# How Does Advanced Technology Solve Unreliability Under Supply Chain Management Using Game Policy?

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Problem Definition, Assumptions, and Notation

#### 3.1. Problem Definition

#### 3.2. Assumptions

- A model of a two-echelon SC for a single type of product is considered under the newsvendor framework. The market demand is variable and is dependent on the service b provided by the retailer.
- In traditional system, only the manufacturer bears the cost for manufacturing, and the total cost of inventory is carried by the retailer. In the consignment policy, the manufacturer provides a contract on a consignment to the retailer under a custom-made production system. The retailer sells the product and sends the money to the manufacturer. The retailer gets a fixed fee from the manufacturer along with a certain amount of commission for selling each product to a customer (Sarkar et al. [32]).
- The total holding cost of the consignment policy is not only spent by the manufacturer or retailer alone. The holding cost is divided into two different segments. One is the operational part, which is carried by the retailer, and the other one is the financial part, which is incurred by the manufacturer.
- The lead time L is not negligible, and the demand during the lead time is random in nature. Any kind of specific probability distribution is not contemplated in the proposed model. The lead time demand follows an unknown probability distribution function with a known value of the standard deviation $\delta $ and mean ($DL$) (Shin et al. [17]).
- As the retailer is not sharing proper information with the manufacturer, an information asymmetry is generated within the SCM. To solve this unreliability and to get the proper information, the manufacturer lets the retailer install the RFID technology (Guchhait et al. [22]). By the tracking facility of the RFID, the manufacturer becomes aware of all the information regarding the demand and the products.
- Along with the VMI, the consignment policy is used by the manufacturer. The retailer gets a commission for selling products and a fixed fee, that is a fixed amount of money for being a business partner of the manufacturer. A positive signed fixed fee denotes the process flow from the manufacturer to the retailer (Sarkar et al. [32]).
- For the asymmetric power of the supply chain players, game theory is used to find the decision maker. Using the Stackelberg game theoretic approach, the manufacturer acts as an SC leader, and the retailer acts as a follower. Thus, the manufacturer is the decision maker and gives the opportunity to the retailer to choose the optimal policy (Sarkar et al. [27]).

#### 3.3. Notation

## 4. Mathematical Model

#### 4.1. Traditional Policy

#### 4.1.1. Retailer’s Model

**Distribution free approach**

**Lemma**

**1.**

- (i)
- The expected overstock quantity is given by the following relation:$$\begin{array}{ccc}\hfill E{({Q}_{r}-a{b}^{\gamma}L)}^{+}& \le & \frac{1}{2}\left[\sqrt{{\delta}^{2}+{({Q}_{r}-a{b}^{\gamma}L)}^{2}}-(a{b}^{\gamma}L-{Q}_{r})\right].\hfill \end{array}$$The upper bound gives the maximum amount of the overstock quantity of the product.
- (ii)
- The expected quantity of shortage is given by the following expression:$$\begin{array}{ccc}\hfill E{(a{b}^{\gamma}L-{Q}_{r})}^{+}& \le & \frac{1}{2}\left[\sqrt{{\delta}^{2}+{({Q}_{r}-a{b}^{\gamma}L)}^{2}}-({Q}_{r}-a{b}^{\gamma}L)\right].\hfill \end{array}$$The upper bound gives the maximum quantity of the shortage amount.Using Equations (4) and (5), the expected profit of the retailer can be written as:$$\begin{array}{c}\hfill \begin{array}{cc}\hfill E\left({\mu}_{r}^{TS}\right)=& p(a{b}^{\gamma}+{Q}_{r})-C\left(L\right)-\omega {Q}_{r}-\frac{{h}_{r}^{TS}}{2}\left[\sqrt{{\delta}^{2}+{({Q}_{r}-a{b}^{\gamma}L)}^{2}}-(a{b}^{\gamma}L-{Q}_{r})\right]\hfill \\ & -\frac{{s}_{r}}{2}\left[\sqrt{{\delta}^{2}+{({Q}_{r}-a{b}^{\gamma}L)}^{2}}-({Q}_{r}-a{b}^{\gamma}L)\right].\hfill \end{array}\end{array}$$

#### 4.1.2. Manufacturer’s Model

#### 4.1.3. Joint Traditional Policy

#### 4.1.4. Solution Methodology

#### 4.2. RFID-Based Consignment Policy

#### 4.2.1. Retailer’s Model

#### 4.2.2. Manufacturer’s Model

#### 4.2.3. RFID Based Joint Consignment Policy

#### 4.2.4. Solution Methodology

## 5. Numerical Experiments

## 6. Sensitivity Analysis

## 7. Managerial Implications

- The chance for the first optimization of the decisions of the follower through the Stackelberg game policy is helpful for the long-term stability of the business between two players of the SCM. This is the strategic policy of the industry manager for a long-term profitable business. Table 3 and Table 4 provide the comparative scenario of the total profit.
- The RFID would help the industry manager reduce the effect of the unreliability of the retailers by tracking the products. The numerical study ensured that RFID implementation earned profit that was more than that of the traditional one. The service was increased in the consignment policy compared to the traditional policy. This implied that the consignment policy was beneficial for the industry.
- The consignment policy proved that the sharing policy of holding cost was effective for the industry. The total profit was increased. RFID provided more security to the industry along with more profit than the traditional policy of business with the Stackelberg policy. Thus, the combination established was more fruitful to the industry than the traditional policy.
- The fixed fee of the consignment policy was really helpful for the retailer. After paying the fixed fee, the joint total profit of the consignment policy was more than the traditional policy (Table 4). This proved that the industry could choose the consignment policy with the fixed fee strategy even if the retailer were unreliable. This earned more profit for the SCM.

