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Innovation of Economic Management Risk Control in Retailer Supply Chain Based on Differential Equation Model

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
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Recibido: 21 Feb 2022
Aceptado: 16 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

The decision equilibrium point of online supply chain finance is realized in a multi-party game with linkage characteristics [1]. The model is embodied as multi-dimensional variables with correlation under comprehensive operational decision-making. The content includes production decisions of core enterprises, financing choices, ordering and sales decisions, pledge rate and interest rate decisions of financing platforms, action choices, and pricing decisions of third-party logistics enterprises. Under this multi-dimensional equilibrium, this paper will study the achievable benefit coordination of individuals and the overall achievable value increase.

1) Manufacturer m, single retailer s, third-party warehousing enterprise l, and financing platform g are mutually independent individuals of interest. They are committed to maximizing the interests of the individual [2]. The value-added generated by all parties as contributors to the benefits brought about by the reallocation of financial resources is shared by multiple parties. All parties should obtain not less than the average industry return, including coverage cost and risk-reward. 2) A single product in the supply chain faces the ransom demand of the market θ ∈ [0, + ∞). The demand in each cycle is independently distributed. They are not exogenous variables controlled by agents and principals. Its distribution function is F (θ) and F (θ) is continuously differentiable. 3) All participants are risk-neutral. The utility function is denoted as U (x) = x. 4) The standard loan contract for financing amount, interest rate, term, and initial margin ratio is ω(F, r, t, ɛ). where r is the financing rate. ɛ is the percentage of the retailer's initial deposit, ɛ ∈ (0, 1). Assume that loanable funds are unconstrained. The financing platform determines the credit limit F0g F_0^g according to the historical transaction data of the supply chain. 5) Incomplete information market [3]. The effort levels of the agents in the principal-agent relationship can be denoted as em, es, el, respectively. The effort cost corresponds to Cm (em), Cs (es), Cl (el).

Model parameters

Other symbols and variables used by the model are defined as follows: c0m c_0^m represents the supplier's product manufacturing cost. p0m p_0^m represents the wholesale price of the product set by the supplier, and 0<p0m<c0m 0 < p_0^m < c_0^m exists under the revenue sharing contract. pts p_t^s represents the retailer's selling price of the product, pts[max(l+p0m,l+c0m),] p_t^s \in \left[{\max \left({l + p_0^m,\,l + c_0^m} \right),\infty} \right] . p˜ts \tilde p_t^s represents the handling price of the product when the sales market is down, p˜ts(0,pts) \tilde p_t^s \in \left({0,p_t^s} \right) . γ represents the average industry rate of return in the competitive market, which is expressed as γm, γs, γl, in the model [4]. l represents the cost of services such as transportation that is shared per product unit. αs represents the retailer's revenue-sharing ratio, then the supplier's revenue-sharing ratio is αm = 1 − αs. In the game between the financing platform and the warehousing company, αl is the revenue sharing ratio obtained by the warehousing company, αm, αs, αl ∈ (0, 1). P represents the probability of observing the corresponding effort level, P ∈ (0, 1]. M0s M_0^s represents the total amount of the retailer's own funds. qts q_t^s represents the retailer's total order quantity. q0s q_0^s represents the number of orders the retailer uses its own funds, q0s{0,(M0s/ptm)} q_0^s \in \left\{{0,\,\left({M_0^s/p_t^m} \right)} \right\} , where [M0s/p0m] \left[{M_0^s/p_0^m} \right] is the largest integer. F0s F_0^s represents the amount of financing requested by the retailer. Fts F_t^s represents the retailer's final financing amount, When F0sF0g F_0^s \le F_0^g , Fts=F0s F_t^s = F_0^s . When F0s>F0g F_0^s > F_0^g , Fts=F0g F_t^s = F_0^g . G1g G_1^g represents the maintenance cost of the online financing platform and payment platform by the financing platform. The retailer's actual sales volume in a single cycle is: S(qts)=Emin(qts,θ)=qts0qtsF(θ)dθ S\left({q_t^s} \right) = E\min \left({q_t^s,\,\theta} \right) = q_t^s - \int_0^{q_t^s} {F\left(\theta \right)d\theta} . The inventory balance is Q(qts)=0qtsF(θ)dθ Q\left({q_t^s} \right) = \int_0^{q_t^s} {F\left(\theta \right)d\theta} . The implicit conditions for retailers facing financial difficulties and credit rationing to apply for online supply chain finance business are as follows: The maximum order quantity that the retailer can achieve with its funds cannot meet the market demand [5]. The out-of-stock loss at this time makes the optimal performance of the supply chain impossible to achieve. The financing amount determined by the retailer according to the order quantity should be the total payment minus its funds. This part of the funds is used for self-ordering and initial deposit payment: F0s=p0mqts(M0sεF0sp0mq0s)F0s=p0m(qts+q0s)M0s1ε F_0^s = p_0^mq_t^s - \left({M_0^s - \varepsilon F_0^s - p_0^mq_0^s} \right) \Rightarrow F_0^s = {{p_0^m\left({q_t^s + q_0^s} \right) - M_0^s} \over {1 - \varepsilon}}

