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Axioms
, Volume 12 (5) – Apr 27, 2023

/lp/multidisciplinary-digital-publishing-institute/sustainable-supply-chain-model-for-defective-growing-items-fishery-utqhcLZd8m

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axioms Article Sustainable Supply Chain Model for Defective Growing Items (Fishery) with Trade Credit Policy and Fuzzy Learning Effect Osama Abdulaziz Alamri Department of Statistics, University of Tabuk, Tabuk 71491, Saudi Arabia; oalmughamisi@ut.edu.sa Abstract: Fundamentally, newborn items that are used commercially, such as chicken, ﬁsh, and small camel, grow day by day in size and also increase their weight. The seller offers a credit policy to the buyer to increase sales for a particular growing item (ﬁsh), and in this paper, it is assumed that the buyer accepts the policy of the trade credit. In this paper, the buyer acquires the newborn items (ﬁsh) from the seller and then sells them when the newborn items have increased their size and weight. From this point of view, the present paper reveals a fuzzy-based supply chain model that includes carbon emissions and a permissible delay in payment for defective growing items (ﬁsh) under the effect of learning where the demand rate is imprecise in nature and is treated as a triangular fuzzy number. Finally, the buyer ’s total proﬁt is optimized with respect to the number of newborn items. A numerical example has been presented for the justiﬁcation of the model. The ﬁndings clearly suggest that the presence of trade credit, learning, and a fuzzy environment have an afﬁrmative effect on the ordering policy. The buyer should order more to avoid higher interest charges after the grace period, which eventually increases their proﬁt, while at the same time, it is also beneﬁcial for the buyer to order less to gain the beneﬁt of the trade credit period. The fuzziness theory controls the uncertainty situation of inventory parameters with the help of a de-fuzziﬁed method. The lower and upper deviation of demand affects the total fuzzy proﬁt. The effect of learning gives a positive response concerning the size of the order and the buyer ’s total fuzzy proﬁt. This means that the decision-maker should be aware of the size of the newborn items, rate of learning, and trade credit period during the supply chain because these directly affect the buyer ’s total fuzzy proﬁt. The impact of the inventory parameter of this model is presented with the help of sensitivity analysis. Citation: Alamri, O.A. Sustainable Supply Chain Model for Defective Keywords: growing items (ﬁsh); trade credit; carbon emissions; learning effect; supply chain; signed Growing Items (Fishery) with Trade distance method; fuzzy environment Credit Policy and Fuzzy Learning Effect. Axioms 2023, 12, 436. https://doi.org/10.3390/ axioms12050436 1. Introduction Academic Editors: Rekha Guchhait This section covers the literature review studies that form the background of the and Mitali Sarkar proposed study and includes an introduction to the proposed study. Received: 18 February 2023 Revised: 31 March 2023 1.1. Literature Review According to the Trade-Credit Policy Model under Various Policies Accepted: 3 April 2023 Supply chain management is a good tool for businesses, industries, ﬁrms, etc., and Published: 27 April 2023 involves stock inventory management. Many researchers relatively worked on supply chains with various strategies for different items. Harris [1] presented the ﬁrst inventory model in 1913 for an ordering policy for deteriorating items. A signiﬁcant amount of inventory models have been extended using the theory of this initial model, with various Copyright: © 2023 by the author. approaches for perishable items or deteriorating items or vegetables, but we have selected Licensee MDPI, Basel, Switzerland. a few literature reviews relative to this paper study. Aggarwal and Jaggi [2] developed a This article is an open access article distributed under the terms and new model with a credit period for deteriorating items and achieved positive effects on the conditions of the Creative Commons total proﬁt. Chu et al. [3] deﬁned a strategy for the credit period for deteriorating items. Attribution (CC BY) license (https:// Abad and Jaggi [4] considered an inventory model with the help of a credit-period policy creativecommons.org/licenses/by/ for the deterioration of items, where the demand rate was a function of the selling price, 4.0/). Axioms 2023, 12, 436. https://doi.org/10.3390/axioms12050436 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 436 2 of 27 and calculated the length of the credit period, which is the most important element for both players. 1.2. Literature Review According to the Trade-Credit Policy and Imperfect Quality Items-Based Model under Various Policies Chung and Liao [5] presented an imperfect quality-based inventory model with decaying items for the ordering strategy, where the seller offers a credit period to his buyer, and both players acquire more proﬁt. Salameh and Jaber [6] proposed a model with imperfect quality items under a screening process. Chung et al. [7], inspired by the model of Chung and Lia [5], proposed an imperfect quality-based model for the two-warehouse system. Various researchers, such as Huang [8], Jaber et al. [9], and Jaggi et al. [10], have worked on imperfect items with various policies. 1.3. Literature Review According to the Imperfect Quality Items, Carbon Emissions, and Growing Items-Based Model under Various Policies Carbon emissions from different sources are very harmful to the environment. An important number of authors have presented models for various strategies regarding carbon emissions from imperfect quality items. Zhang et al. [11] presented a new policy model with growing items regarding carbon emissions, where the amount of carbon emissions was calculated. Tiwari et al. [12] presented a carbon emissions-based model with defective quality items for deteriorating items. Sebatjane and Adetunji [13] proposed an EOQ model with defective products for growing items, calculated the expected total proﬁt for the supply chain under the logistic growth function and split linear. De-la-Cruz- Marquez et al. [14] described a mathematical model for growing defective quality items under the effects of shortages and carbon emissions, where the demand rate depended on the selling price. 1.4. Literature Review According to Imperfect Quality Items, Carbon Emissions, Trade Credit, and Learning Fuzzy Theory-Based Model under Various Policies Most researchers consider the rate of demand as imprecise in nature. In this way, Chang [15] proposed an EOQ model with defective quality items under a fuzzy environ- ment. Rani et al. [16] discussed a fuzzy-based inventory model with carbon emissions for deteriorating items under a green supply chain. The concept of learning has been presented by Wright [17] in the ﬁeld of inventory, which suggested that some factors affect the total inventory cost. Wee and Chen [18] proposed an economic order quantity model for defec- tive items under the effect of shortages. Jayaswal et al. [19] presented an EPQ model under the effect of learning under the credit period. Jayaswal et al. [20] proposed an EOQ model for defective quality items regarding the effect of leaning under a credit period scheme. Alamri et al. [21] proposed an EOQ model with inflation and carbon emissions under the effect of learning for deteriorating items. Jayaswal et al. [22] presented an EOQ model regarding the effect of a leaning and credit financing policy under a cloudy fuzzy environment. 1.5. Introduction of the Proposed Study The present paper deals with a seller–buyer model in which the seller provides new- born growing items (when the weight of the newborn items is raised) to the buyer under a credit policy, where the demand rate is imprecise in nature and treated as a triangular fuzzy number. The seller offers a credit policy scheme to the new client for the increased sale of newborn growing items, which, therefore, generates more revenue. Often, buyers do not have adequate money revenue to conduct business and, therefore, accept the trade credit scheme, which bounds them to multiply more effort in generating sales. When the buyer receives a lot of newborn growing items and has inspected the whole lot with a constant inspection rate, he divides the whole lot into two parts: non-defective and defective. The buyer sells the defective items at a low price and none defective items at a higher price. In this model, the buyer assumes that defective items follow the effect of the learning curve and includes some carbon emission cost due to holding units. Finally, the total fuzzy proﬁt Axioms 2023, 12, 436 3 of 27 is optimized with respect to the number of newborn growing items and de-fuzziﬁed with the help of the signed distance method. The research gap and our research study are shown in the next paragraph. 