Fuzzy Inventory Model for Imperfect Items with Price Discount and Penalty Maintenance Cost

Abstract

The present paper considers the fuzzy economic manufacturing model (FEMM) for an inventory model with an imperfect production process that has been studied along with rework. During the pandemic, it is evident that the products accumulated without a sale, which has increased the maintenance cost of the products. This research paper compares a special sale of products with discount and without discount prices both in the fuzzy environment and in the crisp case. New computing methods based on fuzzy logic are being utilized to enhance identification, decision making, and optimization. A triangular fuzzy number is applied in the economic production quantity to emphasize the importance of optimal manufacturing. The EPQ model’s optimal total cost is obtained in the crisp version. It is to be noted that this model is developed in the fuzzy sense by using the deterioration as a triangular fuzzy number. The applications of this model in the fields are constructing customized industrial machinery or heavy-duty construction equipment, specific chemicals, and processed food. By using MATLAB R2021, a numerical example of the optimal solution is provided. Finally, the present research discusses how changing several parameters affects the optimum total cost.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The extension of the economic production lot size (EPL) model with the production rate as a decision variable was introduced in [1], and they assumed that the quality of production items deteriorates with the increase in the production rate. Cárdenas-Barrón et al. [2] extended the economic production quantity with the planned backorders for a unique production along with the imperfect quality items, and reworking is allowed for imperfect items. Koide and Sandoh [3] discussed the strategies for increasing the day’s revenue in the monopolist firm and dealt with the discount pricing for unsold products using reference effects and inventory constraints. Taleizadeh et al. [4] explained the multi-product economic production quantity model using unique production and backorders. However, in this paper, they introduced a concept of immediate rework without scrapped items left. Also, they compared the result with rework and the result without rework. Taleizadeh et al. [5] extended the manufacturing process for decaying items providing no permanent discount in unit procuring cost, and shortage quantities are determined using closed-form equations. Taleizadeh and Noori-Daryan [6] explained the production inventory in a three-level supply chain for multi-product and also aimed at the rework. Thus, production system with rework and special sale with price discount has recently become interesting research subject. Inspection during production helps to maintain the quality of items produced. This helps to detect the root of defects instantly and is helpful for any industry that wants to improve production, reduce defect rates, reduce rework, and avoid waste. The central inspection team reviews all reports to segregate the damaged goods into the remanufacturing sector. The review of the process should be repeated during production, which results in maintaining the process and the quality of items.

Each unit of inventory plays an economic measure in the books of business. The real-life biodegradable materials should sell in bulk orders with multiple shipments within the predetermined time. Suppliers opt for advertising to promote their products or services if the products remain stagnated. Worldwide, it is estimated that companies spend billions of dollars on advertisements.Please rephrase as follows: "Planning and implementing the stock handling of the consolidated demand is a challenging one. The somatic power of both specifications is also tough to handle. Universally e-commerce sales are chosen to be even higher. The product’s profit is directly related to how much or how little they can reduce the holding cost, shipment cost, price discount with low interference to entry, and undemanding scalability, an attractive return on investment for many industries. A profitable manufacturing quantity replica is developed under an imperfect production process with the inspection team. This team continuously reviews the damaged products and sends them for rework. After the production gets over, a multi-shipment policy is carried out to supply the demand. Here disjoint delivery is applied to practice. Even though some items are likely to be unsold, suppliers promote their products by advertising. Advertisement plays a vital role for both the manufacturers and the customers. It results in increasing business turnaround. These days, advertising can be achieved using various media like television, newspaper, radio, WhatsApp, Facebook, and so on. Advertisement expects to reach many customers. As a result, demand got increased, so backlogging occurs. It is to be noted here that when the rework impact of inventory is great enough, the company should display a positive supply. Food products were probably the first to be kept, being set aside during harvest months for use in winter. Food was prepared in several methods to prevent it from wastage, such as drying, smoking, pickling, or sealing it in water and air-tight containers and storing it in a cool place. Modern refrigeration systems help us to store agricultural items with a slight alteration to their nature. The managerial application of inventory management is to forecast accurate demand and lead the manufacturer to supply enough inventory to meet the demand. The ordered volumes are significantly rising due to holidays or other occasions throughout the year, such as a significant advertisement. Historical and seasonal data can also be used to identify any sales pattern that calls for modifying stock levels at different times of the year.

Due to the imperfect production plan, damaged products are produced, and they can be reworked after the completion of one cycle. Finally, perfect quality items are supplied to the customers. The shortages are fully satisfied by the reworked items. The firm replenishes and sells a product under the scarcity effect of the inventory and then examines a finite horizon periodic combined rework and inventory management model. Scarcity is allowed and fully satisfied with the reworked items. A multi-shipment policy is carried out to supply demand. In general, the researchers have assumed that the demand is fixed. Demand is always influenced by the item’s selling price, existing population, degradation, and how frequently the product is advertised. A few scholars developed inventory models throughout the time that included decaying products, things in short supply, different demand patterns and costs, production of particular items, and combinations of these.

