13], respectively. Basically, Maddah and Jaber [11] and Chang and Ho [13] proposed improved versions for Salameh and Jaber’s [8] and Wee et al.’s [10] inventory models, respectively. For this reason, the proposed inventory model developed in this research work, in fact, converges to Maddah and Jaber’s [11] and Chang and Ho’s [13] inventory models as it was mentioned above.


## 5.1. Theoretical Results

The aim is to find the optimal values for order quantityy, backordering quantity B, and selling price s that maximize the expected total profit per unit of time. As the expected total profit per unit of time function ETPUy,B,s is continuous and twice differentiable with respect to the three decision variables y,B,s on the interval 0,∞, there exists a global maximum on that interval.The necessary conditions which must be satisfied for the optimality of solution that maximizes the expected total profit per unit of time function are∂ETPUy,B,s/∂y=0, ∂ETPUy,B,s/∂B=0, and ∂ETPUy,B,s/∂s=0.The first partial derivative ofETPUy,B,s given in equation (25) with respect to y is expressed as follows:(27)∂ETPUy,B,s∂y=Kπ−ρsny2w11−Ex−hw1E1−x221−Ex−B22y2w11−Ex+π−ρsnw1Exr1−Ex+bB22y2w11−Ex−θ−K⌢π−ρsny2w11−Ex+h⌢w1E1−x221−Ex−B22y2w11−Ex+π−ρsnw1Exr1−Ex=0.The first partial derivative ofETPUy,B,s with respect to B is given by(28)∂TPUy,B,s∂B=−h−1+Byw11−Ex+π−ρsnr−bByw11−Ex−θh⌢−1+Byw11−Ex+π−ρsnr=0.The first partial derivative ofETPUy,B,s with respect to s is(29)∂ETPUy,B,s∂s=π−n+1ρsn−ρnsn−1w11−ExvExw1−p+θp⌢w0−K+θK⌢y−z+θz⌢w1−c+θc⌢αt1+αλln1+βe−λt1−ln1+β−h+θh⌢ryw12Ex+Bw11−Ex=0.By solving equations (27) and (28) simultaneously for the decision variables y and B, one has(30)y=2π−ρsnrK+θK⌢w12h+θh⌢rE1−x2+2π−ρsnEx−h+θh⌢w11−Exr−π−ρsn2/h+b+θh⌢r,(31)B=h+θh⌢yw11−Exr−π−ρsnh+b+θh⌢r.The sufficient conditions for the optimality of the solution that maximizeETPUy,B,s are given in Appendix C.


## 5.2. Algorithm for Finding the Optimal Solution

The following Algorithm1 is proposed taking into account the theoretical results and constrains.Algorithm 1:Algorithm to find the optimal solution.
Step 1. Provide the input parameters of the inventory system.Step 2. Compute the selling prices′, the order quantity y′, and the backordering quantity B′ by solving simultaneously equations (29)–(31).Step 3. If the optimality conditions are satisfied, then go to Step 4. Else, go to Step 8.Step 4. Ifp≤s≤π/ρ1/n, then go to Step 7. Else, go to Step 5.Step 5. Ifs>π/ρ1/n, then set s′=π/ρ1/n, determine the order quantity y′ with equation (30) and the backordering quantity B′ with equation (31), and go to Step 6. Else, set s′=p and calculate the order quantity y′ with equation (30) and the backordering quantity B′ with equation (31) and go to Step 6.Step 6. Calculate the expected total profit per unit of timeETPUy′,B′,s′ with equation (25).Step 7. Report the solution:y∗,B∗,s∗=y′,B′,s′ and ETPU∗y∗,B∗,s∗=ETPUy′,B′,s′.Step 8. Stop.