1.5.1. Introduction of the Proposed Study’s Background, Perspective, Research Gap, and Our Contribution As mentioned above, it can clearly be seen that little attention was paid to managing growing items (ﬁshes) during the business deal. We tried to promote a sustainable supply chain model with carbon emissions and learning effects under a fuzzy environment and trade credit scheme in which the supply chain players get more proﬁt under some realistic situation. Basically, the present study has been developed for the ﬁshery where sellers purchase newborn growing items and then sometimes sell them when the weight of the newborn items is raised. We have discussed a few relatable literature articles that synchronize with the present study’s background, such as Rezaei [23], which presented an inventory model for the growing items where the demand rate has been taken as a linear function. In this order, Sebatjane and Adetunji [13] developed a mathematical model for the growing items with imperfect quality items. Mittal and Sharma [24] proposed a growing items-based inventory model under a credit scheme. Renowned authors such as Rezaei [23], Sebatjane and Adetunji [13], and Mittal and Sharma [24] were not oriented toward the application of the learning curve, fuzzy theory as well as carbon emissions concept. We were motivated by the study of Mittal and Sharma [24] and from the literature review of Jayaswal et al. [25] to Rajeswari et al. [26], and their contributions have been described in Table 1. The present study tried to ﬁll the research gap using the concept of learning, credit scheme, fuzzy concept, and carbon emissions. We considered some selected research work related to our work, as included in Table 1. Table 1 is treated as a contribution of authors-related research work, and in the end, our work will be justiﬁed. We assumed the inventory problem in which demand for newborn items is considered a fuzzy variable to handle the uncertainty of the market. The present study helps decision-makers to make more stable decisions in an uncertain environment. It is also evident that the learning process will facilitate decision-makers in their decision-making prerogatives, which helps bring about higher proﬁt. We have discussed the subsequent realistic problems in our proposed model: (i) How buyer ’s total fuzzy proﬁt and order quantity get affected by trade credit policy under fuzzy environment. (ii) What is the impact of the learning rate on the buyer ’s total fuzzy proﬁt under a fuzzy environment? (iii) What is the impact of the number of shipments on the buyer ’s total fuzzy proﬁt under a fuzzy environment? (iv) How the buyer ’s total fuzzy proﬁt gets affected by changes in various parameters (distance, feeding cost, ordering cost, selling price, etc.). (v) How the buyer ’s total fuzzy proﬁt gets affected by changes in lower and upper deviation of demand rate. Table 1. Description of the author ’s contribution, research gap, and present study. Imperfect Trade Carbon Learning Fuzzy Authors Growing Items Quality Items Credit Policy Emissions Effect Environment Wright [17] 3 Salameh and Jaber [6] 3 Jaber et al. [9] 3 3 Sebatjane and 3 3 Adetunji [13] Mittal and 3 3 Sharma [24] Axioms 2023, 12, 436 4 of 27 Table 1. Cont. Imperfect Trade Carbon Learning Fuzzy Authors Growing Items Quality Items Credit Policy Emissions Effect Environment Jayaswal et al. [19] 3 3 3 3 Alamri et al. [21] 3 3 3 Jayaswal et al. [22] 3 3 3 3 Rezaei [23] 3 Mittal and 3 3 3 Sharma [24] Pattnaik [25] 3 3 Rajeswari et al. [26] 3 Mahapatra et al. [27] 3 3 Taheri and 3 3 Mirzazadeh [28] Dinagar and 3 3 Manvizhi [29] Garg et al. [30] 3 3 Kuppulakshmi 3 3 et al. [31] Jayaswal and 3 3 3 Mittal [32] Chung and 3 3 Huang [33] Sulak [34] 3 3 Shekarian et al. [35] 3 3 3 Kazemi et al. [36] 3 3 Present study 3 3 3 3 3 3 The contribution made by the researcher in the present paper is the inclusion of trade credit policy and learning fuzzy theory for the growing items (Fishery) shown at the bottom in Table 1. 1.5.2. Outlook of Present Study through Flowchart The presentation of the ﬂowchart is a short review of any proposed work. We tried to summarize the methodology of the proposed study through a ﬂowchart, as shown Axioms 2023, 12, x FOR PEER REVIEW 5 of 29 in Figure 1. Figure 1. Representation of model’s flowchart. Figure 1. Representation of model’s ﬂowchart. 2. Assumptions and Definitions 2.1. Assumption The following assumptions have been considered in this paper and are given below: ➢ The demand rate has been considered imprecise in nature and treated as a triangular fuzzy number (Alsaedi et al. [37]). The triangular fuzzy number is used to fuzzify the model, and the signed distance method is applied to defuzzify the model. ➢ Let us consider that the seller provides growing items to the buyer, and the lot has some defective growing items (Sebatjane and Adetunji [13]). It is also assumed that the seller offers a trade credit period to the buyer, and the buyer accepts the policy of trade credit (Aggarwal and Jaggi [2]). ➢ It is assumed that the buyer inspects the whole lot received at a constant screening rate; after that, the buyer separates the whole lot received from the seller into two categories, defective and non-defective items, and then sells them at different selling prices (Salameh and Jaber [6]). It is also considered that the selling price of good quality is greater than that of defective quality items (poorer) (Jaggi et al. [38]). The fraction of defective growing items follows the S-shape learning curve (Jaber et al. [9]). All defective growing items are sold in different markets (Mittal and Sharma [24]). ➢ To avoid shortages within screening time. ➢ It is assumed that the rework or lack of replacement of growing defective items after the delivery of the lot. ➢ The demanded items are capable of growing prior to being slaughtered (Sebatjane and Adetunji [13]) and include the feeding cost. The cost of feeding the items is proportional to the weight gained by the items (Mittal and Sharma [24]). ➢ It is supposing that when the transaction of the newborn items shifts from one place to another, a lot of carbon units emit due to transportation, which is very harmful to the environment and incorporates emission cost for low carbon units (Guru et al. [39]). Axioms 2023, 12, 436 5 of 27 2. Assumptions and Deﬁnitions 2.1. Assumption The following assumptions have been considered in this paper and are given below: â The demand rate has been considered imprecise in nature and treated as a triangular fuzzy number (Alsaedi et al. [37]). The triangular fuzzy number is used to fuzzify the model, and the signed distance method is applied to defuzzify the model. â Let us consider that the seller provides growing items to the buyer, and the lot has some defective growing items (Sebatjane and Adetunji [13]). It is also assumed that the seller offers a trade credit period to the buyer, and the buyer accepts the policy of trade credit (Aggarwal and Jaggi [2]). â It is assumed that the buyer inspects the whole lot received at a constant screening rate; after that, the buyer separates the whole lot received from the seller into two categories, defective and non-defective items, and then sells them at different selling prices (Salameh and Jaber [6]). It is also considered that the selling price of good quality is greater than that of defective quality items (poorer) (Jaggi et al. [38]). The fraction of defective growing items follows the S-shape learning curve (Jaber et al. [9]). All defective growing items are sold in different markets (Mittal and Sharma [24]). â To avoid shortages within screening time. â It is assumed that the rework or lack of replacement of growing defective items after the delivery of the lot. â The demanded items are capable of growing prior to being slaughtered (Sebatjane and Adetunji [13]) and include the feeding cost. The cost of feeding the items is proportional to the weight gained by the items (Mittal and Sharma [24]). â It is supposing that when the transaction of the newborn items shifts from one place to another, a lot of carbon units emit due to transportation, which is very harmful to the environment and incorporates emission cost for low carbon units (Guru et al. [39]). 2.2. Some Basic Deﬁnitions Below are some useful deﬁnitions required for developing the present paper. Deﬁnition 1. Suppose that if the universal set is S and any set L on S, the fuzzy set can be n o e e deﬁned on S and it is mentioned by L and where L = l, m l : l 2 S , m is a membership e e L L function that is deﬁned from S to [0,1]. If the three numbers with the condition b < b < b , then 1 2 3 the triplet (b , b , b ) are known as triangular fuzzy numbers; if their membership function will be 1 2 3 bb b 2 [b , b ] < 1 2 b b 2 1 b b m = b 2 [b , b ] L 2 3 b b > 3 2 0 Otherwise Deﬁnition 2. If we take any number, let us say, z and 0 2 S, then the signed distance can be calculated from the number z to 0, which can be represented by d(z, 0) = z and also for the negative of any number, the signed distance from z to 0, we can represent it such that ( ) d(z, 0) = z where the negative sign represents the direction of z and it will be in the opposite direction of positive z. It is considered that W is the collection of fuzzy sets Z, which is deﬁned on S. The a cut, Z(a) = [Z (a ), Z (a )], exits 8 a 2 [0.1] then Z(a) = [Z (a ), Z (a )], L U L U Z (a ) and Z (a ) are the continuous function and deﬁned on a. The a cut, Z(a) can be L U represented by Z(a) = [ [Z (a ) , Z (a ) ]. 0a1 L U a a e e e Deﬁnition 3. It is supposed that, Z2 W, then the signed distance of Z from 0 can be deﬁned below [Z (a ) + Z (a )]da L U d(z, 0) = (1) 2 Axioms 2023, 12, 436 6 of 27 e e Deﬁnition 4. If Z = z , z , z represents the triangular fuzzy number, then a cut of Z is ( ) 1 2 3 Z(a) = [Z (a ), Z (a )], where Z (a ) = z + (z z )a and C (a ) = z (z z )a. The L U L 1 2 1 U 3 1 2 e e signed distance of Z to 0 is (z + 2z + z ) 2 3 d Z, 0 = (2) 3. Model Formulation 3.1. S-Shape Learning Curve The effects of learning operate as a signiﬁcant function for reducing the inventory cost and also optimizing the total proﬁt of the inventory system. Figure 2 graphically represents Axioms 2023, 12, x FOR PEER REVIEW 7 of 29 the S-shape learning curve with the help of the given data below and in the form of a Axioms 2023, 12, x FOR PEER REVIEW 7 of 29 formula (Jaber et al. [9]) P(n) = , a > 0, g > 0, where b represents the parameter of bn g+e learning and n is a shipment (shown in Figure 3). Figure 2. Membership function of a triangular fuzzy number. Figure 2. Membership function of a triangular fuzzy number. Figure 2. Membership function of a triangular fuzzy number. Figure 3. S-Shape learning curve. Figure 3. S-Shape learning curve. Figure 3. S-Shape learning curve. 