An EPL model with reliability-dependent randomly imperfect production system over different uncertain finite time horizons

Abstract

Imperfect economic production lot size (EPL) models are considered over different types of uncertain finite time horizons with stock-dependent demand, reliability dependent defective rate and random out-of-control state. Generally in EPL models, defective production starts after the passage of some time from production commencement. So occurrence of defective production is random and imposed here through a chance constraint. Reliability of a machinery system affects on the defective rate and production cost to produce an item. Here unit production cost depends on reliability and production rate and part of it is taken against the environment protection cost. Both linear and non-linear production dependent forms of quality are considered. The problems are formulated as total cost minimization problems with crisp, random, fuzzy, fuzzy-random, rough and fuzzy-rough constraints and solved using generalized reduced gradient method (GRG). Several special cases are derived and numerical experiments are performed to illustrate the general and particular models.

Keywords

Imperfect production random defective rate stock-dependent demand chance constraints reliability uncertain time horizon

1 Introduction

As per Levin et al. [19], normally large piles of goods displayed in a super market leads the customers to buy more. It is a common belief that the massive displays of items in stores through electronic media are used as “psychic stock” to stimulate more sales of the items. For this reason, several authors [22 , 25] presented inventory models with stock dependent demand of the form D = c + dq or D = dq^β. Recently, Behret and Kahraman [1] analysed single period inventory models with discrete demand under fuzzy environment.

In the literature, there are some EPL models [10, 30] for imperfect units. Salameh and Jaber [32] studied the EOQ/EPQ (Economic Production Quantity) models for items with imperfect quality and proposed discount sales for them. Chiu [7] extended the work of Hayek and Salameh [15] and examined an EPQ model with defective items and rework of repairable items immediately. Sana [33] and Sarkar et al. [34] presented EPL models with random imperfect production process and connected process reliability with imperfectness respectively. Recently Chen [5] studied an integrated problem with production preventive maintenance, inspection and inventory for an imperfect production process.

In EPL models, normally unit production cost of a manufacturing system depends on the combination of different production factors such as raw materials, technical knowledge, resources, productionprocedure, wear and tear of machineries, firm size, quality of product, environmental pollution etc. Khouja and Mehrez [17] assumed a unit production cost involving raw material, labour/energy, and wear and tear costs. After that, several authors [9, 24] have implemented this in their EPL models. Sarkar et al. [35] considered the unit production cost as a function of reliability parameter in imperfect production process with safety stock. In the same year, they [34] investigated a model by introducing optimal reliability, production lot size and safety stock. But, till now, none considered the cost to be incurred by the manufacturer against the measures for environment protection.

When one or more parameters are described by random variables, the problem is expressed as a chance constraint. Charnes and Cooper [4] were first to develop the Chance Constraint technique. The problem with fuzzy parameters was solved by Liu and Iwamura [21] in the form of a chance constraint type. It has been used for various applications in several directions [27]. Similarly following Katagiri et al. [16] and Liu [20], the problems with fuzzy-random, rough and fuzzy-rough parameters can be reduced to crisp ones with the help of constraints like chance constraints.

The assumption of infinite planning horizon [8, 14] is not realistic due to several reasons such as variation of inventory costs, change in product specifications and designs, technological changes, etc. Moreover, for seasonal products like fruits, vegetables, warm garments, etc., business period is not infinite, rather fluctuates with each season. Therefore, it is better to estimate this type of finite time horizon as random or fuzzy or fuzzy-random or rough or fuzzy-rough in nature. An EOQ model [26] and inventory models [22] with stock dependent demand were developed over the random planning horizon. Some more research papers are also available in this direction [13, 31].

In the learning situation, Biskup [2] pointed out that repeated processing of similar tasks improves workers skills, e.g., workers are able to perform better, deal with machine operations or softwares, or handle raw materials and components at a faster pace. This concept of learning was also introduced by Cheng and Wang [6] in the field of scheduling. Recently, Biskup [3] presented a comprehensive review of research on scheduling with learning effects. Eren [12] proposed a non-linear mathematical programming model for the single-machine scheduling problem with unequal release and learning effects. But very few researchers have used the concept of learning in EPL with finite time horizon.

Though several inventory models are available with imperfect production, there are some lacunas in the literature. These are:

Most of the above EPL models are developed for infinite planning horizon. But in reality, lifetime of a product is finite and uncertain.

Some researchers have developed inventory models with random planning horizon, till now, none considered the planning horizon of an imperfect production-inventory model as fuzzy, fuzzy-random, rough and fuzzy-rough.

Defective production rate normally increases with the time elapsed from the out-of-control state and the production rate. Sana [33] considered this with constant demand and fully reworked of the defective units. None has investigated this phenomena in conjunction with reliability dependent defective production rate and imprecise time horizon.

Few investigators have considered the out-of-control state (starting point of imperfect production) to be random during the production time, but none has imposed it as a chance constraint for the system.

Unit production cost is normally assumed to be dependent on the raw material cost, development/labour cost and wear and tear cost. Till now, very few have considered development cost as a function of reliability of the system.

Now-a-days environment protection is mandatory in any production system. Till now, none included the cost incurred by management due to the measures taken for environment protection in the unit production cost.

In EPL system with several time cycles, very few introduced learning effect in set-up cost and maintenance cost over the time cycles.

In the present investigation, the above mentioned lacunas are removed. Here, we consider a randomly imperfect single-item production-inventory model over imprecise time horizon with learning effect on set-up cost, stock dependent demand, reliability dependent defective production rate, partially reworked and disposal of defective units, chance-constraint for commencement of imperfect production and variable production cost including environmental protection cost. The problem is formulated as a cost minimization problem with crisp, random, fuzzy, fuzzy-random, rough and fuzzy-rough constraints and solved using generalized reduced gradient method (GRG-LINGO11.0). Several special cases are derived and numerical experiments are performed to illustrate the general and particular models.

The rest of the paper is organized as follows. Following the introduction, in Section 2 preliminaries are briefly reviewed. Model is formulated in Section 3. Some particular cases are derived in Section 4. Solution method is presented in Section 5. Section 6 provides numerical experiments to illustrate the models. Section 7 discusses optimum results and managerial insights. Finally some conclusions are drawn in the last Section 8.

2 Mathematical prerequisites

Following to Liu and Iwamura [21], following Lemmas 1-2 are easily derived.

Lemma 1.If $\tilde{a} = (a_{1}, a_{2}, a_{3})$ be a TFN with 0 < a₁and b is a crisp number, $Pos (\tilde{a} > b) \geq α$ iff $\frac{a_{3} - b}{a_{3} - a_{2}} \geq α$ , where ‘Pos’ represents possibility measure.

Lemma 2.If $\tilde{a} = (a_{1}, a_{2}, a_{3})$ be a TFN with 0 < a₁and b is a crisp number, $Nes (\tilde{a} > b) \geq α$ iff $\frac{b - a_{1}}{a_{2} - a_{1}} \leq 1 - α$ , where ‘Nes’ represents necessity measure.

Fuzzy-random variable [28]: Let R is the set of real numbers, F_c (R) is set of all fuzzy variables and G_c (R) is all of non-empty bounded close interval. In a given probability space (Ω, F, P), a mapping ξ : Ω → F_c (R) is called a fuzzy random variable in (Ω, F, P), if ∀ α ∈(0,1], the set-valued function ξ_α : Ω → G_c (R) defined by ξ_α (ω) = (ξ (ω)) _α = {x|x ∈ R, μ_ξ(ω) (x) ≥ α}, ∀ω ∈ Ω, is F measurable. Different semantics of fuzzy-random variable are also presented by Xu and Zhou [36].

