Inventory management is the foundation for both financial stability and operational efficiency across various industries, including the automobile sector. It includes strategic oversight of the ordering, holding, and utilization of goods and supplies, which is crucial in manufacturing, distribution, and service delivery. Businesses might benefit from inventory management by establishing a plan that automates the process of monitoring and controlling inventory, such as real-time data on inventory states and levels ( Solutions, 2020). Hence, maintaining the ideal quantity stock levels while finding a balance between satisfying the customer demand and reducing excessive cost related to the inventory or stockouts are the keys components of effective inventory management ( Dhiman and Sayal, 2022). Striking a balance between demand and supply is quite important in the automobile industry to achieve customer satisfaction, and maintaining production continuity has heavily depended on the ability to ensure that the right products and components are available when needed and avoid funds being tied up in excess stock ( Square, 2021). Order fulfilment and customer shipping proceed more swiftly when inventory is being handled well. However, inventory must be controlled to reduce the amount of money blocked. It minimizes the excessive tying up of capital in surplus inventories and enhances the company’s financial position ( Tally, 2020).

As the world’s leading electric vehicle manufacturer, Tesla has revolutionized the automotive industry, reshaping perceptions of what is possible in terms of vehicle performance, sustainability, and autonomy. Founded by Elon Musk in 2003, Tesla has quickly become synonymous with cutting-edge technology, sleek design, and commitment to reducing carbon emissions ( Olorunfemi, 2024). Choosing Tesla as the focus of a research project offers a unique opportunity to explore cutting-edge inventory management practices in a fast-paced, technology-driven environment. With its vertically integrated business model, direct-to-consumer sales approach, and ambitious production targets, Tesla provides a rich context for studying inventory management challenges and their implications for organizational performance ( Hagglund, 2023). This project aims to uncover insights into Tesla’s inventory management strategies, offering valuable lessons for the automotive industry amidst its transformation towards electrification and sustainable mobility.

Selecting Tesla as the focus of a research project offers a unique opportunity. In this study, we aim to explore inventory management in the automobile sector through the lens of both traditional crisp systems and more adaptive fuzzy systems. In this study, we aim to narrow the gap between fuzzification and defuzzification techniques to provide information and improve decision-making. Our focus is to identify strategies that businesses can use to maximize profitability. Effective inventory management can improve the overall market competitiveness and support successful operations.

