# An Automated Smart EPQ-Based Inventory Model for Technology-Dependent Products under Stochastic Failure and Repair Rate

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Manufacturing Unreliability

#### 2.2. Smart Production System

#### 2.3. Technology Dependent Demand

#### 2.4. Research Gap

## 3. Problem Statement, Notation, and Assumptions

#### 3.1. Problem Statement

- The cost component $({r}_{1}+{r}_{2}\lambda )$ is the cost of raw material which increases linearly with increasing failure rate.
- The ${c}_{\mu}$ component represents the manufacturing costs, i.e., energy and lobar, and as the production rate increases, the manufacturing cost is equally distributed over a large number of the produced unit. Consequently the per unit manufacturing cost decreases with high production rates.
- The next cost component is $Y\lambda $, which represents the manufacturing yield loss. As the failure rate increases, the downtime increases and the overall costs of system increases. However, with higher production rates, the unit yield loss cost decreases.
- The last cost component is referred to as the tool/die cost, and it is directly proportional to production rate.

#### 3.2. Assumptions

- -
- An automated smart integrated production maintenance policy for an unreliable manufacturing system is considered.
- -
- The unreliability of a manufacturing system is modeled through a system performance-quality parameter which follows a stochastic process as discussed in [33].
- -
- The quality performance parameter of manufacturing system is expressed as a design variable, where it is represented as $\lambda =\frac{Total\phantom{\rule{3.33333pt}{0ex}}number\phantom{\rule{3.33333pt}{0ex}}of\phantom{\rule{3.33333pt}{0ex}}failure}{Total\phantom{\rule{3.33333pt}{0ex}}working\phantom{\rule{3.33333pt}{0ex}}hours}$, and manufacturing quality performance can be increased through investment in advanced production technology and resources.
- -
- During production up-time ${t}_{\mu}$ system failure may occurs randomly, a single-machine and single-product type environment is considered.
- -
- Random failure rate and random repair rate is considered with generalized distributions.
- -
- On manufacturing system failure, corrective repair is started immediately. After corrective repair, manufacturing system is restored back to the same initial operational condition with an extra cost ${R}_{s}\lambda $.
- -
- Upon manufacturing system breakdown, market demand is fulfilled from on-hand inventory, accumulated during production up-time.
- -
- The demand is technology dependent as $(x\left[\tau \right]=a+\gamma {e}^{\eta \tau})$, where a is the potential market demand, $\eta $ is a shape parameter and it shows the effectiveness of product technology $\tau $, whereas $\gamma $ is a scale parameter. Moreover $\tau $ per unit technology investment is a decision variable and $\gamma ,\eta >0$.
- -
- If the on-hand inventory is sufficient to met market demand, then new production cycle is started after complete inventory consumption.
- -
- Market demand is a variable parameter and depend on technology investment $\tau $, where $\tau $ is a decision variable.
- -
- The production rate $\mu $ of smart manufacturing system is a controllable variable and can be varied within given limits $(\mu \in [{\mu}_{min}$$,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\mu}_{max}\left]\right)$. Moreover, the variable production rate is always greater than the variable demand rate as $({\mu}_{min}\ge x\left[\tau \right])$.
- -
- Unit smart production cost is a function of manufacturing quality performance parameter $\lambda $, manufacturing yield Y and variable production rate $\mu $.
- -
- The raw material cost linearly depends on the manufacturing quality performance parameter variable $\lambda $, i.e., (${r}_{1}+{r}_{2}\lambda $).
- -
- As the manufacturing quality performance parameter decreases, the manufacturing yield loss increases, such that higher downtime increases system costs.
- -
- A shortage cost is considered for lost sales due to longer repair times.