Fuzzy logic is a technology that uses computational intelligence to learn knowledge. A new fuzzy profit maximization inventory model with shortages is proposed by Samadi et al. [7]. Demand is considered a power function of price, marketing, and service expenditure. Furthermore, the unit cost is determined as a power function of order quantity. Zhang et al. [8] explained the concept of triangular fuzzy numbers and the defuzzification of triangular fuzzy numbers using different methods. This paper adopts the graded mean integration representation method for defuzzification. Many academicians and researchers have concentrated on demand in this stream, but only a few have proven the best inventory management practices with pricing strategies. According to Kundu et al. [9], fuzzy possibility measures are applied to achieve the optimal cost and optimum inventory level for this model in a fuzzy environment. Anil Kumar et al. [10] developed a cost optimization model for inventory management having fuzzy demand and deterioration with a two-warehouse facility under trade credit. The graded mean integration representation (GMIR) technique is used for defuzzification because of getting the results in crisp versions. Sahoo and Acharya [11] developed an inventory model for deteriorating seasonal items in which demand is price-dependent, and dealers invest in preservation technology to slow down the rate of deterioration in a crisp and fuzzy environment. Traditional ideas about fuzzy parameters fuzziness have not changed through time, but in practice, the system’s fuzziness gradually started to disappear. This analysis motivated the researchers to solve the mathematical model for reworked items during production under the fuzzy framework. The main objective of this study is to increase sales. The main contribution and motivation of the study are that the manufacturers frequently used price reductions to attract customers to buy the products in bulk orders, including the reworked products.

2. Literature Review

Koide and Sandoh [3] discussed the strategies for increasing the day’s revenue in the monopolist firm and also dealt with the discount pricing for unsold products using reference effects and inventory constraints. Taleizadeh et al. [12] proposed a manufacturing model with a unique product production machine providing limited capacity, but no rework allows for the shortage. Taleizadeh et al. [4] investigated a multi-product production quantity using a single machine with the rated production capacity. This paper started with a single product case and extended the concept to the multi-product case with the reworked and scrapped items. Taleizadeh et al. [13] explained a manufacturing process allowing rework, multi-product, and lost sales. A new concept is discussed in this paper. It reduces the manufacturing cost, which undergoes service level and budget limitations. Taleizadeh et al. [14] investigated a multiproduct production quantity using a single machine with the rated production capacity. This paper started with a single product case and extended the concept to the multiproduct case with the reworked and scrapped items.

Cárdenas-Barrón et al. [15] developed the previous work [13] with the same shipment for inventory but presented the optimizing procedure of replenishing a considerable size, including delivery cost. Taleizadeh and Noori-Daryan [6] represented a multiple supplier-single manufacturer-multiple retailer model. Here Stackelberg’s game theory is applied to get the optimal solution. Zhou and Chen [16] dealt with the retrieval process on the inventory control model. In this analysis, they established the multi-echelon stock relationship between the supplier and the customer through the manufacturer. Sekar and Uthayakumar [17] focused on vendor-buyer stock management during production management with no breakdown. The total material cost for vendor-buyer is demonstrated in this model. Chakraborty et al. [18] explained the best replenishment practices for a shop to maximize total profit in an imprecise environment. In supply chain (SC) modeling, Khara et al. [19] introduced the concept of delay in payment and free shipment for the advance payment to obtain the optimum profit function. Branch and bound techniques are used to obtain the optimum replenishment cycle. Kuppulakshmi et al. [20] derived the efficient measure for the fish production process under pentagonal fuzzy number with the multi-shipment. This paper also dealt with the multi-shipment inventory production for decaying items. A new definition of the interval optimization problem in parametric form was introduced by Rahman [21], and he concluded that the established results are justified by proving the inventory problem. Rani et al. [22] assumed that the electronic industry utilizes reverse logistics to collect used products and refurbish them. These products are then sold in the market at a reduced price. They highlight the importance of a green supply chain in the electronic industry and establish a comprehensive, integrated model in the green supply chain.

According to the above research findings, considering system uncertainty is critical in ensuring the industrial sector’s economic and environmental sustainability. The present study fills the research gap by applying this economic production quantity model under the fuzzy framework by considering uncertain parameters, such as deterioration rate, which can be treated as a triangular fuzzy number in order to reduce the total manufacturing cost. With this review, it is straightforward to stay at the forefront of research, remain current with best practices, and evaluate the particular area in fuzzy EPQ models. The novelty of this paper is proved by introducing different cost functions in the production process and proving that the EPQ model attains optimal value. According to the findings, operators should account for flexibility in the input parameter demand and energy to deal with the uncertainties. Table 1 gives an overview of different types of findings in the same area. The contribution of this study with respect to the present literature studies is established in Table 1.

Table 1

Research gap with existing literature.

Author	Inventory control system	Shortage	Deterioration	Fuzzy model	Advertisement	Price discount/delay payments
Kundu et al. [9]	EPQ	No	Yes	Yes	No	No
Rahman [21]	EPQ	No	Yes	No	No	No
Taleizadeh et al. [4]	EPQ/rework	Yes	Yes	No	No	No
Cárdenas-Barrón et al. [2]	EPQ/rework	No	Yes	No	No	No
Sekar and Uthayakumar [17]	EPQ/rework	Yes	Yes	No	No	No
This paper	EPQ/rework	Yes	Yes	Yes	Yes	Yes

Definition 1.