Gnoni et al. (2003) studied how demand uncertainty affects the supply chain performance in automotive braking equipment manufacturing. Using dynamic simulation models, they found that scheduling priorities between the original and aftermarket equipment significantly impacted operational efficiency under various scenarios. This research highlighted that planning tasks differently can improve the car industry’s supply chain when the demand is uncertain. Yildiz et al. (2016) highlighted ways in 2016 to enhance production and inventory management in the automotive industry and developed an optimization model for JIT production, comparing traditional and modern inventory systems. Kozarević and Puška (2018) introduced a method to analyze data, supply chain practices, and performance using fuzzy-logic techniques. By employing fuzzy logic and the FTOPSIS method, qualitative statements about supply chain practices were translated into crisp numerical values, enabling an assessment of their overall impact. Samanta and Al-Araimi (2001) proposed a model for inventory management in medium-sized production systems based on fuzzy logic, seeking to keep inventory levels stable despite changes in demand, while considering how production processes operate. Wang (2009) suggested ways to make the DRP system more accurate in a continuous model by introducing a new method called CRIM using fuzzy variables. Inventory data were made more precise by turning them into closed intervals instead of using exact values and lower costs by considering uncertain demand. The study suggests further exploration of the impact of learning and forgetting on uncertain parameters in supply chains for future research. In 2023, Idrees and Ayesha discussed the importance of inventory forecasting in the supply chain, especially in the automotive industry. The study suggests using fuzzy logic to examine how inventory affects demand. In 2015, Eida and Siti focused on the low implementation level of an Inventory Management System in the automated service industry and proposed a conceptual model using EOQ as a cost-effective solution. This study highlights the importance of efficient inventory management to improve organizational performance in this sector. Max Maxwell in 2022 discussed the challenges faced in production at Tesla in meeting customer needs accurately. Tesla’s use of exponential smoothing to forecast vehicle demand highlights the need for precise forecasting to keep customers satisfied and business running smoothly, but it is still difficult to manage MPS accurately owing to various factors. Ivanchenko et al. (2017) investigated how Tesla manages inventory and how it affects cash flow. This paper mainly discusses how to improve and increase the popularity of electric vehicles. By capturing uncertainties in demand forecasts, fuzzy models allow for more flexible decision making and help optimize inventory levels, ultimately improving inventory control and supply chain performance. Sayal et al. (2018) proposed an inventory model without shortage that is both developed in crisp and fuzzy inventory systems to find the optimized total production cost and determine the optimal order quantity for the inventory where triangular fuzzy numbers are applied here. Sayal et al. (2018) also developed a model for perishable items that contain the demand rate on a ramp type as it increases in the beginning and becomes constant and drops again. Instead, Sayal et al. proposed a mechanism for the optimization of the total cost of the EOQ inventory system in crisps and fuzzy environments using the Weibull Distribution for the ramp type and deterioration rate. Sayal et al. (2019) also reviewed the application of crisps and fuzzy supply chains to examine their effectiveness in enhancing decision making under uncertain conditions. This study establishes inventory system models in crisps and fuzzy terms with the use of Weibull Distribution for demand and deterioration rates to conduct a comparative analysis including shortages and backlogging ( Maxwell, 2022). In 2022, Sayal et al. addressed a deterministic inventory system with a time-dependent demand function and holding cost, in which this model is applied to both crisps and fuzzy models that are valid with relevant numerical examples. In 2022, Sayal et al. developed an optimized inventory model for deteriorating items, addressing both crisp and fuzzy environments with partial backlogging and considered the impact of inflation on the system. Blundell et al. (1993) analyzed consumer demand patterns using microdata and provided insights into household consumption behavior. In 2015, Mike et al. explored the types of fuzzy numbers and their distinct properties, contributing to the understanding of the fuzzy set theory and its applications in decision-making and mathematical modelling. Mondal et al. (2019) proposed crisp and interval inventory models for ameliorating items with Weibull-distributed amelioration and deterioration, utilizing quantum-behaved particle swarm optimization techniques to enhance inventory management efficiency ( Nima, 2017). Olorunfemi (2024) examined the key innovations behind Tesla’s success, focusing on technological disruptions and customer transformation. In 2022, Singh et al. analyzed an inventory model with quadratic demand, incorporating three levels of production to optimize inventory management and improve operational efficiency in dynamic demand environments. Gnoni et al. (2003) studied how demand uncertainty affects the supply chain performance in automotive braking equipment manufacturing. Using dynamic simulation models, they found that scheduling priorities between the original and aftermarket equipment significantly impacted operational efficiency under various scenarios. This research highlighted that planning tasks differently can improve the car industry’s supply chain when demand is uncertain. Sayal et al. (2018) highlighted the importance of fuzzy logic models in managing uncertainties, particularly in terms of inventory management. Olorunfemi (2024) examined Tesla’s innovations in inventory management, including the use of advanced forecasting models. Singh et al. (2022) analyzed an inventory model with quadratic demand, incorporating multiple production levels to optimize inventory management and improve operational efficiency in dynamic demand environments. Kayode Coker (n.d.) emphasizes similar techniques for optimizing supply chains in industries with fluctuating demand. Fleetwood (2018) also examined the role of inventory optimization and cost management, specifically using fuzzy systems to handle real-world uncertainties. Sayal and Dhiman (2022) focus on optimizing inventory systems with inflation considerations and partial backlogging. Jaber (2009) also contributes to this area by discussing how fuzzy logic models can optimize both production and inventory in uncertain environments. Zarandi et al. (2007) also contributed to this field by using a fuzzy-logic-based approach for inventory optimization under uncertain conditions, thereby improving supply chain flexibility. Phair and Warren (2021) also noted that fuzzy expert systems allow for more flexible and adaptive decision-making in inventory control. Recent advancements in automotive supply chain optimization have been highlighted by the CFI Team (2023), who discussed the integration of real-time forecasting techniques to enhance inventory management under uncertain demand conditions. Singh (n.d.) proposed fuzzy inventory models that help manage demand fluctuations within the automotive supply chain. Lutkevich (2021) introduced techniques for forecasting under uncertain conditions, particularly for industries with rapidly changing demand patterns. Kumar et al. (2024) developed a fuzzy production inventory model incorporating carbon emissions and optimized production decisions while considering environmental impacts. Mike Dison Ebinesar (2015) explored the properties of fuzzy numbers and their importance in modelling uncertain demand in inventory systems, whereas Mondal et al. (2019) proposed a crisp and interval inventory model for ameliorating items.