Proposition 1: The retailer's final financing amount Fts F_t^s is determined by its ordering decision qts q_t^s and its capital M0s M_0^s . It is limited by the financing amount F0g F_0^g determined by the financing platform based on its historical transaction data, namely: Fts={p0m(qts+q0s)M0s1εf0g,F0s>F0gF0sF0g F_t^s = \left\{{\matrix{{{{p_0^m\left({q_t^s + q_0^s} \right) - M_0^s} \over {1 - \varepsilon}}} \hfill \cr {f_0^g,F_0^s > F_0^g} \hfill \cr} F_0^s \le F_0^g} \right.

When F0sF0g F_0^s \le F_0^g , the financing amount Fts F_t^s has a linkage relationship with the retailer's transaction decision [6]. The retailer's utility derives from the residual profit from the product's sale. The sales revenue πts \pi _t^s deducts the part of shared revenue, wholesale cost, transportation cost, and interest expense. The retailer's utility function is: Uts=Uts(αsπtsp0mqtslqtsFtsr) U_t^s = U_t^s\left({{\alpha ^s}\pi _t^s - p_0^mq_t^s - lq_t^s - F_t^sr} \right)

The supplier's income comes from two aspects: one is the sales profit πtm=(p0mc0m)qts \pi _t^m = \left({p_0^m - c_0^m} \right)q_t^s generated by the retailer's purchase of products from the supplier. But the supplier may make p0m<c0m p_0^m < c_0^m under a revenue-sharing contract [7]. At this time, the shared benefits obtained by suppliers will become the main source of revenue. The second is the sharing of sales revenue from retailers. Only when the retailer fulfills the contract can the supplier get a corresponding proportion of the sales revenue. The utility function of the supplier is: Utm=Utm(πtm+(1αs)πts) U_t^m = U_t^m\left({\pi _t^m + \left({1 - {\alpha ^s}} \right)\pi _t^s} \right)

In the game between the supplier and the retailer in the supply chain, the supplier expects the retailer to pay the optimal level of sales effort to maximize the individual benefit. Pms P_m^s is the probability that the supplier observes the retailer's effort level as es. The following objective function gives the contract reached by the two: maxes,qts,αs[m=1nPms(es)Utm(πtm+(1αs)πts)Cm(em)]m=1nPms(es)Uts(αsπts+p0mqtslqtsFtsr)Cs(es)Uts¯esargmaxes*{m=1nPms(es)Uts(αsπtsp0mqtslqtsFtsr)Cs(es)Uts¯} \matrix{{\mathop {\max}\limits_{{e^s},\,q_t^s,\,{\alpha ^s}} \left[{\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)U_t^m\left({\pi _t^m + \left({1 - {\alpha ^s}} \right)\pi _t^s} \right) - {C^m}\left({{e^m}} \right)}} \right]} \hfill \cr {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)U_t^s\left({{\alpha ^s}\pi _t^s + p_0^mq_t^s - lq_t^s - F_t^sr} \right) - {C^s}\left({{e^s}} \right) \ge \overline {U_t^s}}} \hfill \cr {{e^s} \in \mathop {\arg \max}\limits_{{e^{{s^*}}}} \left\{{\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)U_t^s\left({{\alpha ^s}\pi _t^s - p_0^mq_t^s - lq_t^s - F_t^sr} \right) - {C^s}\left({{e^s}} \right) \ge \overline {U_t^s}}} \right\}} \hfill \cr}

