Next Article in Journal
Spatial Decay Estimates and Continuous Dependence for the Oldroyd Fluid
Next Article in Special Issue
Leveraging Blockchain for Maritime Port Supply Chain Management through Multicriteria Decision Making
Previous Article in Journal
Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem
Previous Article in Special Issue
Fuzzy Divergence Measure Based on Technique for Order of Preference by Similarity to Ideal Solution Method for Staff Performance Appraisal
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A Sustainable Supply Chain Model with Low Carbon Emissions for Deteriorating Imperfect-Quality Items under Learning Fuzzy Theory

by
Basim S. O. Alsaedi
* and
Marwan H. Ahelali
Department of Statistics, University of Tabuk, Tabuk 71491, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2024, 12(8), 1237; https://doi.org/10.3390/math12081237
Submission received: 16 March 2024 / Revised: 11 April 2024 / Accepted: 14 April 2024 / Published: 19 April 2024

Abstract

:
In this paper, we develop a two-level supply chain model with low carbon emissions for defective deteriorating items under learning in fuzzy environment by using the double inspection process. Carbon emissions are a major issue for the environment and human life when they come from many sources like different kinds of factories, firms, and industries. The burning of diesel and petrol during the supply of items through transportation is also responsible for carbon emissions. When any company, firm, or industry supplies their items through a supply chain by using of transportation in the regular mode, then a lot of carbon units are emitted from the burning of petrol and diesel, etc., which affects the supply chain. Carbon emissions can be controlled by using different kinds of policies issued by the government of a country, and lots of companies have implemented these policies to control carbon emissions. When a seller delivers a demanded lot size to the buyer, as per demand, and the lot size has some defective items, as per consideration, the demand rate is uncertain in nature. The buyer inspects the received whole lot and divides it into two categories of defective and no defective deteriorating items, as well as immediately selling at different price. The fuzzy concept nullifies the uncertain nature of the demand rate. This paper covers two models, assuming two conditions of quality screening under learning in fuzzy environment: (i) the buyer shows the quality screening and (ii) the quality inspection becomes the seller’s responsibility. The carbon footprint from the transporting and warehousing the deteriorating items is also assumed. The aim of this study is to minimize the whole inventory cost for supply chains with respect to lot size and the number of orders per production cycle. Jointly optimizing the delivery lot size and number of orders per production cycle will minimize the whole fuzzy inventory cost for the supply chain and also reduce the carbon emissions. We take two numerical approaches with authentic data (from the literature reviews) for the justification of the proposed model 1 and model 2. Sensitivity observations, managerial insights, applications of these proposed models, and future scope are also included in this paper, which is more beneficial for firms, the industrial sector, and especially for online markets. The impact of the most effective parameters, like learning effect, fuzzy parameter, carbon emissions parameter, and inventory cost are shown in this study and had a positive effect on the total inventory cost for the supply chain.

1. Introduction

Nowadays, many production industries produce a large of number of items for more profit using supply chain processes, and during the production of items, transportation of items, or burning of petrol and diesel by these causes, a lot of carbon units exit in the form of carbon emissions, which damages the ware housing of greening items, environment, and human lifestyle. Supply chain management (SCM)is a good tool for the coordination of customers, buyers, and sellers and also exerts a favorable effect on stock replenishment decisions. Supply chains are also more beneficial for quality improvement, the supply of inventory material, and inventory decision cost information. Carbon emissions affect the supply chain directly or indirectly and cannot be ignored in the supply chain. If the demand for the item is constant or deterministic, then the supply chain runs smoothly and the producer or seller obtains more profit, but on the other hand, if demand for the items is uncertain, then the producer or seller obtains unexpected profit or unexpected loss, and obtaining no profit or no loss may be dependent on that situation. Hence, the analysis of the demand rate of items cannot ignored during the supply of items. During the production of items, each produced item should be good in quality, but this is not generally true, because in the produced lot, some items may not be good due to some technical problems. In this proposed model, it is considered that each lot has some defective items. The seller delivers then demanded lot of deteriorating perfect-quality items to the buyer and the buyer separates the received lot from the seller through an inspection process and divides the whole lot into two categories, where one is good-quality items and other is poor-quality items. The buyer sells both types of items at different prices and also assumes the effect of deterioration. The present scenario incorporates learning concepts, a lock fuzzy environment, carbon emissions, and economic order quantity in this supply chain. We consider that each lot of production has some defective items. With this concept in mind, we develop a supply chain inventory model with low carbon emissions for imperfect deteriorating items under a learning and lock fuzzy environment and present a scenario also assuming the effect of deterioration during the supply chain. The present scenario examines the effect of deterioration, learning, parameters of the lock fuzzy environment, and different types of cost parameters on the joint inventory cost during the supply chain. For the development of the proposed study, we describe a literature review regarding this study in the literature review section.