In conclusion, the literature review examined a wide range of methodologies and how they affect supply chain performance in the automotive sector when there is demand uncertainty. As highlighted by Olvera and Segura (n.d.) the importance of production optimization and adaptive scheduling in inventory management is discussed. This, in order, can help increase operational effectiveness. This review discusses the integration of fuzzy logic approaches and how they might be used to convert qualitative data into exact numerical values.

This study aims to provide a systematic approach for comparing crisp and fuzzy inventory systems in the automotive industry. By developing mathematical models and deriving total cost functions using quadratic functions ( Park, 1987), this methodology addresses key research questions regarding uncertainty and variability. Fuzzification and de-fuzzification techniques were applied in this study. For fuzzification of the variables, the trapezoidal fuzzy number was used, and to defuzzify the variables, the centroid method was applied.

Several assumptions were made to develop a methodology for comparing crisp and fuzzy inventory systems in the automotive industry and deriving a total cost function using quadratic functions.

Assumptions listed below are made for the development of this model considering crisp and fuzzy environments: 1.

The inventory system involves only one item and is not subject to additional inventory.

Notation for crisps system

The notation used in this paper for crisp environment is as follows:

D ( t ) Demand rate dependent on time i.e. D ( t ) = a t 2 + bt + c (days)

– Assume that the values of a, b, and c > 0

T i Period of time expressed in units, i = 1 , 2 , 3

T 4 Length of the replenishment cycle (days)

p Production number of automotive components in a certain amount of time

c 1 Production cost per unit

c 2 Holding cost per unit

c 3 Deterioration cost per unit

c 4 Setup cost per production cycle

c 5 Ordering cost per unit

q ( t ) Inventory level of automotive components at time ‘t’

θ Constant deterioration rate

I 1 Inventory close at time T 1

I 2 Inventory close at time T 2

I 3 Inventory close at time T 3

We assume that production starts at time t = 0 and stops at time t = T 4 . The cycle begins with the opening of the market and ends with its closing. Let the production rate be p and demand rate be a t 2 + bt + c , where a , b , c are positive values with D < p . The inventory level reaches I 1 at time T 1 . In the intervals [ T 1 , T 2 ] and [ T 2 , T 3 ] , the inventory level increases at rates d ( p − D ) and e ( p − D ) respectively, where d and e are constants, with e > d > 1 . The inventory level reached I 2 at time T 2 and I 3 at time T 3 . Production stops at time T 3 . Demand and deterioration caused the inventory level to decrease until it reached zero at time T 4 .

The differential equation for the instantaneous inventory q ( t ) at time t in ( 0 , T 4 ) is given by: dq ( t ) dt = p − [ a t 2 + bt + c ] − θq ( t ) ; for 0 ≤ t ≤ T 1 (1) dq ( t ) dt = d [ p − [ a t 2 + bt + c ] ] − θq ( t ) ; for T 1 ≤ t ≤ T 2 (2) dq ( t ) dt = e [ p − [ a t 2 + bt + c ] ] − θq ( t ) ; for T 2 ≤ t ≤ T 3 (3) dq ( t ) dt = − [ a t 2 + bt + c ] ; for T 3 ≤ t ≤ T 4 (4)

With the initial conditions: q ( 0 ) = 0 , q ( T 1 ) = I 1 , q ( T 2 ) = I 2 , q ( T 3 ) = I 3 , q ( T 4 ) = 0 (5)

