# Supply Chain Contract Choice with Different Behavior Preferences of Retailer

Authors: Yonglong Wang; Xinyu Zheng; Jirong Cai; Yuelong Zheng
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

Retailer may exhibit irrationality when facing the risk of demand uncertainty; therefore, we consider four retailer behavioral preferences: risk neutral (RN), waste aversion (WA), stockout aversion (SA), and stockout-waste aversion (SW). The decision-making and contract choice of upstream and downstream enterprises in cases where demand depends on supplier’s effort are studied. The results show that if the retailer has only SA or RN preferences, then the supplier prefers to choose a wholesale price contract, while the retailer does the opposite, if the retailer has only WA, then the supplier prefers to choose a revenue sharing contract, but the retailer’s contract choice depends on the degree of waste aversion, and if the retailer has SW, then the contract choice of upstream and downstream enterprises is related to the degree of waste aversion and stockout aversion.


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## Body

## 1. Introduction

Because different contracts may bring different benefits to supply chain participants, different participants choose different contracts. The concept of supply chain contract choice comes from the term “prefer” proposed by Cachon and Lariviere [1], who noted that the supplier prefers contracts that is more profitable to him based on different circumstances. However, the research on contract choice is still very limited, and most of the existing literature focuses on the choice between wholesale price contract and revenue-sharing contract, which is also the content of this paper. The reason for choosing the two contracts in this paper is that the two contracts are simple and the two contracts are widely used in real life.To weigh the advantages and disadvantages of the above two contracts, Cachon and Lariviere [1] analyze the contract selection of upstream enterprises when product quantity and retailer effort influence product price. In addition, Pan et al. [2] study the contract choice problem under different dominant agents using a system constructed by two upstream enterprises and a single downstream enterprise or a single upstream enterprise and two downstream enterprises. Xu [3] discusses the contract choice under a three-level tourism supply chain dominated by a travel agency. Jin et al. [4] study the contract choice under different agents who make decisions about effort when demand is affected by retail price and effort and the retailer has financial constraints. Chakraborty et al. [5] consider the situation in which the retailer sells two alternative products at the same time and where the market demand depends on the retail price of the two products, and they study the contract choice problem under different power structures. Lu et al. [6] analyze the influence of advertising effort efficiency and channel member patience on contract choice in the context of downstream enterprises leading and advertising having a dynamic influence. Meng and Huang [7] discuss the contract choice of upstream and downstream enterprises in the case of two upstream logistics service providers.However, the above studies take the participants’ perfect rationality as the premise of their research and do not consider the possibility that the participants may have limited rationality [8]. In fact, there is risk in demand [9], and the behavior of decision makers may be affected by demand [10]. In particular, Schweitzer and Cachon [11] point out that retailers may have a waste aversion preference because they do not like carrying excess inventory, or they may have a stockout aversion preference because they dislike shortages.Therefore, the two behavior preferences are introduced into the supply chain, especially considering that random demand is affected by the supplier’s effort (e.g., advertising, quality, services to enhance the physical store, and after-sales service [12–15]). This paper studies the decision-making and contract choice of upstream and downstream enterprises in the supply chain under different behavior preferences of the retailer.


## 2. Model Assumptions

The model is based on the following assumptions:(1)
The risk-neutral supplier is the leader.(2)
Four retailer behavioral preferences are considered: risk neutral (RN), waste aversion (WA), stockout aversion (SA), and stockout-waste aversion (SW).(3)
Market demand isD=eξ [12], where e is the effort input level of the supplier and ξ is the random disturbance factor. Because of the complexity of the model structure, the model cannot be solved; to simplify the model and obtain more valuable conclusions, the methods of Taylor [12] and Yu and Wan [16] are used, and ξ is assumed to follow the distribution of U0,B.(4)
p>w>c>v.The symbols of this article are summarized in Table1.Table 1
Symbols and meanings.
wWholesale pricepRetail priceeEffortβWaste aversion coefficientRNRisk neutralSAStockout averse preferenceWPWholesale price contractUR−ijUnderj contract, the expected utility obtained by the retailer him or herself when he or she has i preferenceπR−ijUnderjcontract, the expected profit obtained by the retailer him or herself when he or she has i preferencecProduction costsqOrder quantityvSalvage valueλStockout aversion coefficientWAWaste averse preferenceSWStockout-waste averse preferenceRSRevenue-sharing contractπS−ijUnderj contract, the expected profit of the supplier when the retailer has i preference j∈WP,RS and i∈RN,WA,SA,SW


