Hexagonal Fuzzy-Based Review on Imperfect EPQ Models Involving Rework and Lost Sales Penalties

This research discusses imperfect quality reworkable products with penalty lost sale and Hexagonal Fuzzy Numbers (HFN). Production cost, backordered price and penalty bygone sale price could be treated like fuzzy parameters. A Graded Mean Difference Method (GMDM) is applied for the method of defuzzification. While several prior studies have applied fuzzy set approaches in manufacturing systems, this research introduces a novel concept—the use of HFN in modelling the rework process.

Taleizadeh et al. [1] explained an EPQ model where manufacturing imperfect rate sticks to a uniform distribution. Finally, they prove that, they obtained derivation is to be convex and the entire price is minimized. Singh et al. [2] updated a sample to supply chain network with reverse logistics inventory sample providing a remanufacturing of returned items. The solution for this method is solved by hessian matrix to obtain the minimum total average cost. Gani and Sabarinathan [3] investigates an optimal ordering strategy for inventory systems under a fuzzy framework. The main focus is on maximizing the retailer’s profit within a single production cycle, where cost parameters are modeled as fuzzy numbers to address uncertainty. Goyal et al. [4] presented the paper with demand variation based on the time. They made use of genetic algorithm to derive the EPQ sample for reducing the entire relevant amount. Singh et al. [2] designed a production sample using ramp type order rate with trade credit period is allowed in two categories.

Shastri et al. [5] examined the EOQ with two categories of business offers for analyzing how to manage the chain of supplies. A retailer's plays a major role in this model. Three algorithms are discussed to get an optimal replenishment cycle time. Tayal et al. [6] discussed the paper with seasonal products which undergo weibull deterioration with two markets namely primary and alternative. Priyan et al. [7] explained with an EOQ inventory sample which has two kinds of fuzzy numbers namely triangular and trapezoidal. Classical differential optimization technique is adopted for finding an optimal solution. Priyan and Uthayakumar [8] designed the degradation rate in TFN and startup investment is decreased by investing extra amount. Finally, it was shown that the total cost of the production is minimized. Shah and Cardenas-Barron [9] designed an inventory sample for analyzing the concepts of offering the trade credit to the customers and retailers. For retailers, the cash discount and the order link trade credits are allowed.

Kumar et al. [10] analyzed the design of an inventory sample which considered decaying range as weibull. In this paper the optimal cycle lengths as well as output for the profitable maintenance under different cost considerations are derived logically. Singh et al. [11] framed the paper with inventory reliability demand and demand rate based on seasons. This paper aims for optimizing the entire amount of the production process and finding the total quantity. Tayal et al. [12] discussed the manufacturing inventory distribution sample for decaying items. This paper aims to reduce the net amount. The supply chain concept is developed for the suppliers and the buyers.

Sahoo and Dash [13] developed a model addressing fuzzy probabilistic samples, considering fuzzy demand and fuzzy storage capacity for a single period. Their findings revealed that total profit could be maximized in both overstocking and understocking situations. Raula et al. [14] examined a manufacturing model under both crisp and fuzzy environments, where all parameters were expressed as hexagonal fuzzy numbers. Sahoo et al. [15] analyzed a system where holding cost and demand rate were modeled as pentagonal fuzzy numbers, applying the graded mean difference method for defuzzification of total cost.

Jauhari et al. [16] investigated an EPQ inventory model with defective production and fuzzy annual demand, emphasizing shipment optimization between seller and customer. San-Jose et al. [17] introduced an inventory framework dependent on price and time, proposing a novel algorithm to enhance profit per unit time. Agi and Soni [18] formulated a deterministic model in which demand was influenced by product quality and freshness. Kirchi et al. [19] explored upstream and production strategies for both retailers and manufacturers to minimize large-scale production costs.