Theorem 1.Let $\tilde{\bar{ξ}}$ is LR fuzzy random variable, for any ω ∈ Ω, the membership function of $\tilde{\bar{ξ}} (ω)$ is $μ_{\tilde{\bar{ξ}} (ω)} (t) = {\begin{matrix} L (\frac{\bar{ξ} (ω) - t}{ξ_{L}}) for t \leq \bar{ξ} (ω) \\ R (\frac{t - \bar{ξ} (ω)}{ξ_{R}}) for t \geq \bar{ξ} (ω) \end{matrix}$ where the random variable $\bar{ξ} (ω)$ is normally distributed with mean m_ξand standard deviation σ_ξand ξ_L, ξ_Rare the left and right spreads of $\tilde{\bar{ξ}} (ω)$ . The reference functions L: [0,1]→ [0,1], R: [0,1]→ [0,1] satisfies that L(1)=R(1)=0,L(0)=R(0)=1, and both are monotone function. Then ${\begin{matrix} \Pr [Pos {\tilde{\bar{ξ}} (ω) \geq t} \geq δ] \geq γ \\ \Pr [Nec {\tilde{\bar{ξ}} (ω) \geq t} \geq δ] \geq γ \end{matrix}$ are equivalent to $t \leq {\begin{matrix} m_{ξ} + σ_{ξ} Φ^{- 1} (1 - γ) + ξ_{R} R^{- 1} (δ) \\ m_{ξ} + σ_{ξ} Φ^{- 1} (1 - γ) - ξ_{L} L^{- 1} (1 - δ) \end{matrix}$ where Φ is standard normally distributed, δ, γ∈[0,1] are predetermined confidence levels.

Proof. According to definition of possibility [11 , 37] we get, $Pos [\tilde{\bar{ξ}} (ω) \geq t] \geq δ \Leftrightarrow R [\frac{t - \bar{ξ} (ω)}{ξ_{R}}] \leq δ \Leftrightarrow \bar{ξ} (ω) \geq t - ξ_{R} R^{- 1} (δ)$ . So for predetermined level δ, γ ∈ [0, 1] we have, $\begin{matrix} \Pr [Pos {\tilde{\bar{ξ}} (ω) \geq t} \geq δ] \geq γ \\ \Leftrightarrow \Pr [\bar{ξ} (ω) \geq t - ξ_{R} R^{- 1} (δ)] \geq γ \\ \Leftrightarrow \Pr [\frac{\bar{ξ} (ω) - m_{ξ}}{σ_{ξ}} \geq \frac{t - ξ_{R} R^{- 1} (δ) - m_{ξ}}{σ_{ξ}}] \geq γ \\ \Leftrightarrow Φ (\frac{t - ξ_{R} R^{- 1} (δ) - m_{ξ}}{σ_{ξ}}) \leq 1 - γ \\ \Leftrightarrow t \leq m_{ξ} + σ_{ξ} Φ^{- 1} (1 - γ) + ξ_{R} R^{- 1} (δ) \end{matrix}$ Similarly from the measure of necessity [11 , 37] we have, $\begin{matrix} Nes [\tilde{\bar{ξ}} (ω) \geq t] \geq δ \Leftrightarrow L [\frac{\bar{ξ} (ω) - t}{ξ_{L}}] \geq 1 - δ \\ \Leftrightarrow \bar{ξ} (ω) \geq t + ξ_{L} L^{- 1} (1 - δ) \end{matrix}$ So for predetermined level δ, γ ∈ [0, 1] we have, $\begin{matrix} \Pr [Nes {\tilde{\bar{ξ}} (ω) \geq t} \geq δ] \geq γ \\ \Leftrightarrow t \leq m_{ξ} + σ_{ξ} Φ^{- 1} (1 - γ) - ξ_{L} L^{- 1} (1 - δ) \end{matrix}$ The proof is complete.

Trust measure [20]: Let (Λ, Δ, κ, π) be a rough space. The trust measure of event A is denoted by Tr {A} and defined by $Tr {A} = \frac{1}{2} (\underline{Tr} {A} + \bar{Tr} {A})$ , where $\underline{Tr} {A}$ and $\bar{Tr} {A}$ denote the lower and upper trust measures of event A, defined by $\underline{Tr} {A} = \frac{π {A \cap Δ}}{π {Δ}}$ and $\bar{Tr} {A} = \frac{π {A}}{π {Λ}}$ respectively. When the sufficient amount of information is given about the measurement of π for a real life problem, it may be viewed as the Lebesgue measure. More generally, the above form can be considered as $Tr {A} = (1 - η) \underline{Tr} {A} + η \bar{Tr} {A}$ , where 0 < η < 1

Let $\hat{ξ} = ([a, b] [c, d])$ , c ≤ a ≤ b ≤ d be a rough variable and lebesgue measure is used for trust measure of an rough event associated with $\hat{ξ} \geq t$ . Then the trust measure of the rough event $\hat{ξ} \geq t$ is denoted by $Tr {\hat{ξ} \geq t}$ and its function curve is presented below $Tr {\hat{ξ} \geq t} = {\begin{matrix} 0 & for d \leq t \\ \frac{η (d - t)}{(d - c)} & for b \leq t \leq d, \\ \frac{η (d - t)}{d - c} + \frac{(1 - η) (b - t)}{b - a} & for a \leq t \leq b, \\ \frac{η (d - t)}{d - c} + (1 - η) & for c \leq t \leq a, \\ 1 & for t \leq c \end{matrix}$

Theorem 2. Let $\hat{ξ} = ([a, b] [c, d])$ , c ≤ a ≤ b ≤ d be a rough variable and a rough event is $\hat{ξ} \geq t$ . Then $Tr {\hat{ξ} \geq t} \geq α$ iff $t \leq {\begin{matrix} d - \frac{α (d - c)}{η}, & b \leq t \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)}, & a \leq t \leq b \\ d + \frac{(1 - η - α) (d - c)}{η}, & c \leq t \leq a \\ c \end{matrix}$

Proof. For predetermined level α ∈ [0, 1], we have, $\begin{matrix} Tr [\hat{ξ} \geq t] \geq α \\ \Leftrightarrow α \leq Tr [\hat{ξ} \geq t], \\ \Leftrightarrow α \leq {\begin{matrix} \frac{η (d - t)}{(d - c)} & for b \leq t \leq d, \\ \frac{η (d - t)}{d - c} + \frac{(1 - η) (b - t)}{b - a} & for a \leq t \leq b, \\ \frac{η (d - t)}{d - c} + (1 - η) & for c \leq t \leq a, \\ 1 & for t \leq c \end{matrix} \\ [using the definition of trust measure of an event] \\ \Leftrightarrow t \leq {\begin{matrix} d - \frac{α (d - c)}{η}, & b \leq t \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)}, & a \leq t \leq b \\ d + \frac{(1 - η - α) (d - c)}{η}, & c \leq t \leq a \\ c \end{matrix} \end{matrix}$ The proof is complete.

Fuzzy-Rough variable [20]: Fuzzy rough variable is a measurable function from a rough space (Λ, Δ, κ, π) to the set of fuzzy variables. More generally, a fuzzy rough variable is a rough variable taking fuzzy values.

