# The Impact of Lead Time Uncertainty on Supply Chain Performance Considering Carbon Cost

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Inventory Control System Model Considering Stochastic Lead Times

#### 3.1. Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

**Assumption**

**6.**

#### 3.2. Stochastic lead times

**Step 1:**The distribution of random lead time is assumed in the model. MATLAB is used to generate n random numbers representing the lead time in the period; these are stored in an array $L$. There are five presumed types of lead time distributions corresponding to five experiments.

**Step 2:**Next, the array $L$ is introduced into the basic inventory simulation model, and the delay, corresponding to the order quantity $O{Q}_{\mathrm{t}}$ and determined at the end of the simulation period t, is denoted as ${L}_{\mathrm{t}}$.

**Step 3:**Since each ${L}_{\mathrm{t}}$ is generated in step 1, the arrival time of $O{Q}_{\mathrm{t}}$ can be calculated at the beginning of each period and recorded in the arrival time array as: $arrivetime(t)=t+1+{L}_{t}$.

**Step 4:**In each period, the $arrivetime$ array from period $\mathrm{i}$ to period $i-{L}^{+}$ is checked to determine whether $arrivetime(t)$ equals $\mathrm{i}$. If so, the corresponding arrival quantity can be included in the current inventory. Otherwise, it is counted as the work-in-process inventory.

#### 3.3. Models

#### 3.4. System Performance Indicators

#### 3.4.1. The Carbon Cost

#### 3.4.2. Average Inventory Cost

#### 3.4.3. Average Service Level

## 4. Verification and Experimental Design of Simulation Model

#### 4.1. Model Verification

#### 4.2. Model Parameter Analysis

_{S}, α

_{SL}and L

_{t}, L

_{t}, where (z − θ + 1) represents the stable pole of the system. This deduction further verifies and expands the results reported in [44], which concludes that the parameter combination (α

_{S}, α

_{SL}) is the key factor for determining the boundaries among the stability, the stable oscillation, and the exponential oscillation of the system.

#### 4.3. Experiment Design

_{S}and α

_{SL}affect the dynamics of the system, and the various parameter values correspond to different ordering policies. Many researchers have explored the practical impact of the parameter setting on system stability performance with the POUT policy, that is, α

_{S}= α

_{SL}. These researchers argue that POUT policy can improve system stability in a continuous review system [45]. Therefore, this paper adopts the POUT policy, β = α

_{S}= α

_{SL}, to explore the effect of stochastic lead times on the inventory system in the range of [0.1,1.9] for parameter β.

_{L}= 2 and different standard deviations. After a number of trials, five arrays of lead time with standard deviations of 0, 0.5, 1, 1.8, and 2 were screened. The standard deviation of σ

_{L}= 0 represents the fixed lead time. This paper takes the system carbon cost CC, average inventory cost AC, and average service level ASL as the performance evaluation indicators. Under each lead time distribution, the three performance indicators were observed with the help of MATLAB 2016a. We obtained 15 experiments, as depicted in Table 2, and the experimental results are shown with different order policies in Appendix B.

## 5. Analysis of Experimental Results

#### 5.1. Effect of Stochastic Lead Times on System Performance

#### 5.2. Effect of Stochastic Lead Times on Order Decision

_{L}= 1, 1.8, or 2 is also greater than that of fixed lead time. Therefore, β = 0.8 is the better ordering decision criterion than β = 0.2 When parameter β changes from 0.8 to 1.8, CC and AC both increase significantly. The average inventory cost AC increases to 20%. The ASL also increases significantly from 1.80 to 3.6% when β increases. While adopting the order policy of β = 0.8, the inventory system becomes more stable, and its impact on cushioning lead time fluctuation is more efficient than that of policy β = 1.8 Overall, it can be inferred that stochastic lead times does affect the regulation of the ordering decision.