Triangular fuzzy number: a fuzzy number $A=x_1,x_2,x_3$ is considered as a triangular fuzzy number when it possesses the ensuing membership function:$$\begin{array}{}\text{(1)}& {\mu }_A{u}^{\prime }=\begin{array}{cc}0\text{,}& u^{\prime }<x_1\text{,}\\ \frac{u^{\prime }-x_1}{x_2-x_1}\text{,}& x_1\le u^{\prime }\le x_2\text{,}\\ 1\text{,}& u^{\prime }=x_2\text{,}\\ \frac{x_3-u^{\prime }}{x_3-x_2}\text{,}& x_2\le u^{\prime }\le x_3\text{,}\\ 0\text{,}& u^{\prime }>x_3\text{.}\end{array}\end{array}$$

Defuzzification of TFN by the graded mean representation method is given by the following.

If $\tilde{p}$ is a triangular fuzzy number $\tilde{p}=p_1,p_2,p_{3,}\text{,}$ then$$\begin{array}{}\text{(2)}& E\tilde{p}=\frac{p_1+4p_2+p_3}{6}\text{.}\end{array}$$

Selling price-dependent demand for decaying items is considered in this research.

3. Presupposition

This application is performed in the area of inventory control. In this regard, a simple production inventory model is formulated under the following assumptions.

(1) Constant demand and decaying rate are considered.

(2) Backlogging products are supplied to the customers with the reworked items.

(3) Manufacturing items should be more than the demand.

(4) The inspection team audits decaying products.

(5) Machine failure is not allowed during production.

(6) The demand for the products occurs during the period $0,T_2\text{.}$

(7) Advertisement cost is linearly dependent on time $a+b\tau$ during period $\mu ,T_2$, where $a,b$ are constants.

(8) Penalty maintenance cost is applied in the process for the stagnation of products.

(9) During shipment period $\mathrm{\mu },T_2$, two cases are discussed.

(i) With price discount $\eta \ge N$.

(ii) Without price discount $\eta <N$.

(10) A constant rate is allowed for the rework process.

4. Notations

  ${I_p}_1$: operational stock size in production period

  ${I_p}_2\text{:}$ operational stock size during manufacturing process

  ${I_p}_3$: operational inventory level in rework period

  ${I_r}_1$: retrievable stock level in manufacturing process

  ${I_r}_2$: retrievable stock level in non-manufacturing process

  ${I_r}_3$: retrievable stock level in rework process

  ${I_{\tau }}_{P_1}$: complete operational stock level in manufacturing process

  ${I_{\tau }}_{P_2}$: complete operational stock level in non-manufacturing process

  ${I_{\tau }}_{P_3}\text{:}$ complete operational stock level in rework period

  ${I_{\tau r}}_1\text{:}$ complete recoverable stock level in manufacturing process

  ${I_{\tau r}}_2\text{:}$ complete recoverable stock level in non-manufacturing process

  ${I_{\tau r}}_3$: complete recoverable stock level in rework period

  ${I_V}_1\text{:}$ complete recoverable inventory in non-manufacturing process

  $I_{\max R}$: maximum stock level of retrievable products in a production setup

  $I_{\max RR}$: maximum stock level of retrievable products in retrievable process

  $P_r\text{:}$ retrievable process rate (per unit)

  $A_p$: organization setup (cost/setup)

  $A_r$: retrievable manufacturing system (cost/setup)

  $H_P$: operational product holding cost (per unit)

  $H_r$: retrievable item holding cost (per unit)

  $\omega$: decaying rate (per unit)

  $d$: demand rate (per unit)

  $P$: rate of manufacturing

  $K$: cost of manufacturing/unit item

  $c$: backordering cost per unit time

  $\mu$: time period for deterioration

  $\alpha$: percentage of reworkable products

  $r$: price discount per unit time

  $\mathrm{S}\mathrm{H}\mathrm{C}\text{:}$ shortage cost in rework period

  $\mathrm{P}\mathrm{D}$: price discount in rework period

  $\mathrm{I}\mathrm{N}\mathrm{C}$: inspection cost during manufacturing process

  $\mathrm{A}\mathrm{D}\mathrm{C}$: advertisement cost

  $\mathrm{P}\mathrm{M}\mathrm{C}$: penalty maintenance cost in shipment period

  ${\theta }_T$: complete decaying unit

  $T_1$: complete manufacturing period

  $T_2$: complete shipment period

  $T_3$: total time in retrievable period

  $T$: total cycle period

  $n$: number of cycle

  $N$: number of orders

  $I_{\max }\text{:}$ maximum stock level in production period

  $\eta$: number of items purchased at price discount

  $\mathrm{S}\mathrm{P}\mathrm{C}$: shipment cost

  $m$: number of shipment

5. Problem Formulation

The production of the product starts from $\tau =0$, and the stock level is null in this situation. The inspection team continuously scrutinizes the products, and then the products are returned to the retrieval process. In the $0,T_1$ period, supply and manufacturing will start together and the stock level reaches a high $I_{\max }$ this period $\tau =T_1$. During this period, the manufacturing and on-hand stock level will reduce. In the interval $0,T_2$, perfect products are distributed by fixed time interval $T_2=\overline{{\tau }_1}+\overline{{\tau }_2}+\dots +\overline{{\tau }_m}$, as shown in Figure 1.