1.1. Literature Study Based on EOQ, Carbon Emissions, and Supply Chain Management (SCM)

This section covers the basic literature study that helps in the development of this proposed study. In this order, the supply chain is the backbone of the inventory theory because it connects the seller, buyer, and customer, and the manager can observe to see the supply chain and remove unnecessary factors from the supply chain, which affects the selling of items. Glock [1] explained the many problems regarding EOQ through an inventory model and also found an expression for the lot size. Many authors have developed EOQ inventory models with different approaches under carbon emissions. In this order, Luo et al. [2] developed a supply chain inventory model with inventory policies under low carbon emissions for various items and also showed the effect of low carbon emissions, and Das et al. [3] presented new collection of literature reviews of supply chains under carbon emissions and compared old results with recent results of different supply chain inventory models, as well as gave new approach for the supply chain. Kazmi et al. [4] generalized an EOQ model with trade credit policy for imperfect-quality items under carbon emissions and also explained the effect of a trade credit period on the buyer’s profit and seller’s profit. From the motivation of Kazmi et al. [4], Sarkar et al. [5] presented a sustainable supply chain model (SCM) with multiple trade credit policies for the effect of environment issues under a partial backlogging case, and suggested a lot of managerial insights for the investors of shareholders. Taleizadeh et al. [6] suggested an EPQ model with shortages for the production system under carbon emissions and examined the effect of shortages on the EPQ system. Sarkar et al. [7] developed a three-echelon supply chain system with transportation under the impact of carbon emissions and showed the effect of variable transportation on the supply chain, as well as showing the impact of carbon emissions on the joint profit of the supply chain. Sarkar et al. [8] improved the model of Sarkar et al. [7] with the help of multiple trade credit policies for the global sustainable supply chain model under the impact of carbon emissions. Daryanto et al. [9], motivated by the work of Sarkar et al. [8], improved the inventory model with the impact of carbon emissions for deteriorating items under two-level supply chain management. Wahab et al. [10] developed the work of Daryanto et al. [9] by using the international supply chain model for imperfect-quality items and environment issues. In this view, Jauhari et al. [11] improved the fantastic work of Wahab et al. [10] by using unequal lot size policy and carbon emissions under a two-level supply chain model for imperfect-quality items and cooperative policy. Sarkar et al. [12] considered a game theory approach with carbon emissions in the supply chain model under a reduction in ordering cost. Jauhari et al. [13] assumed a stochastic-demand-based supply chain model for imperfect-quality items under carbon emissions and the variations in stochastic demand when changing inventory parameters were also explained deeply. Gautam and Khanna [14] explained a production-based inventory model with carbon emissions under an inspection process and also presented the policy of ordering cost reduction by using of the carbon emissions policy, as well as calculating the joint total profit the supply chain. Tiwari et al. [15] developed a carbon-emissions-based supply chain model for deteriorating imperfect-quality items under an inspection process and showed the effect of deterioration on the joint profit during the supply chain.

1.2. Literature Study Regarding Imperfect-Quality-Based Inventory Model

In general, we see that in many industries, production companies are trying to produce good items, but in reality, this is not true, and they also produce a fixed percentage of defective-quality items. Many authors have worked on the imperfect production system one, like Rosenblatt and Lee [16], who gave a basic inventory model for imperfect-quality item sunder production processes and explained managerial insights for new researchers or investors of shareholders or inspectors of the supply chain, etc. In this order, Porteus [17] derived the optimal lot sizing formula for the ordering policies under a quality improvement scheme and minimized set up costs. Salameh and Jaber [18] developed an inventory model with a screening process and included the screening cost, finding out the cycle length, as well as showing the effect of the cycle on the order size and profit of the inventory system. Huang [19], motivated by the fantastic work of Salameh and Jaber [18], gave a cooperative policy model for sellers and buyers under a discount policy system for defective-quality items. Goyal et al. [20] improved the model of Salameh and Jaber [18] by using a single production system with multiple shipments under an inspection system for defective-quality items, where each lot had a fixed percentage of defective items. In this flow, the research team of Wee et al. [21] developed a production-based inventory model with shortages for deteriorating imperfect-quality items under an inspection process and also presented the effect of shortages on the integrated total profit. In this way, Lee and Kim [22] described a production-based inventory model without shortages for deteriorating imperfect-quality items under an inspection process and also showed the effect of deterioration on the profit for the system. Some renowned authors, Bazan et al. [23], worked on imperfect-quality items under a one-level supply chain model, where items may be of different types like scrap and salvage in a rebate system, and are work of these items also may be possible. Sarkar et al. [24] improved the model of Bazan et al. [23] for a two-level supply chain by using some assumptions like reworking, imperfect-quality items, and an inspection process under transportation costs. Yu and Hsu [25] improved in the production-based inventory model by using the rework policy of returning to the seller. The strategy of the vendor is explained in Figure 1.

1.3. Literature Study Regarding Imperfect-Quality-and Carbon-Emissions-Based Inventory Model

In this subsection, we discuss only research studies which relate to carbon emissions under different polices. The burning of petrol and diesel and transportation of the inventory items are also responsible for emissions. A lot of research on the carbon-emissions-based supply chain has improved rapidly in recent times under different policies. In this order, Beniaafar et al. [26] improved an inventory model for the supply chain by considering the carbon emissions that emit from the transportation of the items to the production place and from the production place to the buyer’s storing place. Wahab et al. [10] considered the two-level supply chain and found the optimal values of shipment and order quantity, and also incorporated the emission cost from the transportation of the items via vehicles. The traveled distance is also responsible for these carbon emissions. Fahimnia et al. [27] analyzed the effect of the emissions of carbon on the supply chain system under different inventory policies. In this way, Bozorgi et al. [28] assumed the strategy of carbon emissions which occur from different sources like electricity, trucks, the transportation of inventory items, and storage of cold inventory items, as well as large freezers. Bozorgi [29] improved the model of Bozorgi et al. [28] for multiple items under limited storage for these items. Ruidas et al. [30], motivated by the above contribution, generalized a model for a three-level supply chain model in which all players benefited through the strategy of the proposed model. Ghosh et al. [31] generalized multiple shipments based on a one-level supply chain for a single set up, where carbon emission costs were included and it was assumed that carbon emissions occurred from the storing of the cold items, transportion of different inventory items, and during the production of items. Toptal and Cetinkaya [32] worked on a model of multiple deliveries of the inventory for the supply chain system under carbon emissions. Bouchery et al. [33] developed a model with carbon emissions and a limited capacity of the vehicle source under the supply chain model. Dwicahvani et al. [34] developed a remanufacturing-based two-level supply chain model by using waste management cost, carbon emission cost, and energy cost. Some authors have worked on the topics of carbon taxation, carbon footprint, and carbon cap policies under different policies. Li et al. [35] proposed a model with carbon taxation and a carbon cap for a two-level supply chain under trade credit policy. Carbon emissions can be reduced by using different policies of government strategies. Wangsa [36] generalized a model with government policies of carbon reduction under various situations of inventory strategies. A lot of authors have worked on the topic of carbon emissions under different polices and their location order [37,38,39,40]. Gosh et al. [41] improved the model by using different policies like shortage backorder, carbon emissions, and carbon taxation, as well as stochastic demand, and also calculated the inventory cost for the supply chain. Ma et al. [42] assumed the strategy of the carbon emissions for a one-level supply chain under some realistic situations like pricing decisions, production of the inventory items, and procurement, and obtained positive responses from the seller and buyer sides. The safety factor and recovery are a better strategy for any inventory system. In this order, Darom et al. [43] generalized a supply chain model with recovery, disturbance risk, and safety factor for the manufacturing system under carbon emissions. Preservation technology is the best policy for reducing the deterioration rate of deteriorating items during leading the inventory. Huang et al. [44] improved the inventory model by using the preservation system for deteriorating items under carbon emissions and production. In this way, Daryanto et al. [45] considered a model with a low carbon emission policy for deteriorating items under a three-level supply chain and showed the impact of the carbon emissions on the supply chain. Kundu and Chakrabarti [46] presented an inventory model with the policy of fewer carbon emissions under an inflationary situation for the supply chain. We included some other authors who worked on carbon emission systems with some realistic situations of inventory and its order location is [47,48,49]. The strategy of the retailer is explained in Figure 2.