From equation (1) , for 0 ≤ t ≤ T 1 , q ( t ) = ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − bt θ 2 − t 2 θ 2 θ 3 (6)

From equation (2) , for T 1 ≤ t ≤ T 2 , q ( t ) = d ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − bt θ 2 − t 2 θ 2 θ 3 (7)

From equation (3) , for T 2 ≤ t ≤ T 3 , q ( t ) = e ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − b t θ 2 − t 2 θ 2 θ 3 (8)

From equation (4), for T 3 ≤ t ≤ T 4 , q ( t ) = − e − tθ ( 2 a e tθ − 2 a e Tθ − b θe tθ + b e Tθ θ − 2 atθ e tθ + 2 aTθ e Tθ + c e tθ θ 2 − c e Tθ θ 2 + b te tθ θ 2 + a e tθ t 2 θ 2 − b e Tθ T θ 2 − a e Tθ T 2 θ 2 θ 3 (9)

Maximum inventory level refers to the highest amount of goods a business keeps in stock at a time. Setting an optimal maximum inventory level to balance customer demand while minimizing costs and risks.

The maximum inventory level during the period ( 0 , T 1 ) is given as follows from q ( T 1 ) = I 1 I 1 = θ 3 T 1 ( b T 1 + 2 a − a T 1 θ − 2 P + Pθ T 1 ) 2 (10)

The maximum inventory level during the period ( T 1 , T 2 ) is given as follows from q ( T 2 ) = I 2 I 2 = d T 2 θ 3 2 [ b T 2 + 2 c − a T 2 θ − 2 P + Pθ T 2 ] (11)

The maximum inventory level during the period ( T 1 , T 2 ) is given as follows from q ( T 3 ) = I 3 I 3 = e T 3 θ 3 2 [ b T 3 + 2 c − a T 3 θ − 2 P + Pθ T 3 ] (12)

For the respective costs in the model: 1.

Production cost PC = D c 1

The production cost over the interval ( 0 , T 4 ) : PC = D c 1 PC = c 1 ( c + b T 4 ε ) (13)

The setup cost over the interval ( 0 , T 4 ) : TS = c 4 (14)

The holding cost over the interval ( 0 , T 4 ) : TH = c 2 ∫ 0 T 4 q ( t ) dt (15) TH = c 2 ∫ 0 T 1 q ( t ) dt + ∫ T 1 T 2 q ( t ) dt + ∫ T 2 T 3 q ( t ) dt + ∫ T 3 T 4 q ( t ) dt (16)

Hence, TH = c 2 [ e − T 1 θ 6 θ 3 { 6 ( − 1 + e T 1 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + e T 1 θ T 1 θ ( − 12 a + 6 bθ − 6 c θ 2 + 6 P θ 2 + θ T 1 ( 6 a − 3 bθ − 2 θ T 1 ) } + e − θ ( T 1 + T 1 ) 6 θ 4 { − 6 d ( e T 1 θ − e T 2 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + d e θ ( T 1 + T 1 ) θ ( T 1 − T 2 ) ( 12 a − b 6 θ + 6 θ 2 c − 6 θ 2 P + θ ( 2 θ T 1 2 + T 1 ( − 6 a + 3 bθ + 2 θ T 2 ) + T 2 ( − 6 a + 3 bθ + 2 θ T 2 ) ) ) } + e − θ ( T 2 + T 3 ) 6 θ 4 { − 6 A ( e T 2 θ − e T 3 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + A e θ ( T 2 + T 3 ) θ ( T 2 − T 3 ) ( 12 a + 6 ( − bθ + c θ 2 − P θ 2 ) + θ ( 2 θ T 2 2 + T 2 ( - 6 a + 3 b θ + 20 T 3 ) + T 3 ( - 6 a + 3 b θ + 2 θ T 3 ) ) } + 1 6 θ 4 { 6 ( − 1 + e θ ( T 4 − T 3 ) ( 2 a − θb + c θ 2 ) + θ ( ( 12 a − 6 bθ + 6 c θ 2 ) T 3 + 3 θ ( − 2 a + bθ ) T 3 2 + 2 a θ 2 T 3 2 + T 4 6 e θ ( T 4 − T 3 ) ( − 2 a + bθ + cθ T 4 ) − T 4 θ 2 ( 6 c + 3 b T 4 + 2 a T 4 2 ) ) } (17) TH = c 2 [ 1 θ 4 { c − p − T 1 2 θ 3 3 + ( p − c ) θ 2 + ( p − c ) T 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + T 2 2 1 6 θ 4 { T 3 2 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } (18)