Chance of Fuzzy Rough event:

Approach-1 [20]: Let $\tilde{\hat{ξ}}$ be a fuzzy rough variable on the rough space (Λ, Δ, κ, π) and $g_{r} (\tilde{\hat{ξ}})$ be the continuous functions, r = 1, 2, . . . , p . Then the chance of the fuzzy rough event $g_{r} (\tilde{\hat{ξ}}) \leq 0 for r = 1, 2, . . ., p .$ be a function from [0,1] to [0,1], which is defined as $\begin{matrix} Ch {g_{r} (\tilde{\hat{ξ}}) & \leq & 0, r = 1, 2, . . ., p} (α) \\ = & \sup {β ∣ Tr {Pos {g_{r} (\tilde{\hat{ξ}}) \\ \leq & 0, r = 1, 2, . . ., p} \geq β} \geq α}, \end{matrix}$ where α, β ∈ [0, 1].

Approach-2 [20]: Let $\tilde{\hat{ξ}}$ be a fuzzy rough variable on the rough space (Λ, Δ, κ, π) and $g_{r} (\tilde{\hat{ξ}})$ be the continuous functions, r = 1, 2, . . . , p . Then the chance of the fuzzy rough event $g_{r} (\tilde{\hat{ξ}}) \leq 0 for r = 1, 2, . . ., p .$ be a function from [0,1] to [0,1], which is defined as $\begin{matrix} Ch {g_{r} (\tilde{\hat{ξ}}) & \leq & 0, r = 1, 2, . . ., p} (α) \\ = & \sup {β ∣ Tr {Nes {g_{r} (\tilde{\hat{ξ}}) \\ < & 0, r = 1, 2, . . ., p} \geq β} \geq α}, \end{matrix}$ where α, β ∈ [0, 1].

Theorem 3.Let $\tilde{\hat{ξ}} = (\hat{ξ} - L, \hat{ξ}, \hat{ξ} + R)$ is a fuzzy rough variable characterized the following membership function, $μ_{\tilde{\hat{ξ}}} (t) = {\begin{matrix} \frac{t - \hat{ξ} + ξ_{L}}{ξ_{L}} & for \hat{ξ} - ξ_{L} \leq t \leq \hat{ξ} \\ \frac{\hat{ξ} + ξ_{R} - t}{ξ_{R}} & for \hat{ξ} \leq t \leq \hat{ξ} + ξ_{R} \\ 0 & otherwise . \end{matrix}$

where ξ_Land ξ_Rare left and right spreads of $\tilde{\hat{ξ}}$ , and $\hat{ξ} = ([a, b] [c, d])$ be a rough variable, characterized by the above mentioned trust measure function, then for an event $\tilde{\hat{ξ}} \geq t$ , ${\begin{matrix} Tr [Pos (\tilde{\hat{ξ}} \geq t) \geq β] \geq α \\ Tr [Nec (\tilde{\hat{ξ}} \geq t) \geq β] \geq α \end{matrix}$ are equivalent to ${\begin{matrix} t \leq {\begin{matrix} d - \frac{α (d - c)}{η} + (1 - β) ξ_{R}, \\ b \leq t - (1 - β) ξ_{R} \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)} + (1 - β) ξ_{R}, \\ a \leq t - (1 - β) ξ_{R} \leq b \\ d + \frac{(1 - η - α) (d - c)}{η} + (1 - β) ξ_{R}, \\ c \leq t - (1 - β) ξ_{R} \leq a \\ c + (1 - β) ξ_{R} \end{matrix} \\ t \leq {\begin{matrix} d - \frac{α (d - c)}{η} - β ξ_{L}, b \leq t + β ξ_{L} \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)} - β ξ_{L}, \\ a \leq t + β ξ_{L} \leq b \\ d + \frac{(1 - η - α) (d - c)}{η} - β ξ_{L}, c \leq t + β ξ_{L} \leq a \\ c - β ξ_{L} \end{matrix} \end{matrix}$

Proof. For the given confidence level α and β in [0, 1] we have, $\begin{matrix} Tr [Pos (\tilde{\hat{ξ}} \geq t) \geq β] \geq α \\ \Leftrightarrow Tr [\frac{\hat{ξ} + ξ_{R} - t}{ξ_{R}} \geq β] \geq α (using Lemma - 1) \\ \Leftrightarrow Tr [\hat{ξ} \geq t - (1 - β) ξ_{R}] \geq α \\ \Leftrightarrow α \leq {\begin{matrix} \frac{η [d - t + (1 - β) ξ_{R}]}{(d - c)}, b \leq t - (1 - β) ξ_{R} \leq d, \\ \frac{η [d - t + (1 - β) ξ_{R}]}{d - c} + \frac{(1 - η) [b - t + (1 - β) ξ_{R}]}{b - a}, \\ a \leq t - (1 - β) ξ_{R} \leq b, \\ \frac{η [d - t + (1 - β) ξ_{R}]}{d - c} + (1 - η), \\ c \leq t - (1 - β) ξ_{R} \leq a, \\ 1, t - (1 - β) ξ_{R} \leq c . \\ (using trust measure .) \end{matrix} \\ \Leftrightarrow t \leq {\begin{matrix} d - \frac{α (d - c)}{η} + (1 - β) ξ_{R}, \\ b \leq t - (1 - β) ξ_{R} \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)} \\ + (1 - β) ξ_{R}, \\ a \leq t - (1 - β) ξ_{R} \leq b \\ d + \frac{(1 - η - α) (d - c)}{η} + (1 - β) ξ_{R}, \\ c \leq t - (1 - β) ξ_{R} \leq a \\ c + (1 - β) ξ_{R} \end{matrix} \end{matrix}$ Similarly for necessity measure with the same confidence level $\begin{matrix} Tr [Nes (\tilde{\hat{ξ}} \geq t) \geq β] \geq α \\ \Leftrightarrow Tr [\hat{ξ} \geq t + β ξ_{L}] \geq α (using Lemma - 2) \\ \Leftrightarrow α \leq {\begin{matrix} \frac{η [d - t - β ξ_{L}]}{(d - c)}, b \leq t + β ξ_{L} \leq d, \\ \frac{η [d - t - β ξ_{L}]}{d - c} + \frac{(1 - η) [b - t - β ξ_{L}]}{b - a}, \\ a \leq t + β ξ_{L} \leq b, \\ \frac{η [d - t - β ξ_{L}]}{d - c} + (1 - η), c \leq t + β ξ_{L} \leq a, \\ 1, t + β ξ_{L} \leq c . (using trust measure .) \end{matrix} \\ \Leftrightarrow t \leq {\begin{matrix} d - \frac{α (d - c)}{η} - β ξ_{L}, b \leq t + β ξ_{L} \leq d \\ \frac{η (b - a) + (1 - η) b (d - c) - α (d - c) (b - a)}{η (b - a) + (1 - η) (d - c)} - β ξ_{L}, \\ a \leq t + β ξ_{L} \leq b \\ d + \frac{(1 - η - α) (d - c)}{η} - β ξ_{L}, c \leq t + β ξ_{L} \leq a \\ c - β ξ_{L} \end{matrix} \end{matrix}$

The proof is complete.

3 Proposed imperfect production inventory model

Assumptions:

Replenishment rate is finite and taken as a decision variable.

Lead time is zero.

Shortages are not allowed.

The inventory system considers a single item and the demand rate is stock-dependent.

The time horizon is finite and the production time is taken as a decision variable.

The production process shifts from the “In-control” state to an “Out-of-control” state at a time, which is a random variable. Imperfect units are produced in this state.

Production of defective units commences at a random time after the commencement of production. Defective rate depends on production rate, reliability of the machinery system producing the item and time duration from the starting of defective units’ production.

The system allows immediate partially reworking for the defective units at a certain cost when they are produced in “out-of-control” state and the defective units which are not reworked, are disposed off by a cost.

