# A Sustainable Green Inventory System with Novel Eco-Friendly Demand Incorporating Partial Backlogging under Fuzziness

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model Formulation

- p—Selling price of unit item.
- ${\mathrm{B}}_{0}$—Ordering cost for each replenishment.
- ${\mathrm{C}}_{{\mathrm{h}}_{1}}$—Holding costs of unit item per unit time.
- ${\mathrm{C}}_{\mathrm{p}}{,\mathrm{C}}_{\mathrm{s}}$—Purchasing cost and Shortage cost of unit item, respectively.
- ${\mathrm{C}}_{\mathrm{l}}$—Opportunity cost due to lack of inventory.
- ${\mathsf{\theta}}_{\mathrm{o}}$—Deterioration rate without preservation.
- ${\mathrm{C}}_{{\mathrm{d}}_{1}}$—Cost of each deteriorated item per unit time.
- $\mathsf{\xi}$—Preservation investment cost per unit time.

- $\mathrm{Y}$—Carbon emissions cap (kg/year).
- k—Deteriorated inventory weight in the warehouse (kg/unit).
- d—Space required for each unit of product (meter/unit).
- ${\mathrm{B}}_{1}$—Carbon emission unit associated with ordering cost (kg/year).
- ${\mathrm{C}}_{{\mathrm{h}}_{2}}$—Carbon emission for holding inventory (kg/year).
- ${\mathrm{C}}_{{\mathrm{d}}_{2}}$—Carbon emission for inventory obsolescence (kg/year).
- $\mathsf{\varphi}$—Carbon tax ($/kg).
- m—Efficiency of greener technology in reducing emission.
- G—Green investment cost per unit time.
- OCE—Total emission associated with ordering cost (kg/order/year).
- HCE—Total emission for holding inventory (kg/year).
- DCE—Total emission of deterioration for inventory (kg/year).

- T—Cycle time in each replenishment.
- ${t}_{1}$—The time at which the inventory level reaches zero.

_{1}, the stock level of the inventory system reaches zero. At the time interval (${t}_{1},$ T), shortages occur, and unsatisfied demand partially backlogs with the backlogging rate $\mathrm{f}$. At shortages, the inventory level is ${\mathrm{Q}}_{2}$. The total reorder quantity of the inventory system is ${\mathrm{Q}=\mathrm{Q}}_{1}{+\mathrm{Q}}_{2}$ as shown in Figure 2.

#### 3.1. Mathematical Analysis of the Proposed Inventory System

#### 3.2. Analysis of the Inventory System under the Green Environment

## 4. Proposed Inventory Model under Pythagorean Fuzziness

#### 4.1. Defuzzification of Pythagorean Fuzzy Number

_{PF}is given by

#### 4.2. Proportional Probability Distribution

_{PF,}which is defined as $h\left(x\right)=\lambda {g}_{1}\left(x\right)+\left(1-\lambda \right){g}_{2}\left(x\right)$, where $0\le \lambda \le 1$.

