# Contract Coordination Strategy of Supply Chain with Substitution under Supply Disruption and Stochastic Demand

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- How does the supply disruption impact the integrated performance?
- (2)
- How does the dual replacement influence the profit of each member and the total profit of the system?
- (3)
- How do the wholesale price contract and buy-back contract coordinate the supply chain system of substitute products?
- (4)
- Which contract strategy could make the channel gross profit recovered?
- (5)
- How does one decide the buy-back price in order to make the channel gross profit recovered?

## 2. Literature Review

## 3. Model Description, Assumptions and Symbols

**Assumption 1**. The system is risk-neutral; market information is symmetrical; and the decision-maker is entirely rational.

**Assumption 2**. The price ${s}_{u}$ of ${A}_{u}$ is not less than the price s of A, namely ${s}_{u}\ge s$.

**Assumption 3**. $s>{w}_{m}>{c}_{m}+{w}_{1}+{w}_{2}>{c}_{m}+{c}_{1}+{c}_{2}$.

**Assumption 4**. ${s}_{u}>{w}_{m}>{c}_{m}+{w}_{s}+{w}_{2}>{c}_{m}+{c}_{s}+{c}_{2}$.

**Assumption 5**. ${w}_{s}>{c}_{s}$, ${w}_{1}>{c}_{1}$, ${w}_{2}>{c}_{2}$, ${c}_{s}>{c}_{2}>{c}_{1}$.

## 4. The Performance of the Supply Chain under the Centralized Decision

**Proposition 1.**

## 5. Analysis on the Decentralized Supply Chain Based on Wholesale Price Contracts

#### 5.1. The Retailer’s Expected Profit and Decision-Making

**Proposition 2.**

#### 5.2. The Expected Profit of the Supplier and Spot Market

#### 5.3. The Expected Profit of Manufacturer

**Proposition 3.**

## 6. Analysis on Decentralized Supply Chain Based on Buy-Back Contracts

#### 6.1. The Retailer’s Expected Profit and Decision-Making

**Proposition 4.**

#### 6.2. The Supplier’s and Spot Market’s Expected Profit

#### 6.3. The Manufacturer’s Expected Profit

**Proposition 5.**

## 7. Numerical Examples

## 8. Conclusions

- (1)
- considering the optimal order quantity of the retailer under both stochastic demand and supply;
- (2)
- giving a more detailed description and quantitative analysis for the influence of substitute products on the performance of the supply chain;
- (3)
- considering a multi-period supply chain model based on both stochastic demand and supply.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of Proposition 2.**

**Proof of Proposition 5.**

## References

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**Figure 2.**Comparison of the profits in two cases: (

**a**) wholesale price; and (

**b**) buy-back contract (p = 0.2).

**Figure 3.**Comparison of the profits in two cases: (

**a**) wholesale price; and (

**b**) buy-back contract (p = 0.3).

**Figure 4.**Comparison of the profits in two cases: (

**a**) wholesale price; and (

**b**) buy-back contract (p = 0.5).

**Figure 5.**Comparison of the channel profit in three case of the centralized supply chain (CSC), the wholesale price contract (WPC) and the buy-back contract (BBC) when $\beta =0.6$.

Decision variables | |
---|---|

Q | Quantity of product A or $Au$ ordered by the retailer |

Parameters | |

h | The unit holding cost |

c | The unit shortage cost |

w | The unit lost cost of supplier ${S}_{1}$ |

s | The unit retail price of product A |

${s}_{u}$ | The unit retail price of product ${A}_{u}$ |

${w}_{m}$ | The unit wholesale price of A and ${A}_{u}$ determined by the manufacturer |

${w}_{i}$ | The unit wholesale price of component ${a}_{i}$ offered by supplier ${S}_{i}$ |

${c}_{m}$ | The unit manufacturing cost of manufacturer |

b | the buy-back price for each unit of unsold products |

${c}_{i}$ | The unit product cost of supplier ${S}_{i}$ |

${w}_{s}$ | The unit wholesale price of component ${a}_{3}$ offered by the spot market |