## 4. Mathematical Model

#### 4.1. Model 1: Random Production Rate Model with Breakdown

#### 4.1.1. Case 1 ( Repair Time Lies within $\frac{{z}_{1}(\mu -x\left[\tau \right])}{x\left[\tau \right]}$ )

#### 4.1.2. Case 2 (Repair Time Exceeds $\frac{{z}_{1}(\mu -x\left[\tau \right])}{x\left[\tau \right]}$)

**The inventory holding cost**

**The smart production cost**

**The shotage cost**

**The set up cost**

**The system repair cost**

#### 4.2. Model 2: Random Production Rate Model without Breakdown

#### 4.3. Integration of the EPQ Models with/without Failure

#### 4.4. Solution Methodology

## 5. Numerical Experiment

#### 5.1. Sensitivity Analysis

#### 5.1.1. Comparative Study

#### 5.1.2. Managerial Insights

- The major insight for the industrial management is to assess uncertain production capacity for automated smart production planning. In a real world production environment, productivity and manufacturing efficiency is compromised generally because of an unreliable and limited production capacity system. Therefore, we provide comprehensive details for the influence of the reliability parameter on production rates and production capacity for optimal production planning.
- As the study is considered for a smart production inventory model, random breakdowns and random repair time is considered, therefore a production manager can easily do the calculations for optimal production quantity and optimal controllable production rates based on random system breakdowns in production planning.
- The unreliable manufacturing system provides optimum values for all decision variables, and if the manufacturing unit is producing smart products, for example high-tech products/electronic products, then our model provides the insight to choose investment options based on manufacturing system reliability.

## 6. Conclusions

- -
- Rather than fixed designed production rates, the model considers a controllable production rate-based automated smart manufacturing system which reduces the per unit production cost. Along with the production rate $\mu $, the per unit production cost also depends on the manufacturing reliability and productivity parameter $\lambda $.
- -
- The manufacturing system productivity and reliability can be increased with an investment in advanced technology, and the smart production technology investment $D\left[\lambda \right]$ is not a constant parameter. Rather, it depends on the failure rate of manufacturing system.
- -
- Concerning product innovation and ever-changing technology advancements, much existing research is considering the approach of remanufacturing and recycling used products to gain two important industrial goals:The problem that all these researchers lack in answering these questions is what are the economical and environmental implications of these growing technological implications on production systems and production efficiency, as production setup and technological development costs are the biggest parts of any production system. Moreover, issues regarding production reliability are currently one of the most important parameters for maintaining an adequate level of access to high-tech products. However, no existing research has considered the influence of smart product innovation and production reliability on manufacturing policies. Therefore, this study investigates the need for considering technological improvement/advancements in the production process and corresponding costs for product innovation. This study outperforms the work of [6] as it considers the demand variability for technology innovation for an unreliable production system, and limits the per-unit technology investment for products $\tau $ as ($<\alpha \%$ ) of the smart production technology development investment $\left(\chi \right)$.
- -
- Due to the stochastic nature of breakdowns and repair times, the study extended the production inventory model to an integrated smart production maintenance model. The study infers the importance of performance quality parameter $\zeta $ for per-unit technological investment for products $\tau $, as even a 10% increase in $\zeta $ gives infeasible results for the proposed unreliable production model.

## Author Contributions

## Conflicts of Interest

## Abbreviations

EOQ | Economic ordering quantity |

EPQ | Economic production quantity |

EMQ | Economic manufacturing quantity |

OPT | Optimum production technology |

AR | Abort/Resume |

NR | No-resumption |

OSE | Overall system effectiveness |

VMI | Vendor managed inventory |

JMI | Jointly managed inventory |

IT | Information technology |

Decision variables | |

$\tau $ | Technology investment for high-techs products ($/unit) |

$\lambda $ | The manufacturing quality performance parameter, where $\lambda \phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}[{\lambda}_{min},\phantom{\rule{3.33333pt}{0ex}}{\lambda}_{max}]$ |

Q | Production quantity (units) |

$\mu $ | Variable rate of smart production, where $\mu \phantom{\rule{3.33333pt}{0ex}}\u03f5\phantom{\rule{3.33333pt}{0ex}}[{\mu}_{min},\phantom{\rule{3.33333pt}{0ex}}{\mu}_{max}]$ (units/unit time) |

Model Parameters | |

Z | Time horizon (years) |

S | Manufacturing setup cost ($/setup) |

$x\left[\tau \right]$ | Technology dependent demand rate (units/unit time) |

$C(\mu ,\lambda )$ | Unit smart production cost, a function of manufacturing quality performance parameter and varia-ble production rate ($/unit/unit time) |

${r}_{1}$ | Raw material cost ($/unit) |

${r}_{2}$ | Raw material increment cost due to low manufacturing quality performance parameter ($/unit) |

