A Study on Two-Warehouse Inventory Systems with Integrated Multi-Purpose Production Unit and Partitioned Rental Warehouse

Abstract

A study of two warehouse inventory systems with a production unit is developed in this article with some constraints which are of practical applicability to optimize the total production cycle and its cost. A production unit evolves in three different states to retain its quality and prolong its lifetime: the state of producing items, the state of reworking the identified defective items, and the state of being idle. It processes the items up to a certain time point. The screening process starts immediately after a product comes out of the production unit. The classified non-defective items are first stored in own warehouse ($OW$), after filling to its maximum capacity, and the remaining items fill in the first block $\left({RW}_1\right)$ of the rental warehouse $\left(RW\right)$. All identified defective items are stored in the second block $\left({RW}_2\right)$ of $RW$. The holding cost of an item is higher in $RW$ than $OW$. All defective items are sent to the production unit for re-do processes as a single lot immediately after the stop of the production and re-do items are stored in ${RW}_1$ to satisfy the demand. The items in the ${RW}_1$ are of higher priority in satisfying the demands after the stop of the production unit in producing new items as to deduce the total cost. Demand is assumed as both time and advertisement dependent and is encouraged once production starts. The deterioration rate differs in both warehouses. No backlog is entertained. The study is directed to achieve optimum total cycle cost towards the attainment of the optimum production time slot and the entire cycle of the system. We have arrived at explicit expressions for the total cost function of the entire production cycle. An analytic optimization process of the discriminant method is employed in the form of an algorithm to arrive at the optimum total cost. It provides a numerical illustration of a specific environment. The implications of the current research work are as follows. The optimum utility of production units in three different states in arriving at the optimum total cost is extensively studied with respect to deterioration, demand, and production rates. It also examined the influence of fluctuating deterioration, demand, and production parameters in arriving at optimum deterioration cost, holding cost, and total cycle cost, as they have important managerial insights. The effect of rental charges on the optimum total cost is examined as the system is used for multi-purpose storage.

1. Introduction

Molding machinery is used in various industries as a production unit for the process of molding raw plastic or material into specific shapes or desired forms. It depends on the material being molded, the complexity of the desired shape, the production volume, and some other factors. A variety of molding machines are available, some of which are as follows: injection molding machines in which the molten plastic is injected into a mold cavity and it is used for producing a wide range of products, which include small components and large automotive parts; extrusion molding machines in which the raw material, usually plastic pellets or rubber, is pushed through a heated extrusion die to create a plastic pipe- or tube-like shape; compression modeling machines in which the preheated material is placed into a mold cavity and then the presser is applied to compress and shape the material to obtain items such as brake pads and electrical materials; blow molding machines, which help to create hollow plastic or glass objects by inflating a hot hollow tube in a mold cavity; rotational molding machines in which the mold is filled with material and then rotated slowly while being heated so that the material melts and coats the interior of the molt to form the desired shape, such as tanks, containers and some playground equipment. Some more molding machines that are under use are thermoforming machines, die-casting machines, vacuum molding machines, transfer molding machines, and injection blow molding machines. These are some of the primary types of molding machinery designed to meet the specific needs of specific product types. The choice of machinery depends on factors such as the desired output, material properties, production time and volume, and other cost considerations.

In general, if the production process prolongs indefinitely, it results in either an increasing percentage of defective items, which means the machine enters out of control, or machine failure occurs due to the generation of thermal effect. It is necessary to follow a certain working schedule, which includes the working state and idle state of the machine. In the case of molding machines, some of the raw materials are reusable after it is identified that the output is defective; in such cases, the re-do processes also continue after the primary production processes are over. In such cases, idle time starts only after the re-do process is over. It motivates us to attempt to analyze the effect of various lengths of production units on optimizing the total cost of the production inventory system in two warehouse storage environments.

Managing two-warehouse inventory systems with an integrated production unit is more complex if we deal with production time-restricted automobile types of machineries used for the production process. Since some types of machines, for example, dye and molding machines, need to be idle or partially working for a certain period of time after producing the item so as to prolong its lifetime and maintain their working efficiency. In general, for each production cycle, the production unit is not allowed to work for the entire cycle. It is stopped after it reaches its assigned task of items produced and kept in idle state, until the start of the next cycle. If not, then it results in an increasing rate of defective items in its production processes and it causes loss to any of the industry irrespective of its scale. Therefore, the processes of cost optimization of any industry which deals with sensitive production units need to be transcendent on the scheduling of production and idle period of the machine in a production cycle. For the purpose of the optimum utility of the production unit, part of its idle time is devoted to re-do the items if any items are identified among its production units in the course of production. In such cases, it is essential to identify the time points such as the stop of the production unit, stop of re-do processes, and idle time durations in each production cycle. Moreover, to entice more customers, the manufacturers, suppliers, and merchants offer a variety of promotions or print product advertisements. As a result, they make use of mainstream media including social media, television, newspapers, movies, and posters. It would be important to investigate functions that more accurately capture the demand rate because it frequently relies on time, just like in actual inventory systems.

This research output will contribute more to the investors who are involved and invest in the two-warehouse inventory management systems of an integrated production unit to optimize the total cost of the system. This article exhibits the following: how the various rates of deterioration in both warehouses influence the optimum production unit slots and optimum total cost, how the parameters in the demand function dominate the optimum production unit slots and optimum total cost, how the production rate and rental cost govern the optimum production unit slots, and optimum total cost.

2. Literature Review and Model Initiation

Table 1. Comparison of our article with the existing relevant literature.

Authors	Number of Warehouses	Demand Nature	Deterioration	Screening	Imperfect Quality	Re-Do Processes	Optimization
Das, S. et al., 2021 [1]	One	Selling price	Constant	NO	NO	NO	Cost
Palanivel, 2014 [2,3]	One	Selling price and cost for advertisement	Continuous probability distribution	NO	NO	NO	Cost
Gautam, P., 2018 [4]	One	Constant	No	Yes	Yes	Yes	Cost
Khara, B., 2021 [5]	One	Constant	No	Yes	Yes	No	Cost
Manna, A.K., 2021 [6]	Two	Constant	Constant	Yes	Yes	Yes	Cost
Sivashankari, C.K., 2014 [7]	One	Constant	No	Yes	Yes	Yes	Cost
Lin, C.S., 2003 [8]	One	Constant	No	Yes	Yes	Yes	Cost
Shah, N.H., 2018 [9]	One	Time and advertisement	Constant		Yes	Yes	Profit
Lin, C.S., 1999 [10]	One	Constant	Exponential		Yes	Yes	Cost
Paul, S.K., 2015 [11]	One	Constant	No	Yes	Rejected	No	Profit
Patra, K., 2018 [12]	One	Advertisement and selling price	No	Yes	Sold with discount price	No	Profit
Sana, S.S., 2010 [13]	One	Constant	No	No	No	No	Cost
Narang, P., 2023 [14]	One	Advertisement, price, and time	Non-instantaneous	Yes	Some % of items are sale at discount price	% are sent for rework	Profit
Mandal, P., 2019 [15]	Two	Stock dependent	No	Yes	Yes	Yes	Cost
Sanjai, M., 2019 [16]	One	Constant	No	Yes	Yes	Yes	Profit
Jauhari, W.A., 2019 [17]	One	Stochastic	No	Yes	Yes	Yes	Cost
Manna, A.K., 2017 [18]	One	Advertisement and time dependent	Production rate dependent	Yes	Sale at reduced cost	No	Profit
Khara, B., 2021 [19]	One	Advertisement	No	No	No	No	Profit
Gautam, P., 2022 [20]	One	Selling price and advertisements.	No	Yes	Yes	Yes	Cost
Lin, T.Y., 2015 [21]	Two	Constant	No	Yes	Yes	Yes	Cost
Singh, S.R., 2013 [22]	Two	Constant	Constant	Yes	Yes	Yes	Cost
Majumder, P., 2016 [23]	Two	Stock	Constant	No	No	No	Cost
Present paper	Multiple	Time and advertisement	Constant	Yes	Yes	Yes	Cost

shows the literature review details for production inventory models.