## 5.3. Special Cases

The proposed inventory model developed in this research work is a general inventory model because it contains the following inventory models as special cases: Sebatjane and Adetunji’s [40] improved inventory model, Maddah and Jaber [11], Chang and Ho [13], Shih [61], Silver [62], and the traditional EOQ inventory models with and without shortages.(i)
Whenn=1, it means that the product has a linear price-dependent demand π−ρs. It is expressed as follows:(32)y=2π−ρsrK+θK⌢w12h+θh⌢rE1−x2+2π−ρsEx−h+θh⌢w11−Exr−π−ρs2/h+b+θh⌢r,(33)B=h+θh⌢yw11−Exr−π−ρsh+b+θh⌢r,(34)π−2ρs−ρw11−ExvExw1−p+θp⌢w0−K+θK⌢y−z+θz⌢w1−c+θc⌢αt1+αλln1+βe−λt1−ln1+β−h+θh⌢ryw12Ex+Bw11−Ex=0.(ii)
When shortages and carbon tax rate are not considered and the demand is not dependent on price (i.e.,b⟶∞,θ⟶0 and ρ⟶0), then an improved version of Sebatjane and Adetunji’s [40] inventory model is obtained. It is shown as follows:(35)y=2πKhw12E1−x2+2πEx/r.(iii)
When shortages are permitted, carbon tax rate is not involved, the demand is not dependent on price, and the products are not of the type of growing items, and thus, the feeding process is not incorporated (i.e.,b>0,θ⟶0,ρ⟶0, and c⟶0), and then, the inventory model of Chang and Ho [13] is determined. It is given by(36)y=2πKhE1−x2+2πEx/r−h1−Ex2/h+b,(37)B=hy1−Exh+b.(iv)
When shortages are not allowed, carbon tax rate is not taken account, the demand is not dependent on price, and the products are not of the type of growing items (b⟶∞,θ⟶0,ρ⟶0, and c⟶0); then, Maddah and Jaber’s [11] inventory model is derived. It is presented as follows:(38)y=2πKhE1−x2+2πEx/r.(v)
When shortages are not tolerable, carbon tax rate is not considered, the demand does not depend on price, the goods are not of the type of growing items, and inspection rate is sufficiently large (i.e.,b⟶∞,θ⟶0,ρ⟶0,c⟶0 and r⟶∞); then, it is converted to the inventory model of Shih [61] and Silver [62]. It is expressed as(39)y=2πKhE1−x2.(vi)
When shortages are permissible, carbon tax rate is not incorporated, the demand is not dependent on price, the products are not of the type of growing items, and the products are of perfect quality (i.e.,b>0,θ⟶0,ρ⟶0,c⟶0, and x=0); then, the traditional inventory model with shortages and with full backordering is found. It is written as follows:(40)y=2πKhh+bb,(41)B=hyh+b.(vii)
When shortages are not allowable, carbon tax rate is not taken into consideration, the demand does not depend on price, the goods are not of type of growing items, and these are of good quality (i.e.,b⟶∞,θ⟶0,ρ⟶0,c⟶0, and x=0); then, it is transformed to the traditional inventory model without shortages proposed by Harris [1]. It is(42)y=2πKh.It is important to remark that the inventory model developed in this research does not reduce to Salameh and Jaber’s [8] and Wee et al.’s [10] inventory models because those inventory models contain a flaw that was identified and corrected by Maddah and Jaber [11] and Chang and Ho [13], respectively. Basically, Maddah and Jaber [11] and Chang and Ho [13] proposed improved versions for Salameh and Jaber’s [8] and Wee et al.’s [10] inventory models, respectively. For this reason, the proposed inventory model developed in this research work, in fact, converges to Maddah and Jaber’s [11] and Chang and Ho’s [13] inventory models as it was mentioned above.