Y | Unit manufacturing yield loss cost due to low manufacturing quality performance parameter ($/unit/unit time) |

$\pi $ | Tool/die cost ($/unit) |

${h}_{c}$ | Inventory carrying cost ($/unit/unit time) |

${R}_{r}$ | System repair cost ($/unit/unit time) |

$D\left[\lambda \right]$ | Manufacturing technology development cost ($) |

$\chi $ | Advanced manufacturing technology investment ($) |

$\varrho $ | Manufacturing quality performance scale parameter (non-negative) |

$\zeta $ | Manufacturing quality performance constant (non-negative) |

${R}_{s}$ | System restoration cost ($/setup) |

${S}_{c}$ | Shortage cost ($/unit/unit time) |

${z}_{1}$ | Stochastic variable indicating manufacturing system failure time (non-negative) |

$\theta $ | Stochastic repair rate (number of repairs/unit time) |

${z}_{2}$ | Stochastic variable indicating repair time upon manufacturing system breakdown (non-negative) |

$g\left({z}_{1}\right)$ | Probability density function of manufacturing system failure time ${z}_{1}$ |

$G\left({z}_{1}\right)$ | Cumulative density function of manufacturing system failure time ${z}_{1}$ |

$g\left({z}_{2}\right)$ | Probability density function of manufacturing system repair time ${z}_{2}$ |

$G\left({z}_{2}\right)$ | Cumulative density function of manufacturing system repair time ${z}_{2}$ |

$E\left[{C}_{1}\right]$ | Expected total cost of system with high manufacturing quality performance variable ($\$/cycle$) |

$E\left[{C}_{2}\right]$ | Expected total cost of system with low manufacturing quality performance variable ($\$/cycle$) |

$E\left[{Z}_{1}\right]$ | Expected cycle length with high quality performance variable ($time$) |

$E\left[{Z}_{2}\right]$ | Expected cycle length with low quality performance variable ($time$) |

$E\left[Z\right]$ | Expected cycle length of manufacturing system ($time$) |

$E\left[T{C}_{1}\right]$ | Expected total cost per unit of time with high manufacturing quality performance variable ($) |

$E\left[T{C}_{2}\right]$ | Expected total cost per unit of time with low manufacturing quality performance variable ($) |

$E\left[TC\right]$ | Expected total cost per unit time of manufacturing system ($) |

## Appendix A

**Proposition**

**A1.**

**Proof.**

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**Figure 6.**Impact of production lot size Q on Expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$.

**Figure 7.**Impact of controllable production rate $\mu $ on expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$.

**Figure 8.**Impact of manufacturing reliability parameter $\lambda $ on expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$.

**Figure 9.**Impact of high-tech investment per unit $\tau $ on Expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$.

**Figure 10.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\mu $ and Q.

**Figure 11.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\mu $ and $\lambda $.

**Figure 12.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\tau $ and $\lambda $.

**Figure 13.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\tau $ and $\mu $.

**Figure 14.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\tau $ and Q.

**Figure 15.**Change in expected total cost $E\left[TC\right(Q,\mu ,\lambda ,\tau \left)\right]$ with varying $\lambda $ and Q.

Author(s) | Manufacturing Reliability | Repair Rate | Production Rate | Unit Production Cost | Technology Development | Demand Rate |
---|---|---|---|---|---|---|

Chakraborty et al. [5] | random | random | constant | NA | NA | constant |

Asghar et al. [6] | stochastic | stochastic | variable | ($\mu ,\lambda $) | ($\mu ,\lambda $) | constant |

Marchi et al. [7] | random | NA | variable | ($\mu $) | NA | constant |

Sarkar et al. [3] | random | NA | variable | ($\mu $) | ($\lambda $) | constant |

Lopes, R. [8] | random | constant | constant | NA | NA | constant |

Marchi et al. [9] | imperfect | constant | variable | ($\mu $) | NA | constant |

Chakraborty et al. [10] | random | random | constant | NA | NA | constant |

Kang et al. [11] | age dependent | constant | random | NA | NA | constant |

Bhunia et al. [12] | flexible (product quality based) | NA | variable | ($\mu $) | NA | variable (selling price + marketing cost) |