1.4. Literature Study Regarding Fuzzy-Environment-and Learning-Concept-Based Inventory Models

In this section, we selected only literature reviews which relate to imperfect-quality items, especially because this is the motivation for the proposed model. Jaggi et al. [50] explained the EOQ with a screening strategy, shortages, and trade credit policy under the fuzzy concept for deteriorating items and also showed the effect of deterioration, shortages, and trade credit time on the buyer’s profit. In this way, Jaggi et al. [51] extended the work of Jaggi et al. [50] by using different patterns of demand rate with the same strategy as the previous model and presented the effect of the behavior of the input inventory parameters on the buyer’s policy. In a similar way, Jaggi et al. [52] derived a fuzzy-based inventory model with a trade credit strategy under shortages and without an inspection strategy and analyzed the effect of fuzzy input with other inventory parameters on the buyer’s profit. Rout et al. [53], motivated by the work of Jaggi et al. [52], developed a new type of inventory model with a fuzzy-2 strategy under are filling process for inventory items. A lot of authors have worked on inspection processes with some realistic situations, and Patro et al. [54] generalized the work of Rout et al. [53] by using the learning concept for imperfect deteriorating-quality items under the fuzzy concept and presented good results for future work through managerial insights. Most authors have developed inventory models by changing the inventory items. Bhavani et al. [55] considered an inventory model with a fuzzy concept for greening items under shortages and also explained managerial insights for the next generation and also for future work. The learning concept is one type of mathematical tool which informs the saturated shape of the lot size or demand size during the ordering policies or supply chain. The concept of learning was first suggested by Wright [56]. Wright [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56] presented many production inventory models by applying the learning concept under some realistic situations. Jayaswal et al. [57] studied the many literature reviews of Wright [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56] and developed an inventory model with a learning concept and cloudy fuzzy environment for good inventory items under a credit policy system. Jayaswal et al. [58] derived a model with human error and backorders under a fuzzy environment and also calculated the lot size for the ordering policies, as well as showed the effect of lot size and backorders on the total inventory cost. Jayaswal et al. [59] generalized a model with a learning effect, inflationary situation, and fuzzy concept under credit policy. Alamari et al. [60] described a credit-policy-based supply chain model with a fuzzy environment under an inspection process and calculated the buyer’s profit for the supply chain. Almari [61] extended the imperfect-based inventory model for growing items under a fuzzy environment and derived a formula for the buyer’s profit. Alsaedi et al. [62] explained a recovery-based green supply chain for imperfect-quality items under learning in a fuzzy environment and showed the effect of inventory input parameters and carbon emissions on the profit of the supply chain model.

1.5. Research Gap and Proposed Study

We studied and analyzed above the literature reviews of these many authors, but many authors did not propose a sustainable supply chain model with a low carbon emissions policy and fuzzy learning concept for imperfect deteriorating items under vendor’s inspection and buyer’s inspection policies, with both players included in the inspection cost. We selected some renowned motivational research studies from the above literature reviews, which are Wright [56], Salameh and Jaber [18], Tiwari et al. [15], Bazan et al. [23], Wangsa [36], Daryanto et al. [45], Jaggi et al. [50], Patro et al. [54], Jayaswal et al. [59], Almari [61], and Alsaedi et al. [62], and their fruitful contributions are shown in Table 1. The work of this proposed model is shown at the bottom of the Table 1 and comparative studies of the authors are also shown in Table 1.

1.6. Our Contribution and Novelty of the Proposed Model

The contribution of this proposed model is that we developed a sustainable supply chain model with low carbon emissions for deteriorating imperfect-quality items under learning fuzzy theory by using a double screening process, learning fuzzy theory, and transportation of the items. We studied a lot of literature reviews in which the authors considered a screening process and also included some carbon emissions costs from the buyer side, where the demand rate was assumed as constant during the supply chain under some realistic situations. Some researchers have calculated the number of shipments and order quantity for the supply chain under carbon taxation for a reduction in carbon emissions. We were motivated by some research and its nice contributions, as presented in Table 1. We know that a lot of carbon units are emitted from the transportation of items from one place to another place and the burning of fuels during transportation, and these are more responsible for carbon emissions. The waste products and storage of deteriorating imperfect-quality items are not recycled. This means that more shipments mean more carbon emissions and less shipments mean less carbon emissions. The aim of this proposed study is achieving low carbon emissions during the supply chain under the proposed assumptions. The novelty of this paper is that the defective imperfect deteriorating items are separated by using a double screening process (the seller inspects the demanded lot before shipment and the buyer inspects the whole received lot) during the supply chain, where the demand rate is considered to be a fuzzy demand rate. Perfect deteriorating items emit less carbon emissions than defective deteriorating items (not recycled), so a double screening process is needed. The number of shipments are minimized by applying fuzzy learning theory for less transportation, and this also minimizes the jointly fuzzy total cost for the supply chain. The learning fuzzy theory is more effective for the design of the lot size, and due to less shipments, the carbon emissions are comparable to other contributions. This proposed model is more applicable for new generations and also for industrial sectors during the supply chain. Finally, by using a double screening process, low carbon emissions and learning fuzzy theory are more effective for the reduction in cost, and this does not ensure a reduction in carbon emissions during the supply chain. A reduction in cost by using such a strategy for both players during supply chain is the uniqueness of the proposed model.
The working methodology of this proposed model can be seen in Figure 3.

