# Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Decision Policies of the Supply Chain with Information Asymmetry

#### 3.1. Problem Description

$c$ | Unit production cost |

$p$ | Unit selling price |

$s$ | Unit salvage price |

$D$ | Random demand |

$f(D)$ | Probability density function (PDF) of D |

$F(D)$ | Cumulative distribution function (CDF) of D |

$\beta $ | Ratio of items that are misplaced among the total physical inventory |

$\eta $ | Risk factor that reflects the degree of risk aversion |

$t$ | Unit variable RFID tag cost |

$L$ | Fixed RFID investment cost |

$E$ | Information sharing cost |

$\pi $ | Expected profit |

$k$ | The retailer’s share of revenue |

$\theta $ | The retailer’s share of RFID cost |

Decision Variables | |

$Q$ | Order quantity |

w | Unit wholesale price |

ε | Scale of RFID investment |

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

#### 3.2. The Optimal Decisions in Case 1: Information Asymmetry about Inventory Errors Exists

_{0}is the order quantity of a retailer who has accurate inventory records.

**Theorem**

**1.**

**Proof.**

#### 3.3. The Optimal Decisions in Case 2: The Retailer Shares Information about Inventory Errors with the Supplier

**Theorem**

**2.**

**Corollary**

**1.**

- (1)
- the retailer orders less if he is more risk-averse,
- (2)
- the supplier obtains less profits if the retailer is more risk-averse.

**Proof.**

#### 3.4. The Benefits of Information Sharing

**Proposition**

**1.**

- (1)
- the supplier lowers her wholesale price, i.e., ${w}_{2}^{\ast}\le {w}_{1}^{\ast}$;
- (2)
- the retailer orders more, i.e., ${Q}_{2}^{\ast}\ge {Q}_{1}^{\ast}$;
- (3)
- the retailer always benefits from information sharing, i.e., ${\pi}_{2}^{\eta}\ge {\pi}_{1}^{\eta}$;
- (4)
- the supplier benefits from information sharing if and only if:$$E\le \{\begin{array}{cc}\hfill \frac{\eta {D}_{\mathrm{max}}{\beta}^{2}(p-s)}{4{(1-\beta )}^{2}},\hfill & \hfill \mathrm{if}\text{}\beta \le \frac{p-c}{2(p-s)}\hfill \\ \hfill \frac{\eta {D}_{\mathrm{max}}{A}_{1}}{4{(1-\beta )}^{2}(p-s)},\hfill & \hfill \mathrm{if}\text{}\frac{p-c}{2(p-s)}\le \beta \le \frac{p-c}{p-s}\hfill \\ \hfill 0,\hfill & \hfill \mathrm{otherwise}\hfill \end{array}.$$

**Proof.**

## 4. Decision Policies of the Supply Chain with RFID Implementation

#### 4.1. The Optimal Decisions in Case 3: RFID Is Implemented in the Supply Chain

_{ε}, and the fixed cost for the retailer (supplier) is denoted as L

_{R}(L

_{S}). We set t

_{ε}= φε

^{2}and L

_{R}= ψ

_{R}ε

^{2}(L

_{S}= ψ

_{S}ε

^{2}), where φ, ψ

_{R}and ψ

_{S}are investment parameters. This quadratic assumption is widely used in the literature to describe the decreasing returns of investment [52,53]. We consider the situation when the variable cost of RFID investment is incurred by the supplier and the two parties bear their fixed costs respectively. This is a prevailing situation when RFID is implemented in the supply chain. For example, in 2005, Wal-Mart required its top 100 suppliers to tag all their pallets and cases and to bear the variable tag cost. RFID implementation also leads to large gains for the suppliers, boosts a long term partner relationship with the retailer, provides timely and accurate information, and helps to speed up the operational processes [46].

