A fuzzy random EPQ model with fuzzy defective rates and fuzzy inspection errors

Abstract

In this paper, we consider an economic production quantity (EPQ) model for imperfect production process under fuzzy random variable demand considering inspection errors. Due to the first stage inspection errors, some proportion of defective items are returned because of dissatisfaction of the customers. In the previous traditional models, the defective rates and the inspection errors follow some probability distributions. However, in real life situation, it is almost impossible to obtain the statistical information precisely. Thus, this study proposes the fuzzy defective rates and the fuzzy inspection errors. In addition, this model interpolates two more stages of inspections, one is after the production run time, and another is after the beginning of the rework process. The purpose of this study is to establish a fuzzy random EPQ model with the fuzzy defective rates and inspection errors. The expected profit per unit time is calculated by using fuzzy random renewal reward theorem. This model maximizes the expected profit per unit time in fuzzy sense. We develop a methodology for finding the global optimal solutions. A numerical example is also provided to illustrate our proposed model. Furthermore, sensitivity analysis is also carried out in order to present some managerial inferences.

Keywords

Inventory imperfect item inspection error fuzzy type-I and type-II error fuzzy random variable

1 Introduction

The economic production quantity (EPQ) is one of the most applicable and effective model in the industry. The EPQ model is mostly used to solve the problem of the optimal lot size and the production run-time for maximization of the expected profit. One of the basic assumptions of the classical economic production quantity model is that, the items produced are of perfect quality. However, in the realistic situations, the manufacturer’s production process is not perfect. Thus, the assumption of perfect quality is not realistic in the industrial applications. Shih [36] was the first, who introduced an inventory model to explore the effect of defective products on the production lot size and the annual cost of the inventory system. Since then, during the last few decades, several researchers have developed EPQ models with imperfect quality of items under various assumptions. The previous existing classic imperfect-quality EPQ model can be mainly categorized into two parts with regard to inspection method of defects connected to the imperfect production process. In the first category, the production system deteriorates in each cycle during the production process and thus produces some defect items. Those defective items are inspected in regular interval with a constant hazard rate. In the second category, the inspection of an entire lot screening of the production process with no deterioration has been performed. The survey on the development of the inventory model with imperfect production can be found in [2 , 35].

Another unrealistic assumption of the EPQ model with imperfect production process is that the inspection of defective items is 100% correct. In an EPQ model with imperfect production process, the inspection process is not often perfect. Considering the survey results of 45 apparel and 65 customer electronic manufacturers Reverse Logistics Executive Council (RLEC), 1999, the average return rates were found to be 19.44% and 8.46% of the apparel and customer respectively and due to defect the corresponding rates were 49.45% and 35.71%, respectively [40]. Therefore, a significant observation is that both of the type-I and type-II inspection errors affects the firm’s profitability. Many researchers [18 , 40] have extensively studied the inventory model, including both type-I and type-II errors in the imperfect inspection process.

In all the above literature, the defective rates, type-I and type-II errors are required to follow some probability distributions. However, in many practical, real-world applications, the statistical information about distribution may be obtained subjectively; that is, the parameters of the distribution are more suitably described by linguistic terms such as “the parameter θ₁ is approximate by a ( $\in ℝ$ )”, “the parameter θ₂ is approximate by $b (\in ℝ)$ ”, rather than by crisp values. Thus, the fuzzy defective rates and inspection errors are much more realistic than commonly used the crisp defective rates and inspection errors. Since it was proposed by Zadeh [41], fuzzy set theory has been widely used in the inventory control. Park [33] introduced fuzzy mathematics into inventory control and developed an economic order quantity model. Vujosevic et al. [39] extended the traditional EOQ model by using the fuzziness of ordering and holding cost. Chen and Wang [11] presented the EOQ model where the demand, ordering cost, inventory cost, and back order were fuzzified as trapezoidal fuzzy numbers. Lin and Yao [20] fuzzified the production quantity into triangular fuzzy numbers in the economic production quantity model. Chang [9] applied the fuzzy set theory to the EOQ model with imperfect quality items, where defective rate and total demand were fuzzified as fuzzy numbers. Baykasoğlu and Göçken [4] analyzed a multi-item fuzzy economic order quantity model. All the parameters of the EOQ model were defined as triangular fuzzy numbers and solved by using four different fuzzy ranking methods. Kazemi et al. [23] considered a fuzzy inventory model with backorders by employing triangular and trapezoidal fuzzy numbers. Baykasoğlu and Göçken [5] developed a fully fuzzy constraint multi-item EOQ model under triangular fuzzy numbers as the parameters of the system. Björk [3] investigated a multi-item fuzzy economic production quantity model with a finite production rate, where the demand is crisp but the cycle time is kept as fuzzy. Mahata and Goswami [31] developed a fuzzy inventory model with imperfect quality and shortage back ordering under fuzzy and crisp decision variables. Recently, Kazemiet al. [24] presented a fuzzy economic order quantity model for imperfect quality by incorporating the learning effect on the fuzzy parameters. Mezei and Bjork [32] proposed a multi-item economic production quantity model with backorders and fuzzy cycletimes.

However, the models specified above fix uncertainty by defining the corresponding variable as either a random or a fuzzy variable, while ignoring the fuzziness and randomness of the variables. But in realistic situations, fuzziness and randomness can be occurred simultaneously. Kwakernaak [29] first incorporated the fuzziness and randomness of an event simultaneously by introducing the fuzzy random variable. Then Puri and Ralescu [34] developed the concept later. Dutta et al. [16] first introduced fuzzy random variable into inventory management and established a single-periodic inventory model under fuzzy random variable demand. Chang et al. [10] proposed an inventory model with mixture of back orders and lost sales. They considered the lead-time as a fuzzy random variable. Dutta et al. [17] developed a continuous review inventory model with fuzzy random variable demand. Bag et al. [1] discussed a production inventory model with flexibility and reliability considerations. The annual demand was treated as fuzzy random variable. Dey and Chakraborty [13] analyzed a periodic review inventory model under fuzzy random variable demand. Dey and Chakraborty [14] extended the model [13] by considering the variable lead-time. Kumar and Goswami [26] studied a fuzzy random continuous review production-inventory system. They extended the min-max distribution free procedure for fuzzy random variable. Dash and Sahoo [12] developed a multi-item single period inventory model under fuzzy random variable demand. Buckley’s concept of minimization of fuzzy numbers is used to obtain the optimal solution.

Above analysis of literature reveals that some studies consider the fuzziness on the imperfect quality of the items without incorporating the randomness and vice versa. There are few numbers of papers on EPQ model (for instances [1, 27]) in the mixed fuzzy random framework, but without consideration of inspection errors. Thus, our intention is to address this research gap of the economic production quantity model.

In this article, we investigate an EPQ model with fuzzy random variable demand. Both the production process and the first stage inspection process are imperfect. Thus, the type-I and type-II errors occur during the inspection process of the entire lot. A fraction of defective items is passed to customers due to the type-II error and come back to the manufacturer with dissatisfaction of customers. Moreover, due to type-I inspection error, a fraction of perfect items are falsely screened out as a defective item. In our model, we propose the fuzzy defective rates, fuzzy type-I, and type-II errors. The membership functions of the expected defective rates, type-I, and type-II errors are constructed by solving a pair of mathematical programming approach. The screened and returned items are managed by rework process. The goal of this study is to establish an EPQ model with the fuzzy defective rates and inspection errors under fuzzy random variable demand. The model has been developed to maximize the expected profit per unit time in fuzzy sense. The expected profit per unit time is derived by using a fuzzy renewal reward theorem. The concavity of the possibilistic mean value of the expected profit function for global optimality is demonstrated and optimal solutions areobtained.

The rest of the paper is organized as follows: Section 2, presents some preliminary concept. In Section 3, we propose the fuzzy proportion of defective rates and inspection errors. In Section 4, we describe the proposed model, the associated assumptions, and notations. The mathematical model is formulated in Section 5. Section 6, provides the solution process of the model. Section 7, presents a numerical example to illustrate the model. Finally, section 8, summarizes the work proposed.