When the sales revenue realized under the order quantity qts q_t^s is sufficient to repay the remaining loan except for the deposit, the retailer performs the contract as scheduled. They can repay the loan principal and interest on schedule: S(qts)pts(1ε)Fts S\left({q_t^s} \right)p_t^s \ge \left({1 - \varepsilon} \right)F_t^s

Substitute the above formula into formula (1) when F0sF0g F_0^s \le F_0^g , we can get: [qts0qtsF(θ)dθ]pts(1ε)p0m(qts+q0s)M0s1ε \left[{q_t^s - \int_0^{q_t^s} {F\left(\theta \right)d\theta}} \right]p_t^s \ge \left({1 - \varepsilon} \right){{p_0^m\left({q_t^s + q_0^s} \right) - M_0^s} \over {1 - \varepsilon}}

When qtsq1s q_t^s \ge \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over q} _1^s , the sales income is enough to repay the loan principal and interest. When qts<q1s q_t^s < \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over q} _1^s , the retailer defaults because it cannot fulfill the agreed sales volume [8]. At this time, the existing sales revenue and initial deposit are all used to repay the loan. Suppliers only get paid for selling products. Substitute equation (1) into equation (5) and use the Lagrangian function to solve it. We can derive from the objective function: pts[1F(qts)]p0mr1ε=p0mγSm=1nPms(es)+(l+c0m) p_t^s\left[{1 - F\left({q_t^s} \right)} \right] - {{p_0^mr} \over {1 - \varepsilon}} = {{p_0^m{\gamma ^S}} \over {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)}}} + \left({l + c_0^m} \right) ptsqtsf(q)+(1αs)pts[1F(qts)]=c0mp0m p_t^sq_t^sf\left(q \right) + \left({1 - {\alpha ^s}} \right)p_t^s\left[{1 - F\left({q_t^s} \right)} \right] = c_0^m - p_0^m

The order quantity qts q_t^s and the paid initial margin ratio ɛ satisfy the relational formula (8). Equation (9) shows that the revenue sharing coefficient αs in the supply chain satisfies the differential equation about the order quantity qts q_t^s . At this time, it can be seen from equation (6) that the retailer's credit risk default boundary under limited financing satisfies the equation: S(q^2s)pts=(1ε)F0g S\left({\hat q_2^s} \right)p_t^s = \left({1 - \varepsilon} \right)F_0^g

Under this financing limit, the principal-agent relationship between the supplier and the retailer is still given by the objective function in Eq. (5). Substitute the suboptimal solution q^ts \hat q_t^s of Proposition 1 and the order quantity, and also use the Lagrangian function to solve it. have to: αs=m=1nPms(es)q^ts(p0m+p0mr+l)+Cs(es)+p0q^tsγsm=1nPms(es)ptsS(q^ts),S(q^ts)(1ε)F0gpts {\alpha ^s} = {{\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)\hat q_t^s\left({p_0^m + p_0^m\,r + l} \right) + {C^s}\left({{e^s}} \right) + {p_0}\hat q_t^s{\gamma ^s}}} \over {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)p_t^sS\left({\hat q_t^s} \right)}}},\,S\left({\hat q_t^s} \right) \ge {{\left({1 - \varepsilon} \right)F_0^g} \over {p_t^s}}

The financing limit F0g F_0^g determined according to historical transaction data, and credit evaluation is common knowledge, and the second optimal solution q^ts \hat q_t^s of the order quantity satisfies q^ts<qts \hat q_t^s < q_t^s . The revenue sharing ratio in the supply chain can be directly obtained from the objective function of equation (5). At this time, the expected profit of the supplier is maximized under the order quantity. This leads to the following propositions:

Proposition 2: When faced with financing constraints (F0s>F0g) \left({F_0^s > F_0^g} \right) , the retailer can only obtain the maximum value F0g F_0^g under the financing limit. At this time, the order batch realizes the suboptimal solution q^ts=F0g/p0m+q0s=F0g/p0m+(M0sεF0g)/p0m \hat q_t^s = F_0^g/p_0^m + q_0^s = F_0^g/p_0^m + \left({M_0^s - \varepsilon F_0^g} \right)/p_0^m .