## 6. Numerical Examples

This section presents and solves some numerical examples in order to illustrate the applicability of the proposed inventory model.Example 1.
This example considers the input parameters of Sebatjane and Adetunji [40]. To solve this inventory problem, it is stated b⟶∞,θ⟶0, and ρ⟶0. The values for the data are π=1000000, ρ=0, θ=0, s=0.05 ZAR/g, v=0.02ZAR/g, K=1000 ZAR/cycle, h=0.04 ZAR/g/year, c=0.2 ZAR/g, p=0.025 ZAR/g, z=0.00025 ZAR/g, r=5256000g/year, α=6870g, β=120λ=40/year, w0=57g, and w1=1500g, and the percentage of imperfect growing items follows a uniform distribution x∼Uγ,δ with the probability density function gx which is given as follows:(43)x∼gx=1δ−γ,γ≤x≤δ,0,otherwise.
Consideringx∼U0,0.04,(44)x∼gx=25,0≤x≤0.04,0,otherwise.
Then,Ex and E1−x2 are computed as follows:(45)Ex=∫γδxfxdx=γ+δ2=0+0.042=0.02,E1−x2=∫γδ1−x2fxdx=γ2+γδ+δ23+1−γ−δ,E1−x2=02+00.04+0.0423+1−0−0.04=0.960533333.
As it was mentioned before, the proposed inventory model generates an improved version of Sebatjane and Adetunji’s [40] inventory model. Additionally, the proposed inventory model also can be used for optimizing the order quantity y and backordering quantity B when the selling price s is given. Obviously, it also optimizes the three decision variables y,B,s. Therefore, the numerical example is solved for different values of δ for the four inventory models: (I) Sebatjane and Adetunji [40] (original version), (II) Sebatjane and Adetunji [40] (improved version), (III) the proposed inventory model when selling price s is given, and (IV) the proposed inventory model when the selling price s is optimized. Table 2 shows the comparison of the optimal solutions among the four inventory models.
From Table2, it is observed that the expected total profit of both the proposed inventory models (see columns III and IV) is greater than that of Sebatjane and Adetunji’s [40] inventory models (see columns I and II).Table 2
A comparison of the optimal solutions of the four inventory models.
IIIIIIIVSebatjane and Adetunji [40] (original), b⟶∞,θ=0, ρ=0Sebatjane and Adetunji [40] (improved inventory model), b⟶∞,θ=0, ρ=0The proposed inventory model,ρ=0,s=0.05The proposed inventory model,ρ=700000γδExy∗ETPU∗y∗ETPU∗y∗B∗ETPU∗y∗B∗s∗ETPU∗00.040.02151.514334641.73151.503934641.11167.907057103.7035518.41138.962351818.670.6940517449132.600.100.05155.288735013.46155.218635009.38171.756656624.7835880.02142.500051509.930.6939366449374.600.200.10161.876035680.21161.558835662.24178.219855663.1636519.49148.542350865.870.6937328449801.700.300.15168.835636413.78168.029336369.20184.655054469.0137209.73154.709150032.070.6935122450261.300.400.20176.159437225.05174.542037137.38190.938053009.2637957.34160.908548973.440.6932723450757.000.500.25183.824438127.53180.980137975.53196.917451252.4738770.55167.017847653.140.6930098451293.2Example 2.
This example uses the input parameters of Wee et al. [10] and Chang and Ho [13]. To solve this inventory system, the following is established: b>0,θ⟶0,ρ⟶0, and c⟶0. The data are as follows: π=50000, ρ=0, θ=0, c=0, s=50 $/unit, v=20$/unit, K=100$/cycle, h=5$/unit/year, b=10$/unit/year p=25$/unit, z=0.5$/unit, r=175200units/year, and x∼U0,0.04. The optimal solution is as follows: ETPU∗y∗,B∗=1213562$/year, y∗=1751.671 units, and B∗=572.2127 units. This solution is the same as in the work of Chang and Ho [13].Example 3.
Now, let us consider some dataset of Sebatjane and Adetunji [40] which is v=0.02 ZAR/g, K=1000ZAR/cycle, c=0.2ZAR/g, p=0.025ZAR/g, z=0.00025ZAR/g, r=5256000g/year, α=6870g, β=120, λ=40/year, w0=57g, w1=1500g, and x∼U0,0.04. Here, the holding cost is h=0.2ZAR/g/year. For the implementation of the proposed inventory model, additional information is required. These data are related to the type of the demand function dependent on the price, the backordering cost, and carbon emissions. Therefore, it is necessary to state the following data: Assume that the demand rate of the growing items follows a polynomial function given by Ds=π−ρsn with values of π=135000, ρ=1050, and n=2. The backordering cost is b= 0.1 ZAR/g/year. The relevant input parameters related to carbon emissions are θ= 0.0045 ZAR/tons, K⌢= 2000 tons/year, h⌢= 0.2 tons/year, c⌢= 0.65 tons/year, p⌢= 0.375 tons/year, and z⌢= 0.005 tons/year. By applying the proposed algorithm, the optimal solution is obtained: ETPU∗y∗,B∗,s∗=584997.4 ZAR/year, y∗=34.26474 units of newborn growing items, B∗=33054.63 g, and s∗=6.555838 ZAR/g. Figures 2–4 demonstrate graphically the concavity property of the expected total profit with respect to pairs of decision variables y and B; y and s; and B and s, respectively.Figure 2
Concavity ofETPUy,B w.r.t. y,BFigure 3
Concavity ofETPUy,s w.r.t. y,sFigure 4
Concavity ofETPUB,s w.r.t. B,s