2 Preliminary concepts

In this section, some required basic concepts of the fuzzy set theory are presented.

Definition 2.1. [42]. A fuzzy number $\tilde{A}$ is said to be a triangular fuzzy number if the membership function $μ_{\tilde{A}} (x)$ satisfies the following conditions:

$μ_{\tilde{A}} (x)$ is a continuous function from $ℝ$ to the closed interval [0, 1],

$μ_{\tilde{A}} (x) = \frac{x - a}{b - a}$ is strictly increasing function on [a, b],

$μ_{\tilde{A}} (x) = 1$ for x = b,

$μ_{\tilde{A}} (x) = \frac{c - x}{c - b}$ is strictly decreasing function on [b, c],

$μ_{\tilde{A}} (x) = 0$ elsewhere,

where a, b, c are real numbers.

Definition 2.2. [42]. The α-cut or α-level set of a fuzzy set $\tilde{A}$ is defined as ${\tilde{A}}_{α} = {x | μ_{\tilde{A}} (x) \geq α}$ and is denoted by the interval ${\tilde{A}}_{α} = [A_{α}^{-}, A_{α}^{+}]$ , ∀α ∈ [0, 1].

2.1 The extension principle

Suppose, f is a function from X to Y, defined by y = f (x), where x = (x₁, x₂, ⋯ , x_n). Then the extension principle,[42] allows us to extend this function to a fuzzy set $\tilde{B}$ in Y by $\tilde{B} = {(y, μ_{\tilde{B}} (y)) | y = f (x)}$ , where $μ_{\tilde{B}} (y) = {\begin{matrix} sup_{y = f (x)} min_{i = 1, 2, \dots, n} {μ_{\tilde{A_{i}}} (x_{i})} & if f^{- 1} (y) \neq φ \\ 0 & otherwise . \end{matrix}$

Definition 2.3. [29, 30]. Let $(Ω, B, ℙ)$ be a probability space, a mapping $\tilde{χ} : Ω ⟶ 𝔽 (ℝ)$ is said to be a fuzzy random variable if ∀α ∈ [0, 1], the two real valued mappings $χ_{α}^{-} : Ω ⟶ ℝ$ and $χ_{α}^{+} : Ω ⟶ ℝ$ are real-valued random variables.

Definition 2.4. [29]. The expectation of the fuzzy random variable $\tilde{χ}$ is define as $\begin{matrix} E (\tilde{χ}) = \int \tilde{χ} d ℙ = {[\int χ_{α}^{-} d ℙ, \int χ_{α}^{+} d ℙ] \\ | 0 \leq α \leq 1} . \end{matrix}$

If $\tilde{χ}$ is a discrete fuzzy random variable such that $p (\tilde{χ} = {\tilde{x}}_{i}) = {\tilde{p}}_{i}; i = 1, 2, \dots,$ then its expectation is defined by $E (\tilde{χ}) = \sum_{i = 1}^{\infty} {\tilde{x}}_{i} {\tilde{p}}_{i}$ .

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2.2 Possibilistic mean value of fuzzy number

Let $\tilde{A}$ be a fuzzy number with α-cut ${\tilde{A}}_{α} = [A_{α}^{-}, A_{α}^{+}]$ , therefore, the possibilistic mean value of $\tilde{A}$ is denoted by $\bar{M} (\tilde{A})$ and defined as [6] $\bar{M} (\tilde{A}) = \int_{0}^{1} α (A_{α}^{-} + A_{α}^{+}) d α .$ (1) Let $\tilde{A}$ and $\tilde{B}$ be two fuzzy numbers where $A_{α} = [A_{α}^{-}, A_{α}^{+}]$ and $B_{α} = [B_{α}^{-}, B_{α}^{+}], α \in [0, 1]$ , then, for ranking fuzzy numbers, we have $\tilde{A} \leq \tilde{B} \Leftrightarrow \bar{M} (\tilde{A}) \leq \bar{M} (\tilde{B})$ .

3 Fuzzy proportion of defective rate and inspection errors

In the conventional EPQ model, the defective rate of items is described as a probability p with corresponding probability density function (or probability mass function) f_{θ
_i} (p). We propose to define the fuzzy proportion of the defective rate and inspection errors, where the parameters of the distribution are known imprecisely.

Let $\tilde{p}$ be the fuzzy proportion of defective rate with corresponding density function $f_{{\tilde{θ}}_{i}} (\tilde{p})$ , then $\tilde{p}$ can be defined as $\tilde{p} = {(p, μ_{\tilde{p}} (p)) | p \in ℙ}$ where $ℙ$ be the crisp probability measure of all possible probability p of defective items, $f_{{\tilde{θ}}_{i}} (\tilde{p})$ is the probability density function with imprecise parameters ${\tilde{θ}}_{i}, i = 1, 2, \dots, n$ and $μ_{\tilde{p}}$ is the corresponding membership function of $\tilde{p}$ .

Based on Zadeh extension principle [42], the membership function $μ_{\tilde{p}}$ can be defined as $μ_{\tilde{p}} (z) = sup_{p \in ℙ} min_{i = 1, 2, \dots, n} {μ_{{\tilde{θ}}_{i}} (θ_{i}) | z = f_{θ_{i}} (p)}$ where, f_{θ
_i} (p) is the crisp probability density function of p with the parameters θ_i, i = 1, 2, ⋯ n.

$E (\tilde{p})$ denotes the expectation of $\tilde{p}$ whose membership function can be defined as $μ_{E (\tilde{p})} (z) = sup_{p \in ℙ} min_{i = 1, 2, \dots, n} {μ_{{\tilde{θ}}_{i}} (θ_{i}) | z = E (p)} .$ The procedure to obtain $μ_{E (\tilde{p})} (z)$ is given in next subsection.

The fuzzy non-defective proportion is denoted by $\tilde{q}$ , and the membership function can be definedas $μ_{\tilde{q}} (x) = {1 - μ_{\tilde{p}} (x) | x \in ℙ} .$

$E (\tilde{q})$ denotes the expectation of $\tilde{q}$ , and the membership function of $E (\tilde{q})$ can be defined as $\begin{matrix} μ_{E (\tilde{q})} (z) = sup_{p \in ℙ} min_{i = 1, \dots n} {μ_{{\tilde{θ}}_{i}} (θ_{i}) | z = 1 \\ - E (p)} (p + q = 1) . \end{matrix}$

3.1 Fuzzy type-I and type-II inspection errors

Conventionally, the type-I and type-II errors of inspection are defined by α = probability that the items screened are resulted defective, but actually those are perfect and β = probability that the items screened are not resulted defective, but actually those are defective. The type-I error α and type-II error β follow the probability density functions (or probability mass function) f_{φ
_i} (α) and f_{φ
_i} (β), respectively, with the parameters φ_i, i = 1, 2, ⋯ , n [21]. In this section, we define the fuzzy proportion of type-I and type-II errors.

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Let $\tilde{α}$ and $\tilde{β}$ be the fuzzy proportion of type-I and type-II errors, respectively as fuzzy probability measure [7]. Let $\tilde{α}$ and $\tilde{β}$ follow the probability density functions $f_{{\tilde{φ}}_{i}} (\tilde{α})$ and $f_{\tilde{φ_{i}}} (\tilde{β})$ , respectively, with imprecise parameters ${\tilde{φ}}_{i}$ , i = 1, 2, ⋯ , n. Then the fuzzy proportion of $\tilde{α}$ and $\tilde{β}$ can be defined as $\tilde{α} = {(α, μ_{\tilde{α}} (α)) | α \in ℚ_{1}}$ and $\tilde{β} = {(β, μ_{\tilde{α}} (β)) | β \in ℚ_{2}}$ where, $ℚ_{1}$ is the classical probability measure of all possible probability α of type-I error and $ℚ_{2}$ is the classical probability measure of all possible probability β of type-II error.