The deterioration cost of finished products: TD = c 3 θ p ∫ 0 T 4 q ( t ) dt (19) TD = c 3 θ p [ ∫ 0 T 1 q ( t ) dt + ∫ T 1 T 2 q ( t ) dt + ∫ T 2 T 3 q ( t ) dt + ∫ T 3 T 4 q ( t ) dt ] (20)

Hence, TD = c 3 θ p [ 1 θ 4 { c − p − T 1 2 θ 3 3 − c θ 2 − cT 1 2 θ 4 2 + p θ 2 + pT 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + 6 d ( T 1 − T 2 ) + ( b θ 3 T 1 + c θ 4 T 1 − p θ 2 T 1 ) } + 1 6 θ 4 { d ( T 2 + T 3 ) ( 6 a T 2 θ 2 − 6 a T 3 θ 2 ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + 1 6 θ 4 { T 3 3 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } ] (21)

The total cost is given by: Total Cost = Setup Cost + Production Cost + Total Holding Cost + Total Deterioration Cost TC = TS + PC + TH + TD (22) TC = [ c 4 + c 1 ( c + b T 4 ε ) + ( c 2 + c 3 θ ) θ 4 [ { c − p − T 1 2 θ 3 3 − c θ 2 − cT 1 2 θ 4 2 + p θ 2 + pT 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + 6 d ( T 1 − T 2 ) + ( b θ 3 T 1 + c θ 4 T 1 − p θ 2 T 1 ) } + 1 6 θ 4 { d ( T 2 + T 3 ) ( 6 a T 2 θ 2 − 6 a T 3 θ 2 ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + 1 6 θ 4 { T 3 3 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } ] (23)

To simplify the total cost equation by letting T 1 = JT 4 , T 2 = Q T 4 , T 3 = K T 4 (24)

This substitution allows us to express the terms T 1 , T 2 , T 3 in terms of the common variable T 4 which can simplify the overall expression and reduce complexity in calculations and analysis.

Therefore, by inserting (24), the total cost is given by T C = [ c 4 + c 1 ( c + b T 4 ε ) + ( c 2 + c 3 θ ) 6 θ 4 ] { 6 c − 6 c θ 2 − c T 4 ( J − 3 T 4 J 2 + 6 e T 4 Q ( Q − K ) ) θ 4 − 6 p + 6 p θ 2 + p θ 2 T 4 ( 6 e T 4 Q ( − Q + K ) − J + 3 T 4 J 2 θ 2 ) + T 4 ( d ( J − Q ) ( 6 + 6 a T 4 ( J + Q ) θ 2 ) ) + T 4 3 J Q θ 3 ( 3 b ( J + Q ) θ + 2 T 4 ( J Q + J 2 θ + Q 2 θ ) ) + θ 2 ( b J θ + 3 b T 4 ( 1 + T 4 K 3 − 2 ( K + e Q ( K − Q ) ) θ + e T 4 2 Q ( Q 2 − K 2 ) K θ 2 + 2 T 4 ( 6 a K − 3 a K − 3 a T 4 K − 2 T 4 K 3 + 3 a e ( Q 2 − K 2 ) + 2 a T 4 θ − θ J 2 + e T 4 3 Q ( Q − K ) K ( Q K θ + Q 2 θ 2 + 2 K 2 θ 2 ) ) ) ) } (25)

To minimize the total relevant cost and investigate the optimal time T 4 as follows, differentiates with respect to T 4 .