**Theorem**

**3.**

**Proof.**

#### 4.2. The Benefits of RFID Implementation

**Proposition**

**2.**

**Proof.**

## 5. Supply Chain Coordination

**Theorem**

**4.**

**Proof.**

- (1)
- $\eta \ge (B-c-{t}_{\epsilon})/(B-s)$. From Equation (14), we get the retailer’s optimal order quantity, as follows:$${Q}_{Co}=\frac{\eta {D}_{\mathrm{max}}(-(p-s)\beta (1-\epsilon )+p-({w}_{Co}+\theta {t}_{\epsilon})/k)}{{(1-\beta +\epsilon \beta )}^{2}(p-s)}.$$When $k(B-(B-c-{t}_{\epsilon})/\eta )-\theta {t}_{\epsilon}={w}_{Co}$, we have ${Q}_{Co}={Q}^{SC}$, which means that the optimal decision of the centralized system is adopted. Substituting Equation (19) into (15), we obtain the supplier’s expected profit after coordination, as follows:$${\pi}_{Co}^{S}=(1-k){\pi}^{SC}-k{Q}^{SC}(B-c-{t}_{\epsilon})(1/\eta -1).$$Comparing ${\pi}_{Co}^{S}$ with ${\pi}_{3}^{S}$, we derive the upper bound of k, ${k}_{U}=\eta /2$, which ensures that the supplier benefits from coordination. Similarly, we derive the lower bound of k, ${k}_{L}={\eta}^{2}/(8-4\eta )$, which ensures that the retailer benefits from coordination. Since the performance of the decentralized supply chain is always improved through coordination, the supplier is able to allocate the profits so that both parties are better off (i.e., $k\in ({k}_{L},{k}_{U})$). Thus, coordination is achieved through the contract.
- (2)
- $\eta <(B-c-{t}_{\epsilon})/(B-s)$. In this case, the optimal order quantity of the retailer is infinite. As the Stackelberg leader, the supplier can limit the retailer’s order quantity to the supply chain’s optimal order quantity (i.e., ${Q}_{Co}={Q}^{SC}$), and negotiate with the retailer about the values of k and θ to make sure that the retailer benefits from coordination. In this situation, coordination is also achieved. $\square $

**Corollary**

**2.**

_{Co}and k are increasing in η.

**Proof.**

_{Co}increases in η. We can also derive that both the upper bound and lower bound of k are increasing in η from Equation (20). $\square $

## 6. Numerical Analysis

_{max}= 300 following the setting used in previous studies [25,45]. Based on observations of empirical studies [3,4], we assume the range of β is from 0 to 0.08.

## 7. Managerial Insights for Decision Makers

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof of Theorem 1.**

**Proof.**

_{max}], as follows:

**Proof of Theorem 3.**

**Proof.**

_{3}and ε, as follows:

_{3}:

_{3}for fixed ε. From the first-order condition, we get ${w}_{3}^{\ast}({\epsilon}^{\ast})$. Substituting ${w}_{3}^{\ast}({\epsilon}^{\ast})$ into Equation (A6), we obtain ${\pi}_{3}^{S}(\epsilon )$. We then take the first- and second-order derivatives of ${\pi}_{3}^{S}(\epsilon )$ with respect to ε, as follows:

**Proof of Proposition 2.**

**Proof.**

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$\mathit{\eta}$ | $\mathit{k}$ | $\mathbf{\Delta}{\mathit{\pi}}_{\mathit{C}\mathit{o}}^{\mathit{S}\mathit{C}}$ | $\mathbf{\Delta}{\mathit{\pi}}_{\mathit{C}\mathit{o}}^{\mathit{R}}$ | $\mathbf{\Delta}{\mathit{\pi}}_{\mathit{C}\mathit{o}}^{\mathit{S}}$ |
---|---|---|---|---|

0.1 | 0.05 | 5788.9412 | 5788.9412 | 0 |

0.2 | 0.05 | 5437.6335 | 2688.2682 | 2749.3652 |

0.3 | 0.05 | 5055.7772 | 1593.6136 | 3462.1636 |

0.4 | 0.05 | 4643.3724 | 977.5521 | 3665.8203 |

0.5 | 0.05 | 4200.4191 | 534.5988 | 3665.8203 |

0.6 | 0.1 | 3726.9173 | 875.7237 | 2851.1936 |

0.7 | 0.1 | 3222.8670 | 386.2204 | 2836.6467 |

0.8 | 0.25 | 2688.2682 | 1313.5856 | 1374.6826 |

0.9 | 0.25 | 2123.1209 | 629.6386 | 1493.4823 |

1 | 0.25 | 1527.4251 | 0 | 1527.4251 |

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Zhu, L.; Lee, C.
Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory. *Appl. Sci.* **2017**, *7*, 676.
https://doi.org/10.3390/app7070676

**AMA Style**

Zhu L, Lee C.
Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory. *Applied Sciences*. 2017; 7(7):676.
https://doi.org/10.3390/app7070676

**Chicago/Turabian Style**

Zhu, Lijing, and Chulung Lee.
2017. "Risk Analysis of a Two-Level Supply Chain Subject to Misplaced Inventory" *Applied Sciences* 7, no. 7: 676.
https://doi.org/10.3390/app7070676