Based on Zadeh extension principle [42], the membership functions $μ_{\tilde{α}}$ and $μ_{\tilde{β}}$ can be defined as $μ_{\tilde{α}} (z) = sup_{α \in ℚ_{1}} min_{i = 1, 2, \dots, n} {μ_{{\tilde{φ}}_{i}} (φ_{i}) | z = f_{φ_{i}} (α)}$ $μ_{\tilde{β}} (z) = sup_{β \in ℚ_{2}} min_{i = 1, 2, \dots, n} {μ_{{\tilde{φ}}_{i}} (φ_{i}) | z = f_{φ_{i}} (β)}$

$E (\tilde{α})$ denotes the expectation of $\tilde{α}$ whose membership function is defined as $μ_{E (\tilde{α})} (z) = sup_{α \in ℚ_{1}} min_{i = 1, 2, \dots, n} {μ_{{\tilde{φ}}_{i}} (φ_{i}) | z = E (α)} .$ Similarly, $E (\tilde{β})$ denotes the expectation of $\tilde{β}$ , and the membership function of $E (\tilde{β})$ is defined as $μ_{E (\tilde{β})} (z) = sup_{β \in ℚ_{2}} min_{i = 1, 2, \dots, n} {μ_{{\tilde{φ}}_{i}} (φ_{i}) | z = E (β)} .$

3.2 Construction of the membership function of $E (\tilde{p})$

The membership function of $E (\tilde{p})$ is given by $μ_{E (\tilde{p})} (z) = sup_{p \in ℙ} min_{i = 1, 2, \dots, n} {μ_{{\tilde{θ}}_{i}} (θ_{i}) | z = E (p)} .$ (2) It is not always possible to find the membership function of $E (\tilde{p})$ in closed form. The closed form of the membership function is not easy to construct because the membership value of $μ_{E (\tilde{p})}$ at a point is the supremum of a set of membership value $μ_{{\tilde{θ}}_{i}} (θ_{i})$ . The membership function $μ_{E (\tilde{p})} (z)$ can be obtained by the following procedure. Without any loss of generality we assume that all the imprecise parameters of the distribution function are fuzzy numbers. Let $f (\tilde{p})$ be the probability density function of $\tilde{p}$ with the imprecise parameters ${\tilde{θ}}_{i}$ , i = 1, 2, ⋯ , n. The α-cut or α-level sets of ${\tilde{θ}}_{i}$ , ∀i are defined as ${\tilde{θ}}_{i}^{α} = [min_{θ_{i}} {θ_{i} \in S ({\tilde{θ}}_{i}) | μ_{{\tilde{θ}}_{i}} (θ_{1}) \geq α}; max_{θ_{i}} {θ_{i} \in S ({\tilde{θ}}_{i}) | μ_{{\tilde{θ}}_{i}} (θ_{i}) \geq α}] = [θ_{i, α}^{-}; θ_{i, α}^{+}],$ where $S ({\tilde{θ}}_{i})$ is the supports of ${\tilde{θ}}_{i}$ for all i = 12, ⋯ , n.

According to (refeq : 2a), $μ_{E (\tilde{p})} (z)$ is the minimum of $μ_{{\tilde{θ}}_{i}} (θ_{i})$ for i = 1, 2, ⋯ , n. We need $μ_{{\tilde{θ}}_{1}} (θ_{1}) \geq α$ , $μ_{{\tilde{θ}}_{2}} (θ_{2}) \geq α$ , ⋯, $μ_{{\tilde{θ}}_{n}} (θ_{n}) \geq α$ , and at least one of $μ_{{\tilde{θ}}_{i}} (θ_{i})$ is equal to α such that z = E (p) to satisfy $μ_{E (\tilde{p})} (z) = α .$ Now, it is sufficient to find the left and right shape functions for finding the membership function of $μ_{E (\tilde{p})}$ , which are the lower and upper bounds of the α-cuts of $E (\tilde{p})$ . We can find the upper and lower bounds of the α-cut of $E (\tilde{p})$ by solving the following pair of mathematical programs: $(LB) {\begin{matrix} (E (\tilde{p}))_{α}^{-} & = min E (p) \\ s . t & θ_{i, α}^{-} \leq θ_{i} \leq θ_{i, α}^{+} \\ i = 1, 2, \dots, n . \end{matrix}$ (3) $(UB) {\begin{matrix} (E (\tilde{p}))_{α}^{+} & = max E (p) \\ s . t & θ_{i, α}^{-} \leq θ_{i} \leq θ_{i, α}^{+} \\ i = 1, 2, \dots, n . \end{matrix}$ (4) where E (p) denotes the conventional expectation of p, that is a function of θ₁, θ₂, ⋯, θ_n.

The crisp interval $[E (\tilde{p}))_{α}^{-}; (E (\tilde{p}))_{α}^{+}]$ represents the α-cut of $E (\tilde{p})$ . Note that all α-cuts make a nested structure with respect to α [22, 42]. Therefore, for two values of α₁ and α₂ such that 0 < α₁ < α₂ ≤ 1, we have $(E (\tilde{p}))_{α_{2}}^{-} \geq (E (\tilde{p}))_{α_{1}}^{-}$ and $(E (\tilde{p}))_{α_{1}}^{+} \geq (E (\tilde{p}))_{α_{2}}^{+}$ . In other words $(E (\tilde{p}))_{α}^{-}$ is increasing and $(E (\tilde{p}))_{α}^{+}$ is decreasing with respect to α. This property confirms the convexity of $E (\tilde{p})$ [22, 42]. If both $(E (\tilde{p}))_{α}^{-}$ and $(E (\tilde{p}))_{α}^{+}$ are invertible with respect to α, then the left and right shape functions can be obtained by $L (z) = [(E (\tilde{p}))_{α}^{-}]^{- 1}$ and $R (z) = [(E (\tilde{p}))_{α}^{+}]^{- 1}$ , respectively, and the membership function can be constructed as $μ_{E (\tilde{p})} (z) = {\begin{matrix} L (z) & for (E (\tilde{p}))_{α = 0}^{-} \leq z \leq (E (\tilde{p}))_{α = 1}^{-} \\ 1 & for (E (\tilde{p}))_{α = 1}^{-} \leq z \leq (E (\tilde{p}))_{α = 1}^{+} \\ R (z) & for (E (\tilde{p}))_{α = 1}^{+} \leq z \leq (E (\tilde{p}))_{α = 0}^{+} . \end{matrix}$

Otherwise, the numerical solution for $(E (\tilde{p}))_{α}^{-}$ and $(E (\tilde{p}))_{α}^{+}$ at different level of α can be collected together to construct the approximate shapes of L (z) and R (z). In the similar manner, other membership functions of $E (\tilde{q}), E (\tilde{α}),$ and $E (\tilde{β})$ can also be derived.