2. Problem Definition, Assumption, and Notations

2.1. Problem Definition

The present paper deals with a manufacturer and a retailer through a supply chain model, in which single deteriorating items are produced and sold in one channel under an inspection process, where the demand for the item is imprecise in nature and carbon emissions policies are also included. We developed a supply chain model with low carbon emissions for deteriorating imperfect-quality items under a learning and luck fuzzy environment. In this model, supposing that a retailer demands n deliveries of items and each deliveryhas an equal lot size, s a y Y , then the manufacturer produces n Y units of the item during each production cycle. In this scenario, we developed two models on the basis of the inspection process under a learning fuzzy environment with the carbon emissions policy. (i) The first model is an extension of the good contribution of Tiwari et al. [5] by using the fuzzy concept, learning concept, variable and fixed inspection costs, and also including the transportation cost, which depends on the weight and distance. During the first model, the buyer inspects the whole received lot from the vendor side under a learning fuzzy environment. (ii) The second scenario is the extension of the first model by using another inspection cost, learning effect, and carbon emissions cost to minimize the system inventory cost. During the second model, the seller inspects the whole lot and no defective products are delivered to the buyer under the learning fuzzy environment. In both models, we assume that the defective products after inspection are sold at a low cost without any extra cost and also include the carbon emissions in the both scenarios. We solved some problems during the supply chain with carbon emissions policies under the learning fuzzy environment when the demand rate is imprecise in nature. The problems are given below in each model:
  • What is the optimal shipment during the supply chain with a learning effect under a fuzzy environment for deteriorating items?
  • What is the optimal cycle time during the supply chain with a learning effect under a fuzzy environment for deteriorating items?
  • What is the minimum fuzzy inventory cost during the supply chain with a learning effect under a fuzzy environment for imperfect deteriorating items?

2.2. Assumptions

We make some assumptions for the development of the proposed model and follow some assumptions of the renowned authors who have worked in this field. The assumptions are given below:
  • This scenario is based on the single-setup multiple-deliveries (SSMD) policy.
  • In this scenario, we considered the buyer’s demand rate to be imprecise in nature and follow the triangular fuzzy number.
  • The fuzzy learning effect involves the lower and upper deviation of the demand rate.
  • The production rate of the product is known and constant from the manufacturer side and the manufacturer prepares n Y units of product in each production cycle to minimize the set time and system inventory cost, and also delivers the product in an equal lot size of the product at a fixed time interval [50].
  • The replenishment of the product is instantaneous.
  • The deterioration rate of the product is fixed and equal for the manufacturer and retailer.
  • In model one, the retailer inspects whole lot with a fixed screening rate and the manufacturer also inspects the whole lot in the same manner as the retailer to guarantee the best facility.
  • The percentage of imperfect-quality items ( P ) has a uniform distribution, where 0 α < β < 1 .
  • The defective items are sold at a discounted price without any extra cost, and products are always available during the quality inspection as γ > D .
  • The fixed inspection cost per cycle is constant, whether performed by the buyer or the manufacturer.
  • The concept of the carbon units/emissions of the carbon comes from the storage of inventory, burning of fuel, consumption of electricity, and transportation of the inventory products.
  • The concept of shortages is not allowed in this model.
  • The consumption of extra fuel depends on the truck loads, which is linearly dependent on the truckloads and represented in Figure 4. We adopted the policy of Hariga et al. [64].

3. Model Development

We divide the whole model in two parts on the basis of the inspection/screening process.