#### 5.3. Summary of Experimental Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Agi, M.A.; Nishant, R. Understanding influential factors on implementing green supply chain management practices: An interpretive structural modelling analysis. J. Environ. Manag.
**2017**, 188, 351–363. [Google Scholar] [CrossRef] - Diabat, A.; Kannan, D.; Mathiyazhagan, K. Analysis of enablers for implementation of sustainable supply chain management–A textile case. J. Clean. Prod.
**2014**, 83, 391–403. [Google Scholar] [CrossRef] - Giusti, R.; Iorfida, C.; Li, Y.; Manerba, D.; Musso, S.; Perboli, G.; Yuan, S. Sustainable and De-Stressed International Supply-Chains Through the SYNCHRO-NET Approach. J. Sustain.
**2019**, 11, 1083. [Google Scholar] [CrossRef] - Boute, R.N.; Van Mieghem, J.A. Global Dual Sourcing and Order Smoothing: The Impact of Capacity and Lead Times. J. Manag. Sci.
**2015**, 61, 2080–2099. [Google Scholar] [CrossRef] - Galal, N.M.; El-Kilany, K.S. Sustainable agri-food supply chain with uncertain demand and lead time. J. Int. J. Simul. Model.
**2016**, 15, 485–496. [Google Scholar] - Disney, S.M.; Maltz, A.; Wang, X.; Warburton, R.D.H. Inventory management for stochastic lead times with order crossovers. Eur. J. Oper. Res.
**2016**, 248, 473–486. [Google Scholar] [CrossRef] - Thorsen, A.; Yao, T. Robust inventory control under demand and lead time uncertainty. J. Ann. Oper. Res.
**2017**, 157, 207–236. [Google Scholar] [CrossRef] - Alhaj, M.A.; Svetinovic, D.; Diabat, A. A carbon-sensitive two-echelon-inventory supply chain model with stochastic demand. Resour. Conserv. Recycl.
**2016**, 108, 82–87. [Google Scholar] [CrossRef] - Giusti, R.; Manerba, D.; Bruno, G.; Tadei, R. Synchromodal logistics: An overview of critical success factors, enabling technologies, and open research issues. J. Transp. Res. Part E: Logist. Transp. Rev.
**2019**, 129, 92–110. [Google Scholar] [CrossRef] - Chao, C.; Zhihui, T.; Baozhen, Y. Optimization of two-stage location–routing–inventory problem with time-windows in food distribution network. J. Ann. Oper. Res.
**2019**, 273, 111–134. [Google Scholar] [CrossRef] - Schiffer, M.; Schneider, M.; Laporte, G. Designing sustainable mid-haul logistics networks with intra-route multi-resource facilities. J. Eur. J. Oper. Res.
**2018**, 265, 517–532. [Google Scholar] [CrossRef] - Zhao, Y.; Xue, Q.; Zhang, X. Stochastic Empty Container Repositioning Problem with CO2 Emission Considerations for an Intermodal Transportation System. J. Sustain.
**2018**, 10, 4211. [Google Scholar] [CrossRef] - Das, C.; Jharkharia, S. Low carbon supply chain: A state-of-the-art literature review. J. Manuf. Technol. Manag.
**2018**, 29, 398–428. [Google Scholar] [CrossRef] - Stopková, M.; Stopka, O.; Ľupták, V. Inventory Model Design by Implementing New Parameters into the Deterministic Model Objective Function to Streamline Effectiveness Indicators of the Inventory Management. J. Sustain.
**2018**, 11, 4175. [Google Scholar] [CrossRef] - Ding, Z.; Xin, J.; Liu, Z.; Long, R.; Xu, Z.; Cao, Q. Factors affecting low-carbon consumption behavior of urban residents: A comprehensive review. J. Resour. Conserv. Recycl.
**2018**, 132, 3–15. [Google Scholar] [CrossRef] - Song, J.; Leng, M. Analysis of the single-period problem under carbon emissions policies. In Handbook of Newsvendor Problems; Springer: New York, NY, USA, 2012; pp. 297–313. [Google Scholar]
- Chen, X.; Benjaafar, S.; Elomri, A. The carbon-constrained EOQ. J. Oper. Res. Lett.
**2013**, 41, 172–179. [Google Scholar] [CrossRef] - Hovelaque, V.; Bironneau, L. The carbon-constrained EOQ model with carbon emission dependent demand. Int. J. Prod. Econ.
**2015**, 164, 285–291. [Google Scholar] [CrossRef] - Benjaafar, S.; Li, Y.; Daskin, M. Carbon footprint and the management of supply chains: Insights from simple models. J. Trans. Autom. Sci. Eng.
**2013**, 10, 99–116. [Google Scholar] [CrossRef] - Diabat, A.; Al-Salem, M. An integrated supply chain problem with environmental considerations. Int. J. Prod. Econ.
**2015**, 164, 330–338. [Google Scholar] [CrossRef] - Park, S.J.; Cachon, G.P.; Lai, G.; Seshadri, S. Supply chain design and carbon penalty: Monopoly vs. monopolistic competition. J. Prod. Oper. Manag.
**2015**, 24, 1494–1508. [Google Scholar] - Konur, D.; Campbell, J.F.; Monfared, S.A. Economic and environmental considerations in a stochastic inventory control model with order splitting under different delivery schedules among suppliers. J. Omega
**2017**, 71, 46–65. [Google Scholar] [CrossRef] - Xu, Z.; Pokharel, S.; Elomri, A.; Mutlu, F. Emission policies and their analysis for the design of hybrid and dedicated closed-loop supply chains. J. Clean. Prod.