Alfares and Ghaithan [20] provided a survey of EOQ and EPQ models under varying holding cost structures. Pakhira et al. [21] considered an EOQ model incorporating memory effects in a fuzzy environment, where defuzzification was carried out using the graded integration approach. Mahapatra et al. [22] proposed a fuzzy inventory model with trade credit and permissible delay in payment, employing symmetric triangular fuzzy numbers for fuzzification and $\lambda$-integral for defuzzification. Kuppulakshmi et al. [23] extended an EPQ framework to estimate annual fish production and single-period supply to retailers through an efficient allocation process. Dong et al. [24] designed a production model incorporating multiple pricing strategies to achieve the minimum total cost. Vadivel et al. [25] presented an EPQ-style production planning model extended to a single-stage system with rework, explicitly incorporating hexagonal fuzzy numbers to model uncertainty and showing cost reductions under imperfect production.

Kumar et al. [26] compared two EPQ variants (with and without shortages) that include rework and scrap; the paper uses fuzzy modelling and references hexagonal fuzzy approaches in its methodology. The conference PDF contains the model formulations and numerical examples. Yiğit [27] developed hexagonal type-2 fuzzy set methods for inventory decisions (safety stock). While this paper focuses on safety-stock and HT2FS, its methods and HFN extensions are highly relevant for EPQ models that require robust ranking/defuzzification under ambiguity. Hemalatha and Annadurai [28] compared triangular, trapezoidal, pentagonal and hexagonal fuzzy numbers for inventory optimization (EOQ); useful for methodology and defuzzification comparison when applying HFNs to EPQ/rework problems. Arora et al. [29] used hexagonal fuzzy numbers to fuzzify cost parameters in a remanufacturing/production model (includes remanufacturing/rework aspects) and applies graded mean integration for defuzzification. Shows HFN use for remanufacturing + environmental constraints. Arora et al. [30] developed an EPQ-type optimization that explicitly treats imperfect quality, rework and repair; although several fuzzy representations are used across the literature, this paper ties imperfect products + rework into production planning and is useful background for direct HFN extensions.

The fundamental definitions are adopted from Zadeh (1965).

Definition 1.1

A hexagonal fuzzy number $\tilde{\Delta}=\left(a_1, a_2, a_3, a_4, a_5, a_6\right)$ is represented with membership function $\mu_{\breve{A}}$.

\[ \mu_{\breve{A}} = \begin{cases} L_1(\mathfrak{X})=\frac{1}{2}\left(\frac{\mathfrak{X}-m_1}{m_2-m_1}\right), & m_1 \leq \mathfrak{X} \leq m_2 \\ L_2(\mathfrak{X})=\frac{1}{2}+\frac{1}{2}\left(\frac{\mathfrak{X}-m_2}{m_3-m_2}\right), & m_2 \leq \mathfrak{X} \leq m_3 \\ 1, & m_3 \leq \mathfrak{X} \leq m_4 \\ R_1(\mathfrak{X})=1-\frac{1}{2}\left(\frac{\mathfrak{X}-m_4}{m_5-m_4}\right), & m_4 \leq \mathfrak{X} \leq m_5 \\ R_2(\mathfrak{X})=\frac{1}{2}\left(\frac{m_6-\mathfrak{X}}{m_6-m_5}\right), & m_5 \leq \mathfrak{X} \leq m_6 \\ 0, & \text { otherwise } \end{cases} \]

$\tilde{\Delta}=\left(a_1, a_2, a_3, a_4, a_5, a_6\right)$ is the $\propto$ cut, $0 \leq \propto \leq 1$ is $\Delta(\propto)=\left[\Delta_L(\propto), \Delta_R(\propto)\right]$

$\Delta_{L_1}(\propto)={m}_1+\left({m}_2-{m}_1\right) \propto=L_1^{-1}(\propto)$

$\Delta_{L_2}(\propto)={m}_2+\left({m}_3-{m}_2\right) \propto=L_2^{-1}(\propto)$

$ \Delta_{R_1}(\propto)={m}_3+\left({m}_3-{m}_4\right) \propto=R_1^{-1}(\propto) $

