We introduce loss aversion into the decision framework of the newsvendor model. By introducing the loss aversion coefficient

In recent years, the study about the newsvendor model has attracted the attentions of many researchers and it has been applied to many fields, such as production plan and yield management [

In the real world, it is known that some decision makers are more averse to the losses coming from the decisions than being attracted to the same sized profits from the decisions, which is referred to as “loss aversion” in the decision theory. In fact, loss aversion is both intuitively attractive and well supported in various fields (e.g., finance and economics). For example, the empirical studies by Shapira [

In light of the above successful applications of loss aversion, this paper then introduces loss aversion into the decision framework of the newsvendor model. First, we present the gain obtained in the selling season and the loss comes from the excess order when the selling season is due for the newsvendor separately and then introduce a novel utility function for the loss-averse newsvendor by introducing the loss aversion coefficient

The rest of this paper is organized as follows. In the following section, we give a detailed description on the loss-averse newsvendor model and present some preliminaries about VaR and CVaR. Section

In this section, we will give a detailed description on the loss-averse newsvendor model studied in this paper and present some basic knowledge about the CVaR measure in financial risk management.

For the newsvendor model, suppose that the market demand

In the following, we first obtain the optimal order quantity for the loss-averse newsvendor to maximize the expected utility. However, in recent years, some unpredictable disasters (e.g., earthquakes and economic crisis) disrupt the supply chain operations repeatedly, and this makes the retailers in reality become more sensitive to the market demand and more averse to the risk originating from the fluctuation in the market demand. Then, many researchers paid attention to the risk analysis and risk control in the newsvendor model by introducing various risk measures [

Before Conditional Value-at-Risk (CVaR) measure was introduced, Value-at-Risk (VaR) measure is widely used in the risk management in finance. Here, for a decision

In this section, we will introduce different objectives about the utility function

For the loss-averse newsvendor problem, since the realized market demand

For the loss-averse newsvendor model, the optimal order quantity for a risk-neutral newsvendor to maximize the expected utility

For a given order quantity

By Theorem

By Theorem

For the loss-averse newsvendor model, the optimal order quantity

It is not surprising to see that this result holds, and this result also holds in the classical newsvendor model when the newsvendor selects an optimal order quantity to maximize his/her expected profit. Here, it is pointed out that, for any fixed

For the loss-averse newsvendor model, the optimal order quantity

By this result, if the loss-averse newsvendor becomes more loss-averse to the loss from the excess order, he/she will order less products. Particularly, let

In this subsection, for the loss-averse newsvendor model, we obtain the optimal order quantity decision for a loss-averse newsvendor to maximize his/her expected utility

For the order quantity

For the loss-averse newsvendor model, the optimal order quantity for a loss-averse newsvendor to maximize his/her CVaR about

See the Appendix.

Here, it is easily checked that if it satisfies

For the loss-averse newsvendor model, the optimal order quantity

In the loss-averse newsvendor model, by Corollary

For the loss-averse newsvendor model, the optimal order quantity

This result shows that, to reduce the risk coming from the fluctuation in the market demand, the loss-averse newsvendor will order less products if he/she becomes more loss-averse.

For the loss-averse newsvendor model, the optimal order quantity

The confidence level

As mentioned above, if the confidence level

For the loss-averse newsvendor model, the expected utility

By (

By this result, in the loss-averse newsvendor model, if the risk-averse newsvendor decreases his/her order quantity to reduce the risk coming from the fluctuation in the market demand, he/she will expect a lower utility. This verifies that high return follows high risk, while low risk means low return.

In this section, we will give two examples to show the results obtained in Section

For the loss-averse newsvendor model, suppose the market demand

First, let

Optimal order quantities

Optimal order quantities

Optimal order quantities

By Figures

Further, let

Optimal order quantities

Then, let

Optimal order quantities

Finally, let

Expected utilities

For the loss-averse newsvendor model, suppose the market demand

First, let

Optimal order quantities

Optimal order quantities

Optimal order quantities

By Figures

Further, let

Optimal order quantities

Then, let

Optimal order quantities

To summarize this section, the numerical results and sensitivity analysis confirm that the results obtained in Section

In the newsvendor model, some newsvendors are more averse to the losses (comeing from the excess order or lost sales) when the selling season is due than they are attracted to the same sized gains obtained in the selling season, which can be seen as the loss aversion in the newsvendor model. However, the study about the influence of loss aversion on the optimal order quantity decisions of the newsvendor model is very few. Then, this paper contributes to the study about the optimal order quantity decisions of such a loss-averse newsvendor model. By introducing the loss aversion coefficient

Some extensions of this research are possible. For example, in this paper, the shortage penalty for the lost sales is not considered in defining the utility function for the loss-averse newsvendor. However, the loss from the lost sales that ranges from profit loss on the scale to some unspecific loss of goodwill of the customers has an important influence on the utility of the newsvendor, and then a possible extension is to integrate the shortage penalty for the lost sales into the definition of the utility function for the loss-averse newsvendor and then consider the optimal order quantities of the loss-averse newsvendor with different objectives about such a utility.

For a realized market demand

Then, by the result in Section

Obviously, it satisfies

Then, if it satisfies

Then, the optimal solution

Based on the analysis above, it is clear that, for any fixed

Then, to solve problem

(i) Consider

In this case, it follows from (

Then by (

(ii) Consider

In this case, it follows from (

Then by (

Then it follows from (

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the editor and the anonymous referee for their valuable suggestions and comments, which help us to improve this paper greatly. This research is supported by the Natural Science Foundation of Shandong Province with Grant ZR2014GQ005.