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

RFID | Radio Frequency IDentification |

SCM | Supply Chain Management |

SC | Supply Chain |

## Appendix A

#### Notation

**Decision variables**

${Q}_{r}$ | retailer’s order quantity (units) |

b | service by the retailer |

L | lead time (weeks) |

**Parameters**

p | each product’s retail price ($/unit) |

${h}_{r}^{TS}$ | retailer’s holding cost for traditional policy ($/unit/unit time) |

${s}_{r}$ | retailer’s shortage cost ($/unit) |

${h}_{r}^{CP}$ | retailer’s holding cost for consignment policy ($/unit/unit time) |

${s}_{r}^{CP}$ | retailer’s shortage cost for consignment policy ($/unit) |

${h}_{m}^{TS}$ | manufacturer’s holding cost for traditional policy ($/unit/unit time) |

${s}_{m}$ | manufacturer’s shortage cost ($/unit) |

${h}_{m}^{CP}$ | manufacturer’s holding cost for consignment policy ($/unit/unit time) |

${s}_{m}^{CP}$ | manufacturer’s shortage cost for consignment policy ($/unit) |

k | manufacturing cost ($/unit) |

$\omega $ | wholesale price ($/unit) |

$\delta $ | standard deviation |

$\u03f5$ | cost of one passive RFID tag ($/unit) |

T | fixed cost of RFID implementation ($) |

a | scaling parameter |

$\gamma $ | shape parameter |

$C\left(L\right)$ | lead time crashing cost |

A | fixed cost for retailer given by the manufacturer ($) |

$\theta $ | commission for each item sold ($/unit) |

${c}_{i}$ | unit crashing cost ($/day) |

${g}_{i}$ | normal duration of lead time |

${f}_{i}$ | minimum duration of lead time |

**Other notation**

${\mu}_{r}^{TS}$ | profit of retailer under traditional system |

${\mu}_{r}^{CP}$ | profit of retailer under consignment policy |

${\mu}_{m}^{CP}$ | profit of manufacturer under consignment policy |

${\mu}_{m}^{TS}$ | profit of manufacturer under traditional system |

${\mu}_{j}^{CP}$ | joint profit under consignment policy |

${x}^{+}$ | maximum value of x and 0 |

$E(.)$ | mathematical expectation |

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Distribution | Consignment | Improvement | ||||
---|---|---|---|---|---|---|

Author(s) | SCM | Free | Unreliability | RFID | Policy | of |

Approach | Service | |||||

Dias et al. [28] | √ | √ | ||||

Gallego and Moon [11] | √ | √ | ||||

Guchhait et al. [22] | √ | √ | √ | √ | ||

Hefeeda and Ahmadi [29] | √ | √ | ||||

Kim and Glock [30] | √ | √ | ||||

Moon et al. [16] | √ | √ | ||||

Ouyang et al. [13] | √ | √ | ||||

Ru and Wang [14] | √ | |||||

Sarac et al. [31] | √ | √ | ||||

Sarkar et al. [32] | √ | √ | √ | |||

Scarf [10] | √ | √ | ||||

Shin et al. [17] | √ | √ | ||||

This research | √ | √ | √ | √ | √ | √ |

Lead time Component i | Normal Duration g_{i} | Minimum Duration f_{i} | Unit Crashing Cost c_{i} |
---|---|---|---|

1 | 20 days | 6 days | 0.4 |

2 | 20 days | 6 days | 1.2 |

3 | 16 days | 9 days | 5.0 |

${\mathit{Q}}_{\mathit{r}}^{*}$ | ${\mathit{L}}^{*}$ | ${\mathit{b}}^{*}$ | Expected Profit | |
---|---|---|---|---|

Joint | $38.698$ | 3 | $1.593$ | $1054.952$ |

${\mathit{Q}}_{\mathit{r}}^{*}$ | ${\mathit{L}}^{*}$ | ${\mathit{b}}^{*}$ | Expected Profit | |
---|---|---|---|---|

Joint | $43.529$ | 3 | $1.660$ | $1253.106$ |

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**MDPI and ACS Style**

Sardar, S.K.; Sarkar, B.
How Does Advanced Technology Solve Unreliability Under Supply Chain Management Using Game Policy? *Mathematics* **2020**, *8*, 1191.
https://doi.org/10.3390/math8071191

**AMA Style**

Sardar SK, Sarkar B.
How Does Advanced Technology Solve Unreliability Under Supply Chain Management Using Game Policy? *Mathematics*. 2020; 8(7):1191.
https://doi.org/10.3390/math8071191

**Chicago/Turabian Style**

Sardar, Suman Kalyan, and Biswajit Sarkar.
2020. "How Does Advanced Technology Solve Unreliability Under Supply Chain Management Using Game Policy?" *Mathematics* 8, no. 7: 1191.
https://doi.org/10.3390/math8071191