$ \Delta_{R_2}(\propto)={m}_6+\left({m}_6-{m}_5\right) \propto=R_2^{-1}(\propto) $

Hence,

$ L^{-1}(\propto)=\frac{L_1^{-1}(\propto)+L_2^{-1}(\propto)}{2}=\frac{{m}_1+{m}_2+\left({m}_3-{m}_1\right) \propto}{2} $

$ R^{-1}(\propto)=\frac{R_1^{-1}(\propto)+R_2^{-1}(\propto)}{2}=\frac{{m}_5+{m}_6+\left({m}_4-{m}_6\right) \propto}{2} $

Definition 1.2

$\tilde{\Delta}=\left(a_1, a_2, a_3, a_4, a_5, a_6\right)$ is HFN then the graded mean representation (GMDM) of $\tilde{\Delta}$ is defined as

$\mathrm{P}(\tilde{\Delta})=\frac{1}{12}\left[{m}_1+3 {m}_2+2 {m}_3+2 {m}_4+3 {m}_5+{m}_6\right] $

Definition 1.3

Suppose $\tilde{A}=\left({m}_1, {m}_2, {m}_3, {m}_4, {m}_5, {m}_6\right)$ and $\tilde{B}=\left({n}_1, {n}_2, {n}_3, {m}_4, {n}_5, {n}_6\right)$ are two HFN's ${m}_1$, ${m}_2$, ${m}_3$, ${m}_4$, ${m}_5$, ${m}_6$, ${n}_1$, ${n}_2$, ${n}_3$, ${n}_4$, ${n}_5$, ${n}_6$ are all real numbers.

1. $\tilde{A}+\tilde{B}=\left({m}_1+{n}_1, {m}_2+{n}_2, {m}_3+{m}_3, {m}_4+{n}_4, {m}_5+{n}_5, {m}_6+{n}_6\right).$

2. $\tilde{A} * \tilde{B}=\left({m}_1 {n}_1, {m}_2 {n}_2, {m}_3 {n}_3, {m}_4 {n}_4, {m}_5 {n}_5, {m}_6 {n}_6\right).$

3. $-\tilde{B}=\left(-{n}_6,-{n}_5,-{n}_4,-{n}_3,-{n}_2,-{n}_1\right).$

4. $\frac{1}{\tilde{B}}=\tilde{B}^{-1}=\left(\frac{1}{{n}_6}, \frac{1}{{n}_5}, \frac{1}{{n}_4}, \frac{1}{{n}_3}, \frac{1}{{n}_2}, \frac{1}{{n}_1}\right).$

5. $\frac{\tilde{A}}{\tilde{B}}=\left(\frac{{m}_1}{{m}_6}, \frac{{m}_2}{{m}_5}, \frac{{m}_3}{{m}_4}, \frac{{m}_4}{{m}_3}, \frac{{m}_5}{{m}_2}, \frac{{m}_6}{{m}_1}\right).$

6. Let $\propto \in R$, then

$\alpha * \tilde{A} = \begin{cases} (\alpha m_1, \alpha m_2, \alpha m_3, \alpha m_4, \alpha m_5, \alpha m_6), & \alpha \geq 0 \\ (\alpha n_1, \alpha n_2, \alpha n_3, \alpha n_4, \alpha n_5, \alpha n_6), & \alpha < 0 \end{cases}$

• A single machine operating with multiple setups is considered.

• The planning horizon is finite.

• Defective items generated during production can be reworked.

• Backlogging is permitted, and unmet demand is treated as a penalty for lost sales.

• Some defective units may be discarded as scrap during the rework process.

• In a fuzzy environment, factors such as production cost, backlogging cost, and penalty for lost sales are taken into account.

