# Scenario Analysis–Based Decision and Coordination in Supply Chain Management with Production and Transportation Scheduling

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Integrated Scheduling Coordination Model

#### 3.1. Basic Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

#### 3.2. Production Connecting Connection Time (PCCT) and Waiting Time (PCWT)

- (1)
- As scenario 1 shows, if PCCT > PCWT, orders will choose the non-last feasible connecting train instead of the last feasible one, in which ${t}_{sil{l}^{\prime}}^{r}-{t}_{sil{l}^{\prime}}^{w}>0$.
- (2)
- As scenario 2 shows, if PCCT = PCWT, orders will choose the last feasible connecting train; that is, ${t}_{sil{l}^{\prime}}^{r}-{t}_{sil{l}^{\prime}}^{w}=0$.
- (3)
- However, if no feasible train can be chosen, as shown in scenario 3, PCCT is defined as a negative value, and PCWT is a sufficiently large positive value; that is, PCCT < PCWT. Orders will fail to connect; that is, ${t}_{sil{l}^{\prime}}^{r}-{t}_{sil{l}^{\prime}}^{w}<0$. We let ${t}_{sil{l}^{\prime}}^{r}-{t}_{sil{l}^{\prime}}^{w}=-\theta $ to simplify the numerical calculation. $\theta $ is defined as the penalty value of missing the feasible connecting train.

#### 3.3. Production Delivery Connection Time (PDCT) and Waiting Time (PDWT)

- (1)
- If PDCT − PDWT > 0 and PDWT > 0 (scenario 1), orders will be delivered early to the destination, in which case ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}>0$. We let ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}=-\alpha $, and this means the penalty value of delivering early.
- (2)
- If PDCT − PDWT > 0 and PDWT = 0 (scenario 2), orders can be delivered as soon as the connecting train arrives, in which ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}>0$. We let ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}=\mu $ to facilitate the numerical calculation.
- (3)
- However, if PDCT − PDWT < 0, and PDCT < 0 (scenario 3), orders will fail to deliver, and ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}<0$. In this case, ${t}_{{e}_{i}{l}^{\prime}}^{w}$ is a sufficiently large positive value, and ${t}_{{e}_{i}{l}^{\prime}}^{r}$ is a negative value, implying ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}=-M$. We let ${t}_{{e}_{i}{l}^{\prime}}^{r}-{t}_{{e}_{i}{l}^{\prime}}^{w}=-\delta $, and the meaning of $\delta $ is the penalty value of delivering late. Let $\left|\delta \right|\gg \left|\alpha \right|$.

#### 3.4. Model Formulation

## 4. Solution Algorithm and Numerical Experiment

#### 4.1. Genetic Algorithm

#### 4.2. Numerical Test

#### 4.2.1. Sample Network Test

#### 4.2.2. Performance of GA and Convergence Test

#### 4.2.3. Comparisons of Different Objectives

#### 4.2.4. Analysis of Penalty Parameters and GA Parameters

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Symbol Notations

$i,{i}^{\prime}$ | The set of order, $i$, ${i}^{\prime}\in I$; $i\&{i}^{\prime}=$1, 2,…, k |

${Q}_{i}$ | Quantity of order $i$ |

${p}_{i}$ | Processing time of order $i$ |

$j$, ${j}^{\prime}$ | Position or sequence of order $i,j\&{j}^{\prime}=$1, 2,…, k |

${e}_{i}$ | The set of order $i$ destination, $i\in I$ |

$L$ | The set of production lines, $l\in L$, $L=\left\{l\right|l=1,2,\dots ,n\}$, where $n$ is the total number of production lines |

${L}^{\prime}$ | The set of trains in the network, ${l}^{\prime}\in {L}^{\prime}$, ${L}^{\prime}=\left\{{l}^{\prime}\right|{l}^{\prime}=1,2,\dots ,m\}$, where $m$ is the total number of train indexes |

$S({l}^{\prime})$ | The set of stations on line ${l}^{\prime}$, $s\in S({l}^{\prime})$, $S({l}^{\prime})$ = {$s$|$s$ = 1, 2,…, p}, where $p$ is the total number of stations |

${t}_{sl{l}^{\prime}}^{Tra}$ | The order transfer operating time from $l$ to ${l}^{\prime}$ at station $s$ |