Unit production cost is the sum total of per unit material cost, development cost, wear and tear cost and environment protection cost. Here development cost is a function of reliability parameter of the machinery system which is also a decision variable.

A maintenance cost is considered for the machinery system to bring to its initial position by the maintenance operations during the each time gap between the end of production and beginning of next production. Thus the time for maintenance is shorter than the time gap between the end of the production and beginning of next production.

Notations:

The following notation are defined and noted for i-th cycle.

q (t): Inventory level at time t, where q (t) ≥0.

D [q (t)]: The demand rate at time t and D [q (t)] = d₀ + d₁q (t) where d₀ > 0, d₁ is the stock-dependent consumption rate parameter, 0 ≤ d₁ ≤ 1.

P: Controllable production rate in units per unit time (decision variable), where P - D [q (t)] ≥0.

T: Cycle time in appropriate unit.

t₁: Production run-time in each period (decision variable).

C_{s
_i}: Set up cost for i-th cycle which is partly constant and partly decreases in each cycle due to learning effect of the employees and is of the form: Cs_i = C_s0 + C_s1e^−k₁i, where k₁ > 0.

Cm_i: Maintenance cost for the machinery system to bring the system to its original position after the end of each production. For the first cycle no maintenance is required, but for the next cycles, it is increased in each cycle due to the reuse of the system for several times. Maintenance cost for i-th cycle is taken as: Cm_i = C_m0 [1 − e^{−k₂(i−1)}] .

Ch: Holding cost per unit per unit time.

Cd: Cost of disposal for an imperfect unit which is not reworked.

Cr: Cost for rework of an imperfect unit.

C (P, r): Unit production cost which is considered as: $C (P, r) = r_{m} + \frac{g}{P^{δ_{1}}} + η_{1} P^{δ_{2}} + η_{2} P^{δ_{3}}$ , where δ₁, δ₂, δ₃ > 0 and r_m is the material cost per unit item, g is the development cost, defined as $g = g_{1} + g_{2} e^{(1 - f) \frac{r - r_{\min}}{r_{\max} - r}}$ , where r is the reliability parameter of machinery system (decision variable), r_max and r_min are maximum and minimum value of r respectively, f is the feasibility of increasing reliability, g₁ is total labour/energy costs per unit time in a production system which is equally distributed over the unit item and independent of reliability parameter r, g₂ is technology, resource and design complexity costs for production. So, $(\frac{g}{P^{δ_{1}}})$ decreases with increases of P. The third term η₁P^δ
₂ is the wear and tear cost, proportional to the positive power of production rate P and the fourth term η₂P^δ
₃ is environment protection cost, proportional to the positive power of production rate P.

$\frac{1}{f (P)}$ : The mean and standard deviation of the random variable τ. Here, f (P) is an increasing function of P and the mean time of failure, 1/f (P) is a decreasing function of P.

τ: An exponential random variable that depends on P and denotes the time at which the process shifts to the “out-of-control” state from in-control variable. The distribution function of ‘out-of-control’ state is G (τ) =1 − e^−f(P)τ such that $\int_{0}^{\infty} dG (τ) = f (P) \int_{0}^{\infty} e^{- f (P) τ} d τ = 1$ .

λ (t′, τ, P, r): Percentage of defective units produced at time t when the machine is in the ‘out-of-control’ state. Here λ (t′, τ, P, r) is defined as λ (t′, τ, P, r) = αP^βe^(1−r)t′ (t′ − τ) ^γ. where β ≥ 0,γ ≥ 0 and t′ ≥ τ. Generally speaking, the percentage of defective units increases with increase of production rate and production-run time. The formulation of the function λ (t′, τ, P, r) shows that it is an increasing function of production rate and production-run time simultaneously and a decreasing function with respect to reliability parameter. Here t′ is the time measured from the commencement of production in each cycle and varies between (0,T). This is assumed in this way as the machinery system is brought back to its original condition by its proper maintenance in each cycle after each production run.

θ: Percentage of rework of defective units.

N: Defective units in a production cycle.

S_q: Expected production lot size (or inventory) of good units (without defective units) at the end of production period.

m: Total number of cycles which is a decision variable.

H: Finite time horizon.

3.1 Model-1: (with stock-dependent demand)

In this production process for the i-th cycle, production starts at a rate P from time t = (i − 1) T and runs up to time t = (i − 1) T + t₁. The inventory piles up, during the time span [0, t₁] adjusting demand D [q (t)] in the market and the production process stocks good quality Sq units at time t = t₁ and this stock depletes satisfying the demand in the market and it reaches at zero level at time iT (cf. Fig. 1units) with a defective rate). This production system produces perfect units up to a certain time τ (i.e., in-control state), after that, the production system shifts to an “out-of-control” state. In this “out-of-control” state, some of the produced units are of non-conforming quality (i.e., defective units) with a defective rate λ (t, τ, P, r) = αP^βe^{(1−r)[t−(i−1)T]} [t − (i − 1) T − τ] ^γ and some of these defective units are in a condition to rework immediately when they are produced. Here we assume that after the end of one production run, the machinery system is maintained against a cost and brought back to its original good condition for the next production. Thus the maintenance time is shorter than the production lay off time, T − t₁. The governing differential equations for the i-th cycle are

$\frac{dq (t)}{dt} = {\begin{matrix} P - D [q (t)], (i - 1) T \leq t \leq (i - 1) T + τ \\ P - D [q (t)] - (1 - θ) λ (t, τ, P, r) P, \\ (i - 1) T + τ \leq t \leq (i - 1) T + t_{1} \\ - D [q (t)], (i - 1) T + t_{1} \leq t \leq iT \end{matrix}$ (1)

with the boundary conditions q ((i − 1) T) =0 = q (iT) and the continuity conditions of q (t) at t = (i − 1) T + τ and t = (i − 1) T + t₁.

Using the above boundary and continuity conditions, the solutions of the above differential equation are given by,

$\begin{matrix} q (t) \\ = {\begin{matrix} \frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} [t - (i - 1) T]}), \\ (i - 1) T \leq t \leq (i - 1) T + τ \\ \frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} [t - (i - 1) T]}) - (1 - θ) α P^{β + 1} \\ [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! (t - (i - 1) T - τ)^{γ - j}}{(γ - j)! u^{j + 1}} e^{(u - d_{1}) [t - (i - 1) T]} \\ + \frac{(- 1)^{γ} γ! e^{- d_{1} [t - (i - 1) T]}}{u^{γ + 1}} (e^{u [t - (i - 1) T]} - e^{u τ})], \\ (i - 1) T + τ \leq t \leq (i - 1) T + t_{1}, \\ \frac{d_{0}}{d_{1}} (e^{d_{1} (iT - t)} - 1), (i - 1) T + t_{1} \leq t \leq iT \end{matrix} \end{matrix}$ (2) where u = d₁ + 1 − r.