Article [14] speaks on the single imperfect production system with a re-do of some fraction of defective items and the demand rate is influenced by time, advertisement, and price. The remaining defective items are sold for minimum cost. The non-linear profit function is optimized by genetic algorithm coding. A defective production inventory model is discussed in [20] by a profitable rework strategy on defective items and optimizes the batch size of the production and selling price of the end product. In addition, both energy consumption during the production process and carbon emission cost is considered in optimizing the system cost. The demand rate is influenced by selling prices and advertisements. An integrated inventory system for imperfect production is presented in article [24] by taking demand rate, levels of inventory, selling prices, and advertising frequency into account. It is searched for the lower total cost to both suppliers and vendors.

Mathematical analysis of the production inventory model with the consideration of production system reliability was carried out by [1]. The trade credit policy is implemented and Taylor’s approximation technique is used to optimize the arrived non-linear problem. In order to increase consumer demand and lower deterioration costs, [25] suggests a price discount approach using a Stackelberg-type game concept to address a two-echelon supply system with unpredictable demand. The combined purchasing production inventory system has been the subject of 102 published papers between 1992 and 2021. Article [26] reviews the papers which focus on the complex data type, temporal dynamics, and optimization approach.

Article [5] emphasizes the retailers’ partial advance payment with a free transport facility on a two-echelon integrated production inventory. The retailers’ profit is maximized by the branch and bound technique to obtain an integer solution of the model to find the optimum number of production and replenishment cycles along with the reliability of the system. The retailer and manufacturer interacting with an imperfect production inventory model was developed in [19], with the consideration of reliability factors on the breakable products’ production unit. The effect of the increase in average profit by advertisement is proven by appropriate numerical examples. Article [27] analyzes how quality and sustainability affect sales of perishable goods, and it finds that strict pricing reduces profitability demand fluctuations depending on consumer habits, and quality deteriorates over time.

The imperfect production inventory model is formulated in [16] to attain the optimal minimum total cost by optimizing the production lot size of the system. Two models are discussed with different policies. The first one is with a rework on defective items in the same cycle and no shortages are allowed. The second model is with a rework on defective items but shortages are allowed. The model was validated by numerical examples with the help of Microsoft Visual basic code. In order to maximize profit, project [28] has designed a combined system of inventory prices and control for consumable goods with shortages. It will concentrate on the bestselling prices, repayment terms, and order sizes.

Implementing carbon emissions into inventory modeling helps companies to make informed decisions that balance economic efficiency with environmental sustainability. It helps in identifying emission reduction opportunities, optimizing supply chain operations, and aligning with broader sustainability objectives. The vendor and buyer supply chain management of the production inventory model was developed by [4], with the consideration of one of the greenhouse gas emission carbon emission costs. The individual and joint decisions of the investors are considered in optimizing the cost of the system. Studies [29,30] propose a mixed-integer linear programming model for optimizing the blood supply chain, addressing challenges such as carbon emissions and obsolete products in a real-world case study.

The integrated production inventory model with two states of production units is considered by [6] to maximize the average profit of the system. The production of the number of imperfect items by the production unit while it is out of control is production system random and it depends on rate of the production and the length of out-of-control state of the production unit. Due importance is given to minimizing environmental pollution and the development cost to improve the system reliability is considered. In [9,31], the authors analyzed the EPQ inventory model with reliability as a decision variable in order to maximize the total profit. Some of the hurdles which disturb the production system to produce defective items are considered and the demand is assumed to be influenced by time and advertisement. A single product imperfect production inventory with advertisement and selling price-dependent demand as well as demand depreciation rate and item imperfect production rate-dependent risk function is discussed in [12,32]. The profit is maximized and the risk is minimized as well.

Article [18] speaks of the classical EPQ model of a production inventory system with imperfect production units. A dynamic advertisement-rate-dependent demand rate is considered and the production rate is influenced by the defective rate. The produced items are screened to identify the defective items at a constrained rate. Some important parameters in [33] that impact the optimal total profit are analyzed in the sensitive analysis section. A single-stage production inventory with a rescheduling of production policy after one or two disruptions is modeled by [11]. The model is solved for optimizing the profit by pattern search and by genetic algorithm and has developed a dynamic solution approach for multiple disruption cases. The study [34,35] concludes the SVSB model for food goods, which takes into account the exponential quality degradation and uses the grey wolf optimizer algorithm to maximize joint total profit, exceeding Indonesia’s genetic algorithm. The classical EPQ model of deterministic production inventory with three probabilistically varying deterioration cost functions was studied by [2] with demand influenced by both product cost and its advertisement. Model [36] includes more parameters which include raw and labor forms in the formulation of production cost. The objective of minimizing the manufacturers’ total inventory cost was achieved with a valid numerical example. In [3], an EPQ model was developed and studied the effect of time–value money and money inflation on minimizing the total cost of the inventory induced from optimum cycle length and production quantity. The authors of [37,38] have included: production cost-dependent selling price, variable replenishment, deterioration function as continuous and probabilistic in nature, and increasing linear holding cost function in their model. In article [7], the single-stage production inventory model is developed to obtain optimum quantity where all imperfect items are reworked in the same cycle and achieve optimum total cost. The model is numerically verified and validated by Microsoft Visual Basic code.

The integrated production inventory model is developed by [17] to minimize the expected total cost of the system with the assumption of linearly varying lead time which depends on the size of the production lot and the demand is stochastic in nature. The model in [39] has simultaneously taken the optimality of all safety factors, ordering quantity, production batch, delivery of the frequency, and quality of the production processes into account. The suggested algorithm is verified by a numerical example and emphasizes that the added quality improvement affects the expected total cost in production inventory decision making. The economic manufacturing model in [40] is used in inventory management to determine the optimal production quantity that minimizes the total cost of manufacturing and holding inventory. This model considers several factors, including the setup cost, holding cost, demand rate, production rate, and setup time. By balancing the setup costs and holding costs, the model helps to determine the most cost-effective manufacturing quantity.

An economic manufacturing model with imperfect production units is developed by [13]. The optimal system total cost is arrived at by optimizing safety stock, production rate, and lot size of the production with two types of production policies. A single warehouse-defective production inventory system is analyzed in [8] with batch production and a rework on defective items. The production system is periodically inspected and restored, which minimizes the expected total cost of the system. The model is further extended by the following restrictions: rate of defective as a function of the system setup cost, proportion of non-constant defective items, and limited raw material capacity. In some special cases, the production system may too be subject to deteriorate and produce an incomplete end product as a result. Article [10] speaks on such a production inventory system with constrained raw materials for production and the assumption of batch production. The total cost function is optimized and the model is examined for the cases by considering that the defective items are not constant and the setup cost-dependent defective rate.