3.2. Model Description 3.2. Model Description Suppose that the buyer orders newborn items, say P at the staring of the growing Suppose that the buyer orders newborn items, say 𝑃 at the staring of the growing 3.2. Model Description cycle, and the newborn items have some weight S . Now the total weight of inventory is cycle, and the newborn items have some weight 𝑆 0 . Now the total weight of inventory is Suppose that the buyer orders newborn items, say 𝑃 at the staring of the growing Y = PS and it is also considered that after time t its weight is S then the whole weight 𝑌 0 = 𝑃 𝑆 0 and it is also considered that after time 𝑡1 its weight is 𝑆 1 then the whole weight 0 0 1 1 cycle, and the newborn items have some weight 𝑆 . Now the total weight of inventory is of the inventory as Y = PS . After the whole lot is received, the buyer inspects the whole of the inventory as 𝑌 1 = 𝑃 𝑆 1 . After the whole lot is received, the buyer inspects the whole 1 1 𝑌 = 𝑃 𝑆 and it is also considered that after time 𝑡 its weight is 𝑆 then the whole weight 0 0 1 1 lot during the time period 𝒕 with the screening rate 𝑅 and divides the whole lot into two of the inventory as 𝑌 = 𝑃 𝑆 . After the whole lot is received, the buyer inspects the whole 1 1 categories: one is the good quality items and the other is the defective quality items. It is lot during the time period 𝒕 with the screening rate 𝑅 and divides the whole lot into two assumed the percentage of defective items in the whole lot is 𝑑 (𝑛 ) and the whole defective categories: one is the good quality items and the other is the defective quality items. It is items are 𝑑 (𝑛 )𝒀 while the non-defective items are (1 − 𝑑 (𝑛 ))𝒀 . The buyer sales the 𝟏 𝟏 assumed the percentage of defective items in the whole lot is 𝑑 (𝑛 ) and the whole defective defective items at low prices during the time period 𝒕 and the good quality at a high items are 𝑑 (𝑛 )𝒀 while the non-defective items are (1 − 𝑑 (𝑛 ))𝒀 . The buyer sales the 𝟏 𝟏 price. The carrying costs are included for the time period of consumption time, and we defective items at low prices during the time period 𝒕 and the good quality at a high have described it in Figure 4 briefly. The accept ed policy of trade credit is also described price. The carrying costs are included for the time period of consumption time, and we in Figure 5. We included some costs such as purchasing cost ( = 𝑃 𝑃 𝑆 ) , ordering cost 𝑐 0 have described it in Figure 4 briefly. The accept ed policy of trade credit is also described 2 2 2 ( ) ( ) 𝑃 (1−𝑑 𝑛 ) 𝑃 𝑆 𝑑 𝑛 in Figure 5. We included some costs such as purchasing cost ( = 𝑃 𝑃 𝑆 ) , ordering cost ( = 𝐾 ) , inventory carrying cost (𝐻𝐼𝐶 = ℎ( + ), i𝑐 nspec 0 tion cost( = 2𝐷 2 𝑅 2 2 2 𝑃 (1−𝑑 (𝑛 )) 𝑃 𝑆 𝑑 (𝑛 ) (𝐼 𝑌 )= , 𝐾 ca ) , rb ino vn ent emis ory ca sio rr n yi n co g st cos ( t 𝐶𝐸𝐶 (𝐻𝐼𝐶 ==2 ℎ𝑑 (𝐸 𝑇 ) due +to carb) o,n i nspec units ti oex n co it st d( urin= g 𝑠 1 1 𝑡 2𝐷 𝑅 transportation and feeding cost (𝐹𝐶 ) . The buyer obtains the whole revenue ( = ( ) 𝐼 𝑌 ) , carbon emission cost 𝐶𝐸𝐶 = 2 𝑑 𝐸 𝑇 due to carbon units exit during 𝑠 1 1 𝑡 𝑆 𝑃 𝑆 (1 − 𝑑 (𝑛 ))+ 𝑆 𝑃 𝑆 𝑑 (𝑛 ) )from the other sources. The whole profit is given below : 𝑔 1 𝑑 1 transportation and feeding cost (𝐹𝐶 ) . The buyer obtains the whole revenue ( = 𝑆 𝑃 𝑆 (1 − 𝑑 (𝑛 ))+ 𝑆 𝑃 𝑆 𝑑 (𝑛 ) )from the other sources. The whole profit is given below : 𝑔 1 𝑑 1 𝑇𝑅 𝑐𝑖 𝑇𝑅 𝑐𝑖 𝐼𝐶 𝑂𝐶 𝐼𝐶 𝑂𝐶 𝑃𝐶 𝑃𝐶 Axioms 2023, 12, 436 7 of 27 lot during the time period t with the screening rate R and divides the whole lot into two categories: one is the good quality items and the other is the defective quality items. It is assumed the percentage of defective items in the whole lot is d(n) and the whole defective items are d(n)Y while the non-defective items are (1 d(n))Y . The buyer sales the defec- 1 1 tive items at low prices during the time period t and the good quality at a high price. The carrying costs are included for the time period of consumption time, and we have described it in Figure 4 brieﬂy. The accepted policy of trade credit is also described in Figure 5. We included some costs such as purchasing cost (PC = P PS ), ordering cost (OC = K), inven- c 0 Axioms 2023, 12, x FOR PEER REVIEW 2 2 2 8 of 29 P (1d (n)) P S d(n) tory carrying cost ( I HC = h + , inspection cost ( IC = I Y ), carbon s 1 2D R emission cost (CEC = 2 d E T ) due to carbon units exit during transportation and feed- 1 ci Axioms 2023, 12, x FOR PEER REVIEW 8 of 29 ing cost (FC). The buyer obtains the whole revenue T R = S PS (1 d(n)) + S PS d(n) g 1 d 1 from the other sources. The whole proﬁt is given below: Figure 4. Model presentation under logistic growth function. Figure 4. Model presentation under logistic growth function. Figure 4. Model presentation under logistic growth function. Figure 5. Model presentation under logistic growth function and credit policy scheme. Figure 5. Model presentation under logistic growth function and credit policy scheme. Figure 5. Model presentation under logistic growth function and credit policy scheme. The whole buyer’s profit 𝜉 (𝑃 )= whole sales revenue( )− whole inventory cost( ) The whole buyer’s profit 𝜉 (𝑃 )= whole sales revenue( )− whole inventory cost( ) 𝜉 (𝑃 ) = (𝑆 𝑃 𝑆 (1 − 𝑑 (𝑛 ))+ 𝑆 𝑃 𝑆 𝑑 (𝑛 ) ) 𝑔 1 𝑑 1 2 2 2 ( ) ( ) 𝑃 (1 − 𝑑 𝑛 ) 𝑃 𝑆 𝑑 𝑛 (3) ( ) ( ) − (𝑃 𝑃 𝑆 + 𝐾 + ℎ( + )+ 𝐼 𝑃 𝑆 + 2 𝑑 𝑒 𝑇 + 𝐹𝐶 ) 𝜉 𝑃 = (𝑆 𝑃 𝑆 1 − 𝑑 (𝑛 ) + 𝑆 𝑃 𝑆 𝑑 (𝑛 ) ) 𝑐 0 𝑠 1 1 𝑐𝑖 𝑔 1 𝑑 1 2𝐷 𝑅 2 2 2 𝑃 (1 − 𝑑 (𝑛 )) 𝑃 𝑆 𝑑 (𝑛 ) (3) − (𝑃 𝑃 𝑆 + 𝐾 + ℎ( In Equatio+ n (3), the fee) d+ ing 𝐼 𝑃cos𝑆 t (+ FC2) 𝑑 ca𝑒n𝑇 be + ca𝐹𝐶lcula ) ted with the help of the growth 𝑐 0 𝑠 1 1 𝑐𝑖 2𝐷 𝑅 linear function (𝑆 ), which gives In Equation (3), the feeding cost (FC) can be calculated with the help of the growth (4) = 𝑓 𝑃 ∫ 𝑆 𝑑𝑡 linear function (𝑆 ), which gives 𝑑 𝑡 From Equation (4), the value of feeding cost (FC) putting in Equation (3), and we (4) = 𝑓 𝑃 ∫ 𝑆 𝑑𝑡 𝑑 𝑡 obtain From Equation (4), the value of feeding cost (FC) putting in Equation (3), and we obtain 𝐹𝐶 𝐹𝐶 𝑇𝐶 𝑇𝑅 𝑇𝐶 𝑇𝑅 Axioms 2023, 12, 436 8 of 27 The whole buyer ’s proﬁt x P = whole sales revenue T R whole inventory cost TC ( ) ( ) ( ) x(P) = S PS (1 d(n)) + S PS d(n) 1 d 1 (3) 2 2 2 P (1d (n)) P S d(n) P PS + K + h + + I PS + 2 d eT + FC c 0 s 1 1 ci 2D R In Equation (3), the feeding cost (FC) can be calculated with the help of the growth linear function (S ), which gives FC = f P S dt (4) d t From Equation (4), the value of feeding cost (FC) putting in Equation (3), and we obtain x P = S PS 1 d n + S PS d n ( ) ( ( )) ( ) g 1 1 (5) 2 2 2 2 R P S d(n) P (1d (n)) t P PS + K + h + + I PS + 2 d eT + f P S dt c 0 s 1 1 ci d t 2D R 0 For the calculation of the buyer ’s whole proﬁt, the feeding function is needed for the growing items, and its value is different for different items. To calculate the feeding cost, the growth function is needed. Here, we discuss the logistic growth functions for the calculation of feeding costs. The logistic growth function is one type of growth function that relates the weight of the items with time and has three parameters; one is the weight under an asymptotic situation (B), the second is the constant of integration (l), and the third is the exponentially growth rate (m). The growth function of the growing items is deﬁned and brieﬂy given in Sebatjane and Adetunji [13]. From Figure 4, the weight of an item rises slowly in the initial stage and steadily rises over time, up to the maturity level; after that, the weight gain reduces. At the initial stage, the stock of the inventory is Y (= PS ). The inventory level at the maturity time t is 0 0 1 Y = PS . The whole items screened up to time t . ( ) 1 1 2 S = (6) mt 1 + le From Equation (6), we can calculate for t and we obtain B 1 log lS l t = (7) Hence, the feeding cost (FC) can be calculated from Figure 3, and we obtain B 1 log ( ) lS l Z Z m B FC = f P S dt = f P dt d d mt 1 + le 0 0 B 1 log ( ) lS l Z 1 FC = f P dt mt 1 + le mt B 1 + le FC = f P B t + log (8) d 1 m 1 + l The buyer ’s whole proﬁt under the logistic growth function from Equation (5) and Equation (8); we obtain Axioms 2023, 12, 436 9 of 27 x (P) = S PS (1 d(n)) + S PS d(n) LGF 1 d 1 2 2 2 2 P (1d (n)) P S d(n) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci 2D R (9) mt B 1+le + f P B t + log . d 1 m 1+l The seller offers the credit ﬁnancing period to the buyer, and the buyer accepts the scheme of the trade credit policy. The buyer ’s total proﬁt under the trade credit scheme is Axioms 2023, 12, x FOR PEER REVIEW 10 of 29 given in Equation (10) below: x(P) = S PS (1 d(n)) + S PS d(n) + IE g 1 d 1 𝜉 (𝑃 ) = (𝑆 𝑃 𝑆 (1− 𝑑 (𝑛 ))+ 𝑆 𝑃 𝑆 𝑑 (𝑛 )+ IE) 𝑔 1 𝑑 1 2 2 2 2 P S d(n) P (1d (n)) P PS + K + h 2 + +2I PS + 2 d E T c 2 s t 0 1 1 ci 2D R 𝑃 (1 − 𝑑 (𝑛 )) ( ) 𝑃 𝑆 𝑑 𝑛 (10) − (𝑃 𝑃 𝑆 + 𝐾 + ℎ( + )+ 𝐼 𝑃 𝑆 + 2 𝑑 𝐸 𝑇 𝑐 0 𝑠 1 1 𝑡 𝑐𝑖 2𝐷 𝑅 (10) mt B 1+le −𝜇𝑡 + f P B t + log + I P . m B 11++ l 𝑙 𝑒 + 𝑓 𝑃 (B t + log( ))+ 𝐼𝑃 ) 𝑑 1 𝜇 1 + 𝑙 From Equation (10), the values of interest gained (IE) and interest charged (IP) depend From Equation (10), the values of interest gained (IE) and interest charged (IP) on some situations of credit period, which are (i) 0 M t (ii) M t T (iii) T M 2 2 depend on some situations of credit period, which are (i) 0 ≤ 𝑀 ≤ 𝑡 (ii) 𝑀 ≤ 𝑡 ≤ 𝑇 (iii) 2 2 and explained brieﬂy in Figure 5. 