## 5. Results of an Empirical Study

#### 5.1. Results and Discussion

_{o}= 0.1, the deterioration cost per unit time is ${\tilde{C}}_{{d}_{1}}=\$23.6$, the preservation cost per unit time is ξ = $5 and the controlling preservation parameter is ${a}_{1}$ = 0.5, the backlogging rate is f = 0.9, the green investment amount per unit time is G = $6, the carbon cap is given by Y = 800 (kg/year), the tax per kg carbon emission is ϕ = $0.43, the amount of emission associated with setup, holding, and ordering costs are, respectively, ${B}_{1}=150\left(\frac{kg}{setup}\right)$, ${C}_{{h}_{2}}=12\left(\frac{\frac{kg}{unit}}{year}\right)$ ${C}_{{d}_{2}}=14\left(\frac{\frac{kg}{unit}}{year}\right)$, and the other parameters are $a=150$, $b=0.1$ $k=0.2\left(\frac{kg}{unit}\right)$, $d=0.4\left(\frac{meter}{unit}\right)$, $m=0.8$, $\eta =0.2$, $\gamma =2$, $\rho =1.2$ and $\lambda =0.6$, where ${C}_{{h}_{1}}=\left\{\left(10,15,18\right),\left(8,15,20\right)\right\}$, ${C}_{{d}_{1}}=\left\{\left(12,20,38\right),\left(10,20,40\right)\right\},{C}_{l}=\{\left(15,25,35\right),\left(13,25,37\right)$, ${C}_{s}=\left\{\left(20,30,40\right),\left(18,30,42\right)\right\}$, ${C}_{p}=\left\{\left(22,40,48\right),\left(20,40,50\right)\right\}$ and ${B}_{0}=\left\{\left(100,200,300\right),\left(100,200,300\right)\right\}$. The parameter input values are partially taken from previous literature [45].

#### 5.2. Sensitivity Analysis

_{1}and T, also increases, as shown in Figure 5.

_{1}decrease if we increase the holding cost.

_{1}of the inventory system increases, as shown in Figure 6.

_{1}decreases moderately due to the increase in parameters. The total inventory profit is moderately sensitive to the parameters ν and a. It shows that if the impact of the advertisement increases because the product sales increase, the demand rate and profit of the system increase, but it only shows a medium impact. If we increase parameter a, the product’s demand rate moderately increases because the profit of the inventory system increases moderately. The cycle length T and t

_{1}decrease moderately due to parameters ν and a. The remaining parameters show less of an impact on the total profit.

_{1}) increases; the optimal cycle length increases, and optimal profit decreases. The setup process is less sensitive to the total optimal solution.

#### 5.3. Managerial Implication

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

**Proof.**

_{2,2}> 0, the function I(t

_{1},T) is strictly concave and differentiable. Additionally, J(t

_{1}, T) = T strictly positive and an affine function. Now on the basis of Theorem 3.2.5 in Cambini and Martein [46], TAP(t

_{1},T) archives the global maximum value at the unique point, which is obtained from the necessary conditions, i.e., $\left({t}_{1}^{*},{T}^{*}\right)$. This completes the proof. □

## Appendix B

**Pythagorean fuzzy set and its operation:**

_{PF}) is defined as ${A}_{PF}=\left\{\left({a}_{1},{a}_{2},{a}_{3};{a}_{1}^{\prime},{a}_{2},{a}_{3}^{\prime}\right)\right\}$ with the following membership function and non-membership function:

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**Figure 3.**Graphical depiction of concavity of optimal total profit: (

**a**) Without preservation; (

**b**) under preservation.

**Figure 5.**Graphical representation of optimal profit under various inventory cost parameters: (

**a**) Without preservation; (

**b**) under preservation.

**Figure 6.**Graphical representation of optimal profit under various parameters: (

**a**) Without preservation; (

**b**) under preservation.

**Figure 7.**Graphical representation of optimal profit under various emission parameters: (

**a**) Without preservation; (

**b**) under preservation.

Articles | Demand | Preservation Technology | Carbon Emission | Environment |
---|---|---|---|---|

Pal et al. [31] (2015) | Ramp type | NC | NC | Fuzzy |

Mishra [32] (2016) | Constant | Considered | NC | Crisp |

Mahapatra et al. [33] (2019) | Price, advertisement & stock | NC | NC | Fuzzy |

Li et al. [34] (2019) | Price-dependent | Considered | NC | Crisp |

MdMashud et al. [35] (2020) | Price & advertisement | Considered | NC | Crisp |

Pervin et al. [36] (2020) | Price & stock | Considered | NC | Crisp |

De et al. [37] (2020) | Stock dependent | Considered | NC | Fuzzy |

Mashud et al. [38] (2020) | Price-sensitive | Considered | Considered | Crisp |

Das et al. [39] (2021) | Selling price & stock level | Considered | NC | Crisp |

Mohammad et al. [40] (2021) | Price & Stock | NC | NC | Crisp |

Mishra [41] (2021) | Trade credit & price sensitive | Considered | Considered | Crisp |