Risk control and benefits realization of financing platform

The establishment of an online supply chain financing platform is conducive to establishing a multi-level financial service system and promotes the reform of the financial system [9]. The benefits of the SME financing market development in the supply chain are mainly reflected in the following aspects: First, the intermediate business income π1g(ω) \pi _1^g\left(\omega \right) . Second, the income brought by the precipitation funds is π2g(ω) \pi _2^g\left(\omega \right) . The third is the interest income π3g(ω) \pi _3^g\left(\omega \right) of the loan. The utility function of the financing platform is: Utg=Utg(π1g(ω)+π2g(ω)+π3g(ω))G1g U_t^g = U_t^g\left({\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right)} \right) - G_1^g

According to the utility functions of suppliers and retailers in equations (3) and (4), we can get: maxε,rm,s=1nPm,ss(em,es)Utg(π1g(ω)+π2g(ω)+π3g(ω))G1gm,s=1nPm,sg(em,es)Utm(πtm+(1αs)πts)Cm(em)Utm¯m,s=1nPm,sg(em,es)Uts(αsπtsp0mqtslqtsFtsr)Cs(es)Uts¯em,esargmaxem*,es*{m,s=1nPm,sg(em,es)Utg(π1g(ω)+π2g(ω)+π3g(ω))G1G} \matrix{{\mathop {\max}\limits_{\varepsilon,r} \sum\limits_{m,s = 1}^n {P_{m,s}^s\left({{e^m},{e^s}} \right)U_t^g\left({\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right)} \right) - G_1^g}} \hfill \cr {\sum\limits_{m,s = 1}^n {P_{m,s}^g\left({{e^m},{e^s}} \right)U_t^m\left({\pi _t^m + \left({1 - {\alpha ^s}} \right)\pi _t^s} \right) - {C^m}\left({{e^m}} \right) \ge \overline {U_t^m}}} \hfill \cr {\sum\limits_{m,s = 1}^n {P_{m,s}^g\left({{e^m},{e^s}} \right)U_t^s\left({{\alpha ^s}\pi _t^s - p_0^mq_t^s - lq_t^s - F_t^sr} \right) - {C^s}\left({{e^s}} \right) \ge \overline {U_t^s}}} \hfill \cr {{e^m},{e^s} \in \mathop {\arg \max}\limits_{{e^{{m^*}}},\,{e^{{s^*}}}} \left\{{\sum\limits_{m,s = 1}^n {P_{m,s}^g\left({{e^m},{e^s}} \right)U_t^g\left({\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right)} \right) - G_1^G}} \right\}} \hfill \cr} Pm,sg P_{m,s}^g is the probability that the effort level of the supplier and retailer is observed to be em, en, respectively, for the financing platform. Since ω (F, r, t, ɛ) exists, we have the following equation: ω22=π2gr,ω24=π2gε;ω32=π3gr,ω34=π3gε {\omega _{22}} = {{\partial \pi _2^g} \over {\partial r}},\,{\omega _{24}} = {{\partial \pi _2^g} \over {\partial \varepsilon}};{\omega _{32}} = {{\partial \pi _3^g} \over {\partial r}},{\omega _{34}} = {{\partial \pi _3^g} \over {\partial \varepsilon}}

When F0sF0g F_0^s \le F_0^g , the retailer is not effectively constrained by financing. Combining Equation (14) and Equation (15) and using the Lagrangian function to solve it, we can get: ω22+ω32ω24+ω34=1εr {{{\omega _{22}} + {\omega _{32}}} \over {{\omega _{24}} + {\omega _{34}}}} = {{1 - \varepsilon} \over r} Pm,sg(em,es){αsptsS(qts)p0mqtslqtsr[p0m(qts+q0s)M0s]1ε}=Cs(es)+p0mqtsγs P_{m,s}^g\left({{e^m},{e^s}} \right)\left\{{{\alpha ^s}p_t^sS\left({q_t^s} \right) - p_0^mq_t^s - lq_t^s - {{r\left[{p_0^m\left({q_t^s + q_0^s} \right) - M_0^s} \right]} \over {1 - \varepsilon}}} \right\} = {C^s}\left({{e^s}} \right) + p_0^mq_t^s{\gamma ^s}

