Retailer’ Order Problem When Regret Factor is Considered

丁若宸 戴更新

【Abstract】Traditional expected utility theory suggests that decision-makers are completely rational， but a large number of experiments and empirical studies have shown that decision-makers will be influenced by their perception of psychological and other factors， is bounded rationality when making decisions. This paper will be based on regret theory， considering the influence of retailerregret factor on their order quantity in a single cycle，and providing corresponding measures which supplier can take to weaken the influence of retail regret.

【Key words】Surplus regret; Stockout regret; Price subsidy; Optimal order quantity

0 Introduction

With further research， it is found that the individual is only assumed to be rational economic man has some limitations，individual decisions arent only including rational logical reasoning， but also combining individual subjective preferences， values， emotions and other irrational cognitive processes. In recent years， the impact of behavioral factors on the decisions is paid more and more attention.

Regret factor considered in uncertainty is first proposed by Savage. Many researchers find that in many cases peoples patterns of choice violate the traditional expected utility theory，which indicate that there may be some important factors which influence peoples choices have been overlooked. Loomes and Sugden（1982） offer an alternative model which takes feelings of regret and rejoicing into consideration in order to better explain individual behavior under uncertainty. Bell（1982） also makes a number of related studies， he puts regret factor into the utility function and explains some well-known abnormal behaviors， such as Allais paradox. Show that some of the contradictory behaviors of decision-makers are to avoid afterwards regret. After， Zeelenberg （1999） further points out the importance of anticipated regret in decision making under uncertain environment. He defines regret as a negative， cognitively based emotion that we experience when realizing or imagining that our present situation would have been better， had we decided differently.

Regret theory has been used to explain the behavior of consumers in market， such as with respect to customized products， customers who are regret averse prefer standardized products （Syam et al.， 2008）. Cooke （2001） studies whether consumers feel regret in purchase-timing decisions， and the relationship between regret and satisfaction. People will forecast the feel that decisions result in， and use their predictions to guide their choices （Mellers et al.， 1999）. Recently， Diecidue et al.（2012） illustrate how regret affect the purchase decision in a dynamic purchase context when consumer is uncertain about the products valuation and point out that a consumer is more likely to buy forward when more averse to hesitaters regret but more likely to delay the decision when more averse to buyers regret.

Obviously， there are abundant evidences show that regret psychology will impact individual choice under uncertainty. The researchers who are engaged in behavioral operation have took it into account in many cases. Previous studies which are more concentrate on consumer regret and its influence for firm. We intend to apply regret theory into a two-stage supply chain system which is made of a single supplier and a single retailer， planning to explore the effect of regret on retailer optimal order quantity， meanwhile providing some measures which supplier can take to respond to retailer regret.

1 Description of the problem and the basic model

Assume that retailer only have one chance to order， the market demand x is a non-negative， continuous random variables， its probability density function is f（x）， cumulative distribution function is F（x）.

Other assumptions of model are as follow：

p：retailers unit price;

w：retailers unit wholesale price;

q：retailer order quantity;

v：the salvage value of unsold unit commodity;

s：retailer unit loss because of stockout;

α：retailer surplus regret coefficient;

β：retailer stockout regret coefficient;

When x

π（q）=px-wq+v（q-x）-α（w-v）（q-x），x

The retailers goal is Maxπ（q）， his expected profit is

E（π（q））=q（v-w）（1+α）f（x）dx+（p-v+αw-αv）xf（x）dx

+q（p-w+s）（1+β）f（x）dx-（s+βp-βw+βs）xf（x）dx  （1）

=（v-w）（1+α）F（q）+（1+β）（p-w+s）（q）              （2）

=（v+vα-wα-p-s）f（q）-β（p-w+s）f（q）                 （3）

Because， p>w>v， α， β>0， we know （3） is less than zero， now retailer optimal order quantity can be calculated by （2）：

q*=F-1                               （4）

We mainly want to know the optimal quantity will be how to change when retailers regret coefficient is different.

The optimal order quantity should be satisfied ：

（v-w）（1+α）F（q*）+（1+β）（p-w+s）（q*）=0                         （5）

Implicit differentiation on （5）：

=                         （6）

Because <0 ， we can conclude：

Proposition 1 Retailer optimal order quantity is decreasing in α.