**2017**, 142, 4152–4168. [Google Scholar] [CrossRef] - Hoen, K.M.R.; Tan, T.; Fransoo, J.C.; van Houtum, G.J. Effect of carbon emission regulations on transport mode selection under stochastic demand. Flex. Serv. Manuf. J.
**2014**, 26, 170–195. [Google Scholar] [CrossRef] - Sarkar, B.; Ullah, M.; Kim, N. Environmental and economic assessment of closed-loop supply chain with remanufacturing and returnable transport items. J. Comput. Ind. Eng.
**2017**, 111, 148–163. [Google Scholar] [CrossRef] - Arıkan, E.; Fichtinger, J.; Ries, J.M. Impact of transportation lead-time variability on the economic and environmental performance of inventory systems. Int. J. Prod. Econ.
**2014**, 157, 279–288. [Google Scholar] [CrossRef] - Chatfield, D.C.; Kim, J.G.; Harrison, T.P.; Hayya, J.C. The Bullwhip Effect—impact of stochastic lead time, information quality, and information sharing: A simulation study. Prod. Oper. Manag.
**2004**, 13, 340–353. [Google Scholar] [CrossRef] - Duc, T.T.H.; Luong, H.T.; Kim, Y.D. A measure of the Bullwhip Effect in supply chains with stochastic lead time. Int. J. Adv. Manuf. Technol.
**2008**, 38, 1201–1212. [Google Scholar] [CrossRef] - Boute, R.N.; Disney, S.M.; Lambrecht, M.R.; Houdt, B.V. Coordinating lead times and safety stocks under autocorrelated demand. Eur. J. Oper. Res.
**2014**, 232, 52–63. [Google Scholar] [CrossRef] - Saldanha, J.P.; Swan, P. Order Crossover Research: A 60-Year Retrospective to Highlight Future Research Opportunities. Transp. J.
**2017**, 56, 227–262. [Google Scholar] [CrossRef] - Chaharsooghi, S.K.; Heydari, J. LT variance or LT mean reduction in supply chain management: Which one has a higher impact on SC performance. Int. J. Prod. Econ.
**2010**, 124, 475–481. [Google Scholar] [CrossRef] - Li, J.; Liu, L.; Hu, H.; Zhao, Q.; Guo, L. An Inventory Model for Deteriorating Drugs with Stochastic Lead Time. Int. J. Environ. Res. Public Health
**2018**, 15, 2772. [Google Scholar] [CrossRef] - Spiegler, V.L.M.; Naim, M.M. Investigating sustained oscillations in nonlinear production and inventory control models. Eur. J. Oper. Res.
**2017**, 261, 572–583. [Google Scholar] [CrossRef] - Dejonckheere, J.; Disney, S.M.; Lambrecht, M.R.; Towill, D.R. Measuring and avoiding the Bullwhip Effect: A control theoretic approach. Eur. J. Oper. Res.
**2003**, 147, 567–590. [Google Scholar] [CrossRef] - Badole, C.M.; Jain, R.; Rathore, A.P.S.; Nepal, B. Research and opportunities in supply chain modeling: A review. Int. J. Supply Chain Manag.
**2012**, 1, 63–86. [Google Scholar] - Lin, J.; Naim, M.M.; Purvis, L.; Gosling, J. The extension and exploitation of the inventory and order based production control system archetype from 1982 to 2015. Int. J. Prod. Econ.
**2017**, 194, 135–152. [Google Scholar] [CrossRef] - Wang, J.; Wang, X. Complex dynamic behaviors of constrained supply chain systems. J. Syst. Eng. -Theory Pract.
**2012**, 32, 746–751. [Google Scholar] - Cannella, S.; López-Campos, M.; Dominguez, R.; Ashayeri, J.; Miranda, P.A. A simulation model of a coordinated decentralized supply chain. J. Int. Trans. Oper. Res.
**2015**, 22, 735–756. [Google Scholar] [CrossRef] - Cannella, S.; Ciancimino, E. On the bullwhip avoidance phase: Supply chain collaboration and order smoothing. Int. J. Prod. Res.
**2010**, 48, 6739–6776. [Google Scholar] [CrossRef] - Bao, C.; Zhang, S. Cold chain logistics joint distribution path optimization considering carbon emissions. J. Ind. Eng. Manag.
**2018**, 5, 95–100. [Google Scholar] - Li, L.; Yang, Y.; Qin, G. Optimization of Integrated Inventory Routing Problem for Cold Chain Logistics Considering Carbon Footprint and Carbon Regulations. Sustainability
**2019**, 11, 4628. [Google Scholar] [CrossRef][Green Version] - Macchion, L.; Fornasiero, R.; Vinelli, A. Supply chain configurations: A model to evaluate performance in customised productions. Int. J. Prod. Res.
**2017**, 55, 1386–1399. [Google Scholar] [CrossRef] - Disney, S.M. Supply chain aperiodicity, bullwhip and stability analysis with Jury’s inners. IMA J. Manag. Math.
**2008**, 19, 101–116. [Google Scholar] [CrossRef] - Zhou, L.; Naim, M.M.; Disney, S.M. The impact of product returns and remanufacturing uncertainties on the dynamic performance of a multi-echelon closed-loop supply chain. Int. J. Prod. Econ.
**2017**, 183, 487–502. [Google Scholar] [CrossRef] - Bijulal, D.; Venkateswaran, J.; Hemachandra, N. Service levels, system cost and stability of production–inventory control systems. Int. J. Prod. Res.
**2011**, 49, 7085–7105. [Google Scholar] [CrossRef] - Oliveira, E.M.D.; Oliveira, F.L.C. Forecasting mid-long-term electric energy consumption through bagging ARIMA and exponential smoothing methods. Energy
**2018**, 144, 776–788. [Google Scholar] [CrossRef]

