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This article will introduce the minimum requirement and the statue quo reference points into newsvendor model. Then the problem of decision bias can be well explained through researching multiple reference points. Many papers confirm the loss aversion is one of the important factors in the decision-making bias and plays an important role in ordering decisions. We propose crossing failure is more important than the loss aversion depending on the decision maker’s degree of the aversion. In this note, we present situation where minimum requirement and status quo reference point impact the order decision and give a breakthrough in the study of multiple reference points of newsvendor problem.

The traditional newsvendor model is a basic model to analysis inventory problems [

Although we can get the information about optimal order quantity, the real order decision is inconsistent with optimal order quantity under rational conditions. Eeckhoudt et al. (1995) [

Under expected utility theory [

Our main aim is to introduce multi-reference point newsvendor model that analyze order decision bias. The newsvendor model based on MR and SQ reference points extends the conception of loss aversion which further elaborates how crossing failure and loss aversion cause the decision bias in newsvendor problem of multi-reference point. Through a series of analysis, we summarize: 1) Considering MR and SQ reference points, the newsvendor will order less than the risk-neutral newsvendor; 2) If the parameter of failure is big enough, the loss of a sense of failure is much more intense than the loss of a sense of loss; 3) As with traditional concepts, the newsvendor’s optimal order quantity will increase in retail price and decrease in wholesale price; 4) The quantity decreased while there exist MR reference point. And with MR increasing, the newsvendor will decrease optimal order quantity; 5) The bigger the value of the aversion parameter is, the greater the newsvendor decrease.

The contributions of our research are showed from three aspects. First, on the basis of experiment and theory, we introduce multi-reference point theory of behavioral science into newsvendor model so that we can further quantify the problem of order decision bias. Second, we set up newsvendor model based on MR and SQ reference points. And by discussing about the influence of retail price, wholesale price, our research shows why crossing failure is more important than loss aversion. Then we expand our perspective to the supply chain inefficiency and uncoordinated and put forward some new policies responding to this effect.

The following part is organized like this: In Section 2, we briefly review the related newsvendor and reference point literature. In Section 3, we analyze the basic model. In Section 4, we analyze the corresponding parameters. In Section 5, we draw our conclusions and provide some suggestions for future research.

Our decision-making behavior is affected by the reference point. With two experiments, Rosch (1975) [

Research, under conditions of risk and uncertainty in decision-making, has changed a lot. in 1944 Vonnuemna and Mogrnestern [

Kahneman and Tversky (1979) and other papers all emphasize the importance of status quo reference point and ignore other reference points. Yate and Stone (1992) [

However, the theoretical analysis of double reference points is seldom researched. while the research based on multi-reference point is researched in some economic and management field, like Sullivan and Kida (1995) [

Under uncertainty, a newsvendor makes decision through selling short-life-cycle products. For the newsvendor problem, we assume that the acquisition cost from a supplier is w per unit and retail price is p per unit (p > w). We also assume that there is no shortage cost and salvage value. Demand x, a random variable at the time of purchasing from the supplier, is characterized by a probability density function f(x) and a cummunicative distribution function F(x). Then we can get the profit function. We can get the profit function in Equation (1).

The plot of them are shown in

Where

Lemma 1. For any

quantities of realized demand in view of MR and SQ reference points,

Lemma 1 explain the influence of mechanism of MR and SQ reference point. If realized demand x is too low, i.e.,

w. When the ordering cost w is high i.e.,

profit achieving MR. Under the MR reference point, the newsvendor will feel failure. Above the SQ reference point, the newsvendor will obtain more benefits. We normalize this to 1.

In ordering decision-making research literature, loss-aversion and failure-avoiding and target-pursuiting are widely researched. When decision makers make decisions, he is not risk neutral. In Wang and Webster (2009) [

Basing on the MR and the SQ reference points, we consider a simple piecewise linear form of value function. The newsvendor has the following function and plot (

We define

loss-aversion effect under the status quo reference point and an failure-aversion effect under the minimum requirement reference point. And the degree of failure aversion is greater than the degree of loss aversion depending on the value of the aversion parameters

For the newsvendor’s profit function and value function, we can express his expected utility function as

The utility function of newsvendor model based on the MR and the SQ reference points above can be rewritten as

The utility function is the expected profit plus the total expected losses and expected failure losses, biased by the factor of

Theorem 1. For all Q in the definition of the range,

The proof of Theorem is in the Appendix. When we set

duces to

loss. For the part of MR reference point, we also define

overage failure. It shows that marginal overage loss/failure of a certain order quantity respect to the risk-neutral optimal order quantity. The following proposition will elaborate how to influnce decision bias.

Theorem 2. For any aversion parameters

Theorem 2 illustrates that the newsvendor will generate negative bias which usually show ordering too much, Comparing to the risk-neutral newsvendor. From the theorem 2, we can know that the subject certainly decrease the optimal order quantity with respect to the risk-neutral newsvendor.