A literature review on the two-warehouse inventory model reveals that another important factor in the domain of production inventory with integrated production units is the storage of produced items. End products received from the production unit are stored in a single- or multi-storage inventory. In some cases, from the retailer’s point of view, they may go for additional rental warehouses which are nearby their $OW$ for the storage of excess items after filling their own warehouse to meet the demands. The queue disciplines such as last-in-first-out and first-in-last-out are not made out to play a vital role in studying the queuing model in arriving at effective performance measures but also have a crucial role in the inventory model in order to consider the two warehouses’ inventory which includes one as a rental warehouse with high holding cost. It is the better option to follow a last-in-first-out policy in order to release the items in the rental warehouse first to satisfy the demands even if the items in the rental house are stored last after filling their own house. In article [41], a two-warehouse inventory system (as it includes an additional rental warehouse to store the excess items) with time-dependent demand and rework on defective items is discussed. It reveals that production cycle time minimizes the total relevant cost by the generalized reduced gradient method and establishes that the last-in-first-out system is less expensive while the holding cost in RW is higher.

A study is undergone by [42] with the objective of maximizing the profit by arriving at optimal ordering quantity and size of backlog items on a two-warehouse inventory management system of imperfect quality items. To know the influence of money inflation and different payment options in the study of two-warehouse single product inventory in optimizing the total cost, one can refer to articles [43,44]. Managing inventory across multiple warehouses is a common practice for many businesses to improve distribution, reduce shipping costs, and enhance customer service. There are several strategies and models that can be employed to optimize inventory management in a two-warehouse scenario. A two-warehouse integrated production unit inventory model was studied by [15] under the following environment: the production system may produce imperfect products, and also, demand for products depends on the on-hand stock, vendor discount offers on items to the buyer, and the buyer can go to the rental warehouse with infinite capacity once his own warehouse is filled. The average total cost is minimized for the integrated system. All key parameters of their effects are verified extensively.

A study with the objective of minimizing both supplier and retailer total cost of the two-warehouse EPQ inventory model with two levels of trade credit period was carried out by [23]. Two different scenarios are considered based on the offering and acceptance of credit periods and time of exhaustiveness of items in two warehouses. Payment after the credit period is also discussed as an alternate approach. The generalized gradient method of the non-linear optimization technique is employed. A two-warehouse inventory system with imperfect production units is analyzed in [21]. It includes suppliers, quantity discounts, and maintenance policies to keep the production unit under control. An algorithm to obtain the optimum ordering policy is developed and the system behavior is studied with numerical illustrations. In addition, [22] studies a two-warehouse inventory model with defective production items. Money inflation and item decay are considered in order to obtain optimal cost functions and replenishment cycles.

A literature review on inventory models with a screening process reveals that the major fact which decides the steady and exponential growth of any business is the quality maintenance and upgrading of purchasing products or produced products. Therefore, it is essential for any inventory management system to screen the items either once they come out of the production unit or after purchase for sale. We may refer to [45,46,47] for the study of inventory models with decaying items in order to minimize the total cost under the following environment: effect of preservation technology, advertisement demand, selling price and time demand, the partial backlog of shortages, and trade credit. The model in [48] suggests that the SVSB model for food goods, which takes into account the exponential quality degradation and uses the grey wolf optimizer algorithm to maximize joint total profit, exceeding Indonesia’s genetic algorithm.

The optimality profit for the perishable product inventory system is examined by [49] as two separate models with the exclusion of shortages and inclusion of partial backlogs under the assumption of time and an advertisement frequency-influenced linear holding cost. A solution algorithm for maximizing the profit is suggested. A long-term manufacturing inventory structure was established for maximizing the expected overall revenue using Harris–Hawks Optimization, surpassing GA and PSO in [50] which studies the effect of quality degradation. The authors of [51] study and optimize the total revenue by extending the classical and traditional EPQ/EOQ model with the consideration of imperfect quality items in a single warehouse environment and the assumption of selling all the identified defective items at the end of screening processes as a single lot. This model is extended by [52] in two warehouse environments and maximizes the annual total profit of the model.

Paper [53] builds an internet-of-things-related fuzzy theory model, revealing the importance of the influence of the green delivery factor on firm performance. The inventory management in a primary warehouse with a small capacity and a subsidiary warehouse with a high capacity is considered in article [54]. To choose the best thresholds, it employs a sensitivity analysis and a triple-parameter band policy. The study by [55] proposes a production inventory theory for perishable goods, which are stored in two warehouses with the consideration of exponential and sale price-dependent demand. The triangular fuzzy number is used to minimize the overall cost.

In our focused review of the past literature on production models, two-warehouse inventory models and inventory models with screening facility and re-do, no study has reviewed the two-warehouse inventory system with a integrated multi-purpose production unit with immediate screening of items after it gets produced. It has thus urged us to take up the study on deterministic two-warehouse inventory systems where the production unit evolves in three different states, namely, in the production process, in re-do processes, and being idle. All produced items are screened and the identified good items are for sale and items in excesses are stored in both $OW$ and $RW$ for future sale. The partitioned $RW$ is temporarily used to store the identified defective items for a short time until they are sent as a single lot to the production unit for re-do after it stops production. The novel implementation of a three-state production facility, screening facility, and partitioned utility of rental warehouses in traditional two-warehouse inventory helps many investors who own two-warehouse inventories in the above said environment to optimize their total production cycle cost by finding optimal time points of production stop time and production cycle length.

The arrangement of the article is as follows: the model notations, assumptions, and descriptions are discussed in Section 3, a representation of the models by governing equations is made in Section 4, the cost analysis, total cost function, and optimization of total cost are discussed in Section 5, the model validation by numerical illustration and sensitive analysis are in Section 6, and the last section is devoted to a conclusion and future research directions as Section 7.

3. Notations, Assumptions, and Model Descriptions

The notations and assumptions are provided here for modeling purposes.

3.1. Assumptions of the Model

• The production rate of the system is higher than the demand rate.
• The demand rate is higher compared to the rate of rework processes.
• The RW is used not only to store the excess items but also the defective items which are identified by the screening process.
• The RW has two blocks; say RW₁ to store the excess items after filled the OW of its full capacity. Say RW₂ to store the screened and identified defective items which have to be sent to the re-do facility.
• Both production and screening processes are simultaneously taking place.
• After production is completed, the same machine is used for rework processes.
• After completing the screening processes, all the identified defective items are sent to the production unit as a batch for rework.
• Only the non-defective items may be subject to deterioration.
• No deterioration takes place for the identified defective items in RW₂ which are sent for the re-do process as a single slot.
• All defective items are considered to be made as perfect ones after undergoing the re-do process.
• All defective items sent for re-do processes are received back within the interval [t₂,t₃].

3.2. Model Description

A two-warehouse inventory system which has different characteristics depends on various factors such as the type of goods stored, the business model, location, and logistics requirements. A multi-purpose production unit in inventory modeling is a facility that helps to either produce a variety of products or re-do defective products that are identified in its production slot. These production systems have specific characteristics that distinguish them from single-purpose production units.

We considered a multi-purpose production unit in a two-warehouse environment. There are two warehouses. One of them is $OW$ which is of limited storage capacity for storing items which are produced by a multi-purpose production machine. Once the production unit starts producing new items, all the end products are stored in $OW$ first up to its maximum capacity of $W$. Once the $OW$ reaches its maximum level, the remaining items are stored in a $RW$ with higher holding cost. The production stops at a specific time point $t_2$. The production unit starts producing items at the rate of $’P’$.