𝑇 ≤ 𝑀 and explained briefly in Figure 5. Case-1: 0 M t Case-1: 0 ≤ 𝑀 ≤ 𝑡 From Figure 6, the buyer obtains interest gained in the credit period from 0 to From Figure 6, the buyer obtains interest gained in the credit period from 0 to 𝑀 and M and revenue earned on the sale of items. The interest gained (IE) for the buyer revenue earned on the sale of items. The interest gained (IE) for the buyer is calculated is calculated andand equaequal l 𝐷 (𝑀 − D𝑡 ( M ) 𝐼 𝑆 t/2). The I S in /ter 2. est The paiinter d (IP)est for th paid e buy (IP er )isfor calcul the ated buyer from is the cal- time e g 1 𝑒 𝑔 1 𝐷 (𝑇 −𝑀 ) 𝐼 𝑃 𝑝 𝑐 culated from the time period M to T and due to the unsold items, which is equal to ( )( period 𝑀 to 𝑇 and due to the unsold items, which is equal to + 𝑛 𝑡 − 𝑀 − 2 2 D(T M) I P p c 𝑡 )𝐼 𝑃 . + Pd(n)(t M t ) I P . 1 𝑝 𝑐 2 p c Figure 6. Model presentation under logistic growth function for the case 0 M t . Figure 6. Model presentation under logistic growth function for the case 0 ≤ 𝑀 2≤ 𝑡 . Now, the buyer ’s total proﬁt in this scenario is Now, the buyer’s total profit in this scenario is 𝜉 (𝑃 ) = (𝑆 𝑃 𝑆 (1 − 𝑑 (𝑛 ))+ 𝑆 𝑃 𝑆 𝑑 (𝑛 )+ 𝐷 (𝑀 − 𝑡 ) 𝐼 𝑆 /2 ) 1 𝑔 1 𝑑 1 1 𝑒 𝑔 2 2 2 𝑃 (1 − 𝑑 (𝑛 )) 𝑃 𝑆 𝑑 (𝑛 ) − (𝑃 𝑃 𝑆 + 𝐾 + ℎ( + )+ 𝐼 𝑃 𝑆 + 2 𝑑 𝐸 𝑇 𝑐 0 𝑠 1 1 𝑡 2𝐷 𝑅 (11) − 2 B 1 + 𝑙 𝑒 𝐷 (𝑇 − 𝑀 ) 𝐼 𝑃 𝑝 𝑐 + 𝑓 𝑃 (B t + log( ))+ + (𝑛 )(𝑡 − 𝑀 − 𝑡 )𝐼 𝑃 ) 𝑑 1 𝑀 1 𝑝 𝑐 𝜇 1 + 𝑙 2 Case-2: 𝑡 ≤ 𝑀 ≤ 𝑇 In this case, from Figure 6, the buyer earns interest gained and revenue for the sold 𝐷 (𝑀 −𝑡 ) 𝐼 𝑆 1 𝑒 𝑔 items, which are equal to + 𝑃𝑑 (𝑛 )(𝑀 − 𝑡 − 𝑡 )𝐼 𝑆 and the buyer pays 2 1 𝑒 𝑑 𝐷 (𝑇 −𝑀 ) 𝐼 𝑃 𝑝 𝑐 interest for unsold items, which is equal to . 𝑃𝑑 𝜇𝑡 𝑐𝑖 𝑃𝑑 Axioms 2023, 12, 436 10 of 27 x (P) = S PS (1 d(n)) + S PS d(n) + D( M t ) I S /2 1 g 1 d 1 1 e g 2 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c s t 0 1 1 ci 2D R (11) mt D(T M) I P p c B 1+le + f P B t + log + + Pd(n)(t M t ) I P . d 1 M 1 p c m 2 1+l Case-2: t M T In this case, from Figure 6, the buyer earns interest gained and revenue for the sold D( Mt ) I S e g items, which are equal to + Pd(n)( M t t ) I S and the buyer pays interest 2 1 e d D(T M) I P p c for unsold items, which is equal to . Now, the buyer ’s total proﬁt in this scenario is D( Mt ) I S 1 e g x (P) = S PS (1 d(n)) + S PS d(n) + + Pd(n)( M t t ) I S 2 g 1 d 1 2 1 e d 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T (12) c 0 s 1 1 t ci 2D R mt D(T M) I P p c B 1+le + f P B t + log + d 1 m 2 1+l Case-3: t T M The buyer obtains interest gained from the whole up to the time period T and its value D(T) I S e g is + Pd(n)( M t t ) I S + ( M T)DT I S , and the buyer does not give any 2 1 e e g interest paid, which is equal to zero. Now, the buyer ’s total proﬁt in this scenario is D(T) I S e g x (P) = S PS (1 d(n)) + S PS d(n) + + Pd(n)( M t t ) I S + ( M T)DT I S 3 g 1 d 1 2 1 e d e g 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci 2D R (13) mt B 1+le + f P B t + log m 1+l In this order, we are moving in the direction of the fuzzy concept and also have discussed the fuzzify and defuzzify of the buyer ’s total proﬁt for each case in the forthcom- ing section. 4. Proposed Model under Fuzzy Environment In real-world transactions, considering the unstable environment, it would not be easy to calculate the exact value of the inventory parameters. Therefore, the decision- makers calculate the approximate value of these parameters. The triangular fuzzy number represents the uncertainty. In this model, the triangular fuzzy number is used to fuzzify the buyer ’s total proﬁt for each case, and after that, the signed distance method is applied to defuzzify the buyer ’s total fuzzy proﬁt for each case. From the assumption, we considered that the demand rate is imprecise in nature and taken as a triangular fuzzy number, D = (D D , D, D + D ) and the total proﬁt of each case is defuzziﬁed with the help of the l h signed distance method. The buyer’s total profit under case 1 is reduced into a total fuzzy profit from Equation (11) Axioms 2023, 12, 436 11 of 27 e e x (P) = S PS (1 d(n)) + S PS d(n) + D( M t ) I S /2 1 g 1 d 1 1 e g 2 2 2 2 P (1d (n)) P S d(n) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci e R 2D (14) mt D(T M) I P p c B 1+le + f P B t + log + + Pd(n)(t M t ) I P d 1 M 1 p c m 1+l 2 From Equation (14), we defuzziﬁed the buyer ’s total fuzzy proﬁt under case 1, and we obtain e e x (P) = x (P) = d x (P), 0 1 11 1 e e = S PS (1 d(n)) + S PS d(n) + d D, 0 ( M t ) I S /2 g e g 1 d 1 1 2 2 2 2 P S d(n) P (1d (n)) 1 (15) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci e e R 2d(D,0) e e mt d(D,0)(T M) I P p c B 1+le + f P B t + log + + Pd(n)(t M t ) I P p c d 1 M 1 m 1+l 2 Now, using the deﬁnition of the signed distance method from Equation (2) in Equation (15), we obtain x (P) = d x (P), 0 11 1 4D+D D 2 h l = S PS (1 d(n)) + S PS d(n) + ( M t ) I S /2 g 1 1 1 e g 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c 0 4D+D D s 1 1 t ci h l (16) 4D+D D h l (T M) I P mt p c B 1+le + f P B t + log + d 1 m 2 1+l +Pd(n)(t M t ) I P M 1 p c Now, the buyer ’s total fuzzy proﬁt per cycle for the case 1 using Equation (16) 1 PS (1 d(n)) x (P) = (x (P)) where T = 111 11 4D+D D h l 4D+D D h l 4 4D+D D h l x (P) = S PS (1 d(n)) + S PS d(n) + ( M t ) I S /2 111 g 1 d 1 1 e g PS (1d(n)) 4 2 2 2 P (1d (n)) P S d(n) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci 4D+D D h l (17) 4D+D D h l (T M) I P mt p c B 1+le + f P B t + log + d 1 m 1+l 2 +Pd(n)(t M t ) I P M 1 p c Axioms 2023, 12, 436 12 of 27 The buyer’s total profit under case 2 is reduced into a total fuzzy profit from Equation (12) D( Mt ) I S e g x (P) = S PS (1 d(n)) + S PS d(n) + + Pd(n)( M t t ) I S g e 2 1 d 1 2 1 d 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T (18) c s t 0 1 1 ci e R 2D mt D(T M) I P p c B 1+le + f P B t + log + d 1 m 1+l 2 From Equation (18), we defuzziﬁed the buyer ’s total fuzzy proﬁt under case 2, and we obtain e e x (P) = x (P) = d x (P), 0 2 22 2 e e d D,0 ( Mt ) I S ( ) 1 e g = S PS (1 d(n)) + S PS d(n) + + Pd(n)( M t t ) I S g e 1 d 1 2 1 d 2 2 2 (19) P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c 0 s t 1 1 ci e e R 2d D,0 ( ) e e mt d D,0 (T M) I P ( ) p c B 1+le + f P B t + log + m 1+l 2 Now, using the definition of the signed distance method from Equation (2) in Equation (19), we obtain x (P) = d x (P), 0 22 2 4D+D D h l ( Mt ) I S 1 e g = S PS (1 d(n)) + S PS d(n) + 1 d 1 +Pd(n)( M t t ) I S 2 1 e d (20) 2 2 2 2 P S d n P (1d (n)) ( ) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci 4D+D D h l 4D+D D h l (T M) I P mt p c B 1+le + f P B t + log + d 1 m 1+l 2 Now, the buyer ’s total fuzzy proﬁt per cycle for the case 2 using Equation (20) 1 PS (1 d(n)) x P = x P where T = ( ) ( ( )) 222 22 4D+D D h l 4D+D D 4D+D D h l h l ( Mt ) I S 1 e g 4 4 x (P) = = S PS (1 d(n)) + S PS d(n) + 222 g 1 d 1 PS (1d(n)) +Pd(n)( M t t ) I S 2 e 1 d (21) 2 2 2 P 1d n P S d(n) ( ( )) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t 4D+D D ci h l !! 4D+D D h l (T M) I P mt p c B 1+le + f P B t + log + d 1 m 1+l 2 The buyer’s total profit under case 3 is reduced into a total fuzzy profit from Equation (13) Axioms 2023, 12, 436 13 of 27 T I S ( ) e g e e e x (P) = S PS (1 d(n)) + S PS d(n) + D + Pd(n)( M t t ) I S + ( M T) DT I S 3 g 1 d 1 2 1 e d e d 2 2 2 P (1d (n)) P S d(n) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t ci e R (22) 2D mt B 1+le + f P B t + log d 1 m 1+l From Equation (22), we defuzziﬁed the buyer ’s total fuzzy proﬁt under case 3, and we obtain e e e x (P) = x (P) = d(x (P), 0) 3 33 3 (T) I S e g e e = S PS (1 d(n)) + S PS d(n) + d(D, 0) + Pd(n)( M t t ) I S g e 1 d 1 2 1 d e e +( M Td(D, 0))T I S e g (23) 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c s t 0 1 1 ci e R 2d D,0 ( ) mt B 1+le + f P B t + log d 1 m 1+l Now, using the definition of the signed distance method from Equation (2) in Equation (23), we obtain x (P) = d(x (P), 0) 33 3 (T) I S 4D+D D e g h l = S PS 1 d n + S PS d n + + Pd n ( M t t ) I S ( ( )) ( ) ( ) g 1 1 2 1 e d d 4 2 4D+D D h l +( M T) )T I S e g (24) 2 2 2 P S d(n) P (1d (n)) P PS + K + h + + I PS + 2 d E T c 0 4D+D D s 1 1 t ci h l mt B 1+le + f P B t + log d 1 m 1+l Now, the buyer ’s total fuzzy proﬁt per cycle for case 3 using Equation (24): 1 PS (1 d(n)) x (P) = (x (P)) where T = 333 33 4D+D D T h l 4 Axioms 2023, 12, 436 14 of 27 4D+D D h l (T) I S 4 4D+D D e g h l x (P) = = S PS (1 d(n)) + S PS d(n) + 333 1 d 1 4 2 PS (1d(n)) 4D+D D h l +Pd(n)( M t t ) I S + ( M T) )T I S 2 1 e e g ! (25) 2 2 2 P 1d n P S d(n) ( ( )) P PS + K + h + + I PS + 2 d E T c 0 s 1 1 t 4D+D D ci h l !! mt B 1+le + f P B t + log d 1 m 1+l 5. Solution Method The solution methodology has been taken from Mittal and Sharma [24] for case 1, case 2, and case 3. The decision variable (P) has been calculated with the help of the maxima minima test, and the necessary conditions for ﬁnding the decision variable are d(x(P)) = 0. After that, we calculated the value of the decision variable, and the value of the dP d (x(P)) decision variable is optimal if it satisﬁes the sufﬁcient condition 0 at P, which dP also represents the concavity of the buyer ’s total fuzzy proﬁt with respect to the optimal value of the decision variable, then the value of the decision variable (P) can be termed as optimal and it is represented by (P ). The optimal value of the decision variable maximizes the buyer ’s total fuzzy proﬁt. 