## 3. Model Construction

### 3.1. RS Model

Under the RS model, it is assumed that the retailer gives1−ϕ times its own sales revenue to the supplier in exchange for a lower wholesale price w=ϕc [1], where 0<ϕ≤1 and ϕp>v. When the random demand depends on the effort of the supplier, the retailer’s expected sales volume Sq,e, inventory volume Lq,e, and shortage volume Nq,e are(1)Sq,e=Eminq,eξ,Lq,e=Eq−eξ+=q−Sq,e,Nq,e=Eeξ−q+=Eeξ−Sq,e.The expected profits of upstream and downstream enterprises are(2)πS−jRSϕ,e=1−ϕpSq,e−1−ϕcq−ke2,πR−jRSq=ϕpSq,e+vLq,e−ϕcq,where j∈RN,WA,SA,SW.


#### 3.1.1. SW Model

If the retailer has an SW preference, then the expected utility of the retailer is as calculated below:(3)UR−SWRSq=ϕpSq,e+vLq,e−λNq,e−βLq,e−ϕcq.From equation (3), we obtain(4)∂UR−SWRS∂q=qv−β−λ−pϕ+Beλ+ϕp−cBe,∂2UR−SWRS∂q2=pϕ−v+λ+β−Be<0.From equation (3), we can see that UR−SWRS is a concave function of q. Therefore, if ∂UR−SWRS/∂q=0, the unique optimal qSWRS∗ can be obtained:(5)qSWRS∗=Beλ+ϕp−cϕp+λ+β−v.By substitutingqSWRS∗ into equation (2), then(6)maxϕ,eπS−SWRS=Bep1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bec1−ϕλ+ϕp−cϕp+λ+β−v−ke2.From equation (6), we obtain(7)∂πS−SWRS∂e=Bp1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bc1−ϕλ+ϕp−cϕp+λ+β−v−2ke,∂πS−SWRS∂ϕ=Be2cp−cϕ−1ϕp+λ+β−v2+pp−c1−ϕϕp+λ+β−v×2β+λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ+β−vλ+ϕp−c+pp+c1−ϕϕp+λ+β−vλ+ϕp−c+2cϕp+λ+β−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ+2β−2vλ+ϕp−c−pϕp+λ+β−vϕp+ϕc+λ+2β−2vλ+ϕp−c.The optimaleSWRS∗ and ϕSWRS∗ can be obtained by making ∂πS−SWRS/∂e=0 and ∂πS−SWRS/∂ϕ=0, and the optimal expected profits of the upstream and downstream enterprises are(8)πS−SWRS∗=BeSWRS∗p1−ϕSWRS∗pϕSWRS∗+cϕSWRS∗+λ+2β−2vλ+ϕSWRS∗p−c2ϕSWRS∗p−v+λ+β2−Bec1−ϕSWRS∗λ+ϕSWRS∗p−cϕSWRS∗p−v+λ+β−keSWRS∗2,πR−SWRS∗=BeSWRS∗λ+ϕSWRS∗p−cvλ−ϕSWRS∗p−c+ϕSWRS∗p2β+λ+ϕSWRS∗p−c2β+2λ+ϕSWRS∗p2ϕSWRS∗p−v+λ+β2.


#### 3.1.2. WA Model

If the retailer is WA, that is,λ=0 and β>0, then (7) can be used to obtain the optimal eWARS∗ and ϕWARS∗ which are given as(9)4keϕp+β−v2−Bϕ1−ϕp−c2pϕ+2β−2vϕp+β−v=0,Bep−c3pβϕ2+p2ϕ3+v2+β24ϕ−2+vβ4−8ϕ−3pϕ2ϕp+β−v=0.At this point,qWARS∗, πS−WARS∗, and πR−WARS∗ are(10)qWARS∗=BeWARS∗ϕWARS∗p−cpϕWARS∗+β−v,πS−WARS∗=BepϕWARS∗p−c1−ϕWARS∗pϕWARS∗+cϕWARS∗+2β−2v2ϕWARS∗p−v+β2−BecϕWARS∗p−c1−ϕWARS∗ϕWARS∗p−v+β−keWARS∗2,πR−WARS∗=BeϕWARS∗p−cvϕWARS∗c−p+ϕWARS∗p2β+ϕWARS∗p−c2β+pϕWARS∗2ϕWARS∗p−v+β2.