$A$ - Expected setup cost per cycle

$Q$ - Production Quantity per cycle

$\theta$ - Defective production rate

$P$ - Normal production rate

$P^\prime$ - Rework rate of defective item

$D$ - Demand rate (where $P > DP > D$)

$X$ - Proportion of defective items

$f$ - Fraction of demand backordered

$I^{max}$ - Maximum inventory after rework

$R_c$ - Rework Cost per defective items

$D_c$ - Disposal cost of scrabbed items

$H_c$ - Holding cost for reworked items

${H_c}^\prime$ - Anticipated total holding cost for rework able items

$B_c$ - shortage cost

$P_c$ - Production cost

$L_c$ - Lost sales cost

$I$ - Inventory level during production

$B$ - Backordered quantity

$TC (Q, B)$ - Total cost of production process

$E (.)$ - Expected value operator

$\theta^\prime$ - Rework production rate

$\gamma $ - Number of scrapped items during rework

$\tau_1$ - Production time

$\tau_2$ - Idle time (Non production time)

$\tau_3$ - Shortage period

$\tau_4$ - Time to clear backorders

$\tau_5$ - Rework time

The production model is adapted from Taleizadeh et al. [1], which considers a single machine with limited capacity operating under multiple production processes. In the present work, this framework is extended by incorporating fuzzy concepts. The objective is to determine the total cycle time and backordered quantity more effectively by introducing hexagonal fuzzy numbers as parameters. Production commences at $\tau = 0$, and throughout the process, the production rate ($P$) is assumed to be greater than or equal to the demand rate ($D$). The proportion of imperfect items ($\theta$) is illustrated in Figure 1.

The duration of manufacturing cycle length is the union of production, non production, backordered permitted time and rework time. The total cycle time during the production process is given as

The total inventory level during the manufacturing process is defined as

During the time $\tau_1$ the production process takes place and some products are damaged, the demand also satisfied during the same cycle time. Because of deterioration of some products some goods are disposed as a scrap.

The production stops after the time period $\tau_1$ and the demand takes place during the time $\tau_2$:

Due to the demand of the goods, the products are back ordered and this takes place during the time $\tau_3$ and $\tau_4$:

All the rework able items are collected separately after the $n$ cycles, then it will be reworked separately during the period $(\tau_5)$. Waste items during the rework process are separated as scarp.

The total items which is produced during the rework period is given as

The scraped items rate during rework is given by

During the manufacturing time, $\gamma$- rate of scrabbed items are produced

Eq. (11) can be written as

The total quantity of one production is given by

The total cost for $n$-production setup and one rework setup is given by

\begin{aligned} T C(Q, B)= & \textit { Production cost }+ \textit { Mending price for deterioration cost }+ \textit { Disposal cost } \\ & + \textit { Holding cost for non defective items }+ \textit { Rework cost } \\ & + \textit { Holding cost for reworked items }+ \textit { Back ordered cost }+ \textit { Lost sale cost } \\ & + \textit { Aniticipated total setup cost }\end{aligned}

Fuzzy model

In the crisp model, all parameters are considered fixed and determined with certainty. However, in practice, obtaining exact values is often unrealistic. To address this, the problem is examined under a fuzzy environment, where manufacturing cost, penalty for lost sales, and backordering cost are represented as fuzzy numbers.

$\tilde{P_c}=$ fuzzy production cost

$\tilde{B_c}=$ fuzzy backordered cost

$\tilde{L_c}=$ fuzzy lost sale cost

Suppose $\tilde{P}_c=\left(P_1, P_2, P_3, P_4, P_5, P_6\right)$, $\tilde{B_c}=\left(B_1, B_2, B_3, B_4, B_5, B_6\right)$, $\tilde{L_c}=\left(L_1, L_2, L_3, L_4, L_5, L_6\right)$ are non negative hexagonal fuzzy numbers (HFN).

The total production cost is given below:

where,

The defuzzification for hexagonal fuzzy number is defined by the graded mean difference formula is given as

The theoretical findings are validated through a numerical example, with certain parameter values adapted from the study [5]. The adopted values are:$P$ = 4000/cycle, $D$ = 600/cycle, $P^\prime$ = 800/cycle, $A$ = 500/cycle, $f$ = 0.3/cycle, $P_c$ = \$15/unit, $H_c$ = \$5/unit, ${H_c}^\prime$ = \$2/unit, $D_c$ = \$3/unit, $B_c$ = \$10/unit, $L_c$ = \$12/unit, $R_c$ = 1. The parameters $\theta$, $\gamma$ are assumed to follow a uniform distribution, and their respective values are obtained for both perfect and defective items.