Equations (15) and (16) respectively represent the correlation between the unknown coefficient qts q_t^s , αs, ɛ, r determined in the financing platform and the supply chain game. We obtain the relevant unknown coefficients and decision-making results when the financing amount limited by the bank does not have substantial constraints. Simultaneous equation (8), equation (9), equation (15) and equation (16) and solve: {pts[1F(qts)]p0mr1ε=p0mγSm=1nPms(es)+(lc0m)ptsqtsf(qts)+(1αs)pts[1F(qts)]=c0mp0mω22+ω32ω24+ω34=1εrm,s=1nPm,sg(em,es){αsptsS(qts)p0mqtslqtsr[p0m(qts+q0s)M0s]1ε}=Cs(es)+p0mqtsγs \left\{{\matrix{{p_t^s\left[{1 - F\left({q_t^s} \right)} \right] - {{p_0^mr} \over {1 - \varepsilon}} = {{p_0^m{\gamma ^S}} \over {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)}}} + \left({l - c_0^m} \right)} \hfill \cr {p_t^sq_t^sf\left({q_t^s} \right) + \left({1 - {\alpha ^s}} \right)p_t^s\left[{1 - F\left({q_t^s} \right)} \right] = c_0^m - p_0^m} \hfill \cr {{{{\omega _{22}} + {\omega _{32}}} \over {{\omega _{24}} + {\omega _{34}}}} = {{1 - \varepsilon} \over r}} \hfill \cr {\sum\limits_{m,s = 1}^n {P_{m,s}^g\left({{e^m},{e^s}} \right)\left\{{{\alpha ^s}p_t^sS\left({q_t^s} \right) - p_0^mq_t^s - lq_t^s - {{r\left[{p_0^m\left({q_t^s + q_0^s} \right) - M_0^s} \right]} \over {1 - \varepsilon}}} \right\}} = {C^s}\left({{e^s}} \right) + p_0^mq_t^s{\gamma ^s}} \hfill \cr}} \right.

At time F0sF0g F_0^s \le F_0^g the game between suppliers and retailers in the supply chain and the games between financing platforms and suppliers and retailers are carried out simultaneously. We solve the system of equations (17) according to the specific values of the relevant variables. The relevant decision results are as follows: qts=F1(1p0mγsm=1nPms(es)ptsl+c0mptsp0m(ω24+ω34)pts(ω22+ω32))αs=1c0mp0mptsqtsf(qts)pts[1F(qts)] \matrix{{q_t^s = {F^{- 1}}\left({1 - {{p_0^m{\gamma ^s}} \over {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)p_t^s}}} - {{l + c_0^m} \over {p_t^s}} - {{p_0^m\left({{\omega _{24}} + {\omega _{34}}} \right)} \over {p_t^s\left({{\omega _{22}} + {\omega _{32}}} \right)}}} \right)} \hfill \cr {{\alpha ^s} = 1 - {{c_0^m - p_0^m - p_t^sq_t^sf\left({q_t^s} \right)} \over {p_t^s\left[{1 - F\left({q_t^s} \right)} \right]}}} \hfill \cr}

When the wholesale price and the sales price are a certain stable value, the order quantity under the optimal solution of the two sets of game relations can be determined simultaneously [10]. At this time, financing platforms and suppliers can obtain maximum benefits. At this point, the retailer gets the average industry rate of return. The following proposition gives the above conclusion:

Proposition 3: When the financing limit stipulated by the financing platform does not have a substantial impact (F0sF0g) \left({F_0^s \le F_0^g} \right) , the retailer can obtain all the required funds. At this time, the retailer's transaction decision results regarding its financing amount Fts F_t^s , the obtained income distribution coefficient αs, and the financing platform's initial margin rate ɛ and financing interest rate are as follows: qts=F1(1p0mγsm=1nPms(es)ptsl+c0mptsp0m(ω24+ω34)pts(ω22+ω32)) q_t^s = {F^{- 1}}\left({1 - {{p_0^m{\gamma ^s}} \over {\sum\limits_{m = 1}^n {P_m^s\left({{e^s}} \right)p_t^s}}} - {{l + c_0^m} \over {p_t^s}} - {{p_0^m\left({{\omega _{24}} + {\omega _{34}}} \right)} \over {p_t^s\left({{\omega _{22}} + {\omega _{32}}} \right)}}} \right) .