#### 3.1.3. SA Model

If the retailer is SA, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eSARS∗ and ϕSARS∗ which are given as(11)Bp1−ϕλ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ−v2−Bc1−ϕλ+ϕp−cϕp+λ−v−2ke=0,Be2cp−cϕ−1ϕp+λ−v2+pp−c1−ϕϕp+λ−v×λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ−vλ+ϕp−c+pp+c1−ϕϕp+λ−vλ+ϕp−c+2cϕp+λ−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ−2vλ+ϕp−c−pϕp+λ−vϕp+ϕc+λ−2vλ+ϕp−c=0.In the same way as above, the optimalqSARS∗, πS−SARS∗, and πR−SARS∗ can be obtained.


#### 3.1.4. RN Model

If the retailer is RN, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eRNRS∗ and ϕRNRS∗ which is given as(12)Bep−c2ϕ3p2+v24ϕ−2−3pvϕ22v−ϕp3=0,Bϕ1−ϕp−c2ϕp−2v−2keϕp−v22v−ϕp2−ke=0.In the same way as above, the optimalqRNRS∗, πS−RNRS∗, and πR−RNRS∗ can be obtained.


### 3.2. WP Model

If WP is adopted between upstream and downstream enterprises, then the decision models under the four different behavior preferences are as follows.


#### 3.2.1. RN Model

Step 1:maxw,eπS−RNWPw,e=w−cq−ke2Step 2:maxqπR−RNWPq|w,e=pSq,e+vLq,e−wq


#### 3.2.2. WA Model

Step 1:maxw,eπS−WAWPw,e=w−cq−ke2Step 2:maxqUR−WAWPq|w,e=pSq,e+vLq,e−βLq,e−wq


#### 3.2.3. SA Model

Step 1:maxw,eπS−SAWPw,e=w−cq−ke2Step 2:maxqUR−SAWPq|w,e=pSq,e+vLq,e−λNq,e−wq


#### 3.2.4. SW Model

Step 1:maxw,eπS−SWWPw,e=w−cq−ke2Step 2:maxqUR−SWWPq|w,e=pSq,e+vLq,e−λNq,e−βLq,e−wqIf the retailer is SW, let∂UR−SWWP/∂q=0; we can obtain(13)qSWWP∗=Bep−w+λp−v+λ+β.By substitutingqSWWP∗ into πS−SWWPw,e, then(14)maxw,eπ2SSWw,e=Bew−cp−w+λp−v+β+λ−ke2.Let∂πS−SWWP/∂w=0 and ∂πS−SWWP/∂e=0, and we can obtain(15)wSW∗=p+c+λ2,eSWWP∗=Bp−c+λ28kp−v+β+λ.By substitutingwSW∗ and eSWWP∗into qSWWP∗, πS−SWWP, and πR−SWWP, so you can get the result of the fourth row in Table 2. In the same way as above, the Nash equilibrium for each of the other three behavioral preference models under WP can be obtained by using the reverse induction method; therefore, it is omitted. The results are shown in Table 2.Table 2
Optimal value under WP with four behavioral preferences (j∈RN,SA,WA,SW).
PreferenceqjWP∗ejWP∗πjWP∗πjWP∗RNB2p−c3/16kp−v2Bp−c2/8kp−vB2p−c4/64kp−v2B2p−c4/64kp−v2WAB2p−c3/16kp−v+β2Bp−c2/8kp−v+βB2p−c3/64kp−v+β2B2p−v+2βp−c4/64kp−v+β2SAB2p+λ−c3/16kp−v+λ2Bp−c+λ2/8kp−v+λB2p+λ−c4/64kp−v+λ2B2p−c+λ3p2−pv+λ+λ3v−2λ−cp−v+2λ/64kp−v+λ3SWB2p+λ−c3/16kp−v+β+λ2Bp−c+λ2/8kp−v+β+λB2p−c+λ4/64kp−v+β+λ2B2p−c+λ3p2−pv−2β+λ−λ−3v+2β+λ−cp−v+2β+λ/64kp−v+β+λ3