5.1. Solution Method for Case 1 From Equation (17), we can write for the optimal value of P, d(x (P)) = 0 (26) dP After solving Equation (26) and the calculation has been given in Appendix A part ((A1)–(A3)), we obtain the value of P: 2 2 4D+D D 4D+D D 2 4D+D D h l h l h l 2 u 2RK ( M t ) I S + M I P e g p c 4 4 4 P = (27) 4D+D D 2 4D+D D 2 h l 2 h l 2 2 2 I d(n) P S + S Rh(1 d(n)) + 2 hS d(n) + R I P S (1 d(n)) p c 1 p c 4 1 4 1 1 The value of P, from Equation (27) and putting in Equation (28), we obtain 2 2 4D+D D 4D+D D 4D+D D 2 h l h l h l 2 2K P I M S I ( M t ) p g e d x P 4 4 4 ( ( )) = + < 0 (28) 2 3 3 3 dP P S (1 d(n)) (1 d(n))P S (1 d(n))P S 1 1 1 Now, we conclude that the value of P from Equation (26) is optimal because it satisﬁes Equation (28), which represents the concavity of the total fuzzy proﬁt for the case 1, and hence, we can write the optimal value of P , 2 2 4D+D D 4D+D D 2 4D+D D h l h l h l 2 u 2RK ( M t ) I S + M I P 1 e g p c 4 4 4 P = (29) 4D+D D 2 4D+D D 2 2 2 2 h l h l 2 I d(n) P S + S Rh(1 d(n)) + 2 hS d(n) + R I P S (1 d(n)) p c p c 4 1 4 1 1 Axioms 2023, 12, 436 15 of 27 5.2. Solution Method for Case 2 From Equation (21), we can write for the optimal value of P, d(x (P)) = 0 (30) dP We solved Equation (30), and the calculation has been given in Appendix A part ((A4)–(A6)); we obtain the value of P 2 2 4D+D D 4D+D D 4D+D D h l h l h l 2 u 2 RK ( M t ) I S R + M I P R 1 e g p c 4 4 4 P = (31) 4D+D D 2 4D+D D 2 h l 2 2 h l 2 2 d(n) I P S + S Rh(1 d(n)) + 2hS d(n) + R I P S (1 d(n)) e c 1 p c 4 4 1 1 1 Now, we ﬁnd the second derivative of Equation (30), and putting the value of P from Equation (31) in Equation (32) for the justiﬁcation of the optimal value of P, we obtain 2 2 4D+D D 4D+D D 4D+D D 2 h l h l h l 2 2 K P I M S I ( M t ) p g e 1 4 4 4 d (x (P)) = + < 0 (32) 2 3 3 3 d p (1 d(n))P S (1 d(n))P S (1 d(n))P S 1 1 1 From Equation (31), P is the optimal value, and it is represented by P , and Equation (32) denotes the concavity of total fuzzy proﬁt. 2 2 4D+D D 4D+D D 2 4D+D D h l h l h l 2 u 2 RK ( M t ) I S R + M I P R e g p c 4 4 4 P = (33) 4D+D D 2 4D+D D 2 h l 2 2 h l 2 2 d(n) I P S + S Rh(1 d(n)) + 2hS d(n) + R I P S (1 d(n)) e c 1 p c 4 1 1 4 1 5.3. Solution Method for Case 3 From Equation (25), we can write for the optimal value of P, d x P ( ( )) = 0 (34) dP We solved Equation (34), and the calculation has been given in Appendix A part ((A7)–(A9)); we obtain the value of P 4D+D D h l 2R K P = t (35) 4D+D D 2 4D+D D 2 h l 2 h l 2 2 2D d n I S S + S Rh 1 d n + 2 hS d n + r I S S 1 d n ( ) ( ( )) ( ) ( ( )) e 1 p g 4 1 4 1 1 Now, we ﬁnd the second derivative of Equation (34), and putting the value of P from Equation (35) in Equation (36) for the justiﬁcation of the optimal value of P, we obtain 4D+D D h l 2 4RK d (x (P)) = < 0. (36) 2 3 d p p From Equation (34), P is the optimal value, and it is represented by P , and Equation (36) denotes the concavity of total fuzzy proﬁt. 4D+D D h l 2R K P = (37) 4D+D D 2 4D+D D 2 h l 2 h l 2 2D d(n) I S S + S Rh(1 d(n)) + 2 hS d(n) + r I S S (1 d(n)) e d 1 1 p g 4 1 4 1 Axioms 2023, 12, 436 16 of 27 5.4. Algorithm Jayaswal et al. [17] have followed the algorithm process: Step 1: Replace all input parameters in Equations (29), (33), and (37) and calculate the value of P , P , and P . 1 2 3 Step 2: After the calculation of P, we ﬁnd out the value of t and T along the Equations (29), (33), and (37) and compared with credit period M in each case. Step 3: If it satisﬁes the condition 0 < M < t along Equation (29), then calculate the total fuzzy proﬁt from Equation (17); otherwise, move to the next step. Step 4: If it satisﬁes the condition t < M < t along Equation (33), then calculate the 2 3 total fuzzy proﬁt from Equation (21); otherwise, move to the next step. Step 5: If it satisﬁes the condition t < T < M along Equation (37), then calculate the total fuzzy proﬁt from Equation (25). Step 6: In this case, we compare which case is better for this proposed model and use it for sensitivity analysis. 5.5. Numerical Example We have adopted the inventory parameters (Sebatjane and Adetunji [13]) given in Table 2. Table 2. Recommended inventory parameters. Numerical Value of Numerical Value of Inventory Parameter Inventory Parameter Inventory Parameter Inventory Parameter e 1,000,000 g/year Holding cost (h) 0.04 ZAR/g/year Fuzzy demand rate D Upper deviation fuzzy 10,000 g/year Ordering cost K 1000 ZAR/cycle ( ) demand rate (D ) Lower deviation fuzzy 5000 g/year Feeding cost ( f ) 0.2 ZAR/g/year demand rate (D ) Weight of each newborn Weight of each newborn growing 57 g 1500 g growing item (S ) item at slaughtering time (S ) 0 1 Purchasing cost (P ) 0.025 ZAR/g Selling price for good items (S ) 0.05 ZAR/g c g Selling price for defective Inspection cost ( I ) 0.00025 ZAR/g 0.02 ZAR/g items (S ) Inspection rate (R) 5,256,000 g/year Asymptotic weight (B) 6870 g Constant of integration (l) 120 Growth rate (m) 40/year (0.11/day) Carbon emissions per Km due Distance covered in one way 0.00077344 500 Km to transport (E ) (Km) (d ) t 1 Emissions tax rate (T ) 30 $ per Ton Learning rate (b) 0.79 ci Learning supporting Learning supporting 40 999 parameter (a) parameter (g) Number of shipments (n) 5 Interest earned ( I ) 0.05 Trade credit period Interest Interest earned I 0.08 0.363 year earned ( M) Several learning curve models were ﬁtted to the collected data, and the S-shaped logistic learning curve was found to ﬁt well, and it is of the form d(n) = where a, g, bn g+e and b > 0 are model parameters, n is the number of shipments. Axioms 2023, 12, 436 17 of 27 First, test the condition to avoid shortages within the inspection period. The behavior of the inventory level is depicted in Figures 6–8 to avoid shortages during the screening time (t = ), the percentage of defective items is restricted to (1 d(n))Y Dt =) Dd(n) 1 1 2 where Y = PS , t = and d(n) = ; otherwise, shortages will occur. We are 1 1 bn g+e discussing some contributions, such as Sebatjane and Adetunji [13], who calculated the optimal newborn items and the expected proﬁt for the defective items under the Logistic growth function, which are 148 units and USD 34,641.73, respectively. There was no fuzzy Axioms 2023, 12, x FOR PEER REVIEW 18 of 29 learning theory in this model, but the inclusion of fuzzy learning theory has a positive effect. The maximum total fuzzy proﬁt depends on the credit period, which has been discussed in the mathematical formulation. The effect of learning, credit policy, and fuzzy 𝐷 𝑌 𝑑 concept (𝑛 )≤ 1 − had ( a) positive where 𝑌 ef= fect 𝑃 𝑆 on , 𝑡 the= number and 𝑑 of(𝑛 newborn )= items ; and other the wise buyer , sho ’s rta total ges fuzzy will 1 1 2 𝑅 𝑅 𝑔 +𝑒 proﬁt. If, in this study, we had excluded the learning, credit policy, or fuzzy environment, occur. We are discussing some contributions, such as Sebatjane and Adetunji [13], who then the optimal inventory of the number of newborn items would be P = 1890 and the calculated the optimal newborn items and the expected profit for the defective items buyer ’s total proﬁt is x (P ) = $1.03421 10 . From step 6 of the algorithm, we obtain under the Logistic growth function, which are 148 units and USD 34,641.73, respectively. maximum fuzzy proﬁt in case 3, and we have selected case 3 for the sensitivity analysis. There was no fuzzy learning theory in this model, but the inclusion of fuzzy learning The optimal number of newborn items per cycle is calculated in three different cases, and theory has a positive effect. The maximum total fuzzy profit depends on the credit period , the maximum total fuzzy proﬁt with the optimal number of newborn is discussed below in which has been discussed in the mathematical formulation. The effect of learning, credit Table 3. In Figure 7, the model presentation under the logistic growth function for the case policy, and fuzzy concept had a positive effect on the number of newborn items and the t M T is showcased below, and also underneath below Figure 8, describes the model buyer’s total fuzzy profit. If , in this study, we had excluded the learning, credit policy, or presentation under logistic growth function for the case t T M. fuzzy environment, then the optimal inventory of the number of newborn items would be ∗ ∗ ∗ 8 𝑃 = 1890 and the buyer’s total profit is 𝜉 (𝑃 )= $1.03421× 10 . From step 6 of the Table 3. Optimal number of newborn items under different cases. algorithm, we obtain maximum fuzzy profit in case 3 , and we have selected case 3 for the sensitivity analysis. The optimal number of newborn items per cycle is calculated in three Cases M (Year), t (year) and T (Year) Optimal Number of New Born Items Buyer’s Total Fuzzy Proﬁt different cases, and the maximum total fuzzy profit with the optimal number of newborn Case 1, 0 < M < t , 0 < 0.03 < 0.291 P = 1023 x P = $1.05725 10 1 111 1 is discussed below in Table 3. In Figure 7, the model presentation under the logistic growth Case 2, t < M < t , 0.328 < 0.342 < 0.35 P = 1150 x P = $1.07725 10 2 3 2 222 2 function for the case 𝑡 ≤ 𝑀 ≤ 𝑇 is showcased below, and also underneath below Figure Case 3, T < M, 0.007 < 0.363 P = 1214 x P = $1.08725 10 8, describes the model presentation under logistic growth function for the case 𝑡 ≤ 𝑇 ≤ 3 333 3 𝑀 . Figure 7. Model presentation under logistic growth function for the case t M T. Figure 7. Model presentation under logistic growth function for the case 𝑡 ≤ 𝑀 ≤ 𝑇 . 𝑏𝑛 Axioms 2023, 12, x FOR PEER REVIEW 19 of 29 Axioms 2023, 12, 436 18 of 27 Figure 8. Model presentation under logistic growth function for the case t T M. Figure 8. Model presentation under logistic growth function for the case 𝑡 ≤ 𝑇 ≤ 𝑀 . 