The equation is given by: ∂ TC ∂ T 4 = 0 , ∂ 2 TC ∂ T 4 2 > 0 (26)

The notation used in this paper for fuzzy environment is as follows:

D ( t ) Demand rate dependent on time i.e. D ( t ) = a t 2 + bt + c (days)

– Assume that the values of a, b, and c > 0

T i Period of time expressed in units, i = 1 , 2 , 3

T 4 Length of the replenishment cycle (days)

p Production number of automotive components in a certain amount of time

c 1 ~ Production cost per unit

c 2 ~ Holding cost per unit per unit time

c 3 ~ Deterioration cost per unit

c 4 ~ Setup cost per production cycle

c 5 ~ Ordering cost per unit

q ( t ) Inventory level of automotive components at time ‘t’

θ Constant deterioration rate

I 1 Inventory close at time T 1

I 2 Inventory close at time T 2

I 3 Inventory close at time T 3

Fuzzy numbers can be classified into several types based on their shape and properties. There are some common types of fuzzy numbers, such as Triangular Fuzzy Numbers (TFNs), Trapezoidal Fuzzy Numbers (TrFNs), and Gaussian fuzzy numbers etc. ( Mike Dison Ebinesar, 2015)

Owing to the uncertainty in the surrounding environment, it is challenging to precisely define all the specified parameters. Therefore, we assume that some of them, such as production cost ( c 1 ) , holding cost ( c 2 ) , deterioration cost ( c 3 ) , and setup cost ( c 4 ) will fluctuate within certain limits. The costs are considered trapezoidal fuzzy numbers.

Trapezoidal Fuzzy Number:

Let F ~ = ( l , m , n , o ) , L , m < n < o , be a fuzzy set on R = (−∞,∞) it is a trapezoidal fuzzy number if the membership functions of F ~ is defined as: μ F ~ ( λ ) = { 0 , if x < l x − l m − l , if l ≤ x ≤ m 1 , if m < x < n 0 − x 0 − n , if m ≤ x ≤ n 0 , otherwise (28)

The differential equation for the instantaneous inventory q ( t ) at time t in ( 0 , T 4 ) is given by: dq ( t ) dt = p − [ a t 2 + b t + c ] − θ q ( t ) ; for 0 ≤ t ≤ T 1 (29) dq ( t ) dt = d [ p − [ a t 2 + b t + c ] ] − θ q ( t ) ; for T 1 ≤ t ≤ T 2 (30) dq ( t ) dt = e [ p − [ a t 2 + b t + c ] ] − θ q ( t ) ; for T 2 ≤ t ≤ T 3 (31) dq ( t ) dt = − [ a t 2 + b t + c ] ; for T 3 ≤ t ≤ T 4 (32)

With the initial conditions: q ( 0 ) = 0 , q ( T 1 ) = I 1 , q ( T 2 ) = I 2 , q ( T 3 ) = I 3 , q ( T 4 ) = 0 (33)

From equation (29), for 0 ≤ t ≤ T 1 , q ( t ) = ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − bt θ 2 − t 2 θ 2 ) θ 3 (34)

From equation (30), for T 1 ≤ t ≤ T 2 , q ( t ) = d ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − bt θ 2 − t 2 θ 2 ) θ 3 (35)

From equation (31), for T 2 ≤ t ≤ T 3 , q ( t ) = e ( 2 a e − tθ − 2 a − bθ e − tθ + bθ + 2 atθ + c θ 2 e − tθ − c θ 2 − P θ 2 e − tθ + P θ 2 − bt θ 2 − t 2 θ 2 ) θ 3 (36)

From equation (32), for T 3 ≤ t ≤ T 4 , q ( t ) = − e − tθ ( 2 a e tθ − 2 a e Tθ − b θe tθ + b e Tθ θ − 2 atθ e tθ + 2 aTθ e Tθ + c e tθ θ 2 − c e Tθ θ 2 + b te tθ θ 2 + a e tθ t 2 θ 2 − b e Tθ T θ 2 − a e Tθ T 2 θ 2 ) θ 3 (37)

Maximum inventory level I 1 : The maximum inventory level during period ( 0 , T 1 ) is solved as follows from q ( T 1 ) = I 1 I 1 = θ 3 T 1 ( b T 1 + 2 a − a T 1 θ − 2 P + Pθ T 1 ) 2 (38)