We assume that the time between the start of the production unit and the arrival of the first end product to the $OW$ is ignorable. All demands are entertained once the items are stored in $OW$. The screening process stars simultaneously. A part of the $RW$ is used to store the identified defective items for a short span of time until they are sent for re-do processes as a single lot after production is stopped. The utility of the production unit is of multi-purpose and is used to re-do the defective items after production is stopped. Initially, identified non-defective items are stored in the $OW$ which is of limited capacity for accommodating $W$ items. The identified defective items by the production unit up to the time point $t_2$ are stored in a separate block ${RW}_2$ in $RW$. Immediately after a product is identified as non-defective, the arriving demands begin to be satisfied. Time-dependent demand occurs at the rate of $a+b^t$, where $a$ is a constant rate of demand and $b$ is the rate of advertisement which is influenced by time. Excesses of non-defective items are stored in $OW$ until reaching maximum capacity. The non-defective items in the warehouses $OW$ and $RW$ are subject to deterioration with rates ${\phi }_1$ and ${\phi }_2$, respectively. The $OW$ attained its maximum storage capacity at the time point $t_1$. The level of inventory at any point $t$ in the interval $\left[0,t_1\right]$ is dynamic in nature due to the arrival of the identification of non-defective items by screening processes, demand, and deterioration. The production unit is stopped at the time point $t_2$. Identified non-defective items after the time point $t_1$ are stored in block ${RW}_1$ of $RW$. The inventory level in the interval $\left[t_1,t_2\right]$ is governed by demand, deterioration, elimination of defectives, and production.

After production stops, the production unit is engaged with carrying out re-do processes on identified defective items. Items stored in ${RW}_1$ is of priority to sell first to avoid excesses charges. The defective items are made as good quality items by the re-do process and are stored in ${RW}_1$. Items in ${RW}_1$ are decreased and become empty at the time point $t_3$ due to deterioration and demand. From $t_1$ to $t_3$, the items in $OW$ are decreased due to deterioration only. Items from $OW$ are satisfying the demands only after the time point $t_3$. The items in the $OW$ are decreased in the interval [$t_3,t_4$] due to deterioration and demand and reach zero at the time point $t_4$. The diagrammatic representation of the model is depicted in Figure 1.

4. Governing Equations of the Model

Let $I_O\left(t\right)$ represent the inventory level of $OW$ at time t. The following differential equation represents the dynamic inventory level of the $OW$ in the interval $\left[0,t_1\right]$:

$$\frac{d}{dt}{I}_O\left(t\right)=\alpha P-\left(a+b^t\right)-{\phi }_1{I}_O\left(t\right)$$

(1)

We solve (1) by using the condition $I_O\left(0\right)=0$, and we obtain the level of inventory at $t\in \left[0,t_1\right]$ as

$$I_O\left(t\right)=\left(\alpha P-a\right)\left(\frac{{1-e}^{-{\phi }_{1}t}}{{\phi }_1}\right)+\frac{1}{logb+{\phi }_1}\left(-b^t+e^{-{\phi }_{1}t}\right),t\in \left[0,t_1\right]$$

(2)

The screened excesses produced of non-defective items after being filled in $OW$ are stored in block ${RW}_1$ of $RW$. The production unit stops producing the new items at time $t=t_2.$ Dynamic changes in the level of inventory at ${RW}_1$ are influenced by both demand and deterioration. Therefore, the inventory level at time $’t’$ in the interval $\left[t_1,t_2\right]$ is derived as follows:

$$\frac{d}{dt}{I}_{R1}\left(t\right)=\alpha P-\left(a+b^t\right)-{\phi }_2{I}_{R1}\left(t\right)$$

(3)

We arrive at $I_{R1}\left(t\right)$ as

$$I_{R1}\left(t\right)=\left(\alpha P-a\right)\left(\frac{{1-e}^{{\phi }_2\left(t_1-t\right)}}{{\phi }_2}\right)+\frac{1}{logb+{\phi }_2}\left(b{t}_1{e}^{{\phi }_2\left(t_1-t\right)}-b^t\right)$$

(4)

by using the condition $I_{R1}\left(t_1\right)=0$.

As mentioned earlier, all the screened defective items in the production period $\left[0,t_2\right]$ are kept in the block ${RW}_2$ up to time $t_2$ and sent to the production unit as a single batch for re-do processes by the same production unit as a secondary usage. The following differential equation governs the inventory dynamism of defective items in ${RW}_2:$

$$\frac{d}{dt}{I}_{R2}\left(t\right)=\left(1-\alpha \right)P$$

(5)

We obtain

$$I_{R2}\left(t\right)=\left(1-\alpha \right)Pt, t\in \left[0,t_2\right]$$

(6)

by using the condition, $I_{R2}\left(0\right)=0$.

All demands are satisfied by the items from the block ${RW}_1$ of $RW$ immediately after the stopping of the production unit to produce new items. Therefore, the items in $OW$ are decreased only due to the items’ deterioration in the interval $\left[t_1,t_3\right]$.

The level of inventory in $OW$ is governed by the following differential equation:

$$\frac{d}{dt}{I}_O\left(t\right)=-{\phi }_1{I}_O\left(t\right),t\in \left[t_1,t_3\right]$$

(7)

Using the condition, $I_O\left(t_1\right)=W$, we obtain

$$I_O\left(t\right)=W{e}^{{\phi }_1\left(t_1-t\right)}, t\in \left[t_1,t_3\right]$$

(8)

The production system is used for secondary usage to conduct re-do processes once it stops producing new items. All defective items identified by the screening process are sent as a single lot to the production unit for the re-do process.

In the interval $\left[t_2,t_3\right]$, the dynamism of the inventory is governed by the following differential equation and influenced by the occurrence of demand, deterioration, and the arrival of re-do processed items from the production unit.

$$\frac{d}{dt}{I}_{R1}\left(t\right)=R-\left(a+b^t\right)-{\phi }_2{I}_{R1}\left(t\right)$$

(9)

Using the condition $I_{R1}\left(t_3\right)=0$, we obtain the inventory level at $’t’$ in the ${RW}_1$ represented as

$$I_{R1}\left(t\right)=\left(R-a\right)\left(\frac{{1-e}^{{\phi }_2\left(t_3-t\right)}}{{\phi }_2}\right)+\frac{1}{logb+{\phi }_2}\left(-b^t+{bt}_3{e}^{{\phi }_2\left(t_3-t\right)}\right), t\in \left[t_2,t_3\right]$$

(10)

Once the items in the $RW$ become empty, the arriving demands are channelized to $OW$. The items stored in the $OW$ decrease by the occurrence of both demand and deterioration in the interval $\left[t_3,t_4\right]$.

The governing equation representing the inventory level at any $t\in \left[t_3,t_4\right]$ is

$$\frac{d}{dt}{I}_O\left(t\right)=-\left(a+b^t\right)-{\phi }_1{I}_O\left(t\right)$$

(11)

Using the condition $I_O\left(t_4\right)=0$, we obtain

$$I_O\left(t\right)=a\left(\frac{e^{-{\phi }_1\left(t-t_4\right)}-1}{{\phi }_1}\right)+\frac{1}{logb+{\phi }_1}\left(-b^t+{b{t}_{4}e}^{-{\phi }_1\left(t-t_4\right)}\right), t\in \left[t_3,t_4\right]$$

(12)

The total number of items produced up to time $t_2$ is

$$\underset{0}{\overset{t_2}{\int }}Pdt=P{t}_2$$

where the fraction of good quality items is $\alpha P{t}_2$ and the fraction of defective items is $\left(1-\alpha \right)P{t}_2$.