Ouaret et al. [13] | random | constant | flexible | NA | NA | random |

Bag et al. [14] | variable | NA | constant | constant | NA | fuzzy random |

Aldurgam et al. [15] | NA | NA | variable | ($\mu $) | NA | stochastic |

Ouaret et al. [16] | random | constant | variable | NA | NA | random |

Manna et al. [17] | production rate dependent | NA | variable | ($\mu $) | NA | advertisement dependent |

Zhu et al. [18] | NA | NA | constant | ($\varphi $) | NA | (product reliability + warranty period) |

Shah et al. [19] | imperfect | NA | variable | ($\lambda ,\mu $) | NA | time and effort dependent |

Khara et al. [20] | imperfect | NA | constant | ($\lambda ,T,\varphi $) | ($\lambda ,T$) | selling price + $\varphi $ dependent |

Shah et al. [21] | imperfect | NA | constant | ($\lambda ,T$) | ($\lambda ,T$) | ($\varphi ,I\left(t\right)$, selling price) |

Ethem Çanakoğlu, Taner Bilgiç. [22] | NA | NA | NA | NA | fixed | technology dependent |

This study | Stochastic | Stochastic | Variable | ($\mu ,\lambda $) dependent | ($\lambda $) dependent | Technology dependent |

${r}_{1}=1$ ($/unit) | ${r}_{2}=0.3$ ($/unit) | $S=2000$ ($/setup) |

$Y=150$ ($) | ${c}_{\mu}=2500$ ($) | ${R}_{s}=0.0725$ |

$\pi =0.0015$ | ${R}_{r}=500$ ($/unit time) | $\theta =15$ (repairs /unit time) |

${S}_{c}=5$ ($/unit) | $\zeta =1.02$ | $\varrho =0.05$ |

$\chi =500$ ($) | $a=500$ (units/unit time) | $\gamma =0.5$ |

$\eta =0.75$ | ${h}_{c}=0.5$ ($/unit/unit time) |

Parameter | Change | Q | $\mathit{\lambda}$ | $\mathit{\mu}$ | $\mathit{\tau}$ | % Change in $\mathit{E}\left[\mathit{TC}\right]$ |
---|---|---|---|---|---|---|

% | (Units) | (Failures/Unit Time) | (Units/Unit Time) | ($/Unit) | ($/Unit/Unit Time) | |

${c}_{\mu}$ | −20 | 3424.64 | $2.40\times {10}^{-8}$ | 0873.26 | $3.2\times {10}^{-10}$ | −7.92 |

−10 | 3191.81 | $2.19\times {10}^{-7}$ | 0983.87 | $9.9\times {10}^{-16}$ | −3.70 | |

+10 | 2977.92 | $2.89\times {10}^{-7}$ | 1149.03 | $8.5\times {10}^{-16}$ | +3.07 | |

+20 | 2946.28 | $2.00\times {10}^{-8}$ | 1203.95 | $1.01\times {10}^{-14}$ | +5.02 | |

${r}_{1}$ | −20 | 3060.56 | $8.88\times {10}^{-8}$ | 1075.18 | $1.2\times {10}^{-13}$ | −2.78 |

−10 | 3060.64 | $7.85\times {10}^{-8}$ | 1075.15 | $1.2\times {10}^{-14}$ | −1.54 | |

+10 | 3061.87 | $8.30\times {10}^{-8}$ | 1074.71 | $2.2\times {10}^{-16}$ | +1.20 | |

+20 | 3145.10 | $1.10\times {10}^{-8}$ | 1027.73 | $8.7\times {10}^{-15}$ | +2.77 | |

${r}_{2}$ | −20 | 3059.87 | $5.1\times {10}^{-8}$ | 1075.27 | $4.4\times {10}^{-14}$ | −0.11 |

−10 | 3061.85 | $1.2\times {10}^{-7}$ | 1074.72 | $2.1\times {10}^{-13}$ | −0.02 | |

+10 | 3074.20 | $4.5\times {10}^{-8}$ | 1070.24 | $3.6\times {10}^{-16}$ | +0.01 | |

+20 | 3056.48 | $2.1\times {10}^{-9}$ | 1075.55 | $2.6\times {10}^{-10}$ | +0.16 | |