The income distribution coefficient αs is a function of the optimal order quantity. The pricing of the financing platform needs to meet the conditions in conclusion. According to formula (13), calculating the game results between the financing platform and the suppliers and retailers, the pricing decision of the financing platform must meet the conditions: ω22+ω32ω24+ω34=1εr {{{\omega _{22}} + {\omega _{32}}} \over {{\omega _{24}} + {\omega _{34}}}} = {{1 - \varepsilon} \over r}

Income distribution coefficient of third-party warehousing platforms

Warehousing companies have only two actions to choose from, and each action is a one-dimensional variable representing the level of effort. The probability that the financing platform observes that the warehousing enterprise's effort level is egl1 e_g^{l1} , egl2 e_g^{l2} , is Pgl1 P_g^{l1} , Pgl2 P_g^{l2} , respectively. According to different observation results, the financing platform pays different fees to warehousing companies [11]. Where δ is the discount rate, 0 < δ < 0. The reputation loss of the warehousing enterprise is L2g(egl2) L_2^g\left({e_g^{l2}} \right) due to the poor custody of the collateral. The regulatory cost of the financing platform is G2g G_2^g . The effort cost of the warehousing firm is Cl(egli) {C^l}\left({e_g^{li}} \right) , i = 1, 2. In addition, the cost of warehousing enterprises to construct and maintain the logistics information platform is Gl. In the first case, the utility function of the platform in the game between the platform and the warehousing enterprise is: Ug=Ug((1αl)[π1g(ω)+π2g(ω)+π3g(ω)G1g])G2g {U^g} = {U^g}\left({\left({1 - {\alpha ^l}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]} \right) - G_2^g

The utility functions of warehousing companies in the first two cases are: U1l=Ul(αl[π1g(ω)+π2gω+π3gωG1g])GlGl(egl2) U_1^l = {U_l}({\alpha ^l}[\pi _1^g(\omega) + \pi _2^g\omega + \pi _3^g\omega - G_1^g]){G^l} - {G^l}\left({e_g^{l2}} \right) U2l=Ul(δαl[π1g(ω)]+π2g(ω)+π3g(ω)G1g])GlGl(egl2)L2g(egl2) U_2^l = {U_l}(\delta {\alpha ^l}[\pi _1^g(\omega)] + \pi _2^g(\omega) + \pi _3^g(\omega) - G_1^g]) - {G^l} - {G^l}(e_g^{l2}) - L_2^g(e_g^{l2})

The platform hopes that warehousing companies can choose the optimal effort level egl1 e_g^{l1} . The execution of the contract by the warehousing company will allow the platform to reduce the probability of loss. U1l>U2l U_1^l > U_2^l can be known from equations (21) and (22). Profit-driven makes warehousing companies more inclined to choose action egl1 e_g^{l1} . In summary, the contract between the principal and the agent is given by the following objective function: maxαl,egl1g=1nPgl1(egl1)Ul{(1αl)[π1g(ω)+π2g(ω)+π3g(ω)G1g]}G2gg=1nPgl1(egl1)Ul{αl[π1g(ω)+π2g(ω)+π3g(ω)G1g])}GlCl(egl1)Ul¯egl1argmaxegl1*(g=1nPgl1(egl1)Ul{(1αl)[π1g(ω)+π2g(ω)+π3g(ω)G1g])G2g} \matrix{{\mathop {\max}\limits_{{\alpha ^l},e_g^{l1}} \sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right){U^l}\left\{{\left({1 - {\alpha ^l}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]} \right\} - G_2^g}} \hfill \cr {\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right){U_l}\left\{{\left. {{\alpha ^l}\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]} \right)} \right\} - {G^l} - {C^l}\left({e_g^{l1}} \right) \ge \overline {{U_l}}}} \hfill \cr {e_g^{l1} \in \mathop {\arg \max}\limits_{e_g^{l1*}} \left({\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right){U^l}\left\{{\left({1 - {\alpha ^l}} \right)\left. {\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]} \right) - G_2^g} \right\}}} \right.} \hfill \cr}