5. Cost Analysis, Total Cost Function, and Optimization

5.1. Cost Analysis

Cost functions involved in this model to optimize the total cost are as follows:

5.1.1. Production Cost

This price includes all processing, machine, labor, and material expenses.

$$PC=P_{1}P{t}_2$$

(13)

5.1.2. Setup Cost

It is the initial cost of establishing the production system which is a fixed price that does not vary with quantity or time. It covers the cost for setups and changeovers.

$$SPC=S_P$$

(14)

5.1.3. Screening Cost

Inspection is performed at all stages of manufacture to guarantee the customers receive only good products. Defective components are given away for re-do most of the times in many of the situations when they are considered as a new one after being processed. It is represented as the following:

$$SC=S_{C}P{t}_2$$

(15)

5.1.4. Rework Cost

It includes the re-do processing, machine, labor, and material expenses.

$$RWE=R_W\left(1-\alpha \right)P{t}_2$$

(16)

5.1.5. Rent for $RW$

It includes the cost paid for occupying the space to store excess items in $RW$.

$$REC=R_C$$

(17)

5.1.6. Holding Cost

The total holding cost in $OW$ is estimated as follows:

$$H{C}_1=H_O\left\{{\int }_0^{t_1}{I}_O(t)dt+{\int }_{t_1}^{t_3}{I}_O(t)dt+{\int }_{t_3}^{t_4}{I}_O(t)dt\right\}$$

(18)

$${\int }_0^{t_1}{I}_O\left(t\right)dt={\int }_0^{t_1}\left\{\left[\alpha P-a\right]\left[\frac{1-e^{-{\phi }_{1}t}}{{\phi }_1}\right]+\left[\frac{1}{logb+{\phi }_1}\right]\left[-b^t+e^{-{\phi }_{1}t}\right]\right\}dt$$

$${\int }_0^{t_1}{I}_O\left(t\right)dt=\left[\alpha P-a\right]\left\{\frac{t_1}{{\phi }_1}+\left[\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1^2}\right]\right\}+\frac{1}{logb+{\phi }_1}\left\{logb\left(1-b^{t_1}\right)+\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1}\right\}$$

$${\int }_{t_1}^{t_3}{I}_O\left(t\right)dt={\int }_{t_1}^{t_3}W{e}^{{\phi }_1{t}_1}{e}^{-{\phi }_{1}t}dt=\frac{W}{{\phi }_1}\left\{1-e^{-{\phi }_1\left(t_3-t_1\right)}\right\}$$

$${\int }_{t_3}^{t_4}{I}_O\left(t\right)dt={\int }_{t_3}^{t_4}\left(\frac{a}{{\phi }_1}\left(e^{{\phi }_1{(t}_4-t)}-1\right)-\frac{1}{logb+{\phi }_1}\left(b^t-b{t}_4{e}^{{\phi }_1{t}_4}{e}^{-{\phi }_{1}t}\right)\right)dt$$

$$=\frac{a}{{\phi }_1}\left(t_3-t_4+\frac{e^{{\phi }_1{(t}_4-t_3)}}{{\phi }_1}-\frac{1}{{\phi }_1}\right)+\frac{1}{logb+{\phi }_1}\left(logb(b^{t_3}-b^{t_4})+\frac{b{t}_4[e^{{\phi }_1\left(t_4-t_3\right)}-1]}{{\phi }_1}\right)$$

Hence, the holding cost for $OW$ is

$$\begin{gathered} {HC}_1=H_O\left[\left\{(\alpha P-a)\left\{\frac{t_1}{{\phi }_1}+\left[\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1^2}\right]\right\}+\frac{1}{logb+{\phi }_1}\left\{logb(1-b^{t_1})+\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1}\right\}\right\}+\left\{\frac{W}{{\phi }_1}\left\{1-e^{-{\phi }_1(t_3-t_1)}\right\}\right\} \\ +\left\{\frac{a}{{\phi }_1}\left\{t_3-t_4+\frac{e^{{\phi }_1{(t}_4-t_3)}}{{\phi }_1}-\frac{1}{{\phi }_1}\right\}+\frac{1}{logb+{\phi }_1}\left\{logb(b^{t_3}-b^{t_4})+\frac{b{t}_4[e^{{\phi }_1\left\{t_4-t_3\right\}}-1]}{{\phi }_1}\right\}\right\}\right] \end{gathered}$$

(19)

The total holding cost in ${RW}_1$ is estimated as follows:

$${HC}_2=H_1\left[{\int }_{t_1}^{t_2}{I}_{R1}\left(t\right)dt+{\int }_{t_2}^{t_3}{I}_{R1}\left(t\right)dt\right]$$

(20)

$${\int }_{t_1}^{t_2}{I}_{R1}\left(t\right)dt={\int }_{t_1}^{t_2}\left\{\left[\alpha P-a\right]\left[\frac{1-e^{-{\phi }_2\left(t-t_1\right)}}{{\phi }_2}\right]+\left[\frac{1}{logb+{\phi }_2}\right]\left[-b^t+b{t}_1{e}^{-{\phi }_2\left(t-t_1\right)}\right]\right\}dt$$

$$=\frac{\alpha P-a}{{\phi }_2}\left\{\left[t_2-t_1\right]+\frac{1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_1}-b^{t_2}\right)-\frac{b{t}_1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}$$

$${\int }_{t_2}^{t_3}{I}_{R1}\left(t\right)dt={\int }_{t_2}^{t_3}\left\{\frac{a-R}{{\phi }_2}\left[e^{{\phi }_2\left(t_3-t\right)}-1\right]+\frac{1}{logb+{\phi }_2}\left[-b^t+{bt}_3{e}^{{\phi }_2\left(t_3-t\right)}\right]\right\}dt$$

$$=\frac{a-R}{{\phi }_2}\left\{\frac{\left(e^{{\phi }_2\left(t_3-t_2\right)}-1\right)}{{\phi }_2}-t_3+t_2\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_2}-b^{t_3}\right)-\frac{b{t}_3}{{\phi }_2}\left[1-e^{{\phi }_2\left(t_3-t_2\right)}\right]\right\}$$

Hence, the total holding cost in ${RW}_1$ is

$$\begin{gathered} {HC}_2=H_1\left[\left\{\frac{\alpha P-a}{{\phi }_2}\left\{\left[t_2-t_1\right]+\frac{1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_1}-b^{t_2}\right)-\frac{b{t}_1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}\right\} \\ +\left\{\frac{a-R}{{\phi }_2}\left\{\frac{\left(e^{{\phi }_2\left(t_3-t_2\right)}-1\right)}{{\phi }_2}-t_3+t_2\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_2}-b^{t_3}\right)-\frac{b{t}_3}{{\phi }_2}\left[1-e^{{\phi }_2\left(t_3-t_2\right)}\right]\right\}\right\}\right] \end{gathered}$$

(21)

The total holding cost in ${RW}_2$ is

$${HC}_3=H_2{\int }_0^{t_2}{IR}_2\left(t\right)dt$$

(22)

$${HC}_3=\frac{H_2\left(1-\alpha \right)P{t}_2^2}{2}$$

(23)

The total holding cost of a production cycle is

$$THC={HC}_1+{HC}_2+{HC}_3$$

(24)

5.1.7. Deterioration Cost

If there is a cost associated with the disposal of an item because of its deterioration, it is derived as follows:

The deterioration cost in $OW$ is

$${DC}_1={{\phi }_{1}D}_1\left\{{\int }_0^{t_1}{I}_O\left(t\right)dt+{\int }_{t_1}^{t_3}{I}_O\left(t\right)dt+{\int }_{t_3}^{t_4}{I}_O\left(t\right)dt\right\}$$