4 Problem description, assumptions and notations

We consider an economic production quantity inventory system, where the manufacturer’s production process and the first stage inspection process are not perfect. Figure 1 describes the inventory behavior and the flow diagram of the inventory system. During the production run-time T₁, the production house, produces and simultaneously inspects a lot size of y at a rate of P. Because of the imperfections of the production process, the lot size of y contains perfect items of $\tilde{q} y$ together with defective items of $\tilde{p} y$ , where $\tilde{p}$ and $\tilde{q}$ are the fuzzy proportion of defective and non-defective rates, respectively. The fuzzy defective proportion of $\tilde{p}$ follows the probability density function of $f (\tilde{p})$ with imprecise parameters. Since the first stage inspection process of lot size y is also imperfect, the first stage inspector may do both type-Iand type-II errors. Let the proportion of the fuzzy type-I and type-II errors are $\tilde{α} =$ fuzzy probability that the items screened are resulted defective, but actually those are perfect and $\tilde{β} =$ fuzzy probability that the items screened are not resulted defective, but actually those are defective, following the probability density function’s $f (\tilde{α})$ and $f (\tilde{β})$ , respectively, with imprecise parameters. We assume that $\tilde{α}$ and $\tilde{β}$ are independent of the fuzzy defective and non-defective proportion $\tilde{p}$ and $\tilde{q}$ , respectively, as per Liou et al. [21]. So, $\tilde{α} \tilde{q} = \tilde{probability}$ (perfect items ∩ items screened as defects) and $\tilde{β} \tilde{p} = \tilde{probability}$ (defective items ∩ items inspected as perfects). Therefore, the amount of items involving in fuzzy inspection errors is calculated inter-dependently by $\tilde{p}, \tilde{q}, \tilde{α}, \tilde{β}$ , and y. Thus, in a whole lot inspection process during the time T₁, $\tilde{α} \tilde{q} y$ units among the perfect items of $\tilde{q} y$ are falsely screened as defective items due to the fuzzy type-I inspection error. As a result, $(\tilde{q} - \tilde{α} \tilde{q}) y$ units are serviceable items among the perfect items $\tilde{q} y$ .Moreover, due to fuzzy type-II inspection error, $\tilde{β} \tilde{p} y$ units among the defective items of $\tilde{p} y$ are falsely inspected as perfect items and treated as serviceable items, thus $(\tilde{p} - \tilde{β} \tilde{p}) y$ units are successfully screened out among the defective items $\tilde{p} y$ . Therefore, during the time T₁ some proportion of serviceable items (falsely inspected as perfects) of $\tilde{β} \tilde{p} y$ are passed to the customers and other proportion of $\tilde{β} \tilde{p} y$ is incurred on the items of $(P - \tilde{D}) T_{1}$ units.

Thus, the defective items are returned to the manufacturer due to the dissatisfaction of the customers. The returned items are assumed to be fully refunded. During the time ${\tilde{T}}_{2}$ , the proportion of defective items among the serviceable items $(P - \tilde{D}) T_{1}$ is successfully screened out by using second stage inspection process. The defective items $\tilde{β} \tilde{p} y$ are assumed to occur continuously like demand during the time $(T_{1} + {\tilde{T}}_{2})$ . Those screened and returned items $(\tilde{p} + \tilde{α} \tilde{q}) y$ in each cycle will be reworked by the remanufacturing process at a rate of P_R during a rework runtime ${\tilde{T}}_{2}$ right after the production runtime T₁. Since, the production process is not perfect, remanufacturing items $(\tilde{p} + \tilde{α} \tilde{q}) y$ contains non-defective items of amount $\tilde{δ} (\tilde{p} + \tilde{α} \tilde{q}) y$ along with defective items $\tilde{η} (\tilde{p} + \tilde{α} \tilde{q}) y$ . Where the fuzzy proportion of defective rate $\tilde{η}$ follows probability density function $f (\tilde{η})$ and $\tilde{δ}$ is the reworking rate of perfect items. The defective items $\tilde{η} (\tilde{p} + \tilde{α} \tilde{q}) y$ are successfully screened out by employing third stage inspection process and sold out as salvage items at a unit selling price v (< s)/unit after the time $(T_{1} + {\tilde{T}}_{2})$ .

In order to incorporate the both fuzziness and randomness of the demand information, the demand is assumed to be discrete fuzzy random variable $\tilde{D}$ as per Dutta and Chakraborty [16]. We assume that the demand is of the form ${({\tilde{d}}_{1}, p_{1}),$ $({\tilde{d}}_{2}, p_{2}), \dots, ({\tilde{d}}_{n}, p_{n})}$ , where ${\tilde{d}}_{i}$ ’s are triangular fuzzy numbers and p_i’s are their corresponding probabilities for i = 1, 2, ⋯ , n.

The following notations are used to develop the model.

Notations

production rate (unit/unit time)

P _S

production and inspection rate of serviceable items during T₁ (unit/unit time)

P _R

rework rate of screened and returned items (unit/unit time)

P _RS

rework rate of the serviceable items during ${\tilde{T}}_{2}$ (unit/unit time)

\tilde{T}

length of the each cycle

T ₁

length of the production runtime with first stage inspection (decision variable)

{\tilde{T}}_{2}

length of the remanufacturing runtime with second and third stage inspection

{\tilde{T}}_{3}

demand time length

{\tilde{T}}_{4}

depletion time length of the serviceable remanufacturing items

lot size (unit/cycle) (decision variable)

\tilde{D}

demand rate(unit/unit time), which is fuzzy random variable

\tilde{p}

fuzzy proportion of defective rate of y

\tilde{q}

fuzzy proportion of non-defective rate of y

\tilde{α}

fuzzy proportion of type-I error

\tilde{β}

fuzzy proportion of type-II error

\tilde{δ}

fuzzy non-defective proportion of remanufacturing items

\tilde{η}

fuzzy defective proportion of remanufacturing items

fixed setup cost

unit selling price for serviceable items

unit selling price for salvage items

unit production cost

unit holding cost (unit/unit time)

unit rework cost for returned and screened items

unit inspection cost

unit penalty and return cost for items return from customer and lot size of $(1 - \frac{\tilde{D}}{P}) y$

I_i (t)

inventory level in time t, i = 1, 2, ⋯ , 7.

Finally, we considered that no shortages are allowed in each cycle length $\tilde{T}$ . Thus, it is assumed that P ≻ P_S ≻ $\tilde{D}$ , $(\tilde{T} - {\tilde{T}}_{4})$ ≻ T₁, and ${\tilde{T}}_{4}$ ≻ ${\tilde{T}}_{2}$ as per Yoo et al. [40].

Note 1. We consider a production system which is not perfect, i.e. produces some defective items under the fuzzy random framework. The inspection process that screens out the defective items is also imperfect. The probability density functions for the proportion of defective items and the inspection errors are generally taken from the study of past data of a supplier/machine and workers, where these are not dependent on lot size value [25]. Consequently, the inventory model has been studied extensively considering this assumption in the stochastic environment. (for instances [18 , 38]). In this article, we are studying an EPQ model with imperfect production process and inspection errors in the fuzzy random framework.

5 Mathematical model formulation

In this section, we obtain the inventory level I (t) at time t by considering the assumptions and the behavior of the inventory system described in Fig. 1.

The inventory level for serviceable items

During each cycle length $\tilde{T}$ , the governing differential equations of the inventory level of serviceable items are as follows: $\frac{d}{dt} (I_{i} (t)) = {\begin{matrix} P_{S} - \tilde{D} & for 0 \leq t \leq T_{1}, i = 1 \\ P_{RS} - \tilde{D} & for T_{1} \leq t \leq T_{1} + {\tilde{T}}_{2}, i = 2 \\ - \tilde{D} & for T_{1} + {\tilde{T}}_{2} \leq t \leq (T_{1} + {\tilde{T}}_{2} \\ + {\tilde{T}}_{3}), i = 3 \end{matrix}$ with the initial condition I₁ (0) =0.

The solutions of the equation are $I_{1} (t) = (P_{S} - \tilde{D}) t, I_{2} (t) = (P_{RS} - \tilde{D}) t + T_{1} (P_{S} - P_{RS}), I_{3} (t) = - \tilde{D} t + (T_{1} P_{S} - {\tilde{T}}_{2} P_{RS})$ .

Using the condition $I_{3} (T_{1} + {\tilde{T}}_{2} + {\tilde{T}}_{3}) = 0$ , wehave ${\tilde{T}}_{3} = \frac{1}{\tilde{D}} [T_{1} (P_{S} - \tilde{D}) + {\tilde{T}}_{2} (P_{RS} - \tilde{D})] .$ (5)

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The inventory level for screened items

The governing differential equations of the inventory level of screened items are given by $\frac{d}{dt} (I_{i} (t)) = {\begin{matrix} P [\tilde{α} \tilde{q} + (\tilde{p} - \tilde{β} \tilde{p})] & for 0 \leq t \leq T_{1}, i = 4 \\ P_{R} \tilde{η} & for T_{1} \leq t \leq T_{1} + {\tilde{T}}_{2}, \\ i = 5 \end{matrix}$

with the initial conditions I₄ (0) =0 and I₅ (T₁) =0. The solutions of the equation are $I_{4} (t) = P [\tilde{α} \tilde{q} + (\tilde{p} - \tilde{β} \tilde{p})] t, I_{5} (t) = P_{R} \tilde{η} (t - T_{1})$ .