**Figure 1.**Stochastic lead time distributions with different standard deviations: (

**a**) average value ${\mu}_{L}=\text{}1.99$ and standard deviation ${\delta}_{L}=\text{}0.480$; (

**b**) average value ${\mu}_{L}=\text{}2.06$ and standard deviation ${\delta}_{L}=\text{}1.051$; (

**c**) average value ${\mu}_{L}=\text{}2.23$ and standard deviation ${\delta}_{L}=\text{}1.783$; (

**d**) average value ${\mu}_{L}=\text{}2.34$ and standard deviation ${\delta}_{L}=\text{}2.001$.

Decision Variables | Description |

$S{Q}_{\mathrm{t}}$ | The order quantity of manufacturer delivery in period $t$ |

$A{Q}_{\mathrm{t}}$ | The arrival quantity in period $t$ |

$BIN{V}_{\mathrm{t}}$ | The inventory at the beginning of the period in period $t$ |

$TIN{V}_{t}$ | The terminal inventory in period $t$ |

${D}_{t}$ | The retailer market demand in period $t$ |

$SO{Q}_{t}$ | The shortage in period $t$ |

$WI{P}_{t}$ | The work-in-process inventory in period $t$ |

$F{Q}_{t}$ | The demand forecast in period $t$ |

$O{Q}_{t}$ | The order in period $t$ |

${L}_{t}$ | The transport lead time in period $t$ |

Parameters | Description |

$\theta $ | The smoothing factor in forecast |

${D}_{\mu}$ | The average market demand |

${L}_{\mu}$ | The average lead time |

${\alpha}_{S}$ | The parameter of adjustment for inventory discrepancy |

${\alpha}_{SL}$ | The parameter of adjustment for WIP discrepancy |

Performance indicators | Description |

$C{C}_{\mathrm{t}}$ | The carbon cost in period $t$ |

$AC$ | The average inventory cost |

$ASL$ | The average service level |

Performance Indicators | Lead Time Variance | ||||
---|---|---|---|---|---|

σ_{L} = 0 | σ_{L} = 0.5 | σ_{L} = 1 | σ_{L} = 1.8 | σ_{L} = 2 | |

Carbon cost | Experiment 1 | Experiment 4 | Experiment 7 | Experiment 10 | Experiment 13 |

Average inventory cost | Experiment 2 | Experiment 5 | Experiment 8 | Experiment 11 | Experiment 14 |