6. Sensitivity Analysis Table 3. Optimal number of newborn items under different cases . In this section, we discussed the impact of the number of shipments, the impact of learning, credit period, and lower and upper fuzzy deviation of the demand rate on the total fuzzy proﬁt and presented from Tables 4–14 and the graphical representation of trade Cases Optimal Number of New Buyer’s Total Fuzzy Profit credit period, interest gained as learning rate has been shown in Figures 9–12, which are ( ) ( ) 𝑴 𝐲𝐞𝐚𝐫 ,𝒕 (𝐲𝐞𝐚𝐫 ) and 𝑻 𝐲𝐞𝐚𝐫 Born Items the given below: ∗ ∗ 8 𝑃 = 1023 Case 1,0 < 𝑀 < 𝑡 , 0 < 0.03 < 0.291 𝜉 (𝑃 )= $1.05725× 10 2 1 111 1 Table 4. Impact of shipment ∗ on the fuzzy total proﬁt. ∗ ∗ 8 Case 2,𝑡 < 𝑀 < 𝑡 ,0.328 < 0.342< 0.35 𝑃 = 1150 𝜉 (𝑃 )= $1.07725× 10 2 3 2 222 2 ∗ ∗ ∗ 8 Number of Shipments (n) Number of New Born Items (P ) Fuzzy Total Proﬁt x (P ) Case 3,𝑇 < 𝑀 ,0.007 < 0.363 𝑃 = 1214 𝜉 (𝑃 )= $1.08725× 10 3 333 3 3 333 3 1 1193 $1.08710 10 2 1194 $1.08712 10 6. Sensitivity Analysis 3 1197 $1.08716 10 In this section, we discussed the impact of the number of shipments, the impact of 4 1203 $1.08721 10 learning, credit period, and lower and upper fuzzy deviation of the demand rate on the 5 1214 $1.08725 10 total fuzzy profit and p resented from Tables 4–14 and the graphical representation of trade credit period, interest gained as learning rate has been shown in Figures 9–12, which are Table 5. Impact of credit period on the fuzzy total proﬁt. the given below: Credit Period (M) Year Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.1 1211 $1.07475 10 Table 4. Impact of shipment on the fuzzy total profit. 0.2 1213 $1.081 10 ∗ ∗ ∗ 0.3 1214 $1.08725 10 ( ) Number of Shipments (𝒏 ) Number of New Born Items (𝑷 ) Fuzzy Total Profit 𝝃 𝑷 𝟑 𝟑𝟑𝟑 𝟑 1 1193 $1.08710× 10 2 1194 $1.08712× 10 3 1197 $1.08716× 10 4 1203 $1.08721× 10 5 1214 $1.08725× 10 Table 5. Impact of credit period on the fuzzy total profit. Number of New Born Items ∗ ∗ ( ) ( ) Credit Period 𝑴 𝐲𝐞𝐚𝐫 Total Fuzzy Profit 𝝃 𝑷 𝟑𝟑𝟑 𝟑 (𝑷 ) 0.1 1211 $1.07475× 10 0.2 1213 $1.081× 10 0.3 1214 $1.08725× 10 Axioms 2023, 12, 436 19 of 27 Table 6. Impact of learning rate on the total fuzzy proﬁt. Learning Rate (b) Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.3932 1195 $1.08715 10 0.4932 1198 $1.08719 10 0.5932 1201 $1.08721 10 0.6932 1213 $1.08724 10 0.7932 1214 $1.08725 10 0.8932 1214 $1.08725 10 0.9932 1214 $1.08725 10 Table 7. Impact of lower and upper fuzzy deviation of fuzzy demand rate on the total fuzzy proﬁt. Upper Deviation of Lower Deviation Number of Total Fuzzy Proﬁt Fuzzy Demand Rate (D) Demand Rate (D ) of Demand Rate (D ) New Born Items (P ) x (P ) h l 3 333 3 4000 g/year 1,000,000 g/year 2000 g/year 1141 $4.34679 10 6000 g/year 1,000,000 g/year 3000 g/year 1203 $6.52202 10 8000 g/year 1,000,000 g/year 4000 g/year 1210 $8.69729 10 10,000 g/year 1,000,000 g/year 5000 g/year 1214 $1.08725 10 Table 8. Impact of feeding cost on the total fuzzy proﬁt. Feeding Cost (f ) ZAR/g/Year Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) d 3 333 3 0.2 1214 $1.08725 10 0.3 1214 $1.01635 10 0.4 1214 $ Table 9. Impact of holding cost on the total fuzzy proﬁt. Holding Cost (h) ZAR/g/Year Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.04 1214 $1.08725 10 0.06 991 $1.08084 10 0.08 858 $1.07544 10 Table 10. Impact of inspection cost on the total fuzzy proﬁt. Inspection Cost (I ) g/Year Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.00025 1214 $1.08725 10 0.00035 1214 $1.08465 10 0.00045 1214 $1.08205 10 Table 11. Impact of purchasing cost on the total fuzzy proﬁt. Purchasing Cost (P ) ZAR/g Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.025 1214 $1.08725 10 0.035 1214 $1.07737 10 0.045 1214 $1.06749 10 Axioms 2023, 12, 436 20 of 27 Table 12. Impact of selling cost on the total fuzzy proﬁt. Selling Price (S ) for Good Items Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) 3 333 3 0.05 1214 $1.08725 10 Axioms 2023, 12, x FOR PEER REVIEW 21 of 29 0.06 1214 $1.341 10 0.07 1214 $1.59474 10 Table 12. Impact of selling cost on the total fuzzy profit. Number of New Born Total Fuzzy Profit Selling Price (𝑺 ) for Good Items ∗ ∗ ∗ Items (𝑷 ) 𝝃 (𝑷 ) 𝟑 𝟑𝟑𝟑 𝟑 Table 13. Impact of emissions tax rate (T ) on the total fuzzy proﬁt. ci 0.05 1214 $1.08725× 10 0.06 1214 $1.341× 10 Emissions Tax Rate (T ) Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) ci 3 333 3 0.07 1214 $1.59474× 10 30 $ 1214 $1.08725 10 Table 13. Impact of emissions tax rate (𝑇 ) on the total fuzzy prot fi . 35 $ 1214 $1.08719 10 Number of New Born Total Fuzzy Profit 40 $ 1214 $1.08712 10 Emissions Tax Rate (𝑻 ) ∗ ∗ ∗ Items (𝑷 ) 𝝃 (𝑷 ) 𝟑 𝟑𝟑𝟑 𝟑 30 $ 1214 $1.08725× 10 35 $ 1214 $1.08719× 10 Table 14. Impact of travel distance (one-way) on the total fuzzy proﬁt. 40 $ 1214 $1.08712× 10 Emissions Tax Rate (T ) Number of New Born Items (P ) Total Fuzzy Proﬁt x (P ) Table 14. Impact of travel distance (one-way) on the total fuzzy profit. ci 3 3 500 Km 1214 $1.08725 10 Number of New Born Total Fuzzy Profit Emissions Tax Rate (𝑻 ) ∗ ∗ ( ) Items (𝑷 ) 𝝃 𝑷 𝟑 𝟑𝟑𝟑 𝟑 8 600 Km 1214 $1.08642 10 500 Km 1214 $1.08725× 10 700 Km 1214 $1.08511 10 600 Km 1214 $1.08642× 10 700 Km 1214 $1.08511× 10 Axioms 2023, 12, x FOR PEER REVIEW 22 of 29 Figure 9. Effect of the trade credit period on the total fuzzy profit . Figure 9. Effect of the trade credit period on the total fuzzy proﬁt. Figure 10. Effect of the interest gained on the total fuzzy prot fi . Figure 10. Effect of the interest gained on the total fuzzy proﬁt. Figure 11. Effect of the learning rate on the number of newborn items. Figure 12. Effect of the learning rate on the total fuzzy profit . 7. Observation and Managerial Insights 1. From Table 4, we can conclude that if the number of shipments increases, then the number of newborn items and the buyer’s total fuzzy profit increase as well because the percentage of defective items decreases as the number of shipments increases 𝒄𝒊 𝒄𝒊 𝑐𝑖 Axioms 2023, 12, x FOR PEER REVIEW 22 of 29 Axioms 2023, 12, x FOR PEER REVIEW 22 of 29 Axioms 2023, 12, 436 21 of 27 Figure 10. Effect of the interest gained on the total fuzzy prot fi . Figure 10. Effect of the interest gained on the total fuzzy prot fi . Figure 11. Effect of the learning rate on the number of newborn items. Figure 11. Effect of the learning rate on the number of newborn items. Figure 11. Effect of the learning rate on the number of newborn items. Figure 12. Effect of the learning rate on the total fuzzy proﬁt. Figure 12. Effect of the learning rate on the total fuzzy profit . Figure 12. Effect of the learning rate on the total fuzzy profit . 7. Observation and Managerial Insights 7. Observation and Managerial Insights 7. Observation and Managerial Insights 1. From Table 4, we can conclude that if the number of shipments increases, then 1. From Table 4, we can conclude that if the number of shipments increases, then the 1. From Table 4, we can conclude that if the number of shipments increases, then the the number of newborn items and the buyer ’s total fuzzy proﬁt increase as well number of newborn items and the buyer’s total fuzzy profit increase as well because number of newborn items and the buyer’s total fuzzy profit increase as well because because the percentage of defective items decreases as the number of shipments the percentage of defective items decreases as the number of shipments increases the percentage of defective items decreases as the number of shipments increases increases from the mathematical relationship between the number of shipments and the defective percentage. 2. From Table 5, it can be observed that when the credit period increases, the number of newborn items and the buyer ’s total fuzzy proﬁt increase as well. The newborn items and buyer ’s total proﬁt increase due to the presence of the trade credit policy because the buyer obtains more credit period for the selling of items, and its revenue generates more due to interest earned and the selling of items when trade credit period is less than or equal to cycle length. On the other hand, the seller obtains more proﬁt due to interest paid when the buyer does not return borrowed items on or before the ﬁxed credit period. Finally, in this model, the credit policy positively affects the order quantity and the buyer ’s total fuzzy proﬁt. The trade-credit policy can be risky for the seller when the ﬁnancing period is very large. The graphical impact of the trade credit period has been shown in Figures 9 and 10. 3. From Table 6, it can be analyzed that as the learning rate increases with keeping other model parameter constant, the number of newborn items and the buyer ’s total fuzzy proﬁt increase because when the learning rate increases, the percentage of defective items decreases per shipment, but those items cannot be removed from the lot. Initially, the order quantity increases when the learning rate increases, but only for some time, and for some values of the learning rate, the order quantity becomes constant. This speciﬁc value of the learning rate is more important for the ﬁshery Axioms 2023, 12, 436 22 of 27 industry and the order quantity. Learning theory suggests that the buyer obtains more proﬁt on less order quantity. The impact of the learning rate has been shown in Figures 11 and 12. 4. From Table 7, if the lower and upper deviation of the fuzzy demand rate increase, the number of newborn items and the buyer ’s total fuzzy proﬁt increase because the selling of items increases and generates more revenue. The interest earned and interest paid vary due to the lower and upper deviation of the demand rate. The decision-makers can manage the value of the lower and upper deviation of the demand rate for the ﬁshery industry. The fuzzy learning theory is more beneﬁcial for the ﬁshery industry. 5. From Table 8, if the feeding cost increases, the number of newborn items is constant while the buyer ’s total fuzzy proﬁt decreases due to the addition of the feeding cost to the revenue cost. 6. From Table 9, if the holding cost increases, the number of newborn items is constant while the buyer ’s total fuzzy proﬁt decreases because the cost function increases due to the increase in the holding cost. 7. From Table 10, the inspection of the lot must be performed when the lot has some defective items. If the inspection cost ( I ) increases, the number of newborn item is constant, whereas the buyer ’s total proﬁt decreases because the cost function increases. The decision-makers can manage the inspection cost according to the model. 