# Pricing to the Scenario: Price-Setting Newsvendor Models for Innovative Products

## Abstract

**:**

## 1. Introduction

## 2. Newsvendor Models with the OSDT

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Active focus point**: For an order quantity $q$, the active focus point is

**Example**

**1.**

**Passive focus point**: For an order quantity $q$, the passive focus point is

**Apprehensive focus point**: For an order quantity $q$, the apprehensive focus point is

**Daring focus point**: For an order quantity $q$, the daring focus point is

**Comments**: Equations (6)–(9) are from four bi-objective optimization problems as follows: $\underset{x}{\mathrm{max}}\pi (x),\underset{x}{\mathrm{max}}u(x,q)$; $\underset{x}{\mathrm{max}}\pi (x),\underset{x}{\mathrm{min}}u(x,q)$; $\underset{x}{\mathrm{min}}\pi (x),\underset{x}{\mathrm{min}}u(x,q)$ and $\underset{x}{\mathrm{min}}\pi (x),\underset{x}{\mathrm{max}}u(x,q)$. From Equations (6) to (9), there is no other $[\pi (x),\text{}u(x,q)]$ satisfies $\pi (x)>\pi ({x}_{1}^{*}(q))$ and $u(x,q)>u({x}_{1}^{*}(q),q)$; or $\pi (x)>\pi ({x}_{2}^{*}(q))$ and $u(x,q)<u({x}_{2}^{*}(q),q)$; or $\pi (x)<\pi ({x}_{3}^{*}(q))$ and $u(x,q)<u({x}_{3}^{*}(q),q)$; or $\pi (x)<\pi ({x}_{4}^{*}(q))$ and $u(x,q)>u({x}_{4}^{*}(q),q)$. It means that ${x}_{1}^{*}(q)$, ${x}_{2}^{*}(q)$, ${x}_{3}^{*}(q)$ and ${x}_{4}^{*}(q)$ are Pareto optimal solutions of the above four bi-objective optimization problems which are used to seek for the demands with the higher likelihood and satisfaction, the higher likelihood and lower satisfaction, the lower likelihood and satisfaction and the lower likelihood and higher satisfaction, respectively. In other words, for any $q$ no demand can cause an even higher satisfaction with an even higher likelihood than its active focus point ${x}_{1}^{*}(q)$; no demand can provide an even lower satisfaction with an even higher likelihood than its passive focus point ${x}_{2}^{*}(q)$; no demand can lead to an even lower satisfaction with an even lower likelihood than its apprehensive focus point ${x}_{3}^{*}(q)$; no demand can generate an even higher satisfaction with an even lower likelihood degree than its daring focus point ${x}_{4}^{*}(q)$.

**Advantages in phenomena explanation:**Let us consider the following anecdotal evidence. In September 2014, Apple

^{®}released iPhone 6 and iPhone 6 Plus, but the Chinese market was left out the first wave of countries. The iPhone 6 was sold for as much as 10 times the U.S. price in Chinese black market, due to the delayed release. There were many scalpers trying to buy and resell the iPhone 6 in this risky and fragile market [30]. Grothaus [32] observed that some of the scalpers treat it as a “gamble” and just took into account the scenario that they can make profits and “feed their family”. This kind of phenomena in an innovative product market can be explained by the behavior of a daring retailer. Even though some scenario may occur with a low likelihood, the high gain lures him/her to take action. On the other hand, this kind of phenomena is very hard to be explained by lottery-based models, including expected utility models, value at risk models or conditional value at risk models. The reason is that the expression of risk preferences in these models rely on the framework of weighting average, which ignored the importance of some unique and irreplaceable scenario (focus point) in the progress of decision-making.