Corollary 1. For any

which make

As we know, the existence of the MR and SQ reference point will reduce the optimal quantity. But we are not sure the influence degree of MR reference point and SQ reference point. The corollary above expounds the existence of the situation in which the loss of MR part is bigger than the loss of SQ part. In Wang and Webster (2009) [

For the demand distribution, we choose uniform distribution to verify the correctness of the mathematical results. Specific results can be shown in

subjected to the uniform distribution of 0 to 200.

points can well illustrate decision bias. Both of

(8.33, −0.38). When

According to the formula in Corollary 1, when the decision maker very afraid failure,

the value of

smaller than the value of

In our model, we use mathematical analysis to investigate other parameters’s influence on the newsvendor’s optimal order quantity. Then, we apply uniform distribution to test our analysis. Combining with mathematical analysis and numerical analysis to illustrate our view.

Theorem 3. For parameter changes, optimal order quantity

1)

2)

We also get some conclusions about ordering cost w and retail price p. MR as a exogenous reference point is discussed in the following part. For the risk-neutral newsvendor model, the single SQ reference point newsvendor model and the newsvendor model based on MR and SQ reference points, optimal order quantity is increasing in retail price p while decreasing in ordering cost w. Theorem (3) (see Appendix for proof) identify optimal order quantity’s changing direction. Adding in MR and SQ reference points makes the discussion about ordering quantity more complex.

In order to better understand the impact of parameters, we use some numerical examples to illustrate the decision bias. Generally speaking, uniform is widely used in various extended newsvendor models. For uniform distribution, we assume mean demand D/2 = 100. For loss aversion parameter and failure aversion parameter, we choose different

in Therom 1. Raising retail price p will drive up the optimal quantity on SQ and MR reference points. Thus they can make up for the loss of the psychological influence. Likewise,

In our model, the optimal quantity

Comparing to the traditional newsvendor model, which shows optimal order quantity always decreasing in w and increasing in p, the newsvendor model based on MR and SQ reference points gives more conditions. Recent articles confirm the impact of changes in the MR reference point on the optimal order quantity.

Part 3 shows, based on these results, we can understand the MR as an important reference point in explaining the importance of the decision on the bias, to further expand the research of multi-reference points. In random demand of purchasing decision, the decision makers of actual order behavior will be influenced by reference points. This article based on the minimum requirement and status quo reference points build the decision-making behavior of newsvendor model. It explains the mechanism of newsvendor decision-making deviation from the perspective of multiple reference points. Through the analysis of the mathematical model and numerical examples, it verifies the deviation and changing direction.

From the theorem and numerical examples of part 4, we explain, in the current situation based on MR and SQ reference points, how parameters influence the optimal quantity. And we further explain how the minimum requirement and the status quo affect the order decision simultaneously. As previously described to, Agrawal and Seshadri (2000) [

In all, this paper mainly discusses the influence of MR and SQ reference point of the newsvendor’s order decision under stochastic demand. We combine the loss aversion, crossing failure, and use overage failure cost and overage loss cost to analyze the decision bias in the newsvendor problem. Our conclusions show that, adding MR and SQ reference point must decrease the optimal order quantity, which is that the newsvendor order less items. And with the MR reference point increasing, the optimal order quantity will decrease. This situation may illustrate that, when the MR reference point is infinitesimal, the newsvendor do not change his order decision due to feeling nothing about MR reference point. When the MR reference point reach the degree of newsvendor’s feel, a little quantity may decrease. Then with the MR reference point continue increasing, the optimal order quantity will continue decreasing. Finally, when the MR reference point reaches the maximum point, the newsvendor will order nothing. No matter how much the newsvendor orders, he feels negative profit. So ordering is not wise. With wholesale price and retail price increasing, the optimal order quantity will be more or less in some conditions, which is the same as some papers. In another case, the conclusion is opposite.

We also conclude some future research directions about this paper. Setting endogenous reference point will be interesting to see the newsvendor’s psychological perception. Goal reference point can also be added into this model to analyze decision bias. Newsvendor’s decision is no longer a single cycle, but multi-cycle. Around the discussion of the reference point, we can better learn the true impact of decision-making. Psychology experiments and empirical articles have confirmed the existence of the reference points (see [

Zhou, S. (2016) The Newsvendor Problem Based on Multi- ple Reference Points. Open Journal of Social Sciences, 4, 227-240. http://dx.doi.org/10.4236/jss.2016.49019

Proof of Theorem 1.

Since Q is continuous and derivative, we take the first derivative of Function (6) and get:

After taking second derivative of

For all

exists a unique optimal

Since Q is continuous and derivative, we take the first derivative of Function (7) and get:

After taking second derivative of

For all

that satisfies the first-order condition.

Proof of Theorem 3.

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