3.1. Development of Model 1 When the Retailer Inspects Whole Lot

The manufacturer’s inventory level and retailer’s inventory level are presented in Figure 5 with suitable notations. It is supposed that the manufacturer produces R units in one production cycle and also that the deliveries of these units to the retailer occur in n shipment with a fixed lot Y. Then, the retailer inspects the whole received lot during the time period T i and obtains p Y defective items.
The retailer removes all defective items from the level of inventory and also assumes that γ f is the fixed screening cost per shipment. Let us consider that γ u is the unit screening cost, as is suggested by Sarkar et al. [12]. The inventory level of the retailer reduces due the demand of the customer in the time period 0 , T / n and the total screening cost per year.
T S C R = γ f n T + γ u Y n T = n T γ f + γ u Y
From the literature review of Jaggi et al. [53], from the inventory level in the interval 0 , Y / γ and Y / γ , T / n , and from Figure 6,
I d t = Y e θ t + D θ e θ t D θ , t 0 , Y / γ
and,
I d t = Y e θ t + D θ e θ t D θ p Y ,   t Y / γ , T / n
For solving Y from Equation (3), the value of I d t = Y e θ t + D θ e θ t D θ p Y = 0 if, t = T n , then the value of Y (Jaggi et al. [53]),
Y = D e θ T n D θ p θ e θ T n
The inventory level for the retailer from 0 to T n ,
I H C R = n T 0 T n I d t d t
I H C R = n T 0 Y x I d t d t + Y / γ T n I d t d t
If the value of I d t is replaced in Equation (5) from Equations (2) and (3), we obtain,
I H U R = n T 0 Y / γ Y e θ t + D θ e θ t D θ d t + Y / γ T n Y e θ t + D θ e θ t D θ p Y d t
After simplifying Equation (6), we obtain,
I H U R = n T Y θ 1 1 e θ Y t D θ 2 Y θ γ + 1 e Y θ γ 1 1 θ e θ T n 1 e Y θ γ Y + D θ T n Y γ ( p Y + D θ )
As per consideration, the holding units require electrical energy with a fixed amount of carbon units or carbon footprint, so the value of Y from Equation (4) is replaced in Equation (7). Now, the holding cost per unit time is,
I H C R = H d n T D e θ T n D θ p θ e θ T n θ 1 1 e θ D e θ T n D θ p θ e θ T n t D θ 2 D e θ T n D θ p θ e θ T n θ γ + 1 e D e θ T n D θ p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ p θ e θ T n θ γ D e θ T n D θ p θ e θ T n + D θ T n D e θ T n D θ p θ e θ T n γ ( p D e θ T n D θ p θ e θ T n + D θ )
The carbon emission cost for the retailer is,
C E C R = E w n T D e θ T n D θ p θ e θ T n θ 1 1 e θ D e θ T n D θ p θ e θ T n t D θ 2 D e θ T n D θ p θ e θ T n θ γ + 1 e D e θ T n D θ p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ p θ e θ T n θ γ D e θ T n D θ p θ e θ T n + D θ T n D e θ T n D θ p θ e θ T n γ ( p D e θ T n D θ p θ e θ T n + D θ )
The deterioration cost per year for the retailer is,
D C C R = d d n T Y u Y D T n
D C C R = d d n T D e θ T n D θ p θ e θ T n u D e θ T n D θ p θ e θ T n D T n
The ordering cost for the retailer is,
O C R = r c T
The total inventory cost for the retailer, which is the sum of the total screening cost ( T S C R ), holding cost ( I H C R ), carbon emission cost ( C E C R ), deterioration cost D C C R , and ordering cost ( O C R ) is,
Ψ 1 n , T = T S C R + I H C R + C E C R + D C C R + O C R
Ψ 1 n , T = γ f n T + γ u D e θ T n D θ p θ e θ T n n T + H d n T D e θ T n D θ p θ e θ T n θ 1 1 e θ D e θ T n D θ p θ e θ T n t D θ 2 D e θ T n D θ p θ e θ T n θ γ + 1 e D e θ T n D θ p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ p θ e θ T n θ γ D e θ T n D θ p θ e θ T n + D θ T n D e θ T n D θ p θ e θ T n γ ( p D e θ T n D θ p θ e θ T n + D θ ) + E w n T D e θ T n D θ p θ e θ T n θ 1 1 e θ D e θ T n D θ p θ e θ T n t D θ 2 D e θ T n D θ p θ e θ T n θ γ + 1 e D e θ T n D θ p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ p θ e θ T n θ γ D e θ T n D θ p θ e θ T n + D θ T n D e θ T n D θ p θ e θ T n γ ( p D e θ T n D θ p θ e θ T n + D θ ) + d d n T D e θ T n D θ p θ e θ T n u D e θ T n D θ p θ e θ T n D T n + r c T
It is also considered that the expected probability value of the defective-quality products is E [ p ] . The retailer’s total cost is the sum of the ordering, inspection, deteriorating, inventory holding, and emission costs. Therefore, considering the probability of the defective products, the expected total cost per year is assessed from Equation (12).
Ψ 1 E n , T = γ f n T + γ u D e θ T n D θ E [ p ] θ e θ T n n T + H d n T D e θ T n D θ θ E [ p ] e θ T n θ 1 1 e θ D e θ T n D θ E [ p ] θ e θ T n t D θ 2 D e θ T n D θ E p θ e θ T n θ γ + 1 e D e θ T n D θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ E p θ e θ T n θ γ D e θ T n D θ E p θ e θ T n + D θ T n D e θ T n D θ E p θ e θ T n γ ( E [ p ] D e θ T n D θ E [ p ] θ e θ T n + D θ ) + E w n T D e θ T n D θ E [ p ] θ e θ T n θ 1 1 e θ D e θ T n D θ E [ p ] θ e θ T n t D θ 2 D e θ T n D θ E p θ e θ T n θ γ + 1 e D e θ T n D θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ E p θ e θ T n θ γ D e θ T n D θ E p θ e θ T n + D θ T n D e θ T n D θ E p θ e θ T n γ ( E [ p ] D e θ T n D θ E [ p ] θ e θ T n + D θ ) + d d n T D e θ T n D θ E [ p ] θ e θ T n E [ p ] D e θ T n D θ E [ p ] θ e θ T n D T n + r c T