**Example**

**2.**

## 3. Price-Setting Newsvendor Models Based on OSDT

**Active focus point**: For retail price $R$ and production quantity $q$, the active focus point is

**Passive focus point**: For retail price $R$ and production quantity $q$, the passive focus point is

**Apprehensive focus point**: For retail price $R$ and production quantity $q$, the apprehensive focus point is

**Daring focus point**: For retail price $R$ and production quantity $q$, the daring focus point is

## 4. Analysis Results for the OSDT Based Price-Setting Newsvendor Models

**Assumption:**The probability density function $\mathrm{f}(\mathrm{b})$ is a unimodal function defined on the interval $[{\mathrm{b}}_{\mathrm{l}},{\mathrm{b}}_{\mathrm{u}}]$, the mode is ${\mathrm{b}}_{\mathrm{c}}\in ({\mathrm{b}}_{\mathrm{l}},{\mathrm{b}}_{\mathrm{u}})$, $\mathrm{f}({\mathrm{b}}_{\mathrm{l}})=0$ and $\mathrm{f}({\mathrm{b}}_{\mathrm{u}})=0$.

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proposition**

**7.**

^{®}[34] that the luxury manufacturers that build brands on the image and lifestyle are able to withstand bigger competitive pricing differences than manufacturers who build their brands on the price. It future explained that “a well-known luxury manufacturer incorporated the price sensitivity metrics into its overall pricing and assortment strategy in recent years. The strategy has helped boost the company’s profit margins to its highest level.”

## 5. Numerical Example

## 6. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Proposition**

**1**

**Proof**

**of**

**Proposition**

**2**

**Proof**

**of**

**Proposition**

**6**

**Proof**

**of**

**Proposition**

**7**

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satisfaction | |||

higher | lower | ||

likelihood | higher | active focus point | passive focus point |

lower | daring focus point | Apprehensive focus point |

Demands | 350 | 450 | 550 | 650 | 750 |
---|---|---|---|---|---|

likelihood degrees | 0.22 | 0.35 | 1.00 | 0.73 | 0.29 |

Demands | ||||||

350 | 450 | 550 | 650 | 750 | ||

Orders | 350 | 1050 | 650 | 250 | −150 | −550 |

450 | 450 | 1350 | 950 | 550 | 150 | |

550 | −150 | 750 | 1650 | 1250 | 850 | |

650 | −170 | 150 | 1050 | 1950 | 1550 | |

750 | −1350 | −450 | 450 | 1350 | 2250 |

Demands | ||||||

350 | 450 | 550 | 650 | 750 | ||

Orders | 350 | 0.67 | 0.56 | 0.44 | 0.33 | 0.22 |

450 | 0.50 | 0.75 | 0.64 | 0.53 | 0.42 | |

550 | 0.33 | 0.58 | 0.83 | 0.72 | 0.61 | |

650 | 0.17 | 0.42 | 0.67 | 0.92 | 0.81 | |

750 | 0.00 | 0.25 | 0.50 | 0.75 | 1.00 |

Order Quantities | |||||
---|---|---|---|---|---|

350 | 450 | 550 | 650 | 750 | |

Active | 550 | 550 | 550 | 650 | 650 |

Passive | 650 | 650 | 450 | 450 | 550 |

Apprehensive | 750 | 750 | 350 | 350 | 350 |

Daring | 350 | 450 | 750 | 750 | 750 |

Order Quantities | |||||
---|---|---|---|---|---|

350 | 450 | 550 | 650 | 750 | |

Active | 0.44 | 0.64 | 0.83 | 0.92 | 0.75 |

Passive | 0.33 | 0.53 | 0.58 | 0.42 | 0.50 |

Apprehensive | 0.22 | 0.42 | 0.33 | 0.17 | 0.00 |

Daring | 0.67 | 0.75 | 0.61 | 0.81 | 1.00 |

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**MDPI and ACS Style**

Ma, X.
Pricing to the Scenario: Price-Setting Newsvendor Models for Innovative Products. *Mathematics* **2019**, *7*, 814.
https://doi.org/10.3390/math7090814

**AMA Style**

Ma X.
Pricing to the Scenario: Price-Setting Newsvendor Models for Innovative Products. *Mathematics*. 2019; 7(9):814.
https://doi.org/10.3390/math7090814

**Chicago/Turabian Style**

Ma, Xiuyan.
2019. "Pricing to the Scenario: Price-Setting Newsvendor Models for Innovative Products" *Mathematics* 7, no. 9: 814.
https://doi.org/10.3390/math7090814