Among them, the retention utility of warehousing enterprises is reflected in the average income in the industry under the effort cost, namely: Ul¯=(Gl+Cl(egl1))γl \overline {{U_l}} = \left({{G^l} + {C^l}\left({e_g^{l1}} \right)} \right){\gamma ^l} . Solving equation (23), we get: α1l=[Gl+Cl(egl1)](1+γl)g=1nPgl1(egl1)[π1g(ω)+π2g(ω)+π3g(ω)G1g] \alpha _1^l = {{\left[{{G^l} + {C^l}\left({e_g^{l1}} \right)} \right]\left({1 + {\gamma ^l}} \right)} \over {\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]}}}

When αl takes the minimum value, the warehousing enterprise obtains the average income in the industry. The benefit distribution coefficient αl determined by the financing platform when formulating the incentive contract is an interval value [12]. The upper limit of A is the residual income that the platform has under the condition of obtaining the average income in the industry. g=1nPgl1(egl1)(1αl)[π1g(ω)+π2g(ω)+π3g(ω)G1g]G2g=(G1g+G2g)γl \sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right)\left({1 - {\alpha ^l}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right] - G_2^g = \left({G_1^g + G_2^g} \right){\gamma ^l}}

This formula can calculate the maximum value of αl that can be taken under the condition that the third-party warehousing platform can obtain the average income in the industry: α2l=1(G1g+G2g)γl+G2gg=1nPgl1(egl1)[π1g(ω)+π2g(ω)+π3g(ω)G1g] \alpha _2^l = 1 - {{\left({G_1^g + G_2^g} \right){\gamma ^l} + G_2^g} \over {\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]}}}

In summary, the following propositions can be obtained:

Proposition 4: The platform pays the warehousing enterprises corresponding rewards with the income distribution coefficient αl. αl is an interval value. When the distribution income obtained by the warehousing enterprise is the average income in the industry, it takes the minimum value. When the platform uses the income other than the average income in the industry to motivate warehousing enterprises, it takes the maximum value. Choose different incentive levels, namely αl[α1l,α2l] {\alpha ^l} \in \left[{\alpha _1^l,\,\alpha _2^l} \right] , where: α1l=1[(Gl+Cl)(egl1)](1+γl)g=1nPgl1(egl1)[π1g(ω)+π2g(ω)+π3g(ω)G1g],α2l=1(G1g+G2g)γl+G2gg=1nPgl1(egl1)[π1g(ω)+π2g(ω)+π3g(ω)G1g] \alpha _1^l = 1 - {{\left[{\left({{G^l} + {C^l}} \right)\left({e_g^{l1}} \right)} \right]\left({1 + {\gamma ^l}} \right)} \over {\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]}}},\,\alpha _2^l = 1 - {{\left({G_1^g + G_2^g} \right){\gamma ^l} + G_2^g} \over {\sum\limits_{g = 1}^n {P_g^{l1}\left({e_g^{l1}} \right)\left[{\pi _1^g\left(\omega \right) + \pi _2^g\left(\omega \right) + \pi _3^g\left(\omega \right) - G_1^g} \right]}}}

Numerical Simulation
Different financing options for retailers' funds and financing limits

According to formula (1), the data simulation of self-owned funds is carried out. Suppose that market demand θ follows a uniform distribution of [0.1000]. Suppliers determine wholesale prices p0m=16 p_0^m = 16 based on historical sales data and market estimates. The unit product manufacturing cost c0m=20 c_0^m = 20 , pts=28 p_t^s = 28 , s = 2. The average yield in the retailer's industry is 10%. From proposition 3 and formula (20), it can be known that the initial margin ratio and the financing interest rate determined by the financing platform must meet a certain ratio related to the loan contract. Let ɛ = 30%, r 8%. The corresponding changes in the amount of available financing are shown in Table 1.