Y | −20 | 3059.92 | $1.3\times {10}^{-7}$ | 1075.42 | $9.1\times {10}^{-14}$ | −0.02 |

−10 | 3064.73 | $2.2\times {10}^{-8}$ | 1073.37 | $6.1\times {10}^{-15}$ | −0.12 | |

+10 | 3088.96 | $2.9\times {10}^{-8}$ | 1081.51 | $8.2\times {10}^{-14}$ | +0.15 | |

+20 | inf | inf | inf | inf | inf | |

$\theta $ | −20 | 3375.97 | $1.0\times {10}^{-8}$ | 0975.35 | $6.1\times {10}^{-14}$ | +0.81 |

−10 | 3058.88 | $2.1\times {10}^{-8}$ | 1075.57 | $1.2\times {10}^{-13}$ | +0.004 | |

+10 | 3059.67 | $2.8\times {10}^{-8}$ | 1075.51 | $1.2\times {10}^{-10}$ | −0.09 | |

+20 | 3066.45 | $3.9\times {10}^{-8}$ | 1073.06 | $1.9\times {10}^{-14}$ | −0.11 | |

${R}_{r}$ | −20 | 3058.21 | $4.0\times {10}^{-8}$ | 1075.61 | $1.4\times {10}^{-14}$ | −0.01 |

−10 | 3059.87 | $2.8\times {10}^{-7}$ | 1075.43 | $7.2\times {10}^{-16}$ | −0.01 | |

+10 | 3058.48 | $5.4\times {10}^{-8}$ | 1075.59 | $6.3\times {10}^{-15}$ | +0.01 | |

+20 | 3062.35 | $4.0\times {10}^{-8}$ | 1074.54 | $1.6\times {10}^{-15}$ | +0.01 | |

$\pi $ | −20 | 2917.85 | $1.1\times {10}^{-7}$ | 1214.58 | $4.7\times {10}^{-13}$ | −5.25 |

−10 | 2998.08 | $1.1\times {10}^{-7}$ | 1131.27 | $1.8\times {10}^{-13}$ | −2.28 | |

+10 | 3137.53 | $1.2\times {10}^{-8}$ | 1019.07 | $1.1\times {10}^{-13}$ | +2.69 | |

+20 | 3214.55 | $2.8\times {10}^{-7}$ | 0970.72 | $3.7\times {10}^{-16}$ | +4.65 | |

$\gamma $ | −20 | 3237.36 | $6.5\times {10}^{-9}$ | 1012.05 | $6.9\times {10}^{-13}$ | −4.43 |

−10 | 3059.23 | $1.1\times {10}^{-7}$ | 1075.53 | $4.6\times {10}^{-15}$ | −0.02 | |

+10 | 3060.01 | $9.0\times {10}^{-7}$ | 1075.48 | $2.6\times {10}^{-15}$ | +0.004 | |

+20 | 3063.44 | $1.0\times {10}^{-7}$ | 1074.32 | $1.1\times {10}^{-17}$ | +0.004 | |

$\eta $ | −20 | 1916.47 | $6.5\times {10}^{-9}$ | 1161.88 | 0.33 | +2.68 |

−10 | 3063.76 | $4.9\times {10}^{-8}$ | 1074.20 | $1.3\times {10}^{-15}$ | +0.13 | |

+10 | 3066.49 | $5.8\times {10}^{-8}$ | 1073.21 | $3.6\times {10}^{-16}$ | −0.07 | |

+20 | inf | inf | inf | inf | inf | |

${S}_{c}$ | −20 | 3060.39 | $8.2\times {10}^{-8}$ | 1075.25 | $3.1\times {10}^{-14}$ | −0.03 |

−10 | 3059.60 | $3.1\times {10}^{-8}$ | 1075.47 | $3.3\times {10}^{-14}$ | −0.03 | |

+10 | 3060.02 | $4.0\times {10}^{-8}$ | 1075.38 | $1.5\times {10}^{-12}$ | −0.07 | |

+20 | 3062.40 | $4.5\times {10}^{-8}$ | 1074.52 | $1.6\times {10}^{-16}$ | −0.07 | |