[figure(s) omitted; refer to PDF]

During period $\mu ,T_2$, there is no supply, and the inventory level remains the same. So, the manufacturer gives an advertisement for the product with two options,

  Case (i). Purchasing bulk amount of product with price discount $\eta \ge N$.

  Case (ii). Purchasing the product without a price discount $\eta <N\text{.}$

After the advertisement, demand for the product increases and backlogging occurs. During the time $0,T_3$, the retrieval manufacturing begins, as shown in Figure 1. The retrieval stock level reaches $I_{\max RR}$ and the backordered quantity is satisfied by the reworked products. The working process of the production inventory is explained in Figure 2.

[figure(s) omitted; refer to PDF]

5.1. Manufacturing Inventory Level

The stock level of manufacturing products increases moderately from time $0,T_1$, and the retrieval products are grouped apart. The manufacturing stock level is given as$$\begin{array}{}\text{(3)}& \frac{d{I}_{p_1}{\tau }_1}{d{\tau }_1}+\omega I_{p_1}{\tau }_1=1-\alpha P-d,0\le {\tau }_1\le T_1\text{,}\end{array}$$when $I_{p_1}0=0$. The stock level is minimized in the non-manufacturing period $0,T_2$ and is given as$$\begin{array}{}\text{(4)}& \frac{d{I}_{p_2}{\tau }_2}{\mathrm{d}{\tau }_2}+\omega I_{p_2}{\tau }_2=-d,0\le {\tau }_2\le T_2\text{,}\end{array}$$when $I_{p_2}0=0\text{.}$ The stock level in the retrieval time $0,T_3$ is$$\begin{array}{}\text{(5)}& \frac{d{I}_{p_3}{\tau }_3}{\mathrm{d}{\tau }_3}+\omega I_{p_3}{\tau }_3=p_r-d,0\le {\tau }_3\le T_3\text{,}\end{array}$$with the condition that $I_{p_3}0=0$. The effective economic manufacturing model for retrieval items is solved in two cases.

5.2. Case (i)

Buying more products with price discount $\eta \ge N$: the production stock level is obtained by solving (1) which is given as$$\begin{array}{c}\text{(6)}& \frac{d{I}_{p_1}{\tau }_1}{\mathrm{d}{\tau }_1}+\omega I_{p_1}{\tau }_1=1-\alpha P-d\text{,}{I}_{p_1}{\tau }_1=\frac{1-\alpha p-d}{\omega }1-e^{-\omega {\tau }_1}\text{.}\end{array}$$

The entire stock level during the manufacturing time is given as$$\begin{array}{}\text{(7)}& I_{\tau p_1}{\tau }_1=\underset{0}{\overset{{\tau }_1}{{\displaystyle \int }}}\frac{1-\alpha p-d}{\omega }1-e^{-\omega {\tau }_1}d{\tau }_1\text{.}\end{array}$$

The largest powers are negligible like$$\begin{array}{c}\text{(8)}& \frac{{\omega }^3{T_1}^3}{3!}\ll 1\text{,}{I}_{\tau p_1}{\tau }_1=\frac{p1-\alpha -d}{2}{T_1}^2\text{.}\end{array}$$

The non-manufacturing stock level is obtained in the interval $0,T_2$:$$\begin{array}{}\text{(9)}& \frac{d{I}_{p_2}{\tau }_2}{\mathrm{d}{\tau }_2}+\omega I_{p_2}{\tau }_2=-d\text{.}\end{array}$$

The entire stock level during non-manufacturing period is given as$$\begin{array}{c}\text{(10)}& I_{p_2}{\tau }_2={{\displaystyle \int }}_0^{T_2}\frac{d}{\omega }{e}^{-\omega {\tau }_2}-1\mathrm{d}{\tau }_2\text{,}{I}_{\tau p_2}{\tau }_2=\frac{d}{2}{T_2}^2\text{.}\end{array}$$

When $I_{p_1}=I_{p_2}$, ${\tau }_1=T_1$, and ${\tau }_2=0\text{,}$ the following is obtained:$$\begin{array}{}\text{(11)}& \frac{1-\alpha p-d}{\omega }1-e^{-\omega {\tau }_1}=\frac{d}{\omega }{e}^{-\omega {\tau }_2}-1\text{.}\end{array}$$

Then,$$\begin{array}{}\text{(12)}& T_2=T_1-\frac{\omega {T_1}^2}{2}1-\frac{1-\alpha P}{d}\text{.}\end{array}$$

The stock level in the retrieval period is given as$$\begin{array}{c}\text{(13)}& \frac{{\mathrm{d}I}_{p_3}{\tau }_3}{{\mathrm{d}\tau }_3}+\omega I_{p_3}{\tau }_3=p_r-d\text{,}{I}_{\tau p_3}{\tau }_3=\frac{p_r-d}{2}{T_3}^2\text{.}\end{array}$$

The retrieval stock level is given as$$\begin{array}{}\text{(14)}& \frac{d{I}_{r_1}{\tau }_{r_1}}{\mathrm{d}{\tau }_{r_1}}+\omega I_{r_1}{\tau }_{r_1}=\alpha P,0\le {\tau }_{r_1}\le T_1\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{I}_{r_1}0=0\text{.}\end{array}$$