The total defective units during [(i − 1) T + τ, (i − 1) T + t₁] is $N = \int_{(i - 1) T + τ}^{(i - 1) T + t_{1}} λ Pdt$

$\begin{matrix} or, N & = & P \int_{(i - 1) T + τ}^{(i - 1) T + t_{1}} α P^{β} \\ [t - (i - 1) T - τ]^{γ} e^{(1 - r) [t - (i - 1) T]} dt \\ = & α P^{β + 1} [\sum_{j = 0}^{γ} \frac{(- 1)^{j} γ! (t_{1} - τ)^{γ - j} e^{(1 - r) t_{1}}}{(γ - j)! (1 - r)^{j + 1}} \\ - \frac{(- 1)^{γ} γ! e^{(1 - r) τ}}{(1 - r)^{γ + 1}}] \end{matrix}$ (3) Total expected defective units in a production lot size is

$\begin{matrix} E (N) = \int_{0}^{t_{1}} Nd (G (τ)) \\ = α P^{β + 1} [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! e^{(1 - r) t_{1}}}{(γ - j)! (1 - r)^{j + 1}} \\ (\sum_{k = 0}^{γ - j - 1} \frac{(- 1)^{k} (γ - j)! t_{1}^{γ - j - k}}{(γ - j - k)! f^{k + 1} (P)} \\ - \frac{(- 1)^{γ - j + 1} (γ - j)!}{f^{γ - j + 1} (P)} \times [1 - e^{- f (P) t_{1}}]) \\ + \frac{(- 1)^{γ} γ! e^{(1 - r) t_{1}}}{(1 - r)^{γ + 1}} N_{2} - \frac{(- 1)^{γ} γ!}{(1 - r)^{γ + 1}} N_{3}] \end{matrix}$ (4)

where $N_{2} = \int_{0}^{t_{1}} e^{- f (P) τ} d τ = \frac{1 - e^{- f (P) t_{1}}}{f (P)}$ , $N_{3} = \int_{0}^{t_{1}} e^{- v τ} d τ = \frac{1 - e^{- {vt}_{1}}}{v}$ , and v = f (P) − (1 − r).

Now at time t = (i − 1) T + t₁ the expected production lot size without defective units is $\begin{matrix} S_{q} = E [q ((i - 1) T + t_{1})] \\ = \int_{0}^{\infty} q [(i - 1) T + t_{1}] d (G (τ)) \\ = \int_{0}^{\infty} [\frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} t}) - (1 - θ) α P^{β + 1} \\ (\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! (t - τ)^{γ - j}}{(γ - j)! u^{j + 1}} \times e^{(u - d_{1}) t} \\ + \frac{(- 1)^{γ} γ! e^{- d_{1} t}}{u^{γ + 1}} (e^{ut} - e^{u τ}))] d (G (τ)) \\ = \frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} t_{1}}) - (1 - θ) α P^{β + 1} f (P) \\ \times \int_{0}^{t_{1}} (\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! (t - τ)^{γ - j}}{(γ - j)! u^{j + 1}} e^{(u - d_{1}) t} \\ + \frac{(- 1)^{γ} γ! e^{- d_{1} t}}{u^{γ + 1}} (e^{ut} - e^{u τ})) e^{- f (P) τ} d τ \\ = \frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} t_{1}}) - (1 - θ) α P^{β + 1} f (P) \\ \times [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! e^{(u - d_{1}) t_{1}}}{(γ - j)! u^{j + 1}} \\ (\sum_{k = 0}^{γ - j} \frac{(- 1)^{k} (γ - j)! t_{1}^{γ - j - k}}{(γ - j - k)! f^{k + 1} (P)} \\ + \frac{(- 1)^{γ - j + 1} (γ - j)!}{f^{γ - j + 1} (P)} e^{- f (P) t_{1}}) \\ + \frac{(- 1)^{γ} γ! e^{- d_{1} t_{1}}}{u^{γ + 1}} (e^{{ut}_{1}} N_{2} - Q_{3})] \end{matrix}$ (5) $where, Q_{3} = \int_{0}^{t_{1}} e^{- [f (P) - u] τ} d τ = \frac{1 - e^{- [f (P) - u] t_{1}}}{f (P) - u}$

Again from the Equation (2) we get,

$\begin{matrix} S_{q} = E [q ((i - 1) T + t_{1})] = \frac{d_{0}}{d_{1}} (e^{d_{1} (T - t_{1})} - 1) \\ or, T = t_{1} + \frac{1}{d_{1}} log [1 + \frac{d_{1} S_{q}}{d_{0}}] \end{matrix}$ (6)

During the period [(i − 1) T, (i − 1) T + t₁], the inventory which are to be hold, is

$\begin{matrix} Q_{h_{1}} \\ = \int_{(i - 1) T}^{(i - 1) T + t_{1}} q (t) dt \\ = \int_{(i - 1) T}^{(i - 1) T + τ} \frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} [t - (i - 1) T]}) dt \\ + \int_{(i - 1) T + τ}^{(i - 1) T + t_{1}} [\frac{P - d_{0}}{d_{1}} (1 - e^{- d_{1} [t - (i - 1) T]}) \\ - (1 - θ) α P^{β + 1} \\ \times (\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ! [t - (i - 1) T - τ]^{γ - j}}{(γ - j)! u^{j + 1}} \\ e^{(u - d_{1}) [t - (i - 1) T]} \\ + \frac{(- 1)^{γ} γ! e^{- d_{1} [t - (i - 1) T]}}{u^{γ + 1}} (e^{u [t - (i - 1) T]} - e^{u τ}))] dt \\ = \frac{P - d_{0}}{d_{1}} (t_{1} - \frac{1}{d_{1}} (1 - e^{- d_{1} t_{1}})) - (1 - θ) α P^{β + 1} \\ \times [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ!}{(γ - j)! u^{j + 1}} \\ (\sum_{k = 0}^{γ - j} \frac{(- 1)^{k} (γ - j)! (t_{1} - τ)^{γ - j - k} e^{(u - d_{1}) t_{1}}}{(γ - j - k)! (u - d_{1})^{k + 1}} \\ - \frac{(- 1)^{γ - j} (γ - j)! e^{(u - d_{1}) τ}}{(u - d_{1})^{γ - j + 1}}) + \frac{(- 1)^{γ} γ!}{u^{γ + 1}} \\ \times (\frac{e^{(u - d_{1}) t_{1}} - e^{(u - d_{1}) τ}}{u - d_{1}} - \frac{e^{u τ}}{d_{1}} (e^{- d_{1} τ} - e^{- d_{1} t_{1}}))] \end{matrix}$ (7)

During the period [(i − 1) T, (i − 1) T + t₁], the expected quantity of holding units are

$\begin{matrix} E [Q_{h_{1}}] \\ = \int_{0}^{\infty} Q_{h_{1}} d (G (τ)) \\ = \frac{P - d_{0}}{d_{1}} (t_{1} - \frac{1}{d_{1}} (1 - e^{- d_{1} t_{1}})) \\ - (1 - θ) α P^{β + 1} f (P) \\ \times [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ!}{(γ - j)! u^{j + 1}} \\ {\sum_{k = 0}^{γ - j} \frac{(- 1)^{k} (γ - j)! e^{(u - d_{1}) t_{1}}}{(γ - j - k)! (u - d_{1})^{k + 1}} \\ \times (\sum_{s = 0}^{γ - j - k} \frac{(- 1)^{s} (γ - j - k)! t_{1}^{γ - j - k - s}}{(γ - j - k - s)! f^{s + 1} (P)} \\ + \frac{(- 1)^{γ - j - k + 1} (γ - j - k)!}{f^{γ - j - k + 1} (P)} e^{- f (P) t_{1}}) \\ - \frac{(- 1)^{γ - j} (γ - j)!}{(u - d_{1})^{γ - j + 1}} N_{3}} \\ + \frac{(- 1)^{γ} γ!}{u^{γ + 1}} (\frac{e^{(u - d_{1}) t_{1}}}{u - d_{1}} N_{2} \\ - \frac{N_{3}}{u - d_{1}} - \frac{N_{3}}{d_{1}} + \frac{e^{- d_{1} t_{1}}}{d_{1}} Q_{3})] \end{matrix}$ (8)

During the period [(i − 1) T + t₁, iT], the inventory which are to be hold is

$\begin{matrix} Q_{h_{2}} & = & \int_{(i - 1) T + t_{1}}^{iT} q (t) dt \\ = & \int_{t_{1}}^{T} \frac{d_{0}}{d_{1}} (e^{d_{1} (iT - t)} - 1) dt \\ = & \frac{d_{0}}{d_{1}} [\frac{1}{d_{1}} (e^{d_{1} (T - t_{1})} - 1) - (T - t_{1})] \end{matrix}$ (9)

Thus during the period [(i − 1) T, iT], the total expected quantity of holding units can be obtained as,

$E [Q_{h}] = E [Q_{h_{1}}] + Q_{h_{2}}$ (10) where E [Q_{h
₁}] and Q_{h
₂} are given by the Equations (8) and (9) respectively. In i-th cycle [(i − 1) T, iT], the Expected Total cost = Expected holding cost + Rework cost + Disposal cost + Production cost + Set-up cost + Maintenance cost.