$\chi $ | −20 | 2993.77 | $8.9\times {10}^{-8}$ | 1079.94 | $1.8\times {10}^{-13}$ | −0.51 |

−10 | 3084.50 | $4.5\times {10}^{-8}$ | 1056.75 | $1.9\times {10}^{-15}$ | −0.25 | |

+10 | 3093.58 | $3.3\times {10}^{-7}$ | 1072.82 | $1.2\times {10}^{-15}$ | +0.24 | |

+20 | 3126.98 | $4.1\times {10}^{-7}$ | 1070.26 | $1.4\times {10}^{-16}$ | +0.48 | |

$\zeta $ | −20 | 3062.26 | $9.0\times {10}^{-8}$ | 1074.57 | $1.1\times {10}^{-15}$ | −0.02 |

−10 | 3059.98 | $1.7\times {10}^{-7}$ | 1075.39 | $4.6\times {10}^{-18}$ | −0.01 | |

+10 | inf | inf | inf | inf | inf | |

+20 | inf | inf | inf | inf | inf | |

${h}_{c}$ | −20 | 3384.03 | $1.2\times {10}^{-8}$ | 1102.44 | $2.3\times {10}^{-13}$ | −2.85 |

−10 | 3221.49 | $2.3\times {10}^{-8}$ | 1077.35 | $3.3\times {10}^{-15}$ | −1.29 | |

+10 | 2966.62 | $1.7\times {10}^{-8}$ | 1043.72 | $5.4\times {10}^{-15}$ | +1.21 | |

+20 | 2198.90 | $3.6\times {10}^{-5}$ | 1108.43 | 0.04 | +3.13 | |

S | −20 | 2809.52 | $4.5\times {10}^{-7}$ | 1094.69 | $2.1\times {10}^{-14}$ | −1.87 |

−10 | 2939.03 | $7.9\times {10}^{-8}$ | 1084.00 | $1.1\times {10}^{-13}$ | −0.95 | |

+10 | 3213.57 | $4.6\times {10}^{-8}$ | 1061.55 | $8.5\times {10}^{-15}$ | +0.97 | |

+20 | 3356.06 | $1.8\times {10}^{-8}$ | 1043.57 | $3.2\times {10}^{-14}$ | +1.93 | |

${R}_{s}$ | −20 | 3067.79 | $3.6\times {10}^{-8}$ | 1072.58 | $1.4\times {10}^{-14}$ | −0.07 |

−10 | 3093.14 | $2.4\times {10}^{-8}$ | 1063.56 | $5.6\times {10}^{-14}$ | −0.18 | |

+10 | 3062.60 | $5.7\times {10}^{-8}$ | 1074.45 | $1.5\times {10}^{-14}$ | +0.003 | |

+20 | 3195.18 | $1.1\times {10}^{-8}$ | 1016.47 | $2.4\times {10}^{-16}$ | +0.53 |

**Table 4.**Sensitivity analysis for parameters related to manufacturing reliability $F\left(t\right)$ and product technology ($\tau $).

Parameter | % | $\mathit{Manufacturing}$ | $\mathit{\lambda}$ | $\mathit{C}(\mathit{\mu},\mathit{\lambda})$ | $\mathit{\tau}$ | % Change in $\mathit{E}\left[\mathit{TC}\right]$ |
---|---|---|---|---|---|---|

Change | $\mathit{Reliability}\phantom{\rule{3.33333pt}{0ex}}(\%)$ | (Rate) | ($/Unit/Unit Time) | ($/Unit) | ($/Unit/Unit Time) | |

${c}_{\mu}$ | −20 | 100 | $2.40\times {10}^{-8}$ | 4.60 | $3.2\times {10}^{-10}$ | −7.92 |

−10 | 99 | $2.19\times {10}^{-7}$ | 4.70 | $9.9\times {10}^{-16}$ | −3.70 | |

+10 | 99 | $2.89\times {10}^{-7}$ | 5.09 | $8.5\times {10}^{-16}$ | +3.07 | |

+20 | 100 | $2.00\times {10}^{-8}$ | 5.20 | $1.01\times {10}^{-14}$ | +5.02 | |

${r}_{1}$ | −20 | 99 | $8.88\times {10}^{-8}$ | 4.75 | $1.2\times {10}^{-13}$ | −2.78 |