The use of a uniform distribution implies that the manufacturing rate of defective products has an equal probability of occurring across the specified range.

For different values of $a$ and $b$, the production rate of imperfect items and the manufacturing rate of reworking items are calculated and they are shown in Table 1 and Table 2. Taking the first value of the Table 1 and Table 2 and substitute the remaining numerical values in Eq. (15). The total time, back order level, optimum quantity, total cost for the production process , inventory level for the production and rework inventory level are given as

$T=0.2590$, $B=413.68$, $Q=1782$, $TC({Q},{B})=4822600$, $I=1604$, $I^{max}=89.83$

The production rate of reworked items obtained by uniform distribution to get the accurate values of $\theta^\prime$.

Fuzzy model

Let the hexagonal fuzzy numbers be defined as $\tilde{P_c}=\left(5 \; \;10 \;\; 15 \;\; 15 \;\; 20 \;\; 25 \right)$, $\tilde{B_c}=\left(6\;\; 8 \;\;10\;\; 10 \;\;12 \;\; 14\right)$, $\tilde{L_c}=\left(8 \;\; 10 \;\; 12 \;\; 12 \;\; 14 \;\; 16\right).$ Using these parameters, the fuzzy estimates for the system are obtained as: fuzzy total time $\tilde{T}^*=0.2589$, fuzzy backorder level $\tilde{B}^*=412$, fuzzy production quantity $\tilde{Q}^*=1780$, fuzzy total cost $\widetilde{T C}(Q, B)=4823600$, fuzzy inventory level and maximum fuzzy inventory $\tilde{I}^*=1602$, $\tilde{I}^{max*}=89$.

When comparing the crisp and fuzzy models, optimal solutions are derived for cycle time, backorder quantity, and inventory levels during both production and rework. The fuzzy results are defuzzified using the Graded Mean Difference Method (GMDM), which demonstrates the superiority of the fuzzy approach over the crisp case.

From Table 3, it is observed that the setup cost ($A$) rises linearly as the parameter varies between –15% and +15%, which in turn causes the total cost to increase. The demand rate also shows an upward trend within the same range. In addition, the shortage fraction grows linearly from –15% to +15%, and as illustrated in Figure 2, this results in a reduction of the total cost.

Based on the results in Table 4, the production cost ($P_c$) increases linearly as the parameter varies from –15% to +15%, leading to a corresponding rise in total cost. The holding cost ($H_c$) also shows a linear increase over the same range; however, its impact on the total cost is twofold—between –15% and –5% the total cost decreases, while from +5% to +15% it rises linearly. Similarly, the backorder cost grows linearly across the –15% to +15% interval, which in turn elevates the total cost. As illustrated in Figure 3, the rework cost follows the same pattern, increasing linearly with parameter variation, thereby causing the total cost to increase accordingly.

This paper proposed an EPQ model for imperfect production systems incorporating rework, scrap, and lost sales penalties under both crisp and fuzzy environments. The adoption of Hexagonal Fuzzy Numbers and the Graded Mean Deviation Method for defuzzification provided greater flexibility in handling uncertainty, yielding more realistic and cost-effective results than conventional models. The fuzzy-based framework not only minimized total production costs but also improved inventory balance, even when defective items were partially scrapped. Numerical results and sensitivity analyses confirmed the robustness of the approach across varying system parameters. The contributions of this work lie in establishing a structured methodology for managing reworkable items under uncertainty and demonstrating its practical relevance for modern manufacturing. Future research directions include extending the model to multi-stage supply chains, integrating advanced fuzzy and neutrosophic environments, and exploring hybrid decision-making algorithms for large-scale industrial applications.