(25)

$$\begin{array}{c}D{C}_1={\phi }_1{D}_1\left[\left\{(\alpha P-a)\left\{\frac{t_1}{{\phi }_1}+\left[\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1^2}\right]\right\}+\frac{1}{\mathit{log}b+{\phi }_1}\left\{\mathit{log}b\left(1-b^{t_1}\right)+\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1}\right\}\right\}+\left\{\frac{W}{{\phi }_1}\left\{1-e^{-{\phi }_1\left(t_3-t_1\right)}\right\}\right\}\right.\\ \left.+\left\{\frac{a}{{\phi }_1}\left\{t_3-t_4+\frac{e^{{\phi }_1\left(t_4-t_3\right)}}{{\phi }_1}-\frac{1}{{\phi }_1}\right\}+\frac{1}{\mathit{log}b+{\phi }_1}\left\{\mathit{log}b\left(b^{t_3}-b^{t_4}\right)+\frac{b{t}_4\left[e^{{\phi }_1\left(t_4-t_3\right)}-1\right]}{{\phi }_1}\right\}\right\}\right]\end{array}$$

(26)

The deterioration cost in $RW$ is

$${DC}_2={{\phi }_{2}D}_2\left({\int }_{t_1}^{t_2}{I}_{R1}\left(t\right)dt+{\int }_{t_2}^{t_3}{I}_{R1}\left(t\right)dt\right)$$

(27)

$$\begin{gathered} {DC}_2={{\phi }_{2}D}_2\left[\left\{\frac{\alpha P-a}{{\phi }_2}\left\{\left[t_2-t_1\right]+\frac{1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_1}-b^{t_2}\right)-\frac{b{t}_1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}\right\} \\ +\left\{\frac{a-R}{{\phi }_2}\left\{\frac{\left(e^{{\phi }_2\left(t_3-t_2\right)}-1\right)}{{\phi }_2}-t_3+t_2\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_2}-b^{t_3}\right)-\frac{b{t}_3}{{\phi }_2}\left[1-e^{{\phi }_2\left(t_3-t_2\right)}\right]\right\}\right\}\right] \end{gathered}$$

(28)

Hence, the total deterioration cost for the entire production cycle is

$$TDC={DC}_1+{DC}_2$$

(29)

5.2. Total Cost Function and Optimization Algorithm

The entire production cost represents the sum of all of the following costs: production, setup, rework, rental, holding, screening, and deterioration. Therefore, the total cost for the entire production cycle is

$$TC=PC+SPC+SC+RWE+REC+THC+TDC$$

(30)

The total cost for the entire production cycle per unit of time is

$$\begin{array}{c}TC\left(t_2,t_4\right)=\frac{1}{t_4}[P_{1}P{t}_2+S_P+S_{C}P{t}_2+\left\{R_W(1-\alpha )P{t}_2\right\}+R_C+(H_O\\ +{\phi }_1{D}_1)[\left\{(\alpha P-a)\left\{\frac{t_1}{{\phi }_1}+\left[\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1^2}\right]\right\}+\frac{1}{logb+{\phi }_1}\left\{logb(1-b^{t_1})+\frac{1-e^{-{\phi }_1{t}_1}}{{\phi }_1}\right\}\right\}\\ +\left\{\frac{W}{{\phi }_1}\left\{1-e^{-{\phi }_1\left(t_3-t_1\right)}\right\}\right\}\\ +\left\{\frac{a}{{\phi }_1}\left\{t_3-t_4+\frac{e^{{\phi }_1\left(t_4-t_3\right)}}{{\phi }_1}-\frac{1}{{\phi }_1}\right\}+\frac{1}{logb+{\phi }_1}\left\{logb(b^{t_3}-b^{t_4})+\frac{b{t}_4\left[e^{{\phi }_1\left(t_4-t_3\right)}-1\right]}{{\phi }_1}\right\}\right\}]+(H_1\\ +{\phi }_2{D}_2\left)\right[\{\frac{\alpha P-a}{{\phi }_2}\left\{[t_2-t_1]+\frac{1}{{\phi }_2}\left[e^{{\phi }_2\left(t_1-t_2\right)}-1\right]\right\}\\ +\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_1}-b^{t_2}\right)-\frac{b{t}_1}{{\phi }_2}[e^{{\phi }_2\left(t_1-t_2\right)}-1]\right\}\}\\ +\left\{\frac{a-R}{{\phi }_2}\left\{\frac{\left(e^{{\phi }_2\left(t_3-t_2\right)}-1\right)}{{\phi }_2}-t_3+t_2\right\}+\frac{1}{logb+{\phi }_2}\left\{logb\left(b^{t_2}-b^{t_3}\right)-\frac{b{t}_3}{{\phi }_2}\left[1-e^{{\phi }_2\left(t_3-t_2\right)}\right]\right\}\right\}]\\ {+\left\{\frac{H_2(1-\alpha )P{t}_2^2}{2}\right\}]}_{}\end{array}$$

(31)

The total cost function and its concavity is represented in Figure 2.

We optimize the above objective cost function $TC$ of the entire production cycle, which depends on the parameters $t_2^{\ast }$ and $t_4^{\ast }$.

5.3. Optimizing Algorithm

In differential calculus, the theory of optimization of functions of two variables is applied in optimizing the total cost function. In the functioning of any automobile production machine, its idle time is essential to retain its performance efficiency and prolongation of its lifetime.

Let $\left(t_2^{\ast },t_4^{\ast }\right)$ be the desired optimum point which will minimize the objective function $TC\left({t_2,t}_4\right)$. The necessary condition to attain the local maximum or the local minimum points for the cost surface $TC\left({t_2,t}_4\right)$ is ${\left[TC\left({t_2,t}_4\right)\right]}_{t_2}=0$ and ${\left[TC\left({t_2,t}_4\right)\right]}_{t_4}=0$. For negligibly small values of $l_1$ and $l_2$, we have

$$∆=TC\left({t_2+l_1,t}_4+l_2\right)-TC\left({t_2,t}_4\right)$$

After omitting higher powers of $l_1$ and $l_2$ in its Taylors expansion, we obtain $∆$ approximately as

$$∆\approx \frac{1}{2{\left[TC\left({t_2,t}_4\right)\right]}_{t_2{t}_2}}[{\left\{l_1{\left\{TC\left({t_2,t}_4\right)\right\}}_{t_2{t}_2}+l_2{\left\{TC({t_2,t}_4\right\}}_{t_2{t}_4}\right\}}^2+l_2^2\left\{\psi \right\}]$$

(32)

where $\psi =$ [{TC(t₂,t₄)}_t2t2 {TC(t₂,t₄)}_t2t4 − [{TC(t₂,t₄}_t2,t4]²].

From (32), we observe that $∆>0$ if $\psi >0$ for $l_1$ and $l_2$ which are sufficiently small. Optimization is attained by the following steps:

• Step 1: Derive the expressions:${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_2},{\left\{TC\left({t_2,t}_4\right)\right\}}_{t_4}$, ${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_2{t}_2}$, and ${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_4{t}_4}$.
• Step 2: Solve ${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_2}=0$ and ${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_4}=0$ for critical points.
• Step 3: Check and identify the critical points, say $\left(\overline{t_2},\overline{t_4}\right)$ which give the values of both ${\left\{TC\left({t_2,t}_4\right)\right\}}_{t_2{t}_2}>0$ and $\psi >0$.
• Step 4: Among the points, chose a point $\left(\overline{t_2},\overline{t_4}\right)$ which gives a minimum value to $\left({t_2,t}_4\right).$

Such point $\left(\overline{t_2},\overline{t_4}\right)=\left(t_2^{\ast },t_4^{\ast }\right)$ is the required point which minimizes the value of $TC\left({t_2,t}_4\right)$, and the minimum total cost of the production cycle is $TC\left(t_2^{\ast },t_4^{\ast }\right)$.