3.2. Manufacturer Strategy under Cost and Carbon Emissions

When the retailer demands some products under some conditions from the manufacturer, then the manufacturer produces the demanded products n Y with a production rate P and the set up cost per production cycle is m c .
The manufacturer’s set up cost per year for the production ( S c m ) is,
S c m = m c T
The first shipment lot goes ahead as soon as the demand is met and all deliveries arein the time interval T n .
The manufacturer includes the transportation cost, which is the sum of fixed and variable cost. The first part is the fixed transportation, the second part is the transportation cost when a vacant truck goes from the factory of the manufacturer to the retailer’s place and the truck returns to the factory of the manufacturer with the distance being twice the time (coming and going back with the same distance), and the third part depends on the loaded truck, additional fuels, and product weight. We were motivated by the literature reviews of some renowned authors who added the variable transportation cost in their research work, like Wangsaand Wee [65]., Rahman et al. [66], Nie et al. [67], and Swenseth and Godfrey [68],
T c m = m t f + 2 d m t v + d Y w w 2 m t v w 1 T
If the value of Y from Equation (4) is replaced in Equation (15), we obtain,
T c m = n m t f + 2 d m t v + d Y = D e θ T n D θ p θ e θ T n w w 2 m t v w 1 T
During transportation, the delivery distance, actual shipment weight, fuel consumption per kilometer, and CO2 per liter of fuel are also responsible for the carbon emissions; we adopt the policies of Wahab et al. [10]. Considering the quantity of the carbon footprint for the manufacturer, we can calculate this.
Therefore, the amount of the manufacturer’s carbon emissions per year as a result of transportation activity can be derived as follows:
n T 2 d w 1 + d Y w w 2 F e = n T 2 d w 1 + d D ( e θ T n 1 ) θ ( 1 p e θ T n ) w w 2 F e
We studied many literature reviews concerning this study that worked on the theory of supply chain management, especially regarding the manufacturer’s strategy, like Lee and Kim [22], Yang and Wee’s [68], and Jayaswal et al. [59]. The retailer receives the demanded lot, which has some defective items, and for this reason, the retailer uses the policy of inspection. The demand rate of the manufacturer is D 1 p . Therefore, the stock-level functions are obeyed and the work of this strategy is given in Figure 6.
n T I p 1 t 1 = P 1 p D 1 p θ 1 1 e θ t 1 , t 1 [ 0 , T 1 ]
n T I p 2 t 2 = P 1 p D 1 p θ e θ T 2 θ t 2 1 , t 2 [ 0 , T 2 ]
By using the boundary condition I p 1 T 1 = I p 2 0 and adopting the strategy of Misra’s [59] approximation, we can write,
P 1 p D 1 p T 1 1 1 2 θ T 1 = D 1 p T 2 1 + 1 2 θ T 2
After solving the above equation, we obtain,
T 1 D T 2 1 p P D 1 + 1 2 θ T 2
T T 2 1 p P D ( P p P ) + 1 2 D θ T 2
The holding inventory per cycle for the manufacturer is calculated by using the process of Yang and Wee [68],
0 T 1 I p 1 t 1 d t 1 + 0 T 2 I p 2 t 2 d t 2 n 0 T n I d t d t
We solve Equation (22), and after simplifying, then the holding cost per year for the manufacturer is,
h p T P ( D 1 p ) θ T 1 + P D 1 p θ 2 e θ T 1 1 D 1 p θ T 2 D 1 p θ 2 1 e θ T 2 n D θ 2 p e θ T n 1 2 p D x 1 + e 2 θ T n 2 e e θ n + p 1 p p e e θ n + θ T n p 1 + p p e e θ n + e θ T n 1 ) )
The amount of carbon emissions for the manufacturer per year during warehousing activity is,
e e E e T P D / 1 p θ T 1 + P D / 1 p θ 2 e θ T 1 1 D / 1 p θ T 2 D / 1 p θ 1 e θ T 2 n D θ 2 p e θ T n 1 2 ( p D x 1 + e 2 θ T n 2 e θ T n + p 1 e 2 θ T n + p e 2 θ T n p e θ T n + θ T n p 1 + p e θ T n p 2 e θ T n + e θ T n 1 )
Considering the total carbon emissions cost per year for the manufacturer, from Equations (17) and (24), we obtain,
n T 2 d e 1 + d D e θ T n 1 θ 1 p e 2 + E w T P D / 1 p θ T 1 + P D / 1 p θ 2 e θ T 1 1 D / 1 p θ T 2 D / 1 p θ 1 e θ T 2 n D θ 2 u e θ T n 1 2 ( p D x 1 + e 2 θ T n 2 e θ T n + p 1 e 2 θ T n + p e 2 θ T n p e θ T n + θ T n p 1 + p e θ T n p 2 e θ T n + e θ T n 1 )
During the time period T 1 , there is a loss of inventory due to deterioration. The deterioration cost per year for the manufacturer is,
d p T P T 1 n D e θ T n 1 θ ( 1 p e θ T n
By using Equations (18), (23), (25), and (26) and also considering the probability of imperfect-quality items, the total expected cost per year for the manufacturer is,
Ψ 2 E n , T 1 , T 2 = m c T + h p T P ( D 1 E [ p ] ) θ T 1 + P D 1 E [ p ] θ 2 e θ T 1 1 D 1 E [ p ] θ T 2 D 1 E [ p ] θ 2 1 e θ T 2 n D θ 2 E [ p ] e θ T n 1 2 E [ p ] D x 1 + e 2 θ T n 2 e e θ n + E [ p ] 1 E [ p ] E [ p ] e e θ n + θ T n E [ p ] 1 + E [ p ] E [ p ] e e θ n + e θ T n 1 ) )   + n T 2 d e 1 + d D e θ T n 1 θ 1 p e 2 + E w T P D / 1 E [ p ] θ T 1 + P D / 1 E [ p ] θ 2 e θ T 1 1 D / 1 E [ p ] θ T 2 D / 1 E [ p ] θ 1 e θ T 2 n D θ 2 E [ p ] e θ T n 1 2 ( E [ p ] D x 1 + e 2 θ T n 2 e θ T n + E [ p ] 1 e 2 θ T n + E [ p ] e 2 θ T n E [ p ] e θ T n + θ T n E [ p ] 1 + E [ p ] e θ T n E [ p ] 2 e θ T n + e θ T n 1 ) + d p T P T 1 n D e θ T n 1 θ ( 1 E [ p ] e θ T n
The integrated total inventory cost for the supply chain from Equations (13) and (27) is,
Ψ 12 E n , T 1 , T 2 = Ψ 1 E n , T 1 , T 2 + Ψ 2 E n , T 1 , T 2
Ψ 1 E n , T = γ f n T + γ u D e θ T n D θ E [ p ] θ e θ T n n T + H d n T D e θ T n D θ θ E [ p ] e θ T n θ 1 1 e θ D e θ T n D θ E [ p ] θ e θ T n t D θ 2 D e θ T n D θ E p θ e θ T n θ γ + 1 e D e θ T n D θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ E p θ e θ T n θ γ D e θ T n D θ E p θ e θ T n + D θ T n D e θ T n D θ E p θ e θ T n γ ( E [ p ] D e θ T n D θ E [ p ] θ e θ T n + D θ ) + E w n T D e θ T n D θ E [ p ] θ e θ T n θ 1 1 e θ D e θ T n D θ E [ p ] θ e θ T n t D θ 2 D e θ T n D θ E p θ e θ T n θ γ + 1 e D e θ T n D θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D e θ T n D θ E p θ e θ T n θ γ D e θ T n D θ E p θ e θ T n + D θ T n D e θ T n D θ E p θ e θ T n γ ( E [ p ] D e θ T n D θ E [ p ] θ e θ T n + D θ ) + d d n T D e θ T n D θ E [ p ] θ e θ T n E [ p ] D e θ T n D θ E [ p ] θ e θ T n D T n + r c T   + m c T + h p T P ( D 1 E [ p ] ) θ T 1 + P D 1 E [ p ] θ 2 e θ T 1 1 D 1 E [ p ] θ T 2 D 1 E [ p ] θ 2 1 e θ T 2 n D θ 2 E [ p ] e θ T n 1 2 E [ p ] D x 1 + e 2 θ T n 2 e e θ n + E [ p ] 1 E [ p ] E [ p ] e e θ n + θ T n E [ p ] 1 + E [ p ] E [ p ] e e θ n + e θ T n 1 ) )   + n T 2 d e 1 + d D e θ T n 1 θ 1 p e 2 + E w T P D / 1 E [ p ] θ T 1 + P D / 1 E [ p ] θ 2 e θ T 1 1 D / 1 E [ p ] θ T 2 D / 1 E [ p ] θ 1 e θ T 2 n D θ 2 E [ p ] e θ T n 1 2 ( E [ p ] D x 1 + e 2 θ T n 2 e θ T n + E [ p ] 1 e 2 θ T n + E [ p ] e 2 θ T n E [ p ] e θ T n + θ T n E [ p ] 1 + E [ p ] e θ T n E [ p ] 2 e θ T n + e θ T n 1 ) + d p T P T 1 n D e θ T n 1 θ ( 1 E [ p ] e θ T n