The inventory level for returns items

The governing differential equation of the inventory level of return items is obtained by $\frac{d}{dt} (I_{6} (t)) = \begin{matrix} \frac{\tilde{β} \tilde{p} y}{T_{1} + \tilde{T_{2}}} & for 0 \leq t \leq T_{1} + {\tilde{T}}_{2}, \end{matrix}$ with the initial condition I₆ (0) =0. The solution of the equation is $I_{6} (t) = \frac{\tilde{β} \tilde{p} yt}{T_{1} + {\tilde{T}}_{2}} .$

The inventory level for re-workable items

The governing differential equation of the inventory level of re-workable items is as follows: $\frac{d}{dt} (I_{7} (t)) = \begin{matrix} - P_{R} & for T_{1} \leq t \leq T_{1} + {\tilde{T}}_{2}, \end{matrix}$ with the initial condition $I_{7} (T_{1} + {\tilde{T}}_{2}) = 0$ . The solution of the equation is $I_{7} (t) = P_{R} (T_{1} + {\tilde{T}}_{2} - t) .$

5.1 Holding cost

According to Dubois and Prade [15], Bag et al. [1], fuzzy integrations on the differential equations are done and obtained the inventory holding cost.

The inventory holding cost of serviceable items is given by

$\begin{matrix} h [\int_{0}^{T_{1}} I_{1} (t) dt + \int_{T_{1}}^{T_{1} + {\tilde{T}}_{2}} I_{2} (t) dt + \int_{T_{1} + {\tilde{T}}_{2}}^{T_{1} + {\tilde{T}}_{2} + {\tilde{T}}_{3}} I_{3} (t) dt] \\ = h {\frac{T_{1}^{2}}{2} (\frac{P_{S}^{2}}{\tilde{D}} - P_{S}) + \frac{{\tilde{T}}_{2}^{2}}{2} (\frac{P_{RS}^{2}}{\tilde{D}} - P_{RS}) \\ + T_{1} {\tilde{T}}_{2} (\frac{P_{S} P_{RS}}{\tilde{D}} - P_{RS})} . \end{matrix}$ (6)

The inventory holding cost for screened items is obtained as

$\begin{matrix} h [\int_{0}^{T_{1}} I_{4} (t) dt + \int_{T_{1}}^{T_{1} + {\tilde{T}}_{2}} I_{5} (t) dt] \\ = h [P {\tilde{α} \tilde{q} + (\tilde{p} - \tilde{β} \tilde{p})} \frac{T_{1}^{2}}{2} + P_{R} \tilde{η} \frac{{\tilde{T}}_{2}^{2}}{2}] . \end{matrix}$ (7)

The inventory holding cost of returned items is calculated by

$h [\int_{0}^{T_{1} + {\tilde{T}}_{2}} I_{6} (t) dt] = \frac{h}{2} \tilde{β} \tilde{p} y (T_{1} + {\tilde{T}}_{2}) .$ (8) The inventory holding cost of re-workable items is achieved by

$h [\int_{T_{1}}^{T_{1} + {\tilde{T}}_{2}} I_{7} (t) dt] = \frac{h}{2} P_{R} {\tilde{T}}_{2}^{2} .$ (9)

Thus, the total holding cost in each cycle is then given by $\begin{matrix} h [\frac{T_{1}^{2} P_{S}^{2}}{2 \tilde{D}} + \frac{{\tilde{T}}_{2}^{2}}{2} {\frac{P_{RS}^{2}}{\tilde{D}} + 2 (P_{R} - P_{RS})} \\ + T_{1} {\tilde{T}}_{2} (\frac{P_{S} P_{RS}}{\tilde{D}} - P_{RS}) + \frac{y}{2} (T_{1} + {\tilde{T}}_{2}) \tilde{β} \tilde{p}] . \end{matrix}$

5.2 Other cost

The production cost = cy.

The fixed set up cost = K.

The penalty and returned cost = $l (\tilde{β} \tilde{p}) y$ .

Rework cost = $w (\tilde{p} + \tilde{α} \tilde{q}) y$ .

The inspection cost = $i {y + T_{1} (P_{S} - (\tilde{D})) + (\tilde{p} + \tilde{α} \tilde{q}) y}$ .

5.3 Net revenue

The total revenue in each cycle consists of sales revenue of serviceable items and sales revenue from salvage items minus revenue loss due to defect refund. During each cycle length $\tilde{T}$ , the manufacturer obtained the sales revenue $s (\tilde{q} - \tilde{q} \tilde{α}) y$ by selling the perfect serviceable items among the serviceable items $[\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})] y$ (serviceable items including inspection errors) at a unit selling price of s. Moreover, among the screened and returned items $(\tilde{α} \tilde{q} + \tilde{p}) y$ , the manufacturer obtained the sales revenue $s (\tilde{α} \tilde{q} + \tilde{p}) \tilde{δ} y$ by selling the reworked perfect serviceable items $\tilde{δ} (\tilde{α} \tilde{q} + \tilde{p}) y$ at the unit selling price of s, and revenue of $v (\tilde{α} \tilde{q} + \tilde{p}) \tilde{η} y$ through the sales of the salvage items $(\tilde{α} \tilde{q} + \tilde{p}) \tilde{η} y$ at the unit salvage price of v (< s). Thus, the total revenue in each cycle becomes

$\tilde{TG} = s (\tilde{q} - \tilde{q} \tilde{α}) y + (\tilde{α} \tilde{q} + \tilde{p}) y (s \tilde{δ} + v \tilde{η}) .$

6 Optimal solution

The optimal solution maximizes the total expected profit per unit time in fuzzy sense. In this section, we present the solution procedure for the problem of obtaining the optimal lot size and the optimal length of the production runtime. The total profit incurred per production cycle is calculated by $\tilde{R} =$ (total revenue per cycle) - (holding cost per cycle) - ∑(other cost).

Now, the total profit incurred in each cycle can be obtained as

$\begin{matrix} \tilde{R} = (\tilde{q} - \tilde{q} \tilde{α}) ys + (\tilde{α} \tilde{q} + \tilde{p}) y (s \tilde{δ} + v \tilde{η}) - cy - K \\ - l (\tilde{β} \tilde{p}) y - w (\tilde{p} + \tilde{α} \tilde{q}) y - i {T_{1} (P_{S} - \tilde{D})} \\ - i {y + (\tilde{p} + \tilde{α} \tilde{q}) y} - h [\frac{T_{1}^{2} P_{S}^{2}}{2 \tilde{D}} + \frac{{\tilde{T_{2}}}^{2}}{2} {\frac{P_{RS}^{2}}{\tilde{D}} \\ + 2 (P_{R} - P_{RS})} + T_{1} {\tilde{T}}_{2} {\frac{P_{S} P_{RS}}{\tilde{D}} - P_{RS}} \\ - {\frac{y}{2} (T_{1} + T_{2}) \tilde{β} \tilde{p}}] \end{matrix}$ (10)

By plugging the values of $T_{1} = \frac{y}{P}$ , ${\tilde{T}}_{2} = (\frac{\tilde{p} + \tilde{α} \tilde{q}}{P_{R}}) y$ , $P_{S} = {\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})} P$ , and $P_{RS} = \tilde{δ} P_{R}$ in the equation (ref2m), the expected total profit in each cycle can be obtained as $\begin{matrix} E (\tilde{R}) \\ = E (\tilde{q} - \tilde{q} \tilde{α}) ys + E (\tilde{α} \tilde{q} + \tilde{p}) {sE (\tilde{δ}) + vE (\tilde{η})} y \\ - cy - K - lE (\tilde{β} \tilde{p}) y - wE (\tilde{p} + \tilde{α} \tilde{q}) y \\ - iy [1 + {E (\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})) - \frac{1}{P} E (\tilde{D})} \\ + E (\tilde{p} + \tilde{α} \tilde{q})] - {hy}^{2} [\frac{1}{2} E ({\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})}^{2}) E (\frac{1}{\tilde{D}}) \\ + \frac{E (\tilde{p} + \tilde{α} \tilde{q})^{2}}{2} {E ({\tilde{δ}}^{2}) E (\frac{1}{\tilde{D}}) + 2 P_{R}^{- 1} (1 - E (\tilde{δ}))} \\ + {E ((\tilde{p} + \tilde{α} \tilde{q}) {\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})}) E (\tilde{δ}) E (\frac{1}{\tilde{D}}) \\ - \frac{1}{P} E (\tilde{p} + \tilde{α} \tilde{q}) E (\tilde{δ})} \\ + {\frac{E (\tilde{β} \tilde{p})}{2 P} + \frac{E {(\tilde{p} + \tilde{α} \tilde{q}) (\tilde{β} \tilde{p})}}{2 P_{R}}}] . \end{matrix}$

Now, the cycle length is

$\tilde{T} = {(\tilde{q} - \tilde{q} \tilde{α}) + (\tilde{β} \tilde{p}) + (\tilde{α} \tilde{q} + \tilde{p}) \tilde{δ}} \frac{y}{\tilde{D}}$ .