## 3.1. RS Model

Under the RS model, it is assumed that the retailer gives1−ϕ times its own sales revenue to the supplier in exchange for a lower wholesale price w=ϕc [1], where 0<ϕ≤1 and ϕp>v. When the random demand depends on the effort of the supplier, the retailer’s expected sales volume Sq,e, inventory volume Lq,e, and shortage volume Nq,e are(1)Sq,e=Eminq,eξ,Lq,e=Eq−eξ+=q−Sq,e,Nq,e=Eeξ−q+=Eeξ−Sq,e.The expected profits of upstream and downstream enterprises are(2)πS−jRSϕ,e=1−ϕpSq,e−1−ϕcq−ke2,πR−jRSq=ϕpSq,e+vLq,e−ϕcq,where j∈RN,WA,SA,SW.


### 3.1.1. SW Model

If the retailer has an SW preference, then the expected utility of the retailer is as calculated below:(3)UR−SWRSq=ϕpSq,e+vLq,e−λNq,e−βLq,e−ϕcq.From equation (3), we obtain(4)∂UR−SWRS∂q=qv−β−λ−pϕ+Beλ+ϕp−cBe,∂2UR−SWRS∂q2=pϕ−v+λ+β−Be<0.From equation (3), we can see that UR−SWRS is a concave function of q. Therefore, if ∂UR−SWRS/∂q=0, the unique optimal qSWRS∗ can be obtained:(5)qSWRS∗=Beλ+ϕp−cϕp+λ+β−v.By substitutingqSWRS∗ into equation (2), then(6)maxϕ,eπS−SWRS=Bep1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bec1−ϕλ+ϕp−cϕp+λ+β−v−ke2.From equation (6), we obtain(7)∂πS−SWRS∂e=Bp1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bc1−ϕλ+ϕp−cϕp+λ+β−v−2ke,∂πS−SWRS∂ϕ=Be2cp−cϕ−1ϕp+λ+β−v2+pp−c1−ϕϕp+λ+β−v×2β+λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ+β−vλ+ϕp−c+pp+c1−ϕϕp+λ+β−vλ+ϕp−c+2cϕp+λ+β−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ+2β−2vλ+ϕp−c−pϕp+λ+β−vϕp+ϕc+λ+2β−2vλ+ϕp−c.The optimaleSWRS∗ and ϕSWRS∗ can be obtained by making ∂πS−SWRS/∂e=0 and ∂πS−SWRS/∂ϕ=0, and the optimal expected profits of the upstream and downstream enterprises are(8)πS−SWRS∗=BeSWRS∗p1−ϕSWRS∗pϕSWRS∗+cϕSWRS∗+λ+2β−2vλ+ϕSWRS∗p−c2ϕSWRS∗p−v+λ+β2−Bec1−ϕSWRS∗λ+ϕSWRS∗p−cϕSWRS∗p−v+λ+β−keSWRS∗2,πR−SWRS∗=BeSWRS∗λ+ϕSWRS∗p−cvλ−ϕSWRS∗p−c+ϕSWRS∗p2β+λ+ϕSWRS∗p−c2β+2λ+ϕSWRS∗p2ϕSWRS∗p−v+λ+β2.


### 3.1.2. WA Model

If the retailer is WA, that is,λ=0 and β>0, then (7) can be used to obtain the optimal eWARS∗ and ϕWARS∗ which are given as(9)4keϕp+β−v2−Bϕ1−ϕp−c2pϕ+2β−2vϕp+β−v=0,Bep−c3pβϕ2+p2ϕ3+v2+β24ϕ−2+vβ4−8ϕ−3pϕ2ϕp+β−v=0.At this point,qWARS∗, πS−WARS∗, and πR−WARS∗ are(10)qWARS∗=BeWARS∗ϕWARS∗p−cpϕWARS∗+β−v,πS−WARS∗=BepϕWARS∗p−c1−ϕWARS∗pϕWARS∗+cϕWARS∗+2β−2v2ϕWARS∗p−v+β2−BecϕWARS∗p−c1−ϕWARS∗ϕWARS∗p−v+β−keWARS∗2,πR−WARS∗=BeϕWARS∗p−cvϕWARS∗c−p+ϕWARS∗p2β+ϕWARS∗p−c2β+pϕWARS∗2ϕWARS∗p−v+β2.