Rahman et al. [42] (2021) | Price & Stock | Considered | NC | Crisp |

Xu et al. [43] (2020) | Time-varying | NC | Considered | Crisp |

Ruidas et al. [44] (2021) | Price-sensitive | Considered | Considered | Interval-valued |

Present research work | Price, advertisement, stock & eco-friendly | Considered | Considered | Pythagorean Fuzzy |

Environment | ${\mathit{t}}_{1}^{*}\left(\mathbf{Year}\right)$ | ${\mathit{T}}^{*}\left(\mathbf{Year}\right)$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ | ${\mathit{Q}}_{1}^{*}$ | ${\mathit{Q}}_{2}^{*}$ | ${\mathit{Q}}^{*}$ |
---|---|---|---|---|---|---|

Green and preservation | 0.597481 | 0.744975 | 3063.43 | 168.197 | 35.3348 | 203.582 |

Without green and preservation environment | 0.473401 | 0.670989 | 2714.41 | 134.663 | 47.3356 | 181.998 |

$\mathit{G}$ | ${\mathit{t}}_{1}^{*}\left(\mathbf{Year}\right)$ | ${\mathit{T}}^{*}\left(\mathbf{Year}\right)$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ |
---|---|---|---|

0 | 0.592295 | 0.744918 | 3032.58 |

1 | 0.595060 | 0.744936 | 3051.32 |

2 | 0.596135 | 0.744950 | 3057.94 |

3 | 0.596708 | 0.744960 | 3060.99 |

4 | 0.597063 | 0.744967 | 3062.49 |

5 | 0.597305 | 0.744972 | 3063.20 |

6 | 0.597481 | 0.744975 | 3063.43 |

7 | 0.597614 | 0.744978 | 3063.37 |

8 | 0.597718 | 0.744980 | 3063.10 |

Parameter | ${\mathit{t}}_{1}^{*}\left(\mathbf{year}\right)$ | ${\mathit{T}}^{*}\left(\mathbf{year}\right)$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ | ${\mathit{Q}}_{1}^{*}$ | ${\mathit{Q}}_{2}^{*}$ | ${\mathit{Q}}^{*}$ |
---|---|---|---|---|---|---|

$\lambda =0$ | 0.590854 | 0.742191 | 2892.08 | 166.227 | 36.2553 | 202.483 |

$\lambda =0.2$ | 0.593049 | 0.743111 | 2949.15 | 166.880 | 32.9498 | 202.830 |

$\lambda =0.4$ | 0.595258 | 0.744039 | 3006.27 | 167.536 | 35.6430 | 203.179 |

$\lambda =0.6$ | 0.597481 | 0.744975 | 3063.43 | 168.197 | 35.3348 | 203.582 |

$\lambda =0.8$ | 0.599717 | 0.745919 | 3120.63 | 168.862 | 35.0252 | 203.888 |

$\lambda =1$ | 0.601967 | 0.746871 | 3177.88 | 169.532 | 34.7142 | 204.246 |

$\mathit{\omega}\left(\frac{\mathit{\alpha}}{\mathit{\beta}}\right)$ | ${\mathit{t}}_{1}^{*}\left(\mathbf{year}\right)$ | ${\mathit{T}}^{*}\left(\mathbf{year}\right)$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ | ${\mathit{Q}}_{1}^{*}$ | ${\mathit{Q}}_{2}^{*}$ | ${\mathit{Q}}^{*}$ |
---|---|---|---|---|---|---|