Solving the above, it is derived as$$\begin{array}{}\text{(15)}& I_{\tau r_1}{\tau }_{r_1}=\frac{\alpha P}{2}{T_1}^2\text{.}\end{array}$$

Retrieval stock level is denoted by $I_{\max R}$ at beginning.$$\begin{array}{c}\text{(16)}& I_{\max R}=\frac{\alpha P}{\omega }1-e^{-\omega T_1}\text{,}{I}_{\max R}=\alpha P{T}_1-\frac{\omega T_1^2}{2}\text{.}\end{array}$$

In the shipment period, the retrieval stock is$$\begin{array}{}\text{(17)}& {\frac{{\mathrm{d}I}_{r_2}{\tau }_{r_2}}{{\mathrm{d}\tau }_{r_2}}+\omega I_{r_2}{\tau }_{r_2}=0,I}_{r_2}0=0,\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\tau }_{r_2}\le T_2\text{.}\end{array}$$

Initially,$$\begin{array}{}\text{(18)}& I_{r_2}0=I_{\max R}\text{.}\end{array}$$

The entire stock level of the retrieval items is given as$$\begin{array}{c}\text{(19)}& I_{{\tau }_{r_2}}{\tau }_{r_2}={{\displaystyle \int }}_{{\tau }_{r_2}=0}^{n-1T_1+n{T}_2}{I}_{\max }{e}^{-\omega {\tau }_{r_2}}{\mathrm{d}\tau }_{r_2}\text{,}{I}_{V_1}={\displaystyle \sum }_{s=1}^n{{\displaystyle \int }}_{{\tau }_{r_2}=0}^{s-1T_1+s{T}_2}{I}_{\max R}{e}^{-\omega {\tau }_{r_2}}{\mathrm{d}\tau }_{r_2}\text{,}\\ I_{V_1}={\displaystyle \sum }_{s=1}^n{I}_{\max R}s-1T_1+s{T}_2-\frac{\omega {s-1T_1+s{T}_2}^2}{2}\text{.}\end{array}$$

Maximum inventory level is defined as$$\begin{array}{}\text{(20)}& I_{\max RR}{\displaystyle \sum }_{s=1}^n{I}_{\max R}1-\omega s-1T_1+s{T}_2+\frac{{\omega }^2{s-1T_1+s{T}_2}^2}{2}\text{.}\end{array}$$

The stock level of retrieval items is$$\begin{array}{c}\text{(21)}& \frac{d{I}_{r_3}{\tau }_{r_3}}{\mathrm{d}{\tau }_{r_3}}+\omega I_{r_3}{\tau }_{r_3}=-P_r\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}{\tau }_{r_3}\le T_3,I_{r_3}0=0\text{,}{I}_{{{\tau }_r}_3}{\tau }_{r_3}=\frac{P_r}{\omega }{e}^{\omega T_3-{\tau }_{r_3}}-1\text{.}\end{array}$$

The total recoverable items are$$\begin{array}{}\text{(22)}& I_{{{\tau }_r}_3}=\frac{P_{r{T_3}^2}}{2}\text{.}\end{array}$$

Equation (22) can be written as$$\begin{array}{c}\text{(23)}& I_{\max RR}=\frac{P_r}{\omega }{e}^{\omega T_3}-1\text{,}{T}_3=\frac{I_{\max RR}}{P_r}\text{.}\end{array}$$

The entire manufacturing and retrieval stock level is given as$$\begin{array}{c}\text{(24)}& \mathrm{T}\mathrm{o}\mathrm{T}\mathrm{I}=n{I}_{{\tau }_{p_1}}+n{I}_{{\tau }_{p_2}}+I_{{{\tau }_p}_3}\text{,}\mathrm{T}\mathrm{R}\mathrm{I}=n{I}_{{\tau }_{r_1}}+I_{V_1}+I_{{\tau }_{r_3}}\text{,}\\ {\theta }_T=n1-\alpha P{T}_1+P_r{T}_3-nd{T}_1+nd{T}_2+nd{T}_3\text{.}\end{array}$$

5.3. The Cost Function of Manufacturing Process

The process of performing inspections during manufacturing is known as manufacturing assessment. This inspection technique aids in the control of product quality by helping repair flaws as soon as they are discovered. This is important to any manufacturer looking to increase productivity, reduce process variation, and reduce emending.

(1) End-of-line fault is reduced.

(2) Final inspection time and effort are minimized.

(3) It aids in the early detection of problems and the avoidance of common errors.