The cost optimization algorithm is displayed as a flow diagram in Figure 3.

6. Model Validation: Numerical Illustration and Sensitive Analysis

6.1. Numerical Illustration

The reliability, accuracy, and flexibility of any inventory model is ensured by the numerical example with appropriate performance metrics. We consider a particular environment with the following parameter values: production rate of the production unit as $P=70$ items per unit of time, screening rate of identifying the non-defective items as $\alpha =0.86$ per unit of time, rate of re-doing the defective items by the same production unit immediately after the stop of production process as $R=29$, rates of item deterioration on $RW$ and $OW$ are as ${\phi }_1=0.21$ and ${\phi }_2=0.19$, respectively (since the inbuilt facilities in $RW$ is higher than $OW$). The parameter in the demand rate $D\left(t\right)$ is $a=68$ and $b=0.85$. The maximum accommodation of items in $OW$ is $W=200$ items and it is attained at $t_1=3$ unit of time. The production cost for an item in a unit of time is $P_1=30$ money unit, the setup cost per order is $Sp=1800$ money unit, the screening cost per unit of item $Sc=8$ money unit, the rework cost per item in a unit of time is $Rw=15$ money unit, the rental cost for the utility of $RW$ up to the time point $t_3=9$ unit of time is $Re=700$ money unit, the holding cost of an item in $OW$ in a unit of time is $H_o=4$ money unit, the holding cost of an item in $R{W}_1$ in a unit of time is $H_1=3$ money unit. The holding cost of an item in $R{W}_2$ in a unit of time is $H_2=4$ money unit, the deterioration cost of an item in $OW$ in a unit of time is $D_1=6 \mathrm{money} \mathrm{unit}$, the deterioration cost of an item in $RW$ in a unit of time is $D_2=5$ money unit. In the above specified environment, with the help of MATLAB code, the optimum values are arrived as follows: the optimum time points $t_2^{\ast }=6.48$ and $t_4^{\ast }=10.25$ which optimize the total cost as ${TC}^{\ast }=3266.09$ money units for the entire production cycle.

6.2. Sensitive Analysis on Different Environments

The robustness and wholesomeness of any inventory model are ensured only when the sensitivity of the parameters within the reasonable range are analyzed. It helps the researchers to contribute more recommendations and suggestions to the investors in the portfolio of supply chain management. Henceforth, the sensitivities of the deterioration rate of the warehouses’ demand rate, production rate, and screening rate are tested in the processes of optimizing the total cost.

6.2.1. Effect of Various Deterioration Rate in Optimizing Total Production Cycle Cost

We first analyze the sensitivity of the deterioration rate ${\phi }_1$ of the items in the $OW$ by varying it from 0.17 to 0.41 on arrival, ${TC}^{\ast }$ in the same environment which was discussed in numerical illustration Section 6.1. All observations are listed in the first part of

Table 2. Effect of various deterioration rates in RW and OW on optimizing total production cycle cost.

	${\mathit{\phi }}_1\downarrow$	${\mathit{D}\mathit{P}}^{\ast }\downarrow$	${\mathit{t}}_4^{\ast }\downarrow$	${\mathit{D}\mathit{R}}^{\ast }\downarrow$	${\mathit{D}\mathit{I}}^{\ast }\downarrow$	${\mathit{T}\mathit{D}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{H}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{C}}^{\ast }\downarrow$
	0.21	6.48	10.25	2.52	1.25	650.24	597.37	3266.09
	0.25	6.48	10.67	2.52	1.67	684.26	565.88	3189.16
	0.29	6.48	10.75	2.52	1.75	706.83	537.83	3169.27
Part I	0.33	6.48	10.72	2.52	1.72	723.69	514.74	3168.46
	0.37	6.48	10.66	2.52	1.66	737.73	495.98	3174.55
	0.41	6.48	10.59	2.52	1.59	749.07	480.26	3183.02
	0.45	6.48	10.52	2.52	1.52	758.39	466.95	3192.00
	${\phi }_2\downarrow$
	0.20	5.91	10.16	3.09	1.16	665.06	580.43	3122.99
	0.21	5.61	10.11	3.39	1.11	655.71	573.73	3034.31
Part II	0.22	5.45	10.07	3.55	1.07	652.41	557.64	2977.50
	0.23	5.37	10.05	3.63	1.05	649.14	542.70	2940.46
	0.24	5.35	10.04	3.65	1.04	645.60	528.73	2916.29
	0.25	5.34	10.03	3.66	1.03	641.57	515.51	2900.78
	0.26	5.34	10.02	3.66	1.02	638.42	504.62	2878.17

, from which we predict and suggest that an increase in the deterioration rate ${\phi }_1$ causes fluctuation in ${TC}^{\ast }$ with the effect of the increase in total production cycle duration $t_4^{\ast }$ and also it does not have an influence over the optimum production period $t_2^{\ast }$ of the production cycle. As more items deteriorate, the optimum total deterioration cost ${TDC}^{\ast }$ increases, and at the same time, the optimum total holding cost ${THC}^{\ast }$ decreases. Figure 4 clearly exhibits the stability of production and re-do durations and the changes of idle durations of the production unit. Figure 5 illuminates the effect of the decrease in optimum total holding cost and the increase in optimum total deterioration cost and optimum total cost of the entire production cycle due to the fluctuation.

We consider the effect of fluctuating the deterioration rate ${\phi }_2$ of items in the $RW$ by varying it from $0.20$ to $0.26$ on arriving at the ${TC}^{\ast }$ in the same environment. All observations are listed out in part II of Table 2, from which we arrive at the conclusion that the increase in the deterioration rate ${\phi }_2$ causes a decrease in ${TC}^{\ast }$ as the effect of decrease in total production cycle duration $t_4^{\ast }$ and also there is a decrease in optimum production period $t_2^{\ast }$ of the production cycle. As an effect, the ${TDC}^{\ast }$ and ${THC}^{\ast }$ values are decreases. Figure 6 clearly exhibits the influences of variation of deterioration rate ${\phi }_2$ on the slot of production unit. It decreases the production time slot, increases the length of the re-do processing time slot, and decreases the idle time slot of the production unit as well. Figure 7 illuminates the effect of the decreasing trend of optimum total holding cost, optimum total deterioration cost, and optimum total cost of the entire production cycle as an effect of the increase in ${\phi }_2$.

6.2.2. Effect of Various Demand Rate in Optimizing Total Production Cycle Cost

The optimum cost functions and the slot of the production unit are listed in

Table 3. Effect of various demand rates in optimizing total production cycle cost.