Maximum inventory level I 2 : The maximum inventory level during period ( T 1 , T 2 ) is solved as follows from q ( T 2 ) = I 2 I 2 = d T 2 θ 3 2 [ b T 2 + 2 c − a T 2 θ − 2 P + Pθ T 2 ] (39)

Maximum inventory level I 2 : The maximum inventory level during period ( T 1 , T 2 ) is solved as follows from q ( T 3 ) = I 3 I 3 = e T 3 θ 3 2 [ b T 3 + 2 c − a T 3 θ − 2 P + Pθ T 3 ] (40)

For the respective costs in the designed model: 1.

Production cost PC = D c 1 ~

The production cost over the interval ( 0 , T 4 ) : PC = D c 1 ~ PC = c 1 ~ ( c + b T 4 ε ) (41)

The setup cost over the interval ( 0 , T 4 ) : TS = c 4 ~ (42)

The holding cost over the interval ( 0 , T 4 ) : TH = c 2 ~ ∫ 0 T 4 q ( t ) dt (43) TH = c 2 ~ ∫ 0 T 1 q ( t ) dt + ∫ T 1 T 2 q ( t ) dt + ∫ T 2 T 3 q ( t ) dt + ∫ T 3 T 4 q ( t ) dt (44)

Hence, TH = c 2 ~ [ e − T 1 θ 6 θ 3 { 6 ( − 1 + e T 1 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + e T 1 θ T 1 θ ( − 12 a + 6 bθ − 6 c θ 2 + 6 P θ 2 + θ T 1 ( 6 a − 3 bθ − 2 θ T 1 ) } + e − θ ( T 1 + T 1 ) 6 θ 4 { − 6 d ( e T 1 θ − e T 2 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + d e θ ( T 1 + T 1 ) θ ( T 1 − T 2 ) ( 12 a − b 6 θ + 6 θ 2 c − 6 θ 2 P + θ ( 2 θ T 1 2 + T 1 ( − 6 a + 3 bθ + 2 θ T 2 ) + T 2 ( − 6 a + 3 bθ + 2 θ T 2 ) ) ) } + e − θ ( T 2 + T 3 ) 6 θ 4 { − 6 A ( e T 2 θ − e T 3 θ ) ( 2 a − θb + c θ 2 − P θ 2 ) + A e θ ( T 2 + T 3 ) θ ( T 2 − T 3 ) ( 12 a + 6 ( − bθ + c θ 2 − P θ 2 ) + θ ( 2 θ T 2 2 + T 2 ( − 6 a + 3 bθ + 20 T 3 ) + T 3 ( − 6 a + 3 bθ + 2 θ T 3 ) } + 1 6 θ 4 { 6 ( − 1 + e θ ( T 4 − T 3 ) ( 2 a − θb + c θ 2 ) + θ ( ( 12 a − 6 bθ + 6 c θ 2 ) T 3 + 3 θ ( − 2 a + bθ ) T 3 2 + 2 a θ 2 T 3 2 + T 4 6 e θ ( T 4 − T 3 ) ( − 2 a + bθ + cθ T 4 ) − T 4 θ 2 ( 6 c + 3 b T 4 + 2 a T 4 2 ) ) } (45) TH = c 2 ~ [ 1 θ 4 { c − p − T 1 2 θ 3 3 + ( p − c ) θ 2 + ( p − c ) T 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + T 2 2 1 6 θ 4 { T 3 2 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } (46)

The deterioration cost of Finished Products TD = c 3 ~ θ p ∫ 0 T 4 q ( t ) dt (47) TD = c 3 ~ θ p [ ∫ 0 T 1 q ( t ) dt + ∫ T 1 T 2 q ( t ) dt + ∫ T 2 T 3 q ( t ) dt + ∫ T 3 T 4 q ( t ) dt ] (48)