(4) It assists in guaranteeing the effectiveness of manufacturing products. The scrutinizing cost also known as inspection cost (INC) during the manufacturing time is given as$$\begin{array}{c}\text{(25)}& \mathrm{I}\mathrm{N}\mathrm{C}=\frac{I_c{I}_r}{T_1}\underset{0}{\overset{T_1}{{\displaystyle \int }}}Pd{\tau }_1\text{,}\mathrm{I}\mathrm{N}\mathrm{C}=I_c{I}_{r}P\text{.}\end{array}$$

The deterioration of the products starts at the time period $\mu$ (time period for deterioration). During the non-production time $\mu ,T_2$, the demand remains stagnated, so the maintenance cost turns into penalty maintenance cost (PMC) which is given as$$\begin{array}{}\text{(26)}& \mathrm{P}\mathrm{M}\mathrm{C}=\pi T_2-\mu d,\pi =3.141\text{.}\end{array}$$

The supplier wants to promote the product through advertisement during shipment, as supply still needs to grow. An advertising strategy is a plan for communicating with customers and persuading customers to buy a product. Informing the relative benefits to the customer about the products from the alternative pathways leads to minimizing the resulting choices, which will give financial restrictions. In reality, this means that objectives must be defined, the environment must be comprehended, the means must be prioritized, and decisions must be taken depending on available resources. Because resources are always finite, effective product analysis, market categorization, media research, and financial choices result in an optimum strategy, never the perfect plan. The advertisement cost (ADC) is given as$$\begin{array}{}\text{(27)}& \mathrm{A}\mathrm{D}\mathrm{C}=\frac{A{d}_{c}A{d}_r}{T_2}\underset{\mu }{\overset{T_2}{{\displaystyle \int }}}a+b{\tau }_2\mathrm{d}{\tau }_2\text{,}\end{array}$$where a, b are constants.

Deduction in price for large scale in the period $\mu ,T_2$ is$$\begin{array}{c}\text{(28)}& \mathrm{P}\mathrm{D}=\frac{kr}{T_2}{{\displaystyle \int }}_{\mu }^{T_2}d\mathrm{d}{\tau }_2\text{,}\mathrm{P}\mathrm{D}=\frac{\mathrm{k}\mathrm{d}\mathrm{r}{T}_2-\mu }{T_2}\text{.}\end{array}$$

When a customer is ready to buy, stockouts are characterized as the unavailability of specific items or products at the point of purchase. The supply expected exceeds the proportion available. The goods will be backordered, indicating that the order has already been placed, but the item is unavailable now. During the retrieval period, the backordering cost is calculated as$$\begin{array}{c}\text{(29)}& \mathrm{S}\mathrm{H}\mathrm{C}=c\underset{0}{\overset{T_3}{{\displaystyle \int }}}-d\mathrm{d}{\tau }_3\text{,}\mathrm{S}\mathrm{H}\mathrm{C}=-cd{T}_3\text{.}\end{array}$$

The total cost is given as$$\begin{array}{}\text{(30)}& \mathrm{T}\mathrm{C}=\frac{n{A}_P+A_r+H_p\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{I}+H_r\mathrm{T}\mathrm{R}\mathrm{I}+\omega {\theta }_T+\mathrm{I}\mathrm{N}\mathrm{C}+\mathrm{P}\mathrm{M}\mathrm{C}+\mathrm{A}\mathrm{D}\mathrm{C}+\mathrm{P}\mathrm{D}+\mathrm{S}\mathrm{H}\mathrm{C}+m\mathrm{S}\mathrm{P}\mathrm{C}}{n{T}_1+T_2+T_3}\text{.}\end{array}$$

5.4. Case (ii)

Buying the product without price discount $\eta <N$: the total cost is given as$$\begin{array}{}\text{(31)}& \mathrm{T}\mathrm{C}=\frac{n{A}_P+A_r+H_p\mathrm{T}\mathrm{O}\mathrm{T}\mathrm{I}+H_r\mathrm{T}\mathrm{R}\mathrm{I}+\omega {\theta }_T+\mathrm{I}\mathrm{N}\mathrm{C}+\mathrm{P}\mathrm{M}\mathrm{C}+\mathrm{A}\mathrm{D}\mathrm{C}+\mathrm{S}\mathrm{H}\mathrm{C}+m\mathrm{S}\mathrm{P}\mathrm{C}}{n{T}_1+T_2+T_3}\text{.}\end{array}$$

Closed-form solutions cannot be determined because the cost function equation is a non-linear equation and also the second derivative of equation (31) with respect to $T_1$ is very difficult. This implies that the best option cannot be assured. However, practical experiments show that equation (31) is convex for a small value of $T_1$ in this case. Alternative methods can be used to determine the optimal $T_1$ value. The outcomes of the empirical experiment are validated using MATLAB R2021 software. Customer loyalty is promoted via discounts for large buyers and regular consumers. Promotional discounts provide transient benefits such as increased sales, revenue, and profit when implemented appropriately. More units are sold during a short-term discount period, allowing the corporation to minimize inventory and temporarily spur economic growth. Discounts used might often lead to a downward pricing spiral, affecting your capacity to sell the goods at full price. Because they buy in bulk, large merchants can negotiate price discounts from suppliers and then use a discount pricing strategy effectively. So, the total cost function in both cases is analyzed.

6. Numerical Illustrations

This section contains a numerical example to study the feasibility of the proposed integrated model. The values of the parameters in this model have been realistic. Under practical assumptions, the set of equations cannot be solved analytically. The overall goal of the field of numerical analysis is to design and analyze techniques to give approximate but accurate solutions to production management.