### 3.1.3. SA Model

If the retailer is SA, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eSARS∗ and ϕSARS∗ which are given as(11)Bp1−ϕλ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ−v2−Bc1−ϕλ+ϕp−cϕp+λ−v−2ke=0,Be2cp−cϕ−1ϕp+λ−v2+pp−c1−ϕϕp+λ−v×λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ−vλ+ϕp−c+pp+c1−ϕϕp+λ−vλ+ϕp−c+2cϕp+λ−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ−2vλ+ϕp−c−pϕp+λ−vϕp+ϕc+λ−2vλ+ϕp−c=0.In the same way as above, the optimalqSARS∗, πS−SARS∗, and πR−SARS∗ can be obtained.


### 3.1.4. RN Model

If the retailer is RN, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eRNRS∗ and ϕRNRS∗ which is given as(12)Bep−c2ϕ3p2+v24ϕ−2−3pvϕ22v−ϕp3=0,Bϕ1−ϕp−c2ϕp−2v−2keϕp−v22v−ϕp2−ke=0.In the same way as above, the optimalqRNRS∗, πS−RNRS∗, and πR−RNRS∗ can be obtained.


## 3.1.1. SW Model

If the retailer has an SW preference, then the expected utility of the retailer is as calculated below:(3)UR−SWRSq=ϕpSq,e+vLq,e−λNq,e−βLq,e−ϕcq.From equation (3), we obtain(4)∂UR−SWRS∂q=qv−β−λ−pϕ+Beλ+ϕp−cBe,∂2UR−SWRS∂q2=pϕ−v+λ+β−Be<0.From equation (3), we can see that UR−SWRS is a concave function of q. Therefore, if ∂UR−SWRS/∂q=0, the unique optimal qSWRS∗ can be obtained:(5)qSWRS∗=Beλ+ϕp−cϕp+λ+β−v.By substitutingqSWRS∗ into equation (2), then(6)maxϕ,eπS−SWRS=Bep1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bec1−ϕλ+ϕp−cϕp+λ+β−v−ke2.From equation (6), we obtain(7)∂πS−SWRS∂e=Bp1−ϕ2β+λ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ+β−v2−Bc1−ϕλ+ϕp−cϕp+λ+β−v−2ke,∂πS−SWRS∂ϕ=Be2cp−cϕ−1ϕp+λ+β−v2+pp−c1−ϕϕp+λ+β−v×2β+λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ+β−vλ+ϕp−c+pp+c1−ϕϕp+λ+β−vλ+ϕp−c+2cϕp+λ+β−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ+2β−2vλ+ϕp−c−pϕp+λ+β−vϕp+ϕc+λ+2β−2vλ+ϕp−c.The optimaleSWRS∗ and ϕSWRS∗ can be obtained by making ∂πS−SWRS/∂e=0 and ∂πS−SWRS/∂ϕ=0, and the optimal expected profits of the upstream and downstream enterprises are(8)πS−SWRS∗=BeSWRS∗p1−ϕSWRS∗pϕSWRS∗+cϕSWRS∗+λ+2β−2vλ+ϕSWRS∗p−c2ϕSWRS∗p−v+λ+β2−Bec1−ϕSWRS∗λ+ϕSWRS∗p−cϕSWRS∗p−v+λ+β−keSWRS∗2,πR−SWRS∗=BeSWRS∗λ+ϕSWRS∗p−cvλ−ϕSWRS∗p−c+ϕSWRS∗p2β+λ+ϕSWRS∗p−c2β+2λ+ϕSWRS∗p2ϕSWRS∗p−v+λ+β2.