${c}_{s}$ | The unit running cost of spot market |

p | The probability of supply disruption of component ${a}_{1}$ |

β | The probability of substituting ${A}_{u}$ for A |

X | The random market demands of product A |

${\mu}_{{}_{X}}$ | The mean demand |

p | β | ${Q}^{c}$ | $E[{\pi}_{c}^{*}]$ |
---|---|---|---|

$0.2$ | $0.70$ | $91.44$ | $605.92$ |

$0.85$ | $96.43$ | $644.13$ | |

$0.3$ | $0.70$ | $87.3$ | $580.79$ |

$0.85$ | $93.97$ | $628.64$ | |

$0.5$ | $0.70$ | $80.26$ | $540.37$ |

$0.85$ | $89.56$ | $602.09$ |

p | β | ${Q}^{w}$ | $E[{\pi}_{R}^{w*}]$ | $E[{\pi}_{M}^{w*}]$ | $E[{\pi}_{{S}_{1}}^{w*}]$ | $E[{\pi}_{{S}_{2}}^{w*}]$ | $E[{\pi}_{SM}^{w*}]$ | $E[{\pi}_{w}^{*}]$ |
---|---|---|---|---|---|---|---|---|

$0.2$ | $0.70$ | $65.18$ | $175.24$ | $234.65$ | $59.87$ | $65.18$ | $19.55$ | $554.59$ |

$0.85$ | $68.73$ | $189.86$ | $247.43$ | $63.23$ | $68.73$ | $20.62$ | $589.87$ | |

$0.3$ | $0.70$ | $63.61$ | $184.6$ | $216.27$ | $43.25$ | $63.61$ | $28.62$ | $536.35$ |

$0.85$ | $68.65$ | $205.81$ | $233.41$ | $46.68$ | $68.65$ | $30.89$ | $585.44$ | |

$0.5$ | $0.70$ | $60.96$ | $205.03$ | $182.88$ | $12.19$ | $60.96$ | $45.72$ | $506.78$ |

$0.85$ | $68.5$ | $237.64$ | $205.5$ | $13.7$ | $68.5$ | $51.37$ | $576.72$ |

p | β | ${b}^{*}$ | ${Q}^{b}$ | $E[{\pi}_{R}^{b*}]$ | $E[{\pi}_{M}^{b*}]$ | $E[{\pi}_{{S}_{1}}^{b*}]$ | $E[{\pi}_{{S}_{2}}^{b*}]$ | $E[{\pi}_{SM}^{b*}]$ | $E[{\pi}_{b}^{*}]$ |
---|---|---|---|---|---|---|---|---|---|

$0.2$ | $0.70$ | $9.16$ | $91.44$ | $230.44$ | $172.49$ | $84.12$ | $91.44$ | $27.43$ | $605.92$ |

$0.85$ | $9.04$ | $96.43$ | $246.63$ | $183.42$ | $88.72$ | $96.43$ | $28.93$ | $644.13$ | |

$0.3$ | $0.70$ | $8.82$ | $87.3$ | $238.59$ | $156.26$ | $59.36$ | $87.3$ | $39.28$ | $580.79$ |

$0.85$ | $8.73$ | $93.97$ | $258.17$ | $170.41$ | $63.8$ | $93.97$ | $42.29$ | $628.64$ | |

$0.5$ | $0.70$ | 8 | $80.26$ | $255.52$ | $128.34$ | $16.05$ | $80.26$ | $60.2$ | $540.37$ |

$0.85$ | 8 | $89.56$ | $281.15$ | $146.3$ | $17.91$ | $89.56$ | $67.17$ | $602.09$ |

© 2016 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/).

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**MDPI and ACS Style**

Yan, R.; Lu, B.; Wu, J.
Contract Coordination Strategy of Supply Chain with Substitution under Supply Disruption and Stochastic Demand. *Sustainability* **2016**, *8*, 676.
https://doi.org/10.3390/su8070676

**AMA Style**

Yan R, Lu B, Wu J.
Contract Coordination Strategy of Supply Chain with Substitution under Supply Disruption and Stochastic Demand. *Sustainability*. 2016; 8(7):676.
https://doi.org/10.3390/su8070676

**Chicago/Turabian Style**

Yan, Rongfang, Bin Lu, and Jianjun Wu.
2016. "Contract Coordination Strategy of Supply Chain with Substitution under Supply Disruption and Stochastic Demand" *Sustainability* 8, no. 7: 676.
https://doi.org/10.3390/su8070676