For example, insurance companies use numerical programs for actuarial analysis. This paper uses numerical analysis to get the actual value for the uncertain parameters. The following examples have been solved to determine the optimal total cost and production run time. Referring to the numerical data obtained from case study, a numerical illustration has been done. Also, as per case study, $P=$$1500/unit, $P_r=$$900, d = $700, $\omega =0.08$, $A_P=$$9/unit, $A_r=$$1.5/unit, $H_P=$$4.5/unit, k = 5, $H_r=$$0.6/unit$0.6/unit,α = 0.06, π = 3.414, r = 0.1, µ = 0.001, c = 0.15, $A{d}_c=$$0.03/unit, $A{d}_r=0.15$, $I_c=$$1.5/unit, $I_r=4.5$, m = 5, SPC = 100, a = 0.1, N = 250, and b = 5. The manufacturing time and the optimal total cost are given as $T_1$ = 2.4978 and TC = $165360 in case (i) and $T_1$ = 2.4978 and TC = $163690 in case (ii).

7. Fuzzy Environments

The parameters are assumed to be fixed in the crisp case. The accurate value for the proportion of retrieval products, the proportion of scrap items, and the repair period are treated as triangular fuzzy numbers.

$\tilde{\omega }$ = deteriorating rate (per unit). Suppose $\tilde{\omega }={\omega }_1,{\omega }_2,{\omega }_3$ is a non-negative triangular fuzzy number (TFN). The total production cost (TC) is given as$$\begin{array}{}\text{(32)}& E\widetilde{\mathrm{T}\mathrm{C}}=E\widetilde{{\mathrm{T}\mathrm{C}}_1},E\widetilde{{\mathrm{T}\mathrm{C}}_2},E{\widetilde{TC}}_3\text{.}\end{array}$$

The outcome of fuzzy parameter changes in proportion to deteriorating rate (per unit) is given as $\tilde{\omega }=0.060.080.1$. By using the GMIR method, defuzzification is done.$$\begin{array}{}\text{(33)}& E\tilde{\omega }=\frac{{\omega }_1+4{\omega }_2+{\omega }_3}{6}\text{.}\end{array}$$

${\tilde{T}}_1=2.4978$ and $\mathrm{T}{\mathrm{C}}^{\ast }=$$146423 for case (i) and ${\tilde{T}}_1=2.4978$ and $\mathrm{T}\mathrm{C}=$$163715 for case (ii). While comparing the crisp case with the fuzzy case, the optimal minimal total cost is obtained, and the profit is preferably high when compared with the crisp case. The optimal solution is achieved in case (i) under the fuzzy environment than in case (ii).

8. Solution Procedure

Only manufacturing alone is considered for finding the optimal production time $T_1$ (31).

Step 1.

Replace n = 1 in (31), differentiate with respect to $T_1\text{,}$ and apply $\partial TC{T}_1/\partial T_1=0$.

Step 2.

Find the value of $T_1$ for equation (31); increasing the n value gives the corresponding $T_1$ values.

Step 3.

Substitute $T_1$ in equation (31), and we get TC.

Tables 2 and 3 provide the entire total cost of the manufacturing process when the number of cycles increases. As a result, the number of cycles increases production cost decreases, as shown in Figure 3.

Table 2

Optimal total cost with price discount.

n	$T_1$	TC
1	2.498	165360
2	1.767	58503
3	1.443	31967
4	1.251	20865
5	1.119	15006
6	1.022	11477
7	0.947	9155
8	0.886	7532
9	0.836	6343
10	0.793	5440

Table 3

Expected total cost without price discount.

n	$T_1$	TC
1	2.498	163690
2	1.767	57903
3	1.443	31683
4	1.251	20650
5	1.119	14852
6	1.022	11359
7	0.947	9061
8	0.886	7456
9	0.836	6279
10	0.793	5385

[figure(s) omitted; refer to PDF]

8.1. Discussion of Sensitivity Analysis

Table 4 shows that different parameters like P, d, $P_r$, m, $I_c$, $A_p$, $H_p$, $H_r$ are analyzed for sensitivity analysis, and it shows the total cost variations in different criteria. Increasing the parameter shows fluctuations in the total cost, which is visualized in Figure 4.

Table 4

Expected total cost with price discount.

Parameter	−30%	−20%	−10%	10%	20%	30%
A (p)	165350	165350	165350	165360	165370	165370
D	339840	260600	205430	135250	112000	93625
P	53171	82280	119380	221100	287480	365390
I c	150850	155680	160520	170200	175030	179870
m	164640	164880	165120	165600	165840	166080
H p	139310	147990	156680	174040	182730	191410
P r	165210	165280	165330	165380	165390	165400
H r	164740	164950	165150	165560	165770	165970

[figure(s) omitted; refer to PDF]

Table 5 shows that the total production cost variations (without price discount) drastically change the cost functions shown in Figure 5.

Table 5

Calculation of total cost without price discount.

Parameter	−30%	−20%	−10%	10%	20%	30%
A p	363230	363240	363240	363250	363260	363260
D	735120	565810	448310	299560	250540	211940
P	51975	80926	117870	219270	285490	363250
Ic	339020	347090	355170	371330	379400	387480
m	362330	362630	362940	363550	363860	364170
H p	295530	318100	340680	385820	408340	430960
P r	362990	363120	36320	363280	363300	363310
H r	361920	362360	362800	363690	364130	364580

[figure(s) omitted; refer to PDF]

Table 6 shows the total cost variations (with price discount) caused by the extreme change in the PMC and ADC which are shown in Figure 6.