Hence, TD = c 3 ~ θ p [ 1 θ 4 { c − p − T 1 2 θ 3 3 − c θ 2 − cT 1 2 θ 4 2 + p θ 2 + pT 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + 6 d ( T 1 − T 2 ) + ( b θ 3 T 1 + c θ 4 T 1 − p θ 2 T 1 ) } + 1 6 θ 4 { d ( T 2 + T 3 ) ( 6 a T 2 θ 2 − 6 a T 3 θ 2 ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + 1 6 θ 4 { T 3 3 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } ] (49)

The total cost is given by: Total Cost = Setup Cost + Production Cost + Total Holding Cost + Total Deterioration Cost TC = TS + PC + TH + TD (50) TC = [ c 4 ~ + c 1 ~ ( c + b T 4 ε ) + ( c 2 ~ + c 3 ~ θ ) θ 4 [ { c − p − T 1 2 θ 3 3 − c θ 2 − cT 1 2 θ 4 2 + p θ 2 + pT 1 2 θ 4 2 } + 1 6 θ 4 { d ( T 1 + T 2 ) ( 6 a T 1 θ 2 − 6 a T 2 θ 2 ) + d ( T 1 − T 2 ) T 1 T 2 θ 2 ( 3 b T 1 θ 2 + 3 b T 2 θ 2 + 2 T 1 2 θ 2 + 2 T 2 2 θ 2 + 2 T 1 T 2 θ ) + 6 d ( T 1 − T 2 ) + ( b θ 3 T 1 + c θ 4 T 1 − p θ 2 T 1 ) } + 1 6 θ 4 { d ( T 2 + T 3 ) ( 6 a T 2 θ 2 − 6 a T 3 θ 2 ) + d ( T 2 − T 3 ) T 2 T 3 θ 2 ( 3 b θ 2 ( ( T 2 + T 3 ) + 2 θ 2 ( T 2 2 + 2 T 3 2 ) + 2 T 2 T 3 θ ) + 6 d ( T 2 − T 3 ) ( b θ 3 T 2 + c θ 4 T 2 − p θ 2 T 2 ) } + 1 6 θ 4 { T 3 3 θ 2 ( 3 b − 4 a ) + T 4 2 θ 2 ( 3 b − 6 a ) + 4 a T 4 3 θ 3 + θ 2 T 2 T 4 ( 12 a − 6 bθ − 6 a T 4 ) } ] (51)

To simplify the total cost equation by letting T 1 = JT 4 , T 2 = Q T 4 , T 3 = K T 4 (52)

This substitution allows us to express the terms T 1 , T 2 , T 3 in terms of the common variable T 4 which can simplify the overall expression and reduce complexity in calculations and analysis.

Therefore, by inserting (52), the total cost is given by TC = [ c 4 ~ + c 1 ~ ( c + b T 4 ε ) + ( c 2 ~ + c 3 ~ θ ) 6 θ 4 ] { 6 c − 6 c θ 2 − c T 4 ( J − 3 T 4 J 2 + 6 e T 4 Q ( Q − K ) ) θ 4 − 6 p + 6 p θ 2 + p θ 2 T 4 ( 6 e T 4 Q ( − Q + K ) − J + 3 T 4 J 2 θ 2 ) + T 4 ( d ( J − Q ) ( 6 + 6 a T 4 ( J + Q ) θ 2 + T 4 3 JQ θ 3 ( 3 b ( J + Q ) θ + 2 T 4 ( JQ + J 2 θ + Q 2 θ ) ) + θ 2 ( bJθ + 3 b T 4 ( 1 + T 4 K 3 − 2 ( K + eQ ( K − Q ) ) θ + e T 4 2 Q ( Q 2 − K 2 ) K θ 2 + 2 T 4 ( 6 aK − 3 aK − 3 a T 4 K − 2 T 4 K 3 + 3 ae ( Q 2 − K 2 ) + 2 a T 4 θ − θ J 2 + e T 4 3 Q ( Q − K ) K ( QKθ + Q 2 θ 2 + 2 K 2 θ 2 ) ) ) ) } (53)

To minimize the total relevant cost and investigate the optimal time T 4 as follows, differentiates with respect to T 4 .

The equation is given by: ∂ TC ∂ T 4 = 0 , ∂ 2 TC ∂ T 4 2 > 0 (54)