## 3.1.2. WA Model

If the retailer is WA, that is,λ=0 and β>0, then (7) can be used to obtain the optimal eWARS∗ and ϕWARS∗ which are given as(9)4keϕp+β−v2−Bϕ1−ϕp−c2pϕ+2β−2vϕp+β−v=0,Bep−c3pβϕ2+p2ϕ3+v2+β24ϕ−2+vβ4−8ϕ−3pϕ2ϕp+β−v=0.At this point,qWARS∗, πS−WARS∗, and πR−WARS∗ are(10)qWARS∗=BeWARS∗ϕWARS∗p−cpϕWARS∗+β−v,πS−WARS∗=BepϕWARS∗p−c1−ϕWARS∗pϕWARS∗+cϕWARS∗+2β−2v2ϕWARS∗p−v+β2−BecϕWARS∗p−c1−ϕWARS∗ϕWARS∗p−v+β−keWARS∗2,πR−WARS∗=BeϕWARS∗p−cvϕWARS∗c−p+ϕWARS∗p2β+ϕWARS∗p−c2β+pϕWARS∗2ϕWARS∗p−v+β2.


## 3.1.3. SA Model

If the retailer is SA, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eSARS∗ and ϕSARS∗ which are given as(11)Bp1−ϕλ−2v+cϕ+pϕλ+ϕp−c2ϕp+λ−v2−Bc1−ϕλ+ϕp−cϕp+λ−v−2ke=0,Be2cp−cϕ−1ϕp+λ−v2+pp−c1−ϕϕp+λ−v×λ−2v+cϕ+pϕ+2cp1−ϕϕp+λ−vλ+ϕp−c+pp+c1−ϕϕp+λ−vλ+ϕp−c+2cϕp+λ−v2×λ+ϕp−c−2p21−ϕϕp+ϕc+λ−2vλ+ϕp−c−pϕp+λ−vϕp+ϕc+λ−2vλ+ϕp−c=0.In the same way as above, the optimalqSARS∗, πS−SARS∗, and πR−SARS∗ can be obtained.


## 3.1.4. RN Model

If the retailer is RN, that is,λ>0 and β=0, then (7) can be used to obtain the optimal eRNRS∗ and ϕRNRS∗ which is given as(12)Bep−c2ϕ3p2+v24ϕ−2−3pvϕ22v−ϕp3=0,Bϕ1−ϕp−c2ϕp−2v−2keϕp−v22v−ϕp2−ke=0.In the same way as above, the optimalqRNRS∗, πS−RNRS∗, and πR−RNRS∗ can be obtained.


## 3.2. WP Model

If WP is adopted between upstream and downstream enterprises, then the decision models under the four different behavior preferences are as follows.