4. Formulation of the Proposed Model under Fuzzy Environment

As per our assumptions, the demand rate is imprecise in nature and is also treated as a triangular fuzzy number. Equation (28) is converted from the crisp to fuzzy environment, then we obtain,
Ψ 12 ~ n , T 1 , T 2 = γ f n T + γ u D ~ e θ T n D ~ θ E [ p ] θ e θ T n n T + H d n T D ~ e θ T n D ~ θ θ E [ p ] e θ T n θ 1 1 e θ D ~ e θ T n D ~ θ E [ p ] θ e θ T n t D ~ θ 2 D ~ e θ T n D ~ θ E p θ e θ T n θ γ + 1 e D ~ e θ T n D ~ θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D ~ e θ T n D ~ θ E p θ e θ T n θ γ D ~ e θ T n D ~ θ E p θ e θ T n + D ~ θ T n D ~ e θ T n D ~ θ E p θ e θ T n γ ( E [ p ] D ~ e θ T n D ~ θ E p θ e θ T n + D ~ θ ) + E w n T D ~ e θ T n D ~ θ E [ p ] θ e θ T n θ 1 1 e θ D ~ e θ T n D ~ θ E [ p ] θ e θ T n t D ~ θ 2 D ~ e θ T n D ~ θ E p θ e θ T n θ γ + 1 e D ~ e θ T n D ~ θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D ~ e θ T n D ~ θ E p θ e θ T n θ γ D ~ e θ T n D ~ θ E p θ e θ T n + D ~ θ T n D ~ e θ T n D ~ θ E p θ e θ T n γ ( E [ p ] D ~ e θ T n D ~ θ E [ p ] θ e θ T n + D ~ θ ) + d d n T D ~ e θ T n D ~ θ E [ p ] θ e θ T n E [ p ] D ~ e θ T n D ~ θ E [ p ] θ e θ T n D ~ T n + r c T + m c T + h p T P ( D ~ 1 E [ p ] ) θ T 1 + P D ~ 1 E [ p ] θ 2 e θ T 1 1 D ~ 1 E [ p ] θ T 2 D ~ 1 E [ p ] θ 2 1 e θ T 2 n D ~ θ 2 E [ p ] e θ T n 1 2 E [ p ] D ~ x 1 + e 2 θ T n 2 e e θ n + E [ p ] 1 E [ p ] E [ p ] e e θ n + θ T n E [ p ] 1 + E [ p ] E [ p ] e e θ n + e θ T n 1 ) ) + n T 2 d e 1 + d D ~ e θ T n 1 θ 1 p e 2 + E w T P D ~ / 1 E [ p ] θ T 1 + P D ~ / 1 E [ p ] θ 2 e θ T 1 1 D ~ / 1 E [ p ] θ T 2 D ~ / 1 E [ p ] θ 1 e θ T 2 n D ~ θ 2 E [ p ] e θ T n 1 2 ( E [ p ] D ~ x 1 + e 2 θ T n 2 e θ T n + E [ p ] 1 e 2 θ T n + E [ p ] e 2 θ T n E [ p ] e θ T n + θ T n E [ p ] 1 + E [ p ] e θ T n E [ p ] 2 e θ T n + e θ T n 1 ) + d p T P T 1 n D ~ e θ T n 1 θ ( 1 E [ p ] e θ T n
Now, we defuzzify Equation (29) with the help of the signed distance method, and the signed distance method between Ψ 12 ~ n , T 1 , T 2 and 0 ~ is given by,
d Ψ 12 ~ n , T 1 , T 2 , 0 ~ = γ f n T + γ u d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n n T + H d n T d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ θ E [ p ] e θ T n θ 1 1 e θ d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n t d D ~ , 0 ~ θ 2 d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ + 1 e d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n + d D ~ , 0 ~ θ T n d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n γ ( E [ p ] d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n + d D ~ , 0 ~ θ ) + E w n T d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n θ 1 1 e θ d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n t d D ~ , 0 ~ θ 2 d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ + 1 e d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n + d D ~ , 0 ~ θ T n d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E p θ e θ T n γ ( E [ p ] d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n + d D ~ , 0 ~ θ ) + d d n T d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n E [ p ] d D ~ , 0 ~ e θ T n d D ~ , 0 ~ θ E [ p ] θ e θ T n d D ~ , 0 ~ T n + r c T + m c T + h p T P ( d D ~ , 0 ~ 1 E [ p ] ) θ T 1 + P d D ~ , 0 ~ 1 E [ p ] θ 2 e θ T 1 1 d D ~ , 0 ~ 1 E [ p ] θ T 2 d D ~ , 0 ~ 1 E [ p ] θ 2 1 e θ T 2 n d D ~ , 0 ~ θ 2 E [ p ] e θ T n 1 2 E [ p ] d D ~ , 0 ~ x 1 + e 2 θ T n 2 e e θ n + E [ p ] 1 E [ p ] E [ p ] e e θ n + θ T n E [ p ] 1 + E [ p ] E [ p ] e e θ n + e θ T n 1 ) ) + n T 2 d e 1 + d d D ~ , 0 ~ e θ T n 1 θ 1 p e 2 + E w T P d D ~ , 0 ~ / 1 E [ p ] θ T 1 + P d D ~ , 0 ~ / 1 E [ p ] θ 2 e θ T 1 1 d D ~ , 0 ~ / 1 E [ p ] θ T 2 d D ~ , 0 ~ ~ / 1 E [ p ] θ 1 e θ T 2 n d D ~ , 0 ~ θ 2 E [ p ] e θ T n 1 2 ( E [ p ] d D ~ , 0 ~ x 1 + e 2 θ T n 2 e θ T n + E [ p ] 1 e 2 θ T n + E [ p ] e 2 θ T n E [ p ] e θ T n + θ T n E [ p ] 1 + E [ p ] e θ T n E [ p ] 2 e θ T n + e θ T n 1 ) + d p T P T 1 n d D ~ , 0 ~ e θ T n 1 θ ( 1 E [ p ] e θ T n
The signed distance from D ~ to 0 ~ can be defined (Patro et al. [54]) as,
D ~ , 0 ~ = D + 2 D 2 + D 3 4
As per consideration, the demand rate is considered as a triangular fuzzy number and we can write D ~ = D l , D , D + h , where l is a lower deviation and h is an upper deviation of the demand rate. From Equation (31), we can write,
d D ~ , 0 ~ = 4 D + l h 4
Equation (30) can be written by using Equation (32),
d Ψ 12 ~ n , T 1 , T 2 , 0 ~ = γ f n T + γ u D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n n T + H d n T D + l 4 h 4 e θ T n D + l 4 h 4 θ θ E [ p ] e θ T n θ 1 1 e θ D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n t d D ~ , 0 ~ θ 2 D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n θ γ + 1 e D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D + l 4 h 4 e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n + D + l 4 h 4 θ T n D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n γ ( E [ p ] D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n + D + l 4 h 4 θ ) + E w n T D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n θ 1 1 e θ D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n t D + l 4 h 4 θ 2 D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n θ γ + 1 e D + l 4 h 4 e θ T n d D ~ , 0 ~ θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n θ γ D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n + D + l 4 h 4 θ T n D + l 4 h 4 e θ T n D + l 4 h 4 θ E p θ e θ T n γ ( E [ p ] D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n + D + l 4 h 4 θ ) + d d n T D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n E [ p ] D + l 4 h 4 e θ T n D + l 4 h 4 θ E [ p ] θ e θ T n D + l 4 h 4 T n + r c T   + m c T + h p T P ( D + l 4 h 4 1 E [ p ] ) θ T 1 + P D + l 4 h 4 1 E [ p ] θ 2 e θ T 1 1 D + l 4 h 4 1 E [ p ] θ T 2 D + l 4 h 4 1 E [ p ] θ 2 1 e θ T 2 n D + l 4 h 4 θ 2 E [ p ] e θ T n 1 2 E [ p ] D + l 4 h 4 x 1 + e 2 θ T n 2 e e θ n + E [ p ] 1 E [ p ] E [ p ] e e θ n + θ T n E [ p ] 1 + E [ p ] E [ p ] e e θ n + e θ T n 1 ) )   + n T 2 d e 1 + d D + l 4 h 4 e θ T n 1 θ 1 p e 2 + E w T P D + l 4 h 4 / 1 E [ p ] θ T 1 + P D + l 4 h 4 / 1 E [ p ] θ 2 e θ T 1 1 D + l 4 h 4 / 1 E [ p ] θ T 2 D + l 4 h 4 / 1 E [ p ] θ 1 e θ T 2 n D + l 4 h 4 θ 2 E [ p ] e θ T n 1 2 ( E [ p ] D + l 4 h 4 x 1 + e 2 θ T n 2 e θ T n + E [ p ] 1 e 2 θ T n + E [ p ] e 2 θ T n E [ p ] e θ T n + θ T n E [ p ] 1 + E [ p ] e θ T n E [ p ] 2 e θ T n + e θ T n 1 ) + d p T P T 1 n D + l 4 h 4 e θ T n 1 θ ( 1 E [ p ] e θ T n
Now, we develop the present scenario under a learning environment and also consider the learning effect involved in the lower and upper deviation of the demand rate briefly discussed in the forthcoming section.