−10 | 99 | $7.85\times {10}^{-8}$ | 4.80 | $1.2\times {10}^{-14}$ | −1.54 | |

+10 | 100 | $8.30\times {10}^{-8}$ | 5.01 | $2.2\times {10}^{-16}$ | +1.20 | |

+20 | 100 | $1.10\times {10}^{-8}$ | 5.17 | $8.7\times {10}^{-15}$ | +2.77 | |

${r}_{2}$ | −20 | 100 | $5.1\times {10}^{-8}$ | 4.90 | $4.4\times {10}^{-14}$ | −0.11 |

−10 | 99 | $1.2\times {10}^{-7}$ | 4.93 | $2.1\times {10}^{-13}$ | −0.02 | |

+10 | 99 | $4.5\times {10}^{-8}$ | 4.90 | $3.6\times {10}^{-16}$ | +0.01 | |

+20 | 100 | $2.1\times {10}^{-9}$ | 4.93 | $2.6\times {10}^{-10}$ | +0.16 | |

Y | −20 | 99 | $1.3\times {10}^{-7}$ | 4.93 | $9.1\times {10}^{-14}$ | −0.02 |

−10 | 99 | $2.2\times {10}^{-8}$ | 4.94 | $6.1\times {10}^{-15}$ | −0.12 | |

+10 | 100 | $2.9\times {10}^{-8}$ | 4.94 | $8.2\times {10}^{-14}$ | +0.15 | |

+20 | inf | inf | inf | inf | inf | |

$\theta $ | −20 | 100 | $1.0\times {10}^{-8}$ | 5.03 | $6.1\times {10}^{-14}$ | +0.81 |

−10 | 99 | $2.1\times {10}^{-8}$ | 4.94 | $1.2\times {10}^{-13}$ | +0.004 | |

+10 | 100 | $2.8\times {10}^{-8}$ | 4.93 | $1.2\times {10}^{-10}$ | −0.09 | |

+20 | 100 | $3.9\times {10}^{-8}$ | 4.94 | $1.9\times {10}^{-14}$ | −0.11 | |

${R}_{r}$ | −20 | 100 | $4.0\times {10}^{-8}$ | 4.94 | $1.4\times {10}^{-14}$ | −0.01 |

−10 | 99 | $2.8\times {10}^{-7}$ | 4.94 | $7.2\times {10}^{-16}$ | −0.01 | |

+10 | 100 | $5.4\times {10}^{-8}$ | 4.94 | $6.3\times {10}^{-15}$ | +0.01 | |

+20 | 100 | $4.0\times {10}^{-8}$ | 4.94 | $1.6\times {10}^{-15}$ | +0.01 | |

$\gamma $ | −20 | 100 | $6.5\times {10}^{-9}$ | 5.00 | $6.9\times {10}^{-13}$ | −4.43 |

−10 | 99 | $1.1\times {10}^{-7}$ | 4.94 | $4.6\times {10}^{-15}$ | −0.02 | |

+10 | 99 | $9.0\times {10}^{-7}$ | 4.94 | $2.6\times {10}^{-15}$ | +0.004 | |

+20 | 99 | $1.0\times {10}^{-7}$ | 4.94 | $1.1\times {10}^{-17}$ | +0.004 | |

$\eta $ | −20 | 99 | $6.5\times {10}^{-9}$ | 4.89 | 0.33 | +2.68 |

−10 | 100 | $4.9\times {10}^{-8}$ | 4.93 | $1.3\times {10}^{-15}$ | +0.13 | |

+10 | 100 | $5.8\times {10}^{-8}$ | 4.94 | $3.6\times {10}^{-16}$ | −0.07 | |

+20 | inf | inf | inf | inf | inf | |

${S}_{c}$ | −20 | 99 | $8.2\times {10}^{-8}$ | 4.94 | $3.1\times {10}^{-14}$ | −0.03 |

−10 | 100 | $3.1\times {10}^{-8}$ | 4.94 | $3.3\times {10}^{-14}$ | −0.03 | |

+10 | 100 | $4.0\times {10}^{-8}$ | 4.94 | $1.5\times {10}^{-12}$ | −0.07 | |

+20 | 100 | $4.5\times {10}^{-8}$ | 4.94 | $1.6\times {10}^{-16}$ | −0.07 | |

$\chi $ | −20 | 99 | $8.9\times {10}^{-8}$ | 4.94 | $1.8\times {10}^{-13}$ | −0.51 |