$\begin{matrix} i . e . {TC}_{i} (P, t_{1}, r) = Ch . E [Q_{h}] + θ . Cr . E (N) \\ + (1 - θ) . Cd . E (N) + C (P, r) {Pt}_{1} + {Cs}_{i} + {Cm}_{i} \end{matrix}$ (11) Thus the Expected total cost for the all cycle is

$\begin{matrix} TC (P, t_{1}, r, m) \\ = \sum_{i = 1}^{m} {TC}_{i} (P, t_{1}, r) \\ = m [Ch . E [Q_{h}] + θ . Cr . E (N) \\ + (1 - θ) . Cd . E (N) + C (P, r) {Pt}_{1}] \\ + [m C_{s 0} + C_{s 1} \frac{1 - e^{- m k_{1}}}{e^{k_{1}} - 1}] \\ + C_{m 0} [m - \frac{1 - e^{- m k_{2}}}{1 - e^{- k_{2}}}] \end{matrix}$ (12)

3.2 Chance constraint for the out-of-control state

In this production system, it is expected to have total production time greater than the time of beginning of out-of-control state in every cycle. Implementing this, the chance constraint is $\Pr [(t_{1} - τ) \geq ε] \geq x$ (13) where t₁ ≥ 0 and x ∈ (0, 1) is a specified permissible probability. Here $m_{d} [= \frac{1}{f (P)}]$ and $σ [= \frac{1}{f (P)}]$ are the mean and standard deviation of the exponential random variable τ. Then the constraint can be written as $\Pr (\frac{τ - m_{d}}{σ} \leq \frac{t_{1} - ε - m_{d}}{σ}) \geq x$ where $\frac{τ - m_{d}}{σ}$ is a standard normal variate. Considering z, where $\int_{0}^{z} φ (t) dt = x, φ (t)$ , being the standard normal density function, we have $\frac{t_{1} - ε - m_{d}}{σ} \geq z or, t_{1} \geq \frac{1}{f (P)} [1 + z] + ε$ (14) where z is obtained from the normal distribution table for a particular value of x.

3.3 Different types of time horizons

Crisp time horizon: For the crisp finite time horizon we have to consider a constraint as $mT = H$ (15)

Random time horizon: In this consideration, the models remain same as developed above, except the time horizon of the system. Here, $\bar{H}$ is random. For this type of model we impose constraint as (cf. Rao [29]) $\Pr (\bar{H} \geq mT) \geq s$ $or, mT \leq m_{h} + σ_{h} Φ^{- 1} (1 - s)$ (16) where s ∈ (0, 1) is a specified permissible probability. m_h and σ_h are the expectation and standard deviation of normally distributed random variable $\bar{H}$ respectively and Φ⁻¹ (x) denotes inverse function of standard normal distribution of standard normal variate $\frac{\bar{H} - m_{h}}{σ_{h}}$ .

Fuzzy time horizon: If the time horizon $\tilde{H}$ is fuzzy in nature, it can be expressed by the fuzzy constraint $\tilde{H} \geq mT$ which is interpreted in the setting of possibility and necessity theories (cf. Dubois and Prade, [11]). The above constraint reduces to $Pos (\tilde{H} \geq mT) \geq ρ_{1} and Nes (\tilde{H} \geq mT) \geq ρ_{2}$ , where ρ₁ and ρ₂ represent the degree of impreciseness. Let $\tilde{H} = (H_{1}, H_{2}, H_{3})$ be a TFN then, using Lemma-1 and 2 we get $mT \leq {\begin{matrix} (1 - ρ_{1}) H_{3} + ρ_{1} H_{2}, & in Pos . sense \\ (1 - ρ_{2}) H_{2} + ρ_{2} H_{1}, & in Nec . sense \end{matrix}$ (17)

Fuzzy-random time horizon: In this case, the time horizon $\tilde{\bar{H}}$ is fuzzy-random in nature and expressed by the fuzzy-random constraint $\tilde{\bar{H}} \geq mT$ in the setting of possibility and necessity theories (cf. Dubois and Prade, [11]) along with the chance constraint. The above constraint reduces to $\Pr [Pos (\tilde{\bar{H}} \geq mT) \geq ρ_{3}] \geq s_{1}$ and $\Pr [Nes (\tilde{\bar{H}} \geq mT) \geq ρ_{4}] \geq s_{2}$ , where (ρ₃ and ρ₄) and (s₁ and s₂) represent the degree of impreciseness and uncertainty due to randomness respectively. Let $\tilde{\bar{H}} = (\bar{H}, H_{l}, H_{r})$ be L-R fuzzy-random variable then, according to Theorem-1, we get $mT \leq {\begin{matrix} m_{h} + σ_{h} Φ^{- 1} (1 - s_{1}) + R^{- 1} (ρ_{3}) H_{r}, \\ m_{h} + σ_{h} Φ^{- 1} (1 - s_{2}) - L^{- 1} (1 - ρ_{4}) H_{l} \end{matrix}$ (18) where m_h and σ_h are the expectation and standard deviation of normally distributed random variable $\bar{H}$ respectively and Φ⁻¹ (x) denotes inverse function of standard normal distribution of standard normal variate $\frac{\bar{H} - m_{h}}{σ_{h}}$ .

Rough time horizon: If $\hat{H}$ is rough then the rough constraint $\hat{H} \geq mT$ is reduced to the crisp form as $Tr (\hat{H} \geq mT) \geq {tr}_{1} . (using Theorem - 2)$ , i.e.,

$\begin{matrix} mT \\ \leq {\begin{matrix} H_{4} - \frac{{tr}_{1} (H_{4} - H_{3})}{ξ_{1}}, if H_{2} \leq mT \leq H_{4} \\ \frac{ξ_{1} (H_{2} - H_{1}) + (1 - ξ_{1}) H_{2} (H_{4} - H_{3}) - {tr}_{1} (H_{4} - H_{3}) (H_{2} - H_{1})}{ξ_{1} (H_{2} - H_{1}) + (1 - ξ_{1}) (H_{4} - H_{3})}, \\ if H_{1} \leq mT \leq H_{2} \\ H_{4} + \frac{(1 - ξ_{1} - {tr}_{1}) (H_{4} - H_{3})}{ξ_{1}}, if H_{3} \leq mT \leq H_{1} \\ H_{3} \end{matrix} \end{matrix}$ (19) where $\hat{H} = ([H_{1}, H_{2}] [H_{3}, H_{4}]), 0 \leq H_{3} \leq H_{1} \leq H_{2} \leq H_{4}$ , is a rough variable and ξ₁ ∈ (0, 1) and tr₁ ∈ [0, 1] is the confidence level.