### 3.2.1. RN Model

Step 1:maxw,eπS−RNWPw,e=w−cq−ke2Step 2:maxqπR−RNWPq|w,e=pSq,e+vLq,e−wq


### 3.2.2. WA Model

Step 1:maxw,eπS−WAWPw,e=w−cq−ke2Step 2:maxqUR−WAWPq|w,e=pSq,e+vLq,e−βLq,e−wq


### 3.2.3. SA Model

Step 1:maxw,eπS−SAWPw,e=w−cq−ke2Step 2:maxqUR−SAWPq|w,e=pSq,e+vLq,e−λNq,e−wq


### 3.2.4. SW Model

Step 1:maxw,eπS−SWWPw,e=w−cq−ke2Step 2:maxqUR−SWWPq|w,e=pSq,e+vLq,e−λNq,e−βLq,e−wqIf the retailer is SW, let∂UR−SWWP/∂q=0; we can obtain(13)qSWWP∗=Bep−w+λp−v+λ+β.By substitutingqSWWP∗ into πS−SWWPw,e, then(14)maxw,eπ2SSWw,e=Bew−cp−w+λp−v+β+λ−ke2.Let∂πS−SWWP/∂w=0 and ∂πS−SWWP/∂e=0, and we can obtain(15)wSW∗=p+c+λ2,eSWWP∗=Bp−c+λ28kp−v+β+λ.By substitutingwSW∗ and eSWWP∗into qSWWP∗, πS−SWWP, and πR−SWWP, so you can get the result of the fourth row in Table 2. In the same way as above, the Nash equilibrium for each of the other three behavioral preference models under WP can be obtained by using the reverse induction method; therefore, it is omitted. The results are shown in Table 2.Table 2
Optimal value under WP with four behavioral preferences (j∈RN,SA,WA,SW).
PreferenceqjWP∗ejWP∗πjWP∗πjWP∗RNB2p−c3/16kp−v2Bp−c2/8kp−vB2p−c4/64kp−v2B2p−c4/64kp−v2WAB2p−c3/16kp−v+β2Bp−c2/8kp−v+βB2p−c3/64kp−v+β2B2p−v+2βp−c4/64kp−v+β2SAB2p+λ−c3/16kp−v+λ2Bp−c+λ2/8kp−v+λB2p+λ−c4/64kp−v+λ2B2p−c+λ3p2−pv+λ+λ3v−2λ−cp−v+2λ/64kp−v+λ3SWB2p+λ−c3/16kp−v+β+λ2Bp−c+λ2/8kp−v+β+λB2p−c+λ4/64kp−v+β+λ2B2p−c+λ3p2−pv−2β+λ−λ−3v+2β+λ−cp−v+2β+λ/64kp−v+β+λ3


## 3.2.1. RN Model

Step 1:maxw,eπS−RNWPw,e=w−cq−ke2Step 2:maxqπR−RNWPq|w,e=pSq,e+vLq,e−wq


## 3.2.2. WA Model

Step 1:maxw,eπS−WAWPw,e=w−cq−ke2Step 2:maxqUR−WAWPq|w,e=pSq,e+vLq,e−βLq,e−wq


## 3.2.3. SA Model

Step 1:maxw,eπS−SAWPw,e=w−cq−ke2Step 2:maxqUR−SAWPq|w,e=pSq,e+vLq,e−λNq,e−wq


## 3.2.4. SW Model

Step 1:maxw,eπS−SWWPw,e=w−cq−ke2Step 2:maxqUR−SWWPq|w,e=pSq,e+vLq,e−λNq,e−βLq,e−wqIf the retailer is SW, let∂UR−SWWP/∂q=0; we can obtain(13)qSWWP∗=Bep−w+λp−v+λ+β.By substitutingqSWWP∗ into πS−SWWPw,e, then(14)maxw,eπ2SSWw,e=Bew−cp−w+λp−v+β+λ−ke2.Let∂πS−SWWP/∂w=0 and ∂πS−SWWP/∂e=0, and we can obtain(15)wSW∗=p+c+λ2,eSWWP∗=Bp−c+λ28kp−v+β+λ.By substitutingwSW∗ and eSWWP∗into qSWWP∗, πS−SWWP, and πR−SWWP, so you can get the result of the fourth row in Table 2. In the same way as above, the Nash equilibrium for each of the other three behavioral preference models under WP can be obtained by using the reverse induction method; therefore, it is omitted. The results are shown in Table 2.Table 2
Optimal value under WP with four behavioral preferences (j∈RN,SA,WA,SW).
PreferenceqjWP∗ejWP∗πjWP∗πjWP∗RNB2p−c3/16kp−v2Bp−c2/8kp−vB2p−c4/64kp−v2B2p−c4/64kp−v2WAB2p−c3/16kp−v+β2Bp−c2/8kp−v+βB2p−c3/64kp−v+β2B2p−v+2βp−c4/64kp−v+β2SAB2p+λ−c3/16kp−v+λ2Bp−c+λ2/8kp−v+λB2p+λ−c4/64kp−v+λ2B2p−c+λ3p2−pv+λ+λ3v−2λ−cp−v+2λ/64kp−v+λ3SWB2p+λ−c3/16kp−v+β+λ2Bp−c+λ2/8kp−v+β+λB2p−c+λ4/64kp−v+β+λ2B2p−c+λ3p2−pv−2β+λ−λ−3v+2β+λ−cp−v+2β+λ/64kp−v+β+λ3