## 7. Sensitivity Analysis

This section provides a sensitivity analysis in order to investigate the effects of changing the input parameters on the expected total profit per unit of timeETPU∗, order quantity y∗, backordering quantity B∗, and selling price s∗ of the growing items. The sensitivity analysis is performed taking into consideration the data of Example 3 when only one parameter changes at a time and other parameters are kept at their original values. Specifically, the sensitivity analysis is performed to study the effect of scale, sensitivity and power index demand parameters, selling price and expected value of a percent of poorer-quality growing items, setup cost, holding cost, backordering cost, feeding cost, purchasing cost, inspection cost, carbon tax rate, and the amount of carbon emissions caused by the operations of the processes in ordering, holding, feeding, purchasing, and screening. The results of the sensitivity analysis are given in Tables 3–5. Based on the behavioral changes as reflected in Tables 3–5, the following observations are made:(1)
The value ofETPU∗ is highly sensitive to the demand parameters π,ρ,n and less sensitive to other parameters. On the one hand, the higher the value of the scale parameter of demand π, the higher the value of ETPU∗ due to the fact that demand increases; therefore, the sales increase, and this leads to high profits. For this reason, it is suggested for the manager to apply advertising actions in order to boost the demand. On the other hand, when the sensitivity parameter of the demand ρ grows, the ETPU∗ declines because this parameter has a negative impact on the demand, making it drop. For this reason, it is advisable for the decision maker to reduce the selling price. However, a higher value of K induces higher values of s but a lower value for ETPU∗. This means that as the setup cost increases, the company must raise selling price s.(2)
The value ofy is more sensitive to the parameters π,n,K,h,b and less sensitive to other parameters. The higher the value of the parameter b, the smaller the value of y.(3)
The value ofB is more sensitive to the parameters π,n,K,h,b and less sensitive to other parameters. The higher the value of b, the smaller the value of B.(4)
The value ofs is more sensitive to the parameters π,ρ,n and less sensitive to other parameters. The higher the value of the parameter π, the higher the value of s. This means that as demand scale parameter π increases, the company must raise selling price which will also directly impact positively the expected total profit. The higher the value of the parameters ρ,n, the smaller the value of s.(5)
When the expected value of the defective growing itemsEx decreases, the expected total profit also decreases ETPU∗.(6)
Changes on carbon emission parameters have a regular influence on the expected total profitETPU∗.Table 3
Effects of the demand parametersπ,ρ,n, the price of imperfect growing items v, and the percent of the imperfect growing items Ex on the optimal solution.
ParameterValuey∗B∗s∗ETPU∗π8500027.5146826714.185.205486291168.513500034.2647433054.636.555838584997.418500039.6436937996.927.672019940097.923500044.1725442062.768.645232134728828500048.1061945509.169.5194381800598ρ65034.2696133059.168.329802744653.975034.2682833057.927.755266692945.885034.2670333056.767.285369650655.195034.2658633055.676.891773615231.4105034.2647433054.636.555838584997.4n234.2647433054.636.555838584997.4537.9852336485.121.850223202864.7738.8494037274.301.490172170704.71039.5413437903.961.281047151875.71440.0258838343.671.167557141757.1v0.0234.2647433054.636.555838584997.40.0634.2654133055.266.555565585070.70.1034.2660833055.886.555293585144.10.1434.2667533056.506.555020585217.50.1834.2674233057.126.554748585290.8Ex0.0133.9408833076.326.555872584987.50.0234.2647433054.636.555838584997.40.0334.5909533028.816.555804585007.00.0434.9193432998.646.555770585016.40.0535.2497532963.886.555736585025.6Table 4
Impacts of the costs of the inventory systemK,h,b,c,p,z,θ on the optimal solution.
ParameterValuey∗B∗s∗ETPU∗K100034.2647433054.636.555838584997.4120037.5054236180.936.556512584656.6140040.4871439057.416.557133584343.1160043.2635041735.806.557711584051.2180045.8718244252.086.558254583776.9h0.1238.3489730325.676.554933585380.60.1635.8869131977.516.555439585160.00.234.2647433054.636.555838584997.40.2433.0909933786.806.556172584869.70.2832.1862034295.596.556464584765.0b0.134.2647433054.636.555838584997.40.1530.5444525267.196.556552584559.00.228.4314720585.996.557049584259.00.2527.0597117420.146.557416584039.20.326.0941515121.736.557700583870.6c0.0534.2681633057.816.554445585372.30.134.2670233056.756.554909585247.30.1534.2658833055.696.555374585122.30.234.2647433054.636.555838584997.40.2534.2636033053.586.556302584872.4p0.02534.2647433054.636.555838584997.40.0534.2639533053.906.556162584910.20.07534.2631533053.166.556485584823.10.134.2623633052.426.556809584736.00.12534.2615633051.686.557133584648.9z0.0002534.2647433054.636.555838584997.40.000534.2645333054.446.555923584974.40.0007534.2643233054.256.556008584951.50.00134.2641233054.056.556094584928.60.0012534.2639133053.866.556179584905.6θ0.004534.2647433054.636.555838584997.40.0134.4133933258.236.555956584955.20.015534.5609633460.866.556074584913.20.02134.7074633662.556.556192584871.20.026534.8529333863.296.556309584829.3Table 5
Impacts of the carbon emission parametersK⌢,h⌢,c⌢,p⌢,z⌢ on the optimal solution.
ParameterValuey∗B∗s∗ETPU∗K⌢200034.2647433054.636.555838584997.4250034.3029133091.456.555846584993.3300034.3410333128.236.555854584989.3350034.3791033164.966.555862584985.3400034.4171433201.656.555870584981.3h⌢0.234.2647433054.636.555838584997.40.434.2343233074.316.555846584994.20.634.2041133093.816.555854584991.00.834.1741233113.156.555862584987.8134.1443433132.326.555870584984.7c⌢0.6534.2647433054.636.555838584997.41.334.2646833054.576.555865584990.01.9534.2646133054.516.555892584982.72.634.2645433054.456.555919584975.43.2534.2644833054.396.555947584968.1p⌢0.37534.2647433054.636.555838584997.40.7534.2646933054.586.555860584991.51.12534.2646433054.536.555882584985.61.534.2645833054.486.555904584979.71.87534.2645333054.446.555925584973.8z⌢0.00534.2647433054.636.555838584997.40.01534.2647033054.606.555853584993.20.02534.2646733054.566.555869584989.10.03534.2646333054.536.555884584985.00.04534.2645933054.496.555899584980.8With the information of Tables3–5 is constructed Table 6. Table 6 summarizes the sensitivity analysis study in a visual manner which is more helpful for the decision makers due to that these can more easily observe how changes in parameters affect to the decisions variables and the expected total profit per unit of time in order to take the best decisions based on numerical facts.Table 6
Sensitivity analysis of the key parameters of the inventory system.
Parametery∗B∗s∗ETPU∗π↑↑↑↑↑ρ↑↓↓↓↓n↑↑↑↓↓v↑↑↑↓↑K↑↑↑↑↓h↑↓↑↑↓b↑↓↓↑↓c↑↓↓↑↓p↑↓↓↑↓z↑↓↓↑↓θ↑↑↑↑↓K⌢↑↑↑↑↓h⌢↑↓↑↑↓c⌢↑↓↓↑↓p⌢↑↓↓↑↓z⌢↑↓↓↑↓Ex↑↑↓↓↑