$\omega =0.1$ | 0.929017 | 1.31569 | 652.618 | 80.5255 | 29.2729 | 109.798 |

$\omega =0.5$ | 0.857248 | 1.19576 | 858.821 | 90.2861 | 31.0508 | 121.337 |

$\omega =1$ | 0.777642 | 1.06138 | 1185.44 | 104.554 | 33.0861 | 137.640 |

$\omega =2$ | 0.708321 | 0.94251 | 1637.00 | 121.683 | 72.2574 | 193.941 |

$\omega =4$ | 0.677252 | 0.88837 | 1918.93 | 131.567 | 35.2865 | 166.854 |

Parameter | % Change | ${\mathit{t}}_{1}^{*}\left(\mathbf{Year}\right)$ | $\%\mathbf{Change}\mathbf{in}{\mathit{t}}_{1}^{*}$ | ${\mathit{T}}^{*}\left(\mathbf{Year}\right)$ | $\%\mathbf{Change}\mathbf{in}{\mathit{T}}^{*}$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ | % Change in Optimal Profit |
---|---|---|---|---|---|---|---|

−20 | 0.506489 | −15.23 | 0.681956 | −8.46 | 211.998 | −93.08 | |

$p$ | −10 | 0.552553 | −7.52 | 0.714726 | −4.06 | 1725.64 | −43.67 |

10 | 0.641394 | 7.35 | 0.772970 | 3.76 | 4264.10 | 39.19 | |

20 | 0.684374 | 14.54 | 0.798899 | 7.24 | 5355.06 | 74.81 | |

−20 | 0.590467 | −1.17 | 0.752651 | 1.03 | 3090.95 | 0.90 | |

${\tilde{C}}_{l}$ | −10 | 0.594049 | −0.57 | 0.748929 | 0.53 | 3076.90 | 0.44 |

10 | 0.600759 | 0.55 | 0.740787 | −0.56 | 3050.55 | −0.42 | |

20 | 0.603883 | 1.07 | 0.736359 | −1.16 | 3038.28 | −0.82 | |

−20 | 0.684918 | 14.63 | 0.803316 | 7.83 | 3272.55 | 6.73 | |

${\tilde{C}}_{{h}_{1}}$ | −10 | 0.637576 | 6.71 | 0.771261 | 3.53 | 3162.68 | 3.24 |

10 | 0.562924 | −5.78 | 0.722983 | −2.95 | 2973.13 | −2.95 | |

20 | 0.532723 | −10.84 | 0.704282 | −5.46 | 2890.49 | −5.65 | |

−20 | 0.591066 | −1.07 | 0.771057 | 3.50 | 3088.60 | 0.82 | |

${\tilde{C}}_{s}$ | −10 | 0.594565 | −0.49 | 0.756679 | 1.57 | 3074.87 | 0.37 |

10 | 0.599948 | 0.41 | 0.735260 | −1.30 | 3053.74 | −0.32 | |

20 | 0.602064 | 0.77 | 0.727063 | −2.40 | 3045.43 | −0.59 | |

−20 | 0.598489 | 0.17 | 0.745626 | 0.09 | 3066.00 | 0.08 | |

${\tilde{C}}_{{d}_{1}}$ | −10 | 0.597984 | 0.08 | 0.745300 | 0.04 | 3064.71 | 0.04 |

10 | 0.596978 | −0.08 | 0.744651 | −0.04 | 3062.15 | −0.04 | |

20 | 0.596476 | −0.17 | 0.744327 | −0.09 | 3060.87 | −0.08 | |

−20 | 0.644074 | 7.80 | 0.755922 | 1.47 | 5063.67 | 65.29 | |

${\tilde{C}}_{p}$ | −10 | 0.620338 | 3.83 | 0.750409 | 0.73 | 4060.68 | 32.55 |

10 | 0.575438 | −3.69 | 0.739597 | −0.72 | 2071.64 | −32.38 | |

20 | 0.554156 | −7.25 | 0.734258 | −1.44 | 1085.04 | −64.58 |

Parameter | % Change | ${\mathit{t}}_{1}^{*}\left(\mathbf{Year}\right)$ | $\%\mathbf{Change}\mathbf{in}{\mathit{t}}_{1}^{*}$ | ${\mathit{T}}^{*}\left(\mathbf{Year}\right)$ | $\%\mathbf{Change}\mathbf{in}{\mathit{T}}^{*}$ | $\mathit{T}\mathit{A}{\mathit{P}}^{*}\left({\mathit{t}}_{1}^{*},{\mathit{T}}^{*}\right)$ | % Change in Optimal Profit |
---|---|---|---|---|---|---|---|