The retailer's financing amount changes with its funds when there is no financing constraint

Serial number Private capital Financing amount Initial margin Own funds for self-ordering
1 250 1166.67 250 0
2 290 1266.67 290 0
3 400 1222.22 400 0
4 440 1451.71 425.42 0
5 500 1265.71 409.71 90.29
6 550 1294.29 299.29 161.71
7 650 1151.42 245.42 204.57
8 790 965.71 299.71 490.29
9 990 665.71 199.71 790.29
10 1200 265..71 109.71 1090.29
11 1456 0 0 1456
12 1500 0 0 1456
Comparative analysis of benefits on the supply chain under different financing options

When there is no substantial financing limit, the optimal order quantity qts=91 q_t^s = 91 , αs = 0.94 is at this time. When the correlation coefficient takes the above value, the optimal order quantity and the retailer's revenue-sharing coefficient value are determined according to the correlation equation in Proposition 3.

If the financing limit F0g1456 F_0^g \ge 1456 set by the financing platform is set, the financing limit does not constitute a substantial financing constraint on the retailer. At this time, the results shown in Table 2 can be obtained by comparing the retailer's income when financing and not participating in the online supply chain financing.

Benefit Comparison of Retailers Financing or Not Without Financing Constraints

Private capital Financing amount Expected sales Earnings from financing Earnings without financing
250 72 68 426.67 252
280 78 75 450.67 276
400 82 78 478.61 200
440 81 86 508.41 224
500 81 86 516.26 272
550 81 86 521.88 408
650 81 86 522.41 480
780 81 86 548.26 576
880 81 86 572.26 672
1200 81 86 586.22 800
1456 81 86 1082 1082
1500 81 86 1082 1082

At this time, from Proposition 3, it can be known that the revenue sharing coefficient available to suppliers is 0.06. When the revenue-sharing coefficient is constant, the available revenue of the supplier is consistent with the change of the revenue function realized by the retailer.

When the financing limit constitutes a financing constraint for the retailer, the retailer can only realize the suboptimal solution under the financing constraint.

The retailer's available financing and sub-optimal order volume change with its funds under financing constraints

Private capital Ideal financing amount Financing amount under financing limit Order amount
400 1333.33 1280 81
440 1451.43 1280 83
500 1365.71 1280 87
550 1294.29 1280 90

The retailer can only achieve sub-optimal order quantities when the financing limit has a substantially limiting effect. Out-of-stock losses make it impossible to achieve the maximum profit under the optimal order quantity.

Conclusion

The research of this paper explores the feasibility of the “online supply chain finance + revenue-sharing contract” business model. In this context, all parties involved can achieve symbiosis and co-prosperity. The transaction decision and benefit distribution coefficient value determined by the game share the value-added after the financing of the industrial chain is out of trouble. The research theory of the article provides a reference for the financing dilemma of industrial enterprises that needs to be solved in the process of financial system reform and development and industrial transformation and upgrading.

Benefit Comparison of Retailers Financing or Not Without Financing Constraints

Private capital Financing amount Expected sales Earnings from financing Earnings without financing
250 72 68 426.67 252
280 78 75 450.67 276
400 82 78 478.61 200
440 81 86 508.41 224
500 81 86 516.26 272
550 81 86 521.88 408
650 81 86 522.41 480
780 81 86 548.26 576
880 81 86 572.26 672
1200 81 86 586.22 800
1456 81 86 1082 1082
1500 81 86 1082 1082

The retailer's available financing and sub-optimal order volume change with its funds under financing constraints

Private capital Ideal financing amount Financing amount under financing limit Order amount
400 1333.33 1280 81
440 1451.43 1280 83
500 1365.71 1280 87
550 1294.29 1280 90

The retailer's financing amount changes with its funds when there is no financing constraint

Serial number Private capital Financing amount Initial margin Own funds for self-ordering
1 250 1166.67 250 0
2 290 1266.67 290 0
3 400 1222.22 400 0
4 440 1451.71 425.42 0
5 500 1265.71 409.71 90.29
6 550 1294.29 299.29 161.71
7 650 1151.42 245.42 204.57
8 790 965.71 299.71 490.29
9 990 665.71 199.71 790.29
10 1200 265..71 109.71 1090.29
11 1456 0 0 1456
12 1500 0 0 1456

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