Implicit differentiation on （5）， we can also have：

=                         （7）

Because >0， we can conclude：

Proposition 2 Retailer optimal order quantity is increasing in β.

We are more concerned is that when retailers order quantity is too much or too little due to regret factor， what measures can be taken for supplier to weaken the influence of regret.

3 The policy of supplier

3.1 Supplier offer price subsidies

We assume that at the end of sales， retailer still has some unsold commodities， supplier subsidy for each unit is m， and w>v+m， so the retailers profit function is：

π1（q）=px-wq+（v+m）（q-x）-α（w-v-m）（q-x），x

The expected profit is：

E（π1（q））=p-v-m+α（w-v-m）xf（x）dx+q（1+α）（v+m-w）xf（x）dx

+q（p-w+s）（1+β）f（x）dx-s+β（p-w+s）xf（x）dx

Because w>v+m， similarly， at this time the optimal order quantity can be obtained as follow：

v+m-w-α（w-v-m）F（q*）+（p-w+s）（1+β）（q*）=0

q*=F-1

Obviously， q*1>q*， retailers optimal order quantity is larger. So， we can conclude：

Proposition 3 Supplier price subsidies will reduce the influence of retailer's surplus regret on order quantity. That is， when faced with surplus regret retailer， supplier should take price subsidies.

3.2 Consider suppliers punishment

We assume that at the end of sales， retailer still has some products are unsold， supplier will take punitive measure. Punishment amount for per unit of commodity is n， so now the retailers profit function is：

π2（q）=px-wq+（v-n）（q-x）-α（w+n-v）（q-x），x

The expected profit is：

E（π2（q））=p-v+n+α（w+n-v）xf（x）dx+q（1+α）（v-n-w）f（x）dx

+q（p-w+s）（1+β）f（x）dx-s+β（p-w+s）xf（x）dx

=（1+α）（v-n-w）f（x）dx+（1+β）（p-w+s）f（x）dx，    （10）

=（1+α）（v-n-w）f（q）-（1+β）（p-w+s）f（q），           （11）

Obviously， the formula （11） is less than zero， retailer optimal order quantity we can get by （10），

q*=F-1

Similarly，q*2

Proposition 4 Considering the circumstances that supplier will take punitive measures， retailer order quantity becomes smaller. To some extent， weaken the influence of stockout regret， that is to say， in the face of stockout regret retailer， supplier should take punitive measures to reduce the impact of stockout regret on order quantity.

4 Numerrical analysis

Suppose the market demand is uniformly distributed x～U（0，1000）， retailers sales price p=10， wholesale prices is 4， s=15， v=0.5， β=0.5， m=0.8

We can see， retailer optimal order quantity decreased with the increase of surplus regret coefficient， but supplier price subsidies can effectively weaken this reduction， in other words， the impact of retailer surplus regret on order quantity is diminished.

On the other hand， when supplier take punitive measure in respond to the excessive products， suppose the market demand is x～U（0，1000）， retailers sales price p=10， wholesale prices is 4， s=15， v=0.5， α=0.5， n=3

圖2

The figure shows that when supplier take punitive measure， retailer optimal order quantity will be reduced when the stockout regret coefficient is identical， which means that supplier punitive measure effectively reduce the increase of retailer order quantity which is caused by stockout regret.

5 Conclusion

In this paper， regret theory is used to show the effect of retailer regret factor on optimal order quantity in single-period， the study finds that retailer optimal order quantity is decreasing with surplus regret， increasing with stockout regret. Meanwhile， we propose corresponding measures which supplier can take to reduce the influence of regret factor on optimal order quantity. Further research： how will the retailer regret factor affect their order quantity in multi-cycle， and what measures supplier should take to respond to multiple retailer whose regret degree are different， and so on.

【References】

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[3]Diecidue， E.， N. Rudi， W. Tang. Dynamic purchase decisions under regret：Price and availability[J]. Decision Anal.， 2012， 9（1）： 22-30.

[4]Loomes， G.， R. Sugden. Regret theory： An alternative theory of rational choice under uncertainty[J]. Econom. J.， 1982， 92（368）： 805-824.

[5]Mellers， Alan Schwartz， and Ilana Ritov. Emotion-Based Choice[J]. Journal of Experimental Psychology： General， 1999， 9， 128： 332-345.

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