Table 6

Price discount, penalty maintenance cost, and advertisement cost for case (i) with price discount.

n	PD ($)	PMC ($)	ADC ($)
1	349.847	5011	0.026
2	349.790	3659	0.019
3	349.746	3030	0.016
4	349.710	2647	0.014
5	349.677	2381	0.013
6	349.648	2183	0.012
7	349.621	2028	0.011
8	349.596	1903	0.010
9	349.573	1799	0.010
10	349.551	1710	0.011

[figure(s) omitted; refer to PDF]

Table 7 shows that the number of cycles increases, the penalty maintenance cost decreases, and the advertisement cost decreases in the least possible range, as shown in Figure 7.

Table 7

Penalty maintenance cost and advertisement cost for case (i) without price discount.

n	PMC ($)	ADC ($)
1	5012	0.026
2	3660	0.019
3	3031	0.016
4	2647	0.014
5	2381	0.013
6	2184	0.012
7	2029	0.011
8	1903	0.010
9	1799	0.010
10	1710	0.010

[figure(s) omitted; refer to PDF]

Similarly, sensitivity analysis for penalty maintenance cost and advertisement price is deliberate with and without price discount. PMC and price discounts are increased according to the cycles in both cases. After some cycles, the advertisement cost remains the same after achieving regular customers.

9. Sensitivity Analysis

Researchers use sensitivity analysis to determine the effects of different variables in their production models. Sensitivity analysis can help make predictions about uncertain parameters like holding cost, production rate, inspection cost, shipment cost, price discount, and setup cost; some of the variables affect the total cost of the manufacturing process, including the cycle time. The analysis can be refined for future price predictions by making different assumptions or adding different variables. This model can also be used to determine the effect of changes in price rates. In this case, the total cost depends on the above uncertain parameters. Sensitivity analysis allows for forecasting using historical, true data. By studying all the variables and the possible outcomes, important decisions can be made in production management.

(1) The total cost of the manufacturing process increases in all the cost parameters.

(2) The demand, production, and retrieval rate are increased because of the increasing total cost..

(3) Fluctuations in the iteration of the cycle lead to many fluctuations in the promotional price and the cost of production.

(4) Increasing the advertisement cost leads to an increase in total cost.

(5) Penalty maintenance cost is decreased as the number of iterations increases.

10. Conclusion

The economic production quantity with inspection cost with retrieval is executed for case (i) and case (ii). This shows that the total cost without a price discount gives a feasible solution. The demand rate significantly impacts the expected total cost, optimal ordered quantity, and expected total profit. The retrieval process is at the termination of one manufacturing cycle. However, it can be developed that retrieval is done after some time sequence to avoid the loss by selling a large number of deteriorated items. The total manufacturing cost is diminished, and the optimal cycle time and backlogging items are obtained. The penalty maintenance cost leads to advertisement costs and price discounts. This paper uses a profit maximization single objective function to build an EPQ model that integrates industrial reworking, deterioration, backlogging, and multishipment in a fuzzy environment. Through the modeling of rework manufacturing, the traditional concept of considering decaying parameters is assumed as a fuzzy number along with a price discount. However, the researchers explore two separate scenarios based on price discount for supply, including different cost functions, and then divide them into with price discount and without price discount. The researchers calculated the overall profit obtained, the total cost expended, and the total cost generated in the manufacturing process at each stage. It is because of the flexibility of the various cost components involved in the fuzzy system that is used. High financial investment plays a vital role in the warehouse industry. Many containers and trucks are required to transport food products like chili, groundnut, maize, and rice. A very high financial setup is needed for one production plane, including toll, diesel, broker, transport, harbour, and maintenance costs. The cost of the new medium-range container is nearly 1.5 lakhs, and the second-hand vessel the warehouse owners use is nearly 10 K to 0.5 lakhs in North Chennai, Tamil Nadu, India. It is possible to carry 60 tonnes of food items in one production cycle for a new container and 30 tonnes of food items in one production cycle for the second-hand containers. In order to control inaccurate measurements during shipping, the owners will be able to balance the supply and demand of goods. This conclusion validates the overall cost value, but adding transportation costs can alter how fuzzy parameters are discriminated. This comprehensive and more suitable view can alter the warehouse plan (by providing price reductions for large purchases) discerned as optimal in the supply chain, leading to increased cooperation and ultimately turning them more profitable and feasible for all users.

This study presented the first model involving advertisement cost, PMC, and price discount in the production inventory (EPQ) model after the introduction of the EPQ model in 1918. The fluctuating sale prices of commodities and the versatility of all those cost components involved in the manufacturing process affect the total cost of this model. The present research also has scope for future researchers. The presented model in this research can examine the production setup with dual channels under the framework of types 1 and 2 fuzzy sets by considering some uncertain parameters. Finally, sensitivity analysis is discussed for different criteria with the numerical computation proved.

Additional Points

There are a few limitations to this suggested form of the inventory system, and they are as follows. (i) There will be only one item and one stocking point in the inventory system. (ii) Here, the suggested technique is limited to only discounts. (iii) Total cost increases steadily when this model is used in a fuzzy approach.

Consent

Informed consent was obtained from all individual participants included in the study.

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