Hence, the expected cycle length is

$E (\tilde{T}) = y {E (\tilde{q} - \tilde{q} \tilde{α}) + E (\tilde{β} \tilde{p})$ $+ E (\tilde{α} \tilde{q} + \tilde{p}) E (\tilde{δ})} E (\frac{1}{\tilde{D}})$ .

Theorem 6.1.(Fuzzy renewal reward theorem [19] .) Let ${({\tilde{X}}_{1}, {\tilde{Y}}_{1}), ({\tilde{X}}_{2}, {\tilde{Y}}_{2}), \dots}$ be the pair of sequence of independent and identically distributed fuzzy random variables on the probability space $(Ω, 𝔹, ℙ)$ , where ${\tilde{X}}_{n}$ is the inter-arrival time between the (n - 1)th and nth events, and ${\tilde{Y}}_{n}$ is the reward associated with the nth inter-arrival time ${\tilde{X}}_{n}$ , n=1,2,⋯, respectively. In addition, let $\tilde{C} (t)$ denote the total reward earned by the time t. If $E (X_{α}^{-}) < \infty$ , $E (X_{α}^{+}) < \infty$ , $E (Y_{α}^{-}) < \infty$ , and $E (Y_{α}^{+}) < \infty$ for 0 ≤ α ≤ 1, then $lim_{t \to \infty} \frac{E (\tilde{C} (t))}{t} = \frac{E ({\tilde{Y}}_{1})}{E ({\tilde{X}}_{1})}$ (11) where $E (X)_{1, α} = [E (X_{1, α}^{-}); E (X_{1, α}^{+}),]$ and E (Y) _1,α $= [E (Y_{1, α}^{-}); E (Y_{1, α}^{+}),]$ are α-cuts of $E ({\tilde{X}}_{1})$ and $E ({\tilde{Y}}_{1})$ , respectively.

Using the fuzzy renewal reward theorem, the expected unit profit is $E (\tilde{f}) = \frac{E (\tilde{R})}{E (\tilde{T})} = \tilde{X} - \tilde{Y} y^{- 1} - \tilde{Z} y$ , where $\tilde{X}$ , $\tilde{Y}$ , and $\tilde{Z}$ are given by as follows:

$\begin{matrix} \tilde{X} = (\tilde{A})^{- 1} {E (\frac{1}{\tilde{D}})}^{- 1} [E (\tilde{q} - \tilde{q} \tilde{α}) s - c - lE (\tilde{β} \tilde{p}) \\ + E (\tilde{α} \tilde{q} + \tilde{p}) {sE (\tilde{δ}) + vE (\tilde{η})} - wE (\tilde{p} + \tilde{α} \tilde{q}) \\ - i [1 + {E (\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})) - \frac{1}{P} E (\tilde{D})} \\ + E (\tilde{p} + \tilde{α} \tilde{q})]], \end{matrix}$ (12)

$\tilde{Y} = (\tilde{A})^{- 1} K {E (\frac{1}{\tilde{D}})}^{- 1},$ (13) $\begin{matrix} \tilde{Z} = h (\tilde{A})^{- 1} [\frac{1}{2} E ({\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})}^{2}) + \frac{E (\tilde{p} + \tilde{α} \tilde{q})^{2}}{2} \\ \times {E ({\tilde{δ}}^{2}) + 2 P_{R}^{- 1} (1 - E (\tilde{δ})) (E (\frac{1}{\tilde{D}}))^{- 1}} \\ + {\frac{E (\tilde{β} \tilde{p})}{2 P} + \frac{E {(\tilde{p} + \tilde{α} \tilde{q}) (\tilde{β} \tilde{p})}}{2 P_{R}}} (E (\frac{1}{\tilde{D}}))^{- 1} \\ + {E ((\tilde{p} + \tilde{α} \tilde{q}) {\tilde{β} \tilde{p} + (\tilde{q} - \tilde{q} \tilde{α})}) E (\tilde{δ}) \end{matrix}$

$- \frac{1}{P} E (\tilde{p} + \tilde{α} \tilde{q}) E (\tilde{δ}) (E (\frac{1}{\tilde{D}}))^{- 1}}],$ (14) and $\tilde{A} = {E (\tilde{q} - \tilde{q} \tilde{α}) + E (\tilde{β} \tilde{p}) + E (\tilde{α} \tilde{q} + \tilde{p}) E (\tilde{δ})} .$ (15)

Thus, the model can be formalized as the following maximization problem $(P) {\begin{matrix} Max & E (\tilde{f}) = \tilde{X} - \tilde{Y} y^{- 1} - \tilde{Z} y \\ s . t & y > 0 . \end{matrix}$ (16) Including the fuzzy random variable as demand along with fuzzy defective rates and fuzzy inspection errors, we obtained the total expected profit through the entire cycle as $E (\tilde{f}) = \tilde{X} - \tilde{Y} y^{- 1} - \tilde{Z} y .$ (17)

thomoson ok

Therefore, the expected profit per cycle is fuzzy in nature. In order to evaluate the conclusions for decision-making, the fuzzy results are to be transformed into crisp values. The method of developing crisp results from the fuzzy model is called defuzzification. We defuzzified the expected profit function by using the concept of possibilistic mean value. The possibilistic mean value of the expected profit function is denoted by $\bar{M} (E (\tilde{f}))$ and defined as $\bar{M} (E (\tilde{f})) = \int_{0}^{1} α [(E (\tilde{f}))_{α}^{-} + (E (\tilde{f}))_{α}^{+}] d α .$ (18) Now, the α-level set of $E (\tilde{f})$ is given by $(E (\tilde{f}))_{α} = E (\tilde{f_{α}}) = [(E (\tilde{f}))_{α}^{-}, (E (\tilde{f}))_{α}^{+}] \forall α \in [0, 1]$ where, $(E (\tilde{f}))_{α}^{-} = X_{α}^{-} - Y_{α}^{+} y^{- 1} - Z_{α}^{+} y$ (19) $(E (\tilde{f}))_{α}^{+} = X_{α}^{+} - Y_{α}^{-} y^{- 1} - Z_{α}^{-} y .$ (20) Substituting the values of (ref2b) and (ref2c) in the Equation (ref2a), we have the possibilistic mean value of the expected profit

$\begin{matrix} \bar{M} (E (\tilde{f})) = \int_{0}^{1} α [(X_{α}^{-} + X_{α}^{+}) - (Y_{α}^{-} + Y_{α}^{+}) / y] d α \\ - y \int_{0}^{1} α (Z_{α}^{-} + Z_{α}^{+}) d α . \end{matrix}$ (21)

6.1 Determination of the optimal solution

The maximization problem (P) is equivalent to the following conventional optimization problem ofsingle variable $(P^{'}) {\begin{matrix} Max & \bar{M} (E (\tilde{f})) \\ s . t & y > 0 . \end{matrix}$ (22)