Average service level | Experiment 3 | Experiment 6 | Experiment 9 | Experiment 12 | Experiment 15 |

Parameters | Description | Value | |
---|---|---|---|

b | Shortage cost per unit | 9 | (yuan/per unit) |

h | Inventory holding cost per unit | 1 | (yuan/per unit) |

q | Weight of a products | 10 | (kg/per unit) |

cc | Carbon tax | 0.04 | (yuan/per unit) |

P_{0} | No-load fuel consumption of trucks | 0.186 | (L/km) |

P_{c} | Carbon emissions of oil consumption per unit | 2.61 | (kg/L) |

Q | Maximum load weight of truck | 5000 | (kg) |

Ip | Carbon emissions of per unit inventory | 0.5 | (kg/per unit) |

P* | Full-load fuel consumption of trucks of one kilometer | 0.2 | (L/km) |

d | Total mileage of transportation | 1000 | (km) |

σ_{L} | IR (%) | ||
---|---|---|---|

Carbon Cost (CC) | Average Cost (AC) | Average Service Level (ASL) | |

0–0.5 | 0.62 | 15.81 | −1.28 |

0.5–1 | 1.86 | 39.32 | −3.38 |

1–1.8 | 1.05 | 11.04 | −1.58 |

1.8–2 | 1.64 | 2.82 | −0.23 |

AIR (%) | 1.29 | 17.25 | −1.62 |

σ_{L} = 0 | |||

β | CC | AC | ASL |

0.2 | 320.06 | 5045.32 | 0.9997 |

0.8 | 319.58 | 5093.27 | 0.999 |

1.8 | 323.57 | 5518.59 | 0.9966 |

IR1 (%) | –0.15 | 0.95 | –0.07 |

IR2 (%) | 1.25 | 8.35 | –0.24 |

σ_{L} = 0.5 | |||

β | CC | AC | ASL |

0.2 | 324.5 | 5792.86 | 0.9904 |

0.8 | 321.29 | 5842.34 | 0.9871 |

1.8 | 324.57 | 6498.21 | 0.98 |

IR1 (%) | –0.99 | 0.85 | –0.33 |

IR2 (%) | 1.02 | 11.23 | –0.73 |

σ_{L} = 1 | |||

β | CC | AC | ASL |

0.2 | 336.46 | 7762.43 | 0.9693 |

0.8 | 325.53 | 8098.38 | 0.9546 |

1.8 | 329.07 | 9345.13 | 0.9374 |

IR1 (%) | –3.25 | 4.33 | –1.51 |

IR2 (%) | 1.09 | 15.40 | –1.80 |

σ_{L} = 1.8 | |||

β | CC | AC | ASL |

0.2 | 342.39 | 8610.34 | 0.9582 |

0.8 | 327.95 | 8641.42 | 0.9455 |

1.8 | 334.83 | 10,976.46 | 0.9112 |

IR1 (%) | –4.22 | 0.36 | –1.33 |

IR2 (%) | 2.10 | 27.02 | –3.62 |

σ_{L} = 2 | |||

β | CC | AC | ASL |

0.2 | 344.08 | 8653.87 | 0.9575 |

0.8 | 332.82 | 9053.59 | 0.9405 |

1.8 | 339.33 | 11,152.49 | 0.9092 |

IR1 (%) | –3.27 | 4.62 | –1.78 |

IR2 (%) | 1.96 | 23.18 | –3.33 |

© 2019 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**

Li, Z.; Fei, W.; Zhou, E.; Gajpal, Y.; Chen, X. The Impact of Lead Time Uncertainty on Supply Chain Performance Considering Carbon Cost. *Sustainability* **2019**, *11*, 6457.
https://doi.org/10.3390/su11226457

**AMA Style**

Li Z, Fei W, Zhou E, Gajpal Y, Chen X. The Impact of Lead Time Uncertainty on Supply Chain Performance Considering Carbon Cost. *Sustainability*. 2019; 11(22):6457.
https://doi.org/10.3390/su11226457

**Chicago/Turabian Style**

Li, Zhuoqun, Weiwei Fei, Ermin Zhou, Yuvraj Gajpal, and Xiding Chen. 2019. "The Impact of Lead Time Uncertainty on Supply Chain Performance Considering Carbon Cost" *Sustainability* 11, no. 22: 6457.
https://doi.org/10.3390/su11226457