Fuzzy-Rough time horizon: If $\tilde{\hat{H}}$ is fuzzy-rough in nature then the fuzzy-rough constraint $\tilde{\hat{H}} \geq mT$ is reduced to in the following forms which are crisp in nature. $Tr [Pos (\tilde{\hat{H}} \geq mT) \geq ρ_{5}] \geq {tr}_{2}$ and $Tr [Nes (\tilde{\hat{H}} \geq mT) \geq ρ_{6}] \geq {tr}_{2}$ . According to Theorem-3, the above constraints are finally reduced to the following forms. ${\begin{matrix} mT \leq {\begin{matrix} H_{4} - \frac{{tr}_{2} (H_{4} - H_{3})}{ξ_{2}} + (1 - ρ_{5}) H_{R}, \\ if H_{2} \leq mT - (1 - ρ_{5}) H_{R} \leq H_{4} \\ \frac{ξ_{2} (H_{2} - H_{1}) + (1 - ξ_{2}) H_{2} (H_{4} - H_{3}) - {tr}_{2} (H_{4} - H_{3}) (H_{2} - H_{1})}{ξ_{2} (H_{2} - H_{1}) + (1 - ξ_{2}) (H_{4} - H_{3})} \\ + (1 - ρ_{5}) H_{R}, if H_{1} \leq mT - (1 - ρ_{5}) H_{R} \leq H_{2} \\ H_{4} + \frac{(1 - ξ_{2} - {tr}_{2}) (H_{4} - H_{3})}{ξ_{2}} + (1 - ρ_{5}) H_{R}, \\ if H_{3} \leq mT - (1 - ρ_{5}) H_{R} \leq H_{1} \\ H_{3} + (1 - ρ_{5}) H_{R} \end{matrix} \\ and \\ mT \leq {\begin{matrix} H_{4} - \frac{{tr}_{2} (H_{4} - H_{3})}{ξ_{2}} - ρ_{6} H_{L}, \\ if H_{2} \leq mT + ρ_{6} H_{L} \leq H_{4} \\ \frac{ξ_{2} (H_{2} - H_{1}) + (1 - ξ_{2}) H_{2} (H_{4} - H_{3}) - {tr}_{2} (H_{4} - H_{3}) (H_{2} - H_{1})}{ξ_{2} (H_{2} - H_{1}) + (1 - ξ_{2}) (H_{4} - H_{3})} \\ - ρ_{6} H_{L}, if H_{1} \leq mT + ρ_{6} H_{L} \leq H_{2} \\ H_{4} + \frac{(1 - ξ_{2} - {tr}_{2}) (H_{4} - H_{3})}{ξ_{2}} + (1 - ρ_{6}) H_{R}, \\ if H_{3} \leq mT + ρ_{6} H_{L} \leq H_{1} \\ H_{3} - ρ_{6} H_{L} \end{matrix} \end{matrix}$ (20)

where $\tilde{\hat{H}} = (\hat{H} - H_{L}, \hat{H}, \hat{H} + H_{R})$ , $\hat{H} = ([H_{1}, H_{2}]$ [H₃, H₄]) , 0 ≤ H₃ ≤ H₁ ≤ H₂ ≤ H₄, is a fuzzy-rough variable and ξ₂ ∈ (0, 1) and ρ₅, ρ₆ ∈ [0, 1] , tr₂ ∈ [0, 1] are the possibility and trust confidence levels respectively.

3.4 Optimization problem of model-1

Therefore, the problem for the imperfect inventory model is finally reduced to the minimization of the expected total cost given by(12) subject to the Chance constraint(14) and constraints(15−20) for different time horizons. Thus the problem is

$\begin{matrix} Min TC (P, t_{1}, r, m) \\ s . t . & t_{1} \geq \frac{1}{f (P)} [1 + Φ (r)] + ε . \end{matrix}$ (21) and constraints (15–20) for different time horizons.

4 Particular cases

4.1 Model-1.1: (Model-1 with constant demand)

Letting d₁ → 0 in the above Model-1, we have the following reduced necessary expressions: ${\begin{matrix} u = 1 - r, Q_{3} = N_{3}, S_{q} = (P - d_{0}) t_{1} - (1 - θ) E [N], \\ T = t_{1} + \frac{S_{q}}{d_{0}}, E [Q_{h}] = \frac{P - d_{0}}{2} t_{1}^{2} - (1 - θ) α P^{β + 1} f (P) \\ \times [\sum_{j = 0}^{γ - 1} \frac{(- 1)^{j} γ!}{(γ - j)! (1 - r)^{j + 1}} {\sum_{k = 0}^{γ - j} \frac{(- 1)^{k} (γ - j)! e^{(1 - r) t_{1}}}{(γ - j - k)! (1 - r)^{k + 1}} \\ \times (\sum_{s = 0}^{γ - j - k} \frac{(- 1)^{s} (γ - j - k)! t_{1}^{γ - j - k - s}}{(γ - j - k - s)! f^{s + 1} (P)} \\ + \frac{(- 1)^{γ - j - k + 1} (γ - j - k)!}{f^{γ - j - k + 1} (P)} e^{- f (P) t_{1}}) - \frac{(- 1)^{γ - j} (γ - j)!}{(1 - r)^{γ - j + 1}} N_{3}} \\ + \frac{(- 1)^{γ} γ!}{(1 - r)^{γ + 1}} (\frac{e^{(1 - r) t_{1}}}{1 - r} N_{2} - \frac{N_{3}}{1 - r} - t_{1} N_{3})] + \frac{d_{0}}{2} (T - t_{1})^{2} . \end{matrix}$ (22)

Therefore, the problem for the imperfect inventory model with constant demand is finally reduced to the minimization of expected total cost given by (12) with (22) subject to the Chance constraint (14) and constraints (15–20) for different time horizons.

4.2 Model-1.2: (Model-1 with single cycle i.e. infinite horizon)

For m = 1, the Equation (12) reduces to TC (P, t₁, r) = Ch . E [Q_h] + θ . Cr . E (N) + (1 − θ) . Cd . E (N) + C (P, r) Pt₁ + [C_s0 + C_s1e^−k₁] and the expected average total cost is $ATC (P, t_{1}, r) = \frac{TC (P, t_{1}, r)}{T}$ (23) where T is given by Equation (6). Therefore, the production-inventory model for infinite time horizon is finally reduced to the minimization of expected average total cost given by (23) subject to the only one chance constraint given by (14). i.e.

$\begin{matrix} Min ATC (P, t_{1}, r) (= \frac{TC (P, t_{1}, r)}{T}) \\ s . t . & t_{1} \geq \frac{1}{f (P)} [1 + z] + ε \end{matrix}$ (24)

5 Solution methodology

The above non-linear optimization problems are solved by a gradient based non-linear optimization method- Generalized Reduced Gradient Method (cf. Lasdon et al. [18]) using LINGO Solver 11.0 for particular sets of data.

6 Numerical experiments

Input Data: We consider the proposed EPL models (Models-1, 1.1 and 1.2) with following inputs parameters in appropriate units:

α = 0.25, β = 0.25, θ = 0.50, γ = 2, z = 0.40, ε = 0.10, d₀ = 20, d₁ = 0.10, Ch = 3.0, C_s0 = 200,C_s1 = 150, C_m0 = 100, k₁ = 0.80, k₂ = 0.80, Cr = 5.0, Cd = 2.0 and unit production cost as: $C (P, r) = 20 + \frac{g}{P} + 0.10 P + 0.20 P^{1 / 2}$ , where $g = 20 + 40 e^{(1 - 0.75) \frac{r - 0.10}{0.90 - r}}$

For each model, two experiments depending on production quality are performed and the corresponding inputs are presented in Table 1. The input parameters for different time horizons are presented in Table 2.