## 4. Numerical Simulation Analysis

Due to the complexity of the results obtained under the RS model, we cannot analyze them through theoretical derivation. Therefore, in this section, we use the method of numerical simulation to obtain more valuable conclusions. Without loss of generality, we takec=1, v=0.5, k=2, B=50, and p=5.From Figures1–4, we can obtain Lemma 1 and Proposition 1.Figure 1
Effect ofλ on order quantity.Figure 2
Effect ofλ on effort.Figure 3
Effect ofλ on supplier’s expected profit.Figure 4
Effect ofλon retailer’s expected profit.Lemma 1.
If the retailer has SA, (1) under the WP model,qSAWP∗ increases with respect to λ, but qSARS∗ decreases with respect to λ under the RS, (2) under both the WP and RS models, eSAWP∗ and eSARS∗ are increasing with respect to λ, and (3) under both the WP and RS models, πS−SAWP∗ and πS−SARS∗ are increasing with respect to λ, while πR−SAWP∗ and πR−SARS∗ are decreasing with respect to λ.Proposition 1.
If the retailer is SA (RN), then compared with the WP model, (1) the retailer orders more under the RS model, (2) less effort (expected profit) is made by the supplier under the RS model, and (3) the retailer has higher expected profit under the RS model.From Figures5–8, we can obtain Lemma 2 and Proposition 2.Figure 5
Effect ofβ on order quantity.Figure 6
Effect ofβ on effort.Figure 7
Effect ofβ on supplier’s expected profit.Figure 8
Effect ofβ on retailer’s expected profit.Lemma 2.
If the retailer has WA, the following is obtained. (1) Under the WP model, bothqWAWP∗ and eWAWP∗ are decreasing functions of β. However, under the RS model, both qWARS∗ and eWARS∗ are first increasing and then decreasing with an increase in β. (2) Under the WP model, πS−WAWP∗ is a decreasing function of β. However, under the RS model, πS−WARS∗ is first increasing and then decreasing with an increase in β. (3) Under the WP model, πR−WAWP∗ is a decreasing function of β, while under the RS model, with the increase in β, there is a tendency for expected profit to decrease first, then increase, and then decrease.Proposition 2.
If the retailer has WA, then compared with the WP model, (1) the retailer orders more under the RS model, (2) supplier invests more effort under the RS model, (3) the expected profit of the supplier under the RS model is greater, and (4) the size of the retailer’s expected profit between the above two contracts is related toβ; there are thresholds β1 and β2 such that β1<β2, and if β<β1 or β>β2, its expected profit under the RS is greater, whereas when β1<β<β2, the expected profit under the WP model is greater.From Figures9–12, we can obtain Proposition 3.Figure 9
Effect ofλ and β on order quantity.Figure 10
Effect ofλ and β on effort.Figure 11
Effect ofλ and β on the supplier’s expected profit.Figure 12
Effect ofλ and β on the retailer’s expected profit.Proposition 3.
If the retailer has SW, (1) the retailer orders more under the RS model, (2) whenβ is small, then the supplier’s effort (expected profit) under the WP model may be greater than under the RS model, and most of the time it is greater in the RS model, and (3) when β is medium and λ is small, then the retailer’s expected profit under the WP model may be greater, and most of the time it is greater in the RS model.


## 5. Conclusion

In this paper, four situations in which the retailer has behavioral preferences are considered in the case of random demand affected by supplier’s effort. Based on the WP and RS models, the influence of the retailer’s different behavior preferences on the decision-making and contract choice of upstream and downstream enterprises is discussed.The results show the following. (1) When the retailer only has SA (RN), the supplier’s effort investment level under the WP model is higher, and the retailer’s order quantity under the RS model is larger. (2) When the retailer only has WA, the supplier’s effort input level and the retailer’s order quantity under the RS model will be larger. (3) When the retailer has SW, the retailer will order more under the RS model, and the supplier’s effort input level between the two contracts depends on the degree of waste aversion and stockout aversion. (4) When the retailer has only SA (RN), the supplier’s expected profit under the WP model is greater, while the retailer’s expected profit is lower. (5) When the retailer has only WA, the supplier’s expected profit under the RS model is greater, but the expected profit of the retailer between the two contracts depends on the degree of waste aversion. (6) When the retailer has SW, the expected profit of both the retailer and supplier between the two contracts is related to the degree of waste aversion and stockout aversion.Demand is often the private information for retailers [17, 18], so the impact of demand information sharing on supply chain contract choice can be considered in future studies.



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*Source: 7894885-2021-10-30.xml*