## 8. Conclusions and Future Research

This paper develops an inventory model for growing items with imperfect quality and carbon emissions in which the demand rate is price sensitive according to a polynomial function. Furthermore, the shortages are permissible and these are fully backordered. To optimize the expected total profit per unit of time, some theoretical results are derived, and with these results, an effective and efficient solution procedure is developed. It is important to state that the algorithm generates the optimal solution to order quantity, backordering quantity, and selling price which maximizes the expected total profit per unit of time. The effects of carbon tax and the amount of carbon emissions are studied. Some important managerial insights are obtained from numerical examples and sensitivity analysis. The results show that carbon tax has a regular role on the reduction of carbon emissions. It is important to pay attention to quality and ensure that the percentage of defective items is kept to a minimum. Furthermore, if the setup cost increases, the company must raise selling prices, and when the demand elasticity parameter increases, the company must raise selling price which also directly impacts the total profit. Furthermore, the proposed inventory model is a generalized inventory model due the fact that several previously published inventory models are particular cases. It was found that the policy that permits shortages with full backordering is more economical that the one that avoids shortages. Moreover, it was also found that the proposed inventory model outperforms the inventory model of Sebatjane and Adetunji [40].There are several possible extensions of the proposed inventory model that can be explored. These extensions constitute future research endeavors in the inventory management of imperfect growing items. For instance, consider to investigate the effect that only a percentage of defective items can be reworked, and the others must be eliminated immediately. Other research studies that can be conducted are to include new aspects such as stock-dependent demand, nonlinear holding cost, vendor-managed inventory (VMI) with consignment stock (CS), inflation, volume discounts, deterioration, trade credit, supply chain environment, and a vendor-buyer inventory model with multiple shipments, partial backordering, advertising, and multiple products subject to constrains such as space, budget, and time. These are, among others, some interesting and challenging subjects of ongoing future investigation that academicians and researchers would like to study in the future.



---

*Source: 6649048-2021-04-08.xml*