−10 | 99 | $4.5\times {10}^{-8}$ | 4.95 | $1.9\times {10}^{-15}$ | −0.25 | |

+10 | 99 | $3.3\times {10}^{-7}$ | 4.94 | $1.2\times {10}^{-15}$ | +0.24 | |

+20 | 99 | $4.1\times {10}^{-7}$ | 4.94 | $1.4\times {10}^{-16}$ | +0.48 | |

$\zeta $ | −20 | 99 | $9.0\times {10}^{-8}$ | 4.93 | $1.1\times {10}^{-15}$ | −0.02 |

−10 | 99 | $1.7\times {10}^{-7}$ | 4.93 | $4.6\times {10}^{-18}$ | −0.01 | |

+10 | inf | inf | inf | inf | inf | |

+20 | inf | inf | inf | inf | inf | |

${h}_{c}$ | −20 | 100 | $1.2\times {10}^{-8}$ | 4.92 | $2.3\times {10}^{-13}$ | −2.85 |

−10 | 100 | $2.3\times {10}^{-8}$ | 4.94 | $3.3\times {10}^{-15}$ | −1.29 | |

+10 | 100 | $1.7\times {10}^{-8}$ | 4.96 | $5.4\times {10}^{-15}$ | +1.21 | |

+20 | 99 | $3.6\times {10}^{-5}$ | 4.91 | 0.04 | +3.13 | |

$\pi $ | −20 | 99 | $1.1\times {10}^{-7}$ | 4.52 | $4.7\times {10}^{-13}$ | −5.25 |

−10 | 99 | $1.1\times {10}^{-7}$ | 4.75 | $1.8\times {10}^{-13}$ | −2.28 | |

+10 | 99 | $1.2\times {10}^{-8}$ | 5.13 | $1.1\times {10}^{-13}$ | +2.69 | |

+20 | 99 | $2.8\times {10}^{-7}$ | 5.32 | $3.7\times {10}^{-16}$ | +4.65 | |

S | −20 | 99 | $4.5\times {10}^{-7}$ | 4.92 | $2.1\times {10}^{-14}$ | −1.87 |

−10 | 100 | $7.9\times {10}^{-8}$ | 4.93 | $1.1\times {10}^{-13}$ | −0.95 | |

+10 | 100 | $4.6\times {10}^{-8}$ | 4.95 | $8.5\times {10}^{-15}$ | +0.97 | |

+20 | 100 | $1.8\times {10}^{-8}$ | 4.96 | $3.2\times {10}^{-14}$ | +1.93 | |

${R}_{s}$ | −20 | 100 | $3.6\times {10}^{-8}$ | 4.93 | $1.4\times {10}^{-14}$ | −0.07 |

−10 | 100 | $2.4\times {10}^{-8}$ | 4.94 | $5.6\times {10}^{-14}$ | −0.18 | |

+10 | 100 | $5.7\times {10}^{-8}$ | 4.94 | $1.5\times {10}^{-14}$ | +0.003 | |

+20 | 100 | $1.1\times {10}^{-8}$ | 4.99 | $2.4\times {10}^{-16}$ | +0.53 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Asghar, I.; Kim, J.S.
An Automated Smart EPQ-Based Inventory Model for Technology-Dependent Products under Stochastic Failure and Repair Rate. *Symmetry* **2020**, *12*, 388.
https://doi.org/10.3390/sym12030388

**AMA Style**

Asghar I, Kim JS.
An Automated Smart EPQ-Based Inventory Model for Technology-Dependent Products under Stochastic Failure and Repair Rate. *Symmetry*. 2020; 12(3):388.
https://doi.org/10.3390/sym12030388

**Chicago/Turabian Style**

Asghar, Iqra, and Jong Soo Kim.
2020. "An Automated Smart EPQ-Based Inventory Model for Technology-Dependent Products under Stochastic Failure and Repair Rate" *Symmetry* 12, no. 3: 388.
https://doi.org/10.3390/sym12030388