${t}_{s{l}^{\prime}}^{D}$ | The departure time of the last feasible connecting train at station $s$ |

${t}_{s{l}^{\prime}}^{{D}^{\prime}}$ | The departure time of the selected connecting train at station $s$; if the selected connecting train is the last feasible train, ${t}_{s{l}^{\prime}}^{{D}^{\prime}}$ = ${t}_{s{l}^{\prime}}^{D}$ |

${t}_{{e}_{i}{l}^{\prime}}^{A}$ | The arrival time of train ${l}^{\prime}$ to the order $i$ destination |

[${t}_{{e}_{i}}^{beg}$, ${t}_{{e}_{i}}^{end}$] | Delivery time window for order $i$ |

${t}_{e{l}^{\prime}}^{Tra}$ | The order delivery operating time from line ${l}^{\prime}$ at its destination |

$\theta $ | The penalty cost of missing feasible connecting train |

$\alpha $ | The delivery earliness penalty cost for order $i$ |

$\delta $ | The delivery tardiness penalty cost for order $i$ |

$\mu $ | The delivery timely contribution benefit of order $i$ |

$Ca{p}_{{l}^{\prime}}$ | The capacity of train ${l}^{\prime}$ |

${t}_{sil}^{A}$ | The completion time of the order $i$ on production line $l$ at station $s$ |

${t}_{sil{l}^{\prime}}^{r}$ | PCCT, the time difference between the departure time of the last feasible connecting train and the completion time of the order $i$ on the production line $l$ |

${t}_{sil{l}^{\prime}}^{w}$ | PCWT, the waiting time for the order which transfers from the production line to the selected connecting train |

${t}_{sil{l}^{\prime}}^{h}$ | PCT, the production connecting time |

${t}_{{e}_{i}{l}^{\prime}}^{r}$ | PDCT, the difference between the end time of the delivery time window for the order and the selected train’s arrival time at its destination |

${t}_{{e}_{i}{l}^{\prime}}^{w}$ | PDWT, the waiting time for the order until the delivery service allowed by time window |

${t}_{{e}_{i}{l}^{\prime}}^{d}$ | PDT, the production delivery time |

${x}_{il{l}^{\prime}}$ | =1 if the order $i$ on production $l$ choices train ${l}^{\prime}$; = 0 otherwise |

${u}_{ij}$ | =$i$ if the order $i$ is in position $j$ |

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**Figure 1.**Effects of production connecting time (PCT) on production collection. PCCT: production collection connecting time; PCWT: production collection waiting time.

**Figure 2.**Effects of production delivery time (PDT) on production delivery. PDCT: production delivery connection time; PDWT: production delivery waiting time.

Station Line | Departure Time | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|---|

1 | 1 | 3.5 | - | 5 | - | 7.5 | - |

2 | 1.5 | - | 4.5 | - | - | - | - |

3 | 2 | 3 | - | 4.5 | - | - | 6 |

4 | 2.5 | 4.5 | - | 5.5 | - | 8 | - |

5 | 3 | 5 | - | - | 6.5 | - | - |

6 | 3.5 | 4.5 | - | 6.5 | 7.5 | - | |

7 | 4 | - | 5.5 | - | - | - | - |

8 | 4.5 | 5.5 | - | - | 7 | - | - |

Quality/Processing Time | 0.3/1 | 0.4/1 | 0.2/1 | 0.2/1 | 0.1/1 | 0.1/3 | 0.1/5 | 0.1/3 | 0.4/2 | 0.2/4 |
---|---|---|---|---|---|---|---|---|---|---|

Delivery time window | [1,10] | [1,6] | [1,6] | [1,8] | [1,7] | [1,10] | [1,8] | [1,5] | [1,9] | [1,4] |

Production scheduling and train allocated | 10 | 4 | 2 | 8 | 6 | 7 | 3 | 1 | 9 | 5 |

1 | √ | |||||||||

2 | √ | √ | ||||||||

3 | √ | √ | ||||||||

4 | √ | |||||||||

5 | √ | √ | ||||||||

6 | √ | |||||||||

7 | ||||||||||

8 | √ | |||||||||

${t}_{sil{l}^{\prime}}^{r}$ | 2.7 | 2.8 | 2.6 | 0.1 | 1.5 | 2.2 | 1.7 | 0.4 | 0.6 | −0.2 |