8. From Table 11, we see that if the purchasing cost increases, the number of new- born item is constant, and the buyer ’s total fuzzy proﬁt decreases because the cost function increases. 9. From Table 12, when the buyer increases the selling price of good quality items, the number of newborn items is constant and increases the total fuzzy proﬁt because the selling price of good items is more than the purchasing price and generates more revenue due to more sales. 10. From Table 13, when the carbon emission tax rate increases, the number of newborn items is constant, but the buyer ’s total fuzzy proﬁt decreases due to the addition of carbon taxation in the cost function. 11. From Table 14, if the traveling distance increases, then the number of the newborn items is constant, but the buyer ’s total fuzzy proﬁt decreases because the emission cost increases due to the increase in distance. 8. Conclusions The present paper describes a supply chain model with carbon emissions and learning effect for the growing items (ﬁshes) under a fuzzy environment and trade credit policy where the demand rate is taken as a triangular fuzzy number and defective items follow the S-shaped learning curve. The companies or ﬁrms need to optimize the inventory of newborn growing items (ﬁshes, chicks, feeds, and extra). In this paper, we have optimized the buyer ’s total fuzzy proﬁt with respect to the number of newborn growing items (ﬁshes), and the players (seller and buyer) of the supply chain obtained more proﬁt under some realistic situations, such as the inspection of the lot, trade-credit policy, consideration of fuzzy demand rate, and the effect of the learning. We observed and analyzed some results from the behavior of the inventory parameters and brieﬂy discussed them in the observation and managerial insights section. The inventory cost affects the total fuzzy proﬁt of the supply chain’s members. The buyer obtained more proﬁt in case 3 because the buyer does not pay extra interest charges and only earns gained interest as well as no pressure for selling the growing items while both players take beneﬁts during the dealing of that contract under trade credit policy. From case 1 and case 2, we concluded that the buyer faces some difﬁculties in selling the items because he has to pay for the unsold items as per the contracted. From the sensitivity analysis of the inventory parameters, we conclude that the trade credit policy and learning fuzzy theory are more effective tools in this model, and these parameters directly affect the size of the newborn items and the buyer ’s total Axioms 2023, 12, 436 23 of 27 proﬁt. The seller should be aware of the trade credit when it plans to use the policy of trade credit, and without a plan, it can be riskier for the seller. The trade-credit policy is more effective when the demand for newborn items is constant, whereas, in this model, the demand for newborn items is uncertain. The buyer ’s total fuzzy proﬁt is defuzziﬁed with the help of the signed distance method for the nulliﬁcation of the uncertainty of the demand for newborn items. The present model suggests to the decision-makers that, at the time of ordering, one should be aware of the trade credit time, rate of learning, and deviation of lower and upper demand of newborn items. The learning rate suggests a minimum order quantity on which the buyer obtains more proﬁt, and this model is developed especially for the ﬁshery industry. We obtained some decision-making results from the sensitivity analysis of the learning rate when the learning rate is 0.79, and the order quantity and the buyer ’s total fuzzy proﬁt become saturated. The application of this research work is very important wherein a ﬁrm or a company purchases newborn ﬁsh, chicks, feeds, etc., after they are sold in another market when newborns have gained a certain weight. Companies or ﬁrms need to optimize the inventory of newborn growing items. The limitation of the paper is that the demand rate should be imprecise in nature, and the upper and lower deviation of the demand rate should be in the proper range. The rate of learning, shipments, and trade credit period should be checked before the strategy of mathematical modeling because these are changeable under some realistic situations. The proposed model can be improved with the contribution of Sarkar et al. [40], Bachar et al. [41], and Mondal et al. [42] for a closed-loop supply for greening defective items under some realistic situations. We obtained fruitful suggestions from the observation, and when the seller wishes to start a business for growing items, especially for newborn ﬁshes, the seller must use the trade credit scheme to increase the selling of newborn items (ﬁshes) in addition to new clients. The fuzzy learning theory is more effective when the demand for newborn items is uncertain. In the case of the ﬁshery industry, the demand for newborn items is not constant, and sometimes it varies, and in this case, the buyer should use the concept of learning a fuzzy environment. When a seller and buyer connect through a trade credit scheme and generate a supply chain for a long time of the demand for newborn items is uncertain and the carbon units emit during transportation. The contribution of this scenario is more useful because the uncertainty of the demand for newborn items can be nulliﬁed with the help of fuzzy theory, and learning suggests how much order quantity should be for newborn items during the transaction of business. Minimum carbon emission cost reduces the total cost of the supply chain, which results in higher proﬁt for the supply chain partners. The contribution of this paper covers the area of the topic of reputed journals, and it will be very helpful for the ﬁshery industry. Funding: There is no funding support. Data Availability Statement: As per demand, data will be available. Conﬂicts of Interest: The author declares that there is no conﬂict of interest. Notations The description of notations is given below. S Weight for newborn items (g) S Weight for newborn items (g) at any time t. D The rate of demand for items (g per year). D The rate of fuzzy demand for items (g per year). D Upper triangular fuzzy number (g per year). D Lower triangular fuzzy number (g per year). Y Whole weight (g) for inventory at time t. m The rate of exponential growth in per unit time t B The weight (g) in the asymptotic position when growth increases exponentially l The constant of integration for exponential growth function Axioms 2023, 12, 436 24 of 27 Percentage of defective slaughtered items in a whole lot, which follows the learning d(n) curve. Weight (g) at supremum during the ﬁrst growth region for split linear growth function. Weight (g) at supremum during the duration in the growth region for the growth function. P The number of newborn items demanded/cycle (decision variable) P Unit purchasing cost (ZAR per g) s Unit selling cost for goods item (ZAR per g) s Unit selling cost for defective item (ZAR per g) h Unit holding cost for each item (ZAR per g) f Unit feeding cost for the item (ZAR per g) K Ordering cost per growing cycle (ZAR per g per year) I Screening cost for each item (ZAR per g) R Screening rate for each item (g/min) E Carbon emissions per Km due to transport d Distance covered in one way (Km) T Emissions tax rate ($ per Ton) ci IHC Inventory carrying cost (in $) PC Purchasing cost (in $) OC Ordering cost (in $) FC Feeding cost (in $) CEC Carbon emission cost (in $) T R Whole revenue (in $) TC Total Inventory cost for the buyer (in $) M Trade credit period (years). I Interest gained I Interest charged T Cycle length (years) t Growing time period (years) t Time for inspection (years) t Selling time period for items (years) x (P) The buyer ’s total proﬁt without credit policy x (P) The buyer ’s total proﬁt under credit policy for case 1 (0 < M < t ) 1 2 x (P) The buyer ’s total proﬁt under credit policy for case 2 (t < M < t ) 2 2 3 x (P) The buyer ’s total proﬁt under credit policy for case 3 (T M) x (P) The buyer ’s total fuzziﬁed proﬁt under credit policy for case 1 (0 < M < t ) x (P) The buyer ’s total fuzziﬁed proﬁt under credit policy for case 2 (t < M < t ) 22 2 3 x (P) The buyer ’s total fuzziﬁed proﬁt under credit policy for case 3 (T M) x (P) The buyer ’s total defuzziﬁed proﬁt under credit policy for case 1 (0 < M < t ) 111 2 x (P) The buyer ’s total defuzziﬁed proﬁt under credit policy for case 2 (t < M < t ) 222 2 3 x (P) The buyer ’s total defuzziﬁed proﬁt under credit policy for case 3 (T M) Appendix A For case 1: d(x (P)) = 0 dP 4D+D D 4D+D D h l h l K S d(n) PS h(1d(n)) 4 1 4 + h P S 1d n 2 R(1d(n)) ( ( )) (A1) 2 2 4D+D D 4D+D D 4D+D D 2 2 h l h l h l ( Mt ) I P d(n) I P ( M) I P 1 e c p p I PS h(1d(n)) 4 4 p 4 + = 0. 2 2 2P S h(1d(n)) R(1d(n)) 2P S (1d(n)) 1 1 After solving the Equation (A1), we obtain the value of P Axioms 2023, 12, 436 25 of 27 2 2 4D+D D 4D+D D 4D+D D h l h l h l 2 u 2RK ( M t ) I S + M I P 1 e g p c 4 4 4 P = (A2) 4D+D D 2 4D+D D 2 h l 2 h l 2 2 2 I d(n) P S + S Rh(1 d(n)) + 2 hS d(n) + R I P S (1 d(n)) p c 1 p c 4 1 4 1 1 The value of P, from the Equation (A2) and putting in the Equation (A3), we obtain 2 2 4D+D D 4D+D D 4D+D D 2 h l h l 2 h l 2 2K P I M S I ( M t ) p g e 1 d (x (P)) 4 4 4 = + < 0 (A3) 2 3 3 3 dP P S (1 d(n)) (1 d(n))P S (1 d(n))P S 1 1 1 For case 2: d(x (P)) = 0 dP 4D+D D 4D+D D 4D+D D h h l h l h l h S d(n) d(n) I S 1 p d d x P K S 1d n h 4 4 ( ( )) ( ( )) 222 1 = + dP 2 P S (1d(n)) R(1d(n)) R(1d(n)) (A4) 2 2 4D+D D 4D+D D 2 2 h l h l I P ( M) I ( Mt ) S p e g S I 1d n 1 ( ( )) 4 4 1 p + = 0. 