4.1. Formulation of the Proposed Model with Fuzzy Environment under Learning Effect

The learning effect is a cost reduction function and some authors have worked on this concept, like Wright [56], Patro et al. [54], and Jayaswal et al. [57]. The lower and upper deviation of the demand rate are related to learning and follow the formulation of Wright [56]. The mathematical shape of learning was presented by Wright [56] and is mathematically given below:
T Y = T 1 Y l
where T Y is the time to prepare the Y n -th units, T 1 is the time for opening Y , and l is the learning slope. Now, we define the learning in the upper and lower deviation of the demand rate, which is given in Equations (35) and (36).
D h ,   j = D h ,   i ,     j = 1 D h ,   i j 1 365 N l , j > 1
D l ,   j = D l ,   j ,     j = 1 D l ,   j j 1 365 N l , j > 1
Now, the lower and upper deviation of the demand rate follow the effect of learning, then, Equation (33) can be written with the help of Equations (35) and (36), we obtain,
Ψ 12 L ~ n , T 1 , T 2 = γ f n T + γ u 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ E [ p ] θ e θ T n n T + H d n T 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ θ E [ p ] e θ T n θ 1 1 e θ 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ E [ p ] θ e θ T n t 4   D + D h , j D l , j j 1 365 N l 4 θ 2 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ E p θ e θ T n θ γ + 1 e 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ E p θ e θ T n θ γ 1 1 θ e θ T n 1 e 4   D + D h , j D l , j j 1 365 N l e θ T n 4   D + D h , j D l , j j 1 365 N l 4 θ E p θ e θ T n θ γ 4   D + D