Now, we determine the optimal lot size y^*(unit/unit time) to maximize the problem (P′) and get the corresponding optimal length of the production runtime with the first stage inspection $T_{1}^{*}$ . The optimal value of y can be determined from the necessary condition of optimality, i.e., $\frac{d}{dy} (\bar{M} (E (\tilde{f}))) = 0$ . It can also be shown that $\bar{M} (E (\tilde{f}))$ is a concave function of y. We obtain the optimal lot size and get the corresponding optimal length of the production runtime with the first stage inspection as follows $y^{*} = \sqrt{\frac{\int_{0}^{1} α (Y_{α}^{-} + Y_{α}^{+}) d α}{\int_{0}^{1} α (Z_{α}^{-} + Z_{α}^{+}) d α}} .$ (23) $T_{1}^{*} = \frac{y^{*}}{P} = \frac{1}{P} \sqrt{\frac{\int_{0}^{1} α (Y_{α}^{-} + Y_{α}^{+}) d α}{\int_{0}^{1} α (Z_{α}^{-} + Z_{α}^{+}) d α}}$ (24) By taking second derivative of $\bar{M} (E (\tilde{f}))$ , we obtain

$\begin{matrix} \frac{d^{2}}{{dy}^{2}} (\bar{M} (E (\tilde{f}))) = - 2 {\int_{0}^{1} α (Y_{α}^{-} + Y_{α}^{+}) d α} y^{- 3} \\ < 0 \forall y . \end{matrix}$ (25) The sufficient condition of optimality and (ref2d) ensures that the optimal solutions in (23) and (24) are the global optimal solutions.

7 Numerical example

In this section, we consider the following numerical example to demonstrate the model.

A firm produces a product with a production rate 10,000 units per year for industrial clients. The accounting section has estimated that the set up cost per cycle is $100 and each unit costs $20 to manufacture the product during the production runtime. The inventory holding cost is $2 per item per year. Due to imperfect production process, the production produces some defective items. The decision maker confirmed that the fuzzy proportion of imperfect quality items produced is uniformly distributed with the parameters $\tilde{a} = (0.01, 0.02, 0.03)$ and $\tilde{b} = (0.04, 0.05, 0.06)$ . An inspector screens out the imperfect quality items with two types of misclassifications (i.e. type-I and type-II inspection errors) during the production runtime. The unit inspection cost is $1 and the fuzzy proportion of type-I and type-II errors are uniformly distributed with the parameters $\tilde{c} = (0.015, 0.02, 0.025)$ , $\tilde{d} = (0.03, 0.035, 0.04)$ and $\tilde{c^{'}} = (0.012, 0.015, 0.018)$ , $\tilde{d^{'}} = (0.022, 0.025, 0.028)$ ; respectively. The reworkable items are reworked right after regular production at a rate of 1,500 units per year and the fuzzy proportion of imperfect quality of remanufacturing items is uniformly distributed with the parameters $\tilde{a^{'}} = (0.12, 0.16, 0.20)$ and $\tilde{b^{'}} = (0.16, 0.20, 0.24)$ . The unit rework cost is $25 and the unit return and goodwill cost is $5. The unit selling prices of serviceable and salvage items are $55 and $15, respectively, and the annual demand information is given in the Table reftable : demand. Thus, we have:

P = 10, 000 units/year, P_R = 1, 500 units/year, c = $20/unit, w = $25/unit, i = $1/unit, l = $5/unit, h = $2/unit, K = $100/cycle, s = $55/unit, v =$15/unit, $\tilde{p} \sim U (\tilde{a}, \tilde{b})$ (U (a, b) denotes the conventional uniform distribution) where $\tilde{a} =$ (0.01, 0.02, 0.03) and $\tilde{b} = (0.04, 0.05, 0.06)$ , $\tilde{α} \sim$ $U (\tilde{c}, \tilde{d})$ where $\tilde{c} = (0.015, 0.02, 0.025)$ and $\tilde{d} = (0.03, 0.035, 0.04)$ , $\tilde{β} \sim U (\tilde{c^{'}}, \tilde{d^{'}})$ where $\tilde{c^{'}} =$ (0.012, 0.015, 0.018) and $\tilde{d^{'}} = (0.022, 0.025,$ 0.028), $\tilde{η} \sim U (\tilde{a^{'}}, \tilde{b^{'}})$ where $\tilde{a^{'}} = (0.12, 0.16, 0.20)$ and $\tilde{b^{'}} = (0.16, 0.20, 0.24)$ .

Hence, the expectation of fuzzy proportion of defective rates, type-I and type-II errors can be constructed by the proposed procedure described in the Section (refa). It is easy to find

${\tilde{a}}_{α} = [a_{α}^{-}, a_{α}^{+}] = [0.01 + 0.01 α, 0.03 - 0.01 α] .$

${\tilde{b}}_{α} = [b_{α}^{-}, b_{α}^{+}] = [0.04 + 0.01 α, 0.06 - 0.01 α] .$

Thus, following (ref2f) and (ref2g), the mathematical programs for deriving the lower and upper bounds of the α-cuts of $E (\tilde{p})$ are: ${\begin{matrix} (E (\tilde{p}))_{α}^{-} & = min (\frac{a + b}{2}) \\ s . t & 0.01 + 0.01 α \leq a \leq 0.03 - 0.01 α; \\ 0.04 + 0.01 α \leq b \leq 0.06 - 0.01 α . \end{matrix}$ ${\begin{matrix} (E (\tilde{p}))_{α}^{+} & = max (\frac{a + b}{2}) \\ s . t & 0.01 + 0.01 α \leq a \leq 0.03 - 0.01 α; \\ 0.04 + 0.01 α \leq b \leq 0.06 - 0.01 α . \end{matrix}$

The solutions of the above models are $(E (\tilde{p}))_{α}^{-} = 0.025 + 0.01 α$ , $(E (\tilde{p}))_{α}^{+} = 0.045 - 0.01 α$ , respectively. The inverse functions of $(E (\tilde{p}))_{α}^{-}$ and $(E (\tilde{p}))_{α}^{+}$ exist, and the membership function for $E (\tilde{p})$ can be constructed as

$μ_{E (\tilde{p})} (z) = {\begin{matrix} \frac{z - 0.025}{0.01} & for 0.025 \leq z \leq 0.035, \\ 1 & for z = 0.035, \\ \frac{0.045 - z}{0.01} & for 0.035 \leq z \leq 0.045 . \end{matrix}$

The membership functions of $E (\tilde{q})$ , $E (\tilde{α})$ , $E (\tilde{β})$ , $E (\tilde{η})$ , and E (δ) can also be achieved similarly as follows: $μ_{E (\tilde{q})} (z) = {\begin{matrix} \frac{z - 0.955}{0.01} & for 0.955 \leq z \leq 0.965, \\ 1 & for z = 0.965, \\ \frac{0.975 - z}{0.01} & for 0.965 \leq z \leq 0.975 . \end{matrix}$ $μ_{E (\tilde{α})} (z) = {\begin{matrix} \frac{z - 0.0225}{0.005} & for 0.0225 \leq z \leq 0.0275, \\ 1 & for z = 0.0275, \\ \frac{0.0325 - z}{0.005} & for 0.0275 \leq z \leq 0.0375, \end{matrix}$ $μ_{E (\tilde{β})} (z) = {\begin{matrix} \frac{z - 0.017}{0.003} & for 0.017 \leq z \leq 0.020, \\ 1 & for z = 0.020, \\ \frac{0.023 - z}{0.003} & for 0.020 \leq z \leq 0.023, \end{matrix}$

$μ_{E (\tilde{η})} (z) = {\begin{matrix} \frac{z - 0.14}{0.04} & for 0.14 \leq z \leq 0.18, \\ 1 & for z = 0.18, \\ \frac{0.22 - z}{0.04} & for 0.18 \leq z \leq 0.22, \end{matrix}$ and $μ_{E (\tilde{δ})} (z) = {\begin{matrix} \frac{z - 0.78}{0.04} & for 0.78 \leq z \leq 0.82, \\ 1 & for z = 0.82, \\ \frac{0.86 - z}{0.04} & for 0.82 \leq z \leq 0.86 . \end{matrix}$