2 2 2(1d(n))P S 2(1d(n))P S 1 1 We solved the Equation (A4), and we obtain the value of P 2 2 4D+D D 4D+D D 2 4D+D D h l h l h l 2 u 2 RK ( M t ) I S R + M I P R e g p c 4 4 4 P = (A5) 4D+D D 2 4D+D D 2 2 2 2 h l h l 2 d(n) I P S + S hR(1 d(n)) + 2hS d(n) + R I P S (1 d(n)) e c p c 4 1 1 4 1 Now, we ﬁnd the second derivative of Equation (A4), and putting the value of P from Equation (A5) in Equation (A6) for the justiﬁcation of the optimal value of P, we obtain 2 2 4D+D D 4D+D D 4D+D D 2 h l h l h l 2 2 K P I M S I ( M t ) p g e d (x (P)) 4 4 4 = + < 0 (A6) 2 3 3 3 d p (1 d(n))P S (1 d(n))P S (1 d(n))P S 1 1 1 From Equation (A5), P is the optimal value, and it is represented by P , and the Equation (A6) denotes the concavity of total fuzzy proﬁt. 2 2 4D+D D 4D+D D 2 4D+D D h l h l h l 2 RK ( M t ) I S R + M I P R 1 e g p c 4 4 4 P = 4D+D D 4D+D D 2 2 h l 2 2 h l 2 2 d(n) I P S + S hR(1 d(n)) + 2hS d(n) + R I P S (1 d(n)) e c 1 p c 4 1 1 4 1 For case 3: d(x (P)) = 0 which g dP 4D+D D h l h 4D+D D 4D+D D K h l h l d(n) I S d(x (P)) 4 S h(1d(n)) p 333 1 d 4 4 = + d(n) hS 2 1 dP P S (1d(n)) r(1d(n)) R(1d(n)) (A7) S S I (1d(n)) 1 g e = 0. We solved the Equation (A7), and we obtained the value of P 4D+D D h l 2R K t 4 P = (A8) 4D+D D 2 4D+D D 2 2 2 h l h l 2D d(n) I S S + S Rh(1 d(n)) + 2 hS d(n) + r I S S (1 d(n)) e p g d 1 1 4 1 4 1 Axioms 2023, 12, 436 26 of 27 Now, we ﬁnd the second derivative of Equation (A7), and putting the value of P from Equation (A8) in Equation (A9) for the justiﬁcation of the optimal value of P, we obtain 4D+D D h l 2 4RK d (x (P)) = < 0. (A9) 2 3 d p p From Equation (A7), P is the optimal value, and it is represented by P , and Equation (A9) denotes the concavity of total fuzzy proﬁt. 4D+D D h l 2R K t 4 P = (A10) 4D+D D 2 4D+D D 2 2 2 h l h l 2D d(n) I S S + S Rh(1 d(n)) + 2 hS d(n) + r I S S (1 d(n)) e p g d 1 1 4 1 4 1 References 1. Harris, F.W. How many parts to make at once. Factory Mag. Manag. 1913, 10, 135–136. [CrossRef] 2. Aggarwal, S.P.; Jaggi, C.K. Ordering policies of deteriorating items under allowable deferment in payments. J. Oper. Res. Soc. 1995, 46, 658–662. [CrossRef] 3. Chu, P.; Chung, K.J.; Lan, S.P. Economic order quantity of deteriorating items under allowable deferment in payments. Comput. Oper. Res. 1998, 25, 817–824. [CrossRef] 4. Abad, P.L.; Jaggi, C.K. A joint approach for setting unit price and the length of the credit period for a supplier when end demand is price sensitive. Int. J. Prod. Econ. 2003, 83, 115–122. [CrossRef] 5. Chung, K.J.; Liao, J.J. The optimum ordering policy in a DCF analysis for deteriorating items when trade credit depends on the order quantity. Int. J. Prod Econ. 2006, 100, 116–130. [CrossRef] 6. Salameh, M.K.; Jaber, M.Y. Economic production quantity model for items with imperfect quality. Int. J. Prod. Econ. 2000, 64, 59–64. [CrossRef] 7. Chung, K.J.; Her, C.C.; Lin, S.D. A two warehouse inventory model with imperfect quality production process. Comput. Ind. Eng. 2009, 56, 193–197. [CrossRef] 8. Huang, C.K. An integrated vendor–buyer cooperative inventory model for items with imperfect quality. Prod. Planning Control 2002, 13, 355–361. [CrossRef] 9. Jaber, M.Y.; Goyal, S.K.; Imran, M. Economic production quantity model for items with imperfect quality subject to learning effects. Int. J. Prod. Econ 2008, 115, 143–150. [CrossRef] 10. Jaggi, C.K.; Khanna, A.; Mittal, M. Credit ﬁnancing for deteriorating imperfect-quality items under inﬂationary conditions. Int. J. Serv. Oper. Inf. 2011, 6, 292–309. [CrossRef] 11. Zhang, L.; Li, Y.; Tian, X.; Feng, C. Inventory management research for growing items with carbon constrained. In Proceedings of the 2016 35th Chinese Control Conference (CCC), Chengdu, China, 27–29 July 2016; pp. 9588–9593. 12. Tiwari, S.; Daryanto, Y.; Wee, H.M. Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission. J. Clean Pro. 2018, 192, 281–292. [CrossRef] 13. Sebatjane, M.; Adetunji, O. Economic order quantity model for growing items with imperfect quality. Oper. Res. Perspect. 2019, 6, 100088. [CrossRef] 14. De-la-Cruz-Marquez, C.G.; Cardenas-Barron, L.E.; Mandal, B. An inventory model for growing items with imperfect quality when the demand is price sensitive under carbon emissions and shortages. Math. Prob. Eng. 2021, 2021, 1–23. [CrossRef] 15. Chang, H.C. An application of fuzzy sets to the EOQ model with imperfect quality items. Comput. Oper. Res. 2004, 31, 2079–2092. [CrossRef] 16. Rani, S.; Ali, R.; Agarwal, A. Fuzzy inventory model for new and refurbished deteriorating items with cannibalization in green supply chain. Int. J. Syst. Sci. Oper. Logist. 2022, 9, 22–38. 17. Wright, T.P. Factors affecting the cost of airplanes. J. Aero. Sci. 1936, 3, 122–128. [CrossRef] 18. Wee, H.M.; Yu Chen, M.C. Optimum inventory model for items with imperfect quality and shortage backordering. Omega 2007, 35, 7–11. [CrossRef] 19. Jayaswal, M.K.; Mittal, M.; Sangal, I. Ordering policies for deteriorating imperfect quality items with trade-credit ﬁnancing under learning effect. Int. J. Syst. Assur. Eng. Manag. 2021, 12, 112–125. [CrossRef] 20. Jayaswal, M.; Sangal, I.; Mittal, M.; Malik, S. Effects of learning on retailer ordering policy for imperfect quality items with trade credit ﬁnancing. Uncertain Supp. Chain Manag. 2019, 7, 49–62. [CrossRef] 21. Alamri, O.A.; Jayaswal, M.K.; Khan, F.A.; Mittal, M. An EOQ Model with Carbon Emissions and Inﬂation for Deteriorating Imperfect Quality Items under Learning Effect. Sustainability 2022, 14, 1365. [CrossRef] 22. Jayaswal, M.K.; Mittal, M.; Alamri, O.A.; Khan, F.A. Learning EOQ Model with Trade-Credit Financing Policy for Imperfect Quality Items under Cloudy Fuzzy Environment. Mathematics 2022, 10, 246. [CrossRef] 23. Rezaei, J. Economic order quantity for growing items. Int. J. Prod. Econ. 2014, 155, 109–113. [CrossRef] Axioms 2023, 12, 436 27 of 27 24. Mittal, M.; Sharma, M. Economic ordering policies for growing items (poultry) with trade-credit ﬁnancing. Int. J. Appl. Comput. Math. 2021, 7, 39. [CrossRef] 25. Pattnaik, S. Linearly Deteriorating EOQ Model for Imperfect Items with Price Dependent Demand Under Different Fuzzy Environments. Turk. J. Comput. Math Educ. 2021, 12, 5328–5349. 26. Rajeswari, S.; Sugapriya, C.; Nagarajan, D.; Kavikumar, J. Optimization in fuzzy economic order quantity model involving pentagonal fuzzy parameter. Inte. J. Fuz. Sys. 2022, 24, 44–56. [CrossRef] 27. Mahapatra, A.S.; Mahapatra, M.S.; Sarkar, B.; Majumder, S.K. Beneﬁt of preservation technology with promotion and time- dependent deterioration under fuzzy learning. Expert Syst. Appl. 2022, 201, 117169. [CrossRef] 28. Taheri, J.; Mirzazadeh, A. Optimization of inventory system with defects, rework failure and two types of errors under crisp and fuzzy approach. J. Ind. Manag. Optim. 2022, 18, 2289–2318. [CrossRef] 29. Dinagar, D.S.; Manvizhi, M. Single-Stage Fuzzy Economic Inventory Models with Backorders and Rework Process for Imperfect Items. J. Algebraic Stat. 2022, 13, 3008–3020. 30. Garg, H.; Sugapriya, C.; Kuppulakshmi, V.; Nagarajan, D. Optimization of fuzzy inventory lot-size with scrap and defective items under inspection policy. Sot. Comput. 2023, 27, 2231–2250. [CrossRef] 31. Kuppulakshmi, V.; Sugapriya, C.; Kavikumar, J.; Nagarajan, D. Fuzzy Inventory Model for Imperfect Items with Price Discount and Penalty Maintenance Cost. Math. Prob. Engin. 2023, 2023, 1–15. [CrossRef] 32. Jayaswal, M.K.; Mittal, M. Impact of Learning on the Inventory Model of Deteriorating Imperfect Quality Items with Inﬂation and Credit Financing under Fuzzy Environment. Int. J. Fuzzy Syst. Appl. 2022, 11, 1–36. [CrossRef] 33. Chung, K.J.; Huang, Y.F. Retailer ’s optimal cycle times in the EOQ model with imperfect quality and a permissible credit period. Qual. Quant. 2006, 40, 59–77. [CrossRef] 34. Sulak, H. An EOQ model with defective items and shortages in fuzzy sets environment. Int. J. Soc. Sci. Educ. Res. 2015, 2, 915–929. [CrossRef] 35. Shekarian, E.; Olugu, E.U.; Abdul-Rashid, S.H.; Kazemi, N. An economic order quantity model considering different holding costs for imperfect quality items subject to fuzziness and learning. J. Intell. Fuz. Syst. 2016, 30, 2985–2997. [CrossRef] 36. Kazemi, N.; Abdul-Rashid, S.H.; Ghazilla RA, R.; Shekarian, E.; Zanoni, S. Economic order quantity models for items with imperfect quality and emission considerations. Int. J. Syst. Sci. Oper. Logist. 2018, 5, 99–115. [CrossRef] 37. Alsaedi, B.S.; Alamri, O.A.; Jayaswal, M.K.; Mittal, M. A Sustainable Green Supply Chain Model with Carbon Emissions for Defective Items under Learning in a Fuzzy Environment. Mathematics 2023, 11, 301. [CrossRef] 38. Jaggi, C.K.; Goel, S.K.; Mittal, M. Credit ﬁnancing in economic ordering policies for defective items with allowable shortages. Appl. Math. Compt. 2013, 219, 5268–5282. [CrossRef] 39. Gurtu, A.; Jaber, M.Y.; Searcy, C. Impact of fuel price and emissions on inventory policies. Appl. Math. Model. 2015, 39, 1202–1216. [CrossRef] 40. Sarkar, B.; Ganguly, B.; Pareek, S.; Cárdenas-Barrón, L.E. A three-echelon green supply chain management for biodegradable products with three transportation modes. Comput. Ind. Eng. 2022, 174, 108727. [CrossRef] 41. Bachar, R.K.; Bhuniya, S.; Ghosh, S.K.; Sarkar, B. Controllable Energy Consumption in a Sustainable Smart Manufacturing Model Considering Superior Service, Flexible Demand, and Partial Outsourcing. Mathematics 2022, 10, 4517. [CrossRef] 42. Mondal, A.K.; Pareek, S.; Chaudhuri, K.; Bera, A.; Bachar, R.K.; Sarkar, B. Technology license sharing strategy for remanufacturing industries under a closed-loop supply chain management bonding. RAIRO Oper. Res. 2022, 56, 3017–3045. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 27, 2023

**Keywords: **growing items (fish); trade credit; carbon emissions; learning effect; supply chain; signed distance method; fuzzy environment

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