−20 | 0.665700 | 11.42 | 0.868054 | 16.52 | 2041.38 | −33.36 | |

$\nu $ | −10 | 0.629738 | 5.40 | 0.803960 | 7.92 | 2503.65 | −18.27 |

10 | 0.568860 | −4.79 | 0.690794 | −7.27 | 3740.76 | 22.11 | |

20 | 0.543883 | −8.97 | 0.641169 | −13.93 | 4560.23 | 48.86 | |

−20 | 0.657762 | 10.09 | 0.838376 | 12.54 | 2326.92 | −24.04 | |

$a$ | −10 | 0.625239 | 4.65 | 0.787939 | 5.77 | 2692.04 | −12.12 |

10 | 0.573400 | −4.03 | 0.707768 | −5.00 | 3440.25 | 12.30 | |

20 | 0.552230 | −7.57 | 0.675106 | −9.38 | 3821.84 | 24.76 | |

−20 | 0.602878 | 0.90 | 0.745145 | 0.02 | 3032.20 | −1.02 | |

$\varphi $ | −10 | 0.600157 | 0.45 | 0.745046 | 0.01 | 3047.76 | −0.51 |

10 | 0.594846 | −0.44 | 0.744933 | −0.01 | 3079.20 | 0.51 | |

20 | 0.592253 | −0.88 | 0.744918 | −0.02 | 3095.07 | 1.03 | |

−20 | 0.596431 | −0.18 | 0.744955 | −0.003 | 3056.03 | −0.24 | |

$\eta $ | −10 | 0.596955 | −0.09 | 0.744965 | −0.001 | 3059.73 | −0.12 |

10 | 0.598008 | 0.09 | 0.744987 | 0.002 | 3067.14 | 0.12 | |

20 | 0.598537 | 0.18 | 0.745000 | 0.003 | 3070.85 | 0.24 | |

−20 | 0.521267 | −12.76 | 0.592795 | −20.43 | 5632.57 | 83.86 | |

$\delta $ | −10 | 0.555001 | −7.11 | 0.663614 | −10.92 | 4160.98 | 35.83 |

10 | 0.648565 | 8.55 | 0.837693 | 12.45 | 2244.62 | −26.73 | |

20 | 0.708557 | 18.59 | 0.942920 | 26.57 | 1635.08 | −46.63 |

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**MDPI and ACS Style**

Bhavani, G.D.; Meidute-Kavaliauskiene, I.; Mahapatra, G.S.; Činčikaitė, R.
A Sustainable Green Inventory System with Novel Eco-Friendly Demand Incorporating Partial Backlogging under Fuzziness. *Sustainability* **2022**, *14*, 9155.
https://doi.org/10.3390/su14159155

**AMA Style**

Bhavani GD, Meidute-Kavaliauskiene I, Mahapatra GS, Činčikaitė R.
A Sustainable Green Inventory System with Novel Eco-Friendly Demand Incorporating Partial Backlogging under Fuzziness. *Sustainability*. 2022; 14(15):9155.
https://doi.org/10.3390/su14159155

**Chicago/Turabian Style**

Bhavani, G. Durga, Ieva Meidute-Kavaliauskiene, Ghanshaym S. Mahapatra, and Renata Činčikaitė.
2022. "A Sustainable Green Inventory System with Novel Eco-Friendly Demand Incorporating Partial Backlogging under Fuzziness" *Sustainability* 14, no. 15: 9155.
https://doi.org/10.3390/su14159155