${t}_{sil{l}^{\prime}}^{w}$ | 0.2 | 0.3 | 0.1 | 0.1 | 0.5 | 0.7 | 0.2 | 0.4 | 0.6 | ∞ |

${t}_{{e}_{i}{l}^{\prime}}^{r}$ | 1.5 | 0.5 | 0.5 | 1.0 | −0.2 | 2.5 | 0.5 | −2.5 | 2.5 | 0 |

${t}_{{e}_{i}{l}^{\prime}}^{w}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Method | Successful Connections | Successful Deliveries | Objective Function | CPU Time (s) |
---|---|---|---|---|

GA | 9 | 8 | −207.5 | 15.6 |

SA | 9 | 8 | −206.5 | 74.7 |

TS | 9 | 8 | −208.0 | 16.7 |

PSO | 9 | 8 | −206.5 | 27.2 |

Objective | Successful Connections | Successful Deliveries | Total Waiting Time in Both Connection and Delivery Process |
---|---|---|---|

Max PCT | 9 | 7 | 402 |

Max PDT | 4 | 8 | 806.6 |

Max (PCT+PDT) | 9 | 8 | 302 |

Penalty θ | Successful Connections | Successful Deliveries | PCCT | PCWT for Successful Connections | PCT for Missing Connections | PDCT | PDWT for Successful Deliveries | PDT for Missing Deliveries | Objective Function |
---|---|---|---|---|---|---|---|---|---|

10 | 9 | 8 | 15.5 | 2.2 | −10 | 6.5 | 0 | −200 | −116.5 |

20 | 9 | 8 | 15.3 | 2.0 | −20 | 6.5 | 0 | −200 | −126.5 |

30 | 9 | 8 | 15.3 | 2.0 | −30 | 6.5 | 0 | −200 | −136.5 |

40 | 9 | 8 | 15.5 | 2.2 | −40 | 6.5 | 0 | −200 | −146.5 |

50 | 9 | 8 | 15.3 | 2.0 | −50 | 6.5 | 0 | −200 | −156.5 |

60 | 9 | 8 | 15.5 | 2.2 | −60 | 6.5 | 0 | −200 | −166.5 |

70 | 9 | 8 | 15.5 | 2.2 | −70 | 6.5 | 0 | −200 | −176.5 |

80 | 9 | 8 | 15.4 | 2.6 | −80 | 6.5 | 0 | −200 | −186.5 |

90 | 9 | 8 | 15.4 | 2.6 | −90 | 6.5 | 0 | −200 | −197 |

100 | 9 | 8 | 12.8 | 3.0 | −100 | 4 | 0 | −200 | −206.5 |

N\K | 0 | 50 | 100 | 200 | 300 | 400 | 500 |
---|---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

50 | 0 | −207.5 | −207.5 | −207.5 | −207.5 | −207.0 | −207.0 |

100 | 0 | −207.5 | −207.5 | −207.5 | −207.5 | −207.0 | −207.0 |

200 | 0 | −207.5 | −207.5 | −207.5 | −207.5 | −207.0 | −207.0 |

300 | 0 | −207.5 | −207.5 | −207.5 | −207.5 | −207.0 | −207.0 |

400 | 0 | −207.5 | −207.5 | −207.5 | −207.0 | −207.0 | −207.0 |

500 | 0 | −207.5 | −207.5 | −207.5 | −207.0 | −206.5 | −206.5 |

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## Share and Cite

**MDPI and ACS Style**

Jiang, Y.; Zhou, X.; Xu, Q.
Scenario Analysis–Based Decision and Coordination in Supply Chain Management with Production and Transportation Scheduling. *Symmetry* **2019**, *11*, 160.
https://doi.org/10.3390/sym11020160

**AMA Style**

Jiang Y, Zhou X, Xu Q.
Scenario Analysis–Based Decision and Coordination in Supply Chain Management with Production and Transportation Scheduling. *Symmetry*. 2019; 11(2):160.
https://doi.org/10.3390/sym11020160

**Chicago/Turabian Style**

Jiang, Yang, Xiaoye Zhou, and Qi Xu.
2019. "Scenario Analysis–Based Decision and Coordination in Supply Chain Management with Production and Transportation Scheduling" *Symmetry* 11, no. 2: 160.
https://doi.org/10.3390/sym11020160