	$\mathit{a}\downarrow$	${\mathit{D}\mathit{P}}^{\ast }\downarrow$	${\mathit{t}}_4^{\ast }\downarrow$	${\mathit{D}\mathit{R}}^{\ast }\downarrow$	${\mathit{D}\mathit{I}}^{\ast }\downarrow$	${\mathit{T}\mathit{D}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{H}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{C}}^{\ast }\downarrow$
	$52$	5.67	10.71	3.33	1.71	1078.62	984.03	3782.18
	$56$	5.88	10.59	3.12	1.59	975.07	890.93	3660.66
	$60$	6.09	10.48	2.91	1.48	869.15	795.15	3535.01
Part I	$64$	6.29	10.36	2.71	1.36	761.43	761.43	3402.43
	$68$	6.48	10.25	2.52	1.25	650.24	597.37	3266.09
	$72$	6.67	10.14	2.33	1.14	537.26	494.90	3125.12
	$76$	6.84	10.02	2.16	1.02	421.32	389.93	2979.71
	$b\downarrow$
	0.86	5.65	10.39	3.35	1.39	659.59	604.83	3031.46
	0.87	5.15	10.50	3.85	1.50	658.21	602.69	2875.75
	0.88	4.81	10.58	4.19	1.58	653.99	598.41	2764.84
Part II	0.89	4.57	10.65	4.43	1.65	648.97	593.38	2681.60
	0.9	4.39	10.71	4.61	1.71	644.02	588.52	2616.56
	0.91	4.24	10.76	4.76	1.76	640.40	585.19	2564.03
	0.92	4.12	10.81	4.88	1.81	636.38	581.03	2520.92

for varying the constant demand parameter $a$ and the advertisement-influenced demand parameter $b$. From observation of part I in Table 3, as we vary the value of demand parameter $a$ from $52$ to $76$, production slots ${DR}^{\ast }$ and ${DI}^{\ast }$ are decreased, and as a result, both ${TDC}^{\ast }$ and ${THC}^{\ast }$ are decreased which influence the decrease in ${TC}^{\ast }$, and at the same time, ${DP}^{\ast }$ increases. Figure 8 clearly exhibits the production slot variations and Figure 9 illuminates the effect of the decrease in optimum total holding cost, optimum total deterioration cost, and optimum total cost of the entire production cycle.

From observation of part II in Table 3, as we vary the value of demand parameter $b$ from $0.86$ to $0.92$, production slot ${DP}^{\ast }$ decreases but ${DR}^{\ast }$ and ${DI}^{\ast }$ increase, and as a result, both ${TDC}^{\ast }$ and ${THC}^{\ast }$ decrease; thus, influencing a decreased ${TC}^{\ast }$. Figure 10 clearly exhibits the production slot variations and Figure 11 illuminates the effect of a decrease in optimum total holding cost, optimum total deterioration cost, and optimum total cost of the entire production cycle.

6.2.3. Effect of Various Production Rate in Optimizing the Total Production Cycle Cost

In any production inventory model, the production rate is the more essential parameter that influences all other model parameters and cost functions. The effect of changing the production rate $P$ over a range of $65$ to $95$ is shown in

Table 4. Effect of various production rates in optimizing total production cycle cost.

$\mathit{P}\downarrow$	${\mathit{D}\mathit{P}}^{\ast }\downarrow$	${\mathit{t}}_4^{\ast }\downarrow$	${\mathit{D}\mathit{R}}^{\ast }\downarrow$	${\mathit{D}\mathit{I}}^{\ast }\downarrow$	${\mathit{T}\mathit{D}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{H}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{C}}^{\ast }\downarrow$
65	6.87	10.17	2.13	1.17	487.45	447.02	2970.23
70	6.48	10.25	2.52	1.25	650.24	597.37	3266.09
75	6.13	10.42	2.87	1.42	801.61	736.68	3547.50
80	5.81	10.57	3.19	1.57	946.55	870.00	3816.4
85	5.52	10.71	3.48	1.71	1086.09	998.04	4074.33
90	5.26	10.84	3.74	1.84	1220.02	1120.66	4322.54
95	5.01	10.96	3.9	1.96	1351.55	1241.06	4562.03

. The increase in ${DR}^{\ast }$ and ${DI}^{\ast }$ influence the increase in ${TDC}^{\ast },{THC}^{\ast }$, and ${TC}^{\ast }$ even if there is a decrease in ${DP}^{\ast }$. Figure 12 clearly exhibits the production slot variations and Figure 13 illuminates the effect of an increase in optimum total holding cost, optimum total deterioration cost, and optimum total cost of the entire production cycle.

6.2.4. Effect of Various Rental Cost in Optimizing the Total Production Cycle Cost

In the two-warehouse inventory model, the rental cost affects the cost functions. The effect of changing the rental cost $R_e$ over a range from $700$ to $6700$ is shown in

Table 5. Effect of various rental cost of $RW$ in optimizing total production cycle cost.

${\mathit{R}}_{\mathit{e}}\downarrow$	${\mathit{D}\mathit{P}}^{\ast }\downarrow$	${\mathit{t}}_4^{\ast }\downarrow$	${\mathit{D}\mathit{R}}^{\ast }\downarrow$	${\mathit{D}\mathit{I}}^{\ast }\downarrow$	${\mathit{T}\mathit{D}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{H}\mathit{C}}^{\ast }\downarrow$	${\mathit{T}\mathit{C}}^{\ast }\downarrow$
700	6.48	10.25	2.52	1.25	650.24	597.37	3266.09
1700	6.48	10.31	2.52	1.31	657.22	602.42	3363.36
2700	6.48	10.36	2.52	1.36	663.21	606.80	3460.10
3700	6.48	10.42	2.52	1.42	667.45	612.24	3556.36
4700	6.48	10.47	2.52	1.47	677.03	616.91	3652.05
5700	6.48	10.53	2.52	1.53	684.92	622.72	3747.27
6700	6.48	10.58	2.52	1.58	691.68	627.81	3842.02

. The increase in ${DI}^{\ast }$ influences the increase in ${TDC}^{\ast },{THC}^{\ast }$, and ${TC}^{\ast }$. Figure 14 clearly exhibits the rental cost variations and Figure 15 illuminates the effect of the increase in optimum total holding cost, optimum total deterioration cost, and optimum total cost of the entire production cycle.

7. Conclusions and Future Research Directions

The two-warehouse production inventory with three different states of the production unit are called the production slot, re-do slot, and idle slot and produce the new items, re-do the identified defective items, and are in an idle state, respectively. The optimum total production cycle cost is explicitly arrived at with respect to the optimization variables’ total production slot time and total cycle duration. It was achieved in the environment of different deterioration rates in each warehouse with a common demand rate. An analytic expression for the total production cycle cost is arrived at and optimized analytically by the discriminant method. A numerical illustration is provided in a particular sustainable environment. The sensitivity of the parameters, deterioration rates, demand rates, and production rates are analyzed and observations are tabulated with diagrammatic exploration. The importance of maintaining the parameter values in arriving at the optimum total cost are explored in the tables and figures as a suggestion and recommendation which are more essential to any investor who is dealing with deteriorated inventory with an integrated production unit facility. Some of the main observations are an increase in deterioration rate results in a decrease in the optimum total cost of the system as a result of a decrease in holding costs and an increase in deterioration costs; an increase in the constant parameter of the demand function decreases the optimum total cost as the total cycle time decreases; an increase in the advertisement parameter decreases the optimum total cost as the production slot decreases; an increase in production rate improves the optimum total cost; and an increase in rental cost increases the optimum total cost of the cycle.

The work will lead researchers towards the spectrum of an inbuilt service station to enhance the satisfactory level of arriving demands, possible integrated server vacation policy, trade credit policies, money inflation, and partial delay payment of different cost functions. A new venture is gained in the direction of searching for and optimizing the total cost by either fixing the sufficient idle time period for the production system which enables us to retain the quality of the production unit and prolong its lifetime for the $RW$ or essential production time period which boosts the total revenue as we entertain the $RW$ with infinite space or an unavoidable re-do processing time in order to generate money to avoid loss on defective items, as there is no guarantee that any manufacturing machine can produce perfect items 100% of the time.