Optimum Results: With the above parameters and expressions, the Models-1,1.1 and 1.2 are formulated and optimized using LINGO 11.0 software. The corresponding optimum values of production rate (P^*), production run time ( $t_{1}^{*}$ ), number of cycles (m^*), reliability (r^*) for minimum total cost (TC^*), holding (CH^*), rework (CR^*), disposal (CD^*), production ( $C_{p}^{*}$ ), set-up ( $C_{s}^{*}$ ) and maintenance costs ( $C_{m}^{*}$ ) for the total time horizon and the instant of defective production ( $m_{d}^{*}$ ), total expected defective units (E (N^*)) and inventory level of good units( $S_{q}^{*}$ ) for each production cycle are evaluated for different cases and presented in Tables 8.

7 Discussions

From Tables 3 , 4 , 5 , 6 , 7 and 8

Table 3 represents the optimum results of Exp.-1 (i.e. Experiment-1, when quality is linearly production dependent) for the Model-1 with crisp finite time horizon. In this case, minimum total cost is 4001 units for 3 cycles in imperfect production model with 50% rework. This is because, with the increasing of cycle numbers, total holding, rework, disposal and production costs decrease but total set up and maintenance costs increase. Up to 3 cycles, total cost decreases as total decrease in costs for holding, rework, etc dominates over the increase in set-up and maintenance costs but when the total no. of cycles is 4, set-up and maintenance costs dominate over the others, hence total cost increases. For fully rework and no rework 3714 and 4164 units are respective minimum total costs for 3 and 4 number of cycles in crisp finite time horizon models.

Table 4 represents the optimum results of Exps.-1 and 2 (i.e. when quality is linearly and non-linearly production dependent) for the Model-1 with crisp, fuzzy, fuzzy-random, rough and fuzzy-rough finite time horizons. Here, for same type of time horizon as well as rework, the minimum total cost of Exp.-1 is more than that of Exp.-2. For example, minimum total cost TC^* = 3522 units of Exp.-1 is greater than corresponding minimum total cost TC^* = 3466 units of Exp.-2 for Model-1 with fully rework and fuzzy-rough time horizon for which impreciseness is measured in possibility sense.

Table 5 represents the optimum results of Exp.-1 (i.e. when quality is linearly production dependent) for the Model-1.1 with crisp finite time horizon. In this case, minimum total cost is 3852 units for 3 cycles imperfect production model with 50% rework. In this case, the behaviours of the different costs are the same as in Model-1 i.e. total cost initially decreases with cycle numbers and then increases when total cycle no. is 4. This is because, with the increasing of cycle numbers, total holding, rework, disposal and production costs decrease but total set up and maintenance costs increase. For fully rework and no rework, 3584 and 3979 units are the respective minimum total costs for 2 and 3 number of cycles in crisp finite time horizon models.

Table 6 represents the optimum results of Exps.-1 and 2 (i.e. when quality is linearly and non-linearly production dependent) for the Model-1.1 with crisp, fuzzy, fuzzy-random, rough and fuzzy-rough finite time horizons. Here, for same type of time horizon as well as rework, the minimum total cost of Exp.-1 is greater than the minimum total cost of Exp.-2. This behaviour is the same as Model-1.

Table 7 represents the optimum results of Exps.-1 and 2 for the Model-1.2 i.e. imperfect production model with stock-dependent demand and infinite crisp time horizon. In this case, for a same type of rework (say θ = 0.50), the minimum average total cost ATC^* = 815 units of Exp.-1 is greater than the minimum average total cost ATC^* = 801 units of Exp.-2.

It is to be noted from the Tables-3 7 that the mean-time ( $m_{d}^{*}$ ) of the commencement of out-of-control state is less than the production run time for all Models in all experiments. There is a necessary condition for the models imposed by chance constraint as ${md}^{*} \leq t_{1}^{*}$ (§ 3.2).

For all Experiments-1 and 2, the optimal expected total costs of models Model-1 and Model-1.1 with no rework from above tables are more than those of the models corresponding with fully rework. The same behaviour is concluded for all these types experiments for Model-1.2.

From Tables 6, for the Model-1 with stock-dependent demand, the optimal average expected total cost is greater than that of Model-1.1 which is with constant demand. It is observed for all Exps.-1 and 2. Interesting result is that in spite of higher demand (stock-dependent) in the market, production rate is lower to minimize the average cost. Here lower production rates (for Model-1 with stock-dependent demand) increase the average production costs.

In Tables 6, values of T^* are given. As H = mT^* and H is known, T^* changes with m i.e. number of cycle.

Now from the Table 8, it is seen that the production rate and reliability of machinery system which minimizes the unit production cost is quite different from the production rate which minimizes the expected total cost TC for Exp.-1 of same model with different types of finite time horizon. It is interesting to note that the production rate P and reliability r which minimize TC are higher than those of which minimize unit production cost C(P,r). For example, for Model-1 with crisp time horizon due to Exp-1, C (P, r) has the minimum value for C (P^*, r^*) =26.88 units at P^* = 40.33 units, r^* = 0.31, but the corresponding unit production cost C₁ (P, r) attains minimum value (26.05 units) at P^* = 28.17 units and r^* = 0.18.

From Figures 2 , 3 , 4 and 5 :

Considering the optimal value of reliability r as constant, the average total costs for the Model-1.2 due to Exp.-1 are plotted in Fig. 2 against the different values of P and t₁. This figure shows that the objective function is convex.

Figure 3 is obtained by plotting the unit production cost against the different values of production rate and reliability of machinery system. This unit production cost is a convex function against production rate only.

Figure 4 represents total cost against the machinery-system reliability for Exp.-1 of Model-1 with crisp time horizon when m=3 and P, t₁ are treated as variable. In this figure, total cost is a convex function with respect to reliability.

As Figs. 4 and 5 also represents the total cost against the machinery-system reliability for Exp.-1 of Model-1 with crisp time horizon when m = 3 and P, t₁ are the optimal values obtained from Table 3. In this figure, total cost increases with reliability of machinery system.

8 Conclusion

The present investigation presents an EPL model with random imperfect production with reliability dependent defective rate, stock dependent demand rate and rework (may be partially) of the imperfect products over different imprecise finite time horizons. During the production, defective units are produced from out-of-control state. The probability distribution of the beginning of out-of-control state follows an exponential distribution with mean and standard deviation $\frac{1}{f (P)}$ . Here f (P) = a, a + bP or a + bP² where a > 0 and b ≥ 0. It may be extended to other types of increasing function. Several subcases of general model are considered and compared. Finally the problems have been converted into a single-objective inventory problem with imprecise finite time horizon and an appropriate chance constraint for out of control state. Moreover, changes of total cost with respect to reliability are also shown graphically.

In the calculation of percentage of defective units produced at time t, λ (t′, τ, P, r) = αP^βe^(1−r)t′ (t′ − τ) ^γ, γ is an integer for the convenience of calculation. However it can be any positive value and in that case the integrations connecting γ are to be evaluated numerically.

It is a general EPL model in which the necessity of imposition of an out-of-control state constraint is laid down and the above chance constraint can be used for others types of imperfect production-inventory models such as inventory models with trade credit, two warehouses inventory system, EPL model with price discount, etc.

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