Using the Equations (ref2n), (ref2o), and (ref2p), the α-level set of $\tilde{X}$ , $\tilde{Y}$ , and $\tilde{Z}$ are given by as follows: $\begin{matrix} X_{α}^{-} = 137765.7689 + 19707.3735 α \\ + 1104.316 α^{2}, \end{matrix}$ (26) $\begin{matrix} X_{α}^{+} = 181031.9139 - 23609.32945 α \\ + 1215.0997 α^{2}, \end{matrix}$ (27) $\begin{matrix} Y_{α}^{-} = 468598.6614 + 33981.72468 α \\ + 720.7602611 α^{2}, \end{matrix}$ (28) $\begin{matrix} Y_{α}^{+} = 537096.4415 - 34461.51673 α \\ + 669.9671534 α^{2}, \end{matrix}$ (29) $\begin{matrix} Z_{α}^{-} = 0.920850941 + 0.088836448 α \\ + 0.002792343 α^{2}, \end{matrix}$ (30) $\begin{matrix} Z_{α}^{+} = 1.110885086 - 0.101677184 α \\ + 0.003550195 α^{2} . \end{matrix}$ (31)

By plugging-in the values of (ref2q1) , (ref2q4), and (ref2q6) in (ref2b); and the values of (ref2q2) , (ref2q3), and (ref2q5) in (ref2c); the α-level set of the expected annual profit is calculated as

$\begin{matrix} (E (\tilde{f}))_{α}^{-} = (137765.7689 + 19707.3735 α \\ + 1104.316 α^{2}) \\ - (537096.4415 - 34461.51673 α \\ + 669.9671534 α^{2}) y^{- 1} \\ - (1.110885086 - 0.101677184 α \\ + 0.003550195 α^{2}) y . \end{matrix}$ (32)

$\begin{matrix} (E (\tilde{f}))_{α}^{+} = (181031.9139 - 23609.32945 α \\ + 1215.0997 α^{2}) \\ - (468598.6614 + 33981.72468 α \\ + 720.7602611 α^{2}) y^{- 1} \\ - (0.920850941 + 0.088836448 α \\ + 0.002792343 α^{2}) y . \end{matrix}$ (33)

Therefore, the possibilistic mean value of the expected annual profit is obtained from (ref2c1) as follows:

$\begin{matrix} \bar{M} (E (\tilde{f})) = \int_{0}^{1} α [(E (\tilde{f}))_{α}^{-} + (E (\tilde{f}))_{α}^{+}] d α, \\ = 158678.0433 - 503035.3026 y^{- 1} \\ - 1.013173403 y . \end{matrix}$ (34)

The final problem is then constituted in the following manner: ${\begin{matrix} Max & 158678.0433 - 503035.3026 y^{- 1} - 1.013173403 y \\ s . t & y > 0 . \end{matrix}$ (35) The maximization problem given by (ref2l) is a conventional non-linear programming problem of single variable y. Figure 2 graphically illustrates the concavity of the annual profit function and the global optimal solution is: The optimal lot size y^* = 704.6238549 ≈ 705 units/cycle with the production and first stage inspection time $T_{1}^{*} = 0.070462385$ year ≈26 days, and the corresponding annual profit = $157250.231/year.

7.1 The sensitivity analysis

The change in the values of some parameters may appear due to uncertainties and dynamic market condition in any decision-making situation. In a decision-making process, the sensitivity analysis will be of great help for the implications of these changes in the values of parameters. In this subsection, sensitivity analysis has been conducted to study the effect of the cost parameters, fraction of defective items, and two types of errors on the lot size (y^*), production runtime $(T_{1}^{*})$ , and annual profit. The results of sensitivity analysis are carried out in Tables (reftable : 1-reftable :). Based on the results, the managerial phenomena can be interpreted as follows: From Table reftable : 1, it can be stated that as the setup cost (K) increases (decreases), the optimum lot size (y^*) and production runtime $(T_{1}^{*})$ increase (decrease), and as the holding costs (h) increases (decreases), the optimum lot size (y^*) and production runtime $(T_{1}^{*})$ decrease (increase), but the annual profit decreases (increases) in the both cases. It is also a notable observation that changing the production cost (c) during the production runtime $(T_{1}^{*})$ , rework cost (w), inspection cost (i), and penalty cost (l) do not change the optimum lot size (y^*) and production runtime $(T_{1}^{*})$ . But if these costs per unit item increase (decrease), the annual profit decreases (increases). Table reftable : shows the impact of defective rates and inspection errors on the optimal lot size, production runtime, and annual profit. We observe that the optimum lot size (y^*) and production runtime $(T_{1}^{*})$ increase when $\tilde{α}$ , $\tilde{η}$ increase and decrease when $\tilde{p}$ , $\tilde{β}$ increase. Our results indicate that the loss in the annual profit has a direct relation to the increment of the percentage of defects and inspection errors of the lot size. We observed that Type-I error had a much more effect on the annual profit than that of Type-II error. It is noticed that the optimum lot size and production runtime are almost same as the error-free model, but the error-free model attains better values for annual profit function than the error-prone model. This indicates the significance of the human errors in the inspection [25]. The use of multiple inspections is a common practice in the industry to mitigate the effect of errors ([8, 25]).

8 Conclusions

In this study, we have provided a more realistic, optimal economic production model. We discussed the model with imperfect quality items, inspection errors, sales returns, and rework under the fuzzy random variable demand. The traditional EPQ model assumed that the defective rates, type-I, and type-II errors are required to follow some probability distributions. However, in many realistic situations, the parameters of the probability distributions can not be expressed in exact terms. For the sake of practicality, it is generally preferable to describe the parameters of the distribution in linguistic terms, such as “the parameter θ₁ is approximate a ( $\in ℝ$ )”, “the parameter θ₂ is approximate $b (\in ℝ)$ ”, rather than by crisp values. In view of this fact, we have proposed the fuzzy defective rates, and the fuzzy inspection errors. A pair of mathematical programming approach was then used to construct the membership functions of the expected defective rates, the expected type-I, and type-II errors etc. in the model. The fuzzy random renewal reward theorem is used to derive the expected unit profit function. A methodology has been developed such that the total profit is maximized in fuzzy sense. Also, the possibilistic mean value of the expected profit function for the model has been proved to be concave. A numerical example is presented to illustrate the proposed methodology. Finally, the sensitivity analysis has been conducted to obtain some useful managerial insights.

The limitations of the present paper are the fixed production rate, rework rate and setup cost. The above limitations can be relaxed considering the fuzzy random production rate, rework rate and setup cost. On the other hand, there are some useful extensions of our model that can be done in the future study in this field. For example, the present model can be extended by allowing shortage during the production runtime. It would be interesting to incorporate some more realistic features, such as cost fluctuation, quantity discount, multi-item etc. in the model. Moreover, one could investigate the effect of learning in the inspection errors on lot size to enhance the usefulness of the model presented here.

Footnotes

Acknowledgments

The authors would like to express gratitude to the Editor Reza Langari, Associate Editor Cengiz Kahraman and anonymous referees for their helpful and constructive comments that have led to a substantial improvement of the paper.

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A fuzzy random EPQ model with fuzzy defective rates and fuzzy inspection errors

Abstract

Keywords

1 Introduction

2 Preliminary concepts

2.1 The extension principle

2.2 Possibilistic mean value of fuzzy number

3.1 Fuzzy type-I and type-II inspection errors

3.2 Construction of the membership function of E ( p ˜ )

Notations

5 Mathematical model formulation

The inventory level for serviceable items

The inventory level for screened items

The inventory level for returns items

The inventory level for re-workable items

5.3 Net revenue

6 Optimal solution

8 Conclusions

Footnotes

Acknowledgments

References

3.2 Construction of the membership function of $E (\tilde{p})$