# 随机需求下考虑配送时间的第三方物流服务决策Third-Party Logistics Service Decision Considering Delivery Time with Stochastic Demand

DOI: 10.12677/MSE.2019.81007, PDF, HTML, XML, 下载: 376  浏览: 870

Abstract: The paper establishes a service decision model of two levels supply chain secondary system com-posed of a retailer and a 3PL enterprise. The optimal decision of retailers and third-party logistics service providers considering delivery time under stochastic demand information is studied. Through numerical analysis, the mode which third-party logistics companies participate in man-aging retailers’ logistics has an influence on profit of retailers and third-party logistics enterprises under the time-sensitive random demand. The study found that the greater the uncertainty of demand, the third-party logistics companies and retailers are more inclined to make conservative decisions, and the profit will gradually increase with the uncertainty in a certain stage. In addition, this paper also proves that the proportion of the risk-sharing loss has no effect on the optimal de-livery time, but it has a positive impact on the retailer’s service order quantity and profit, and has a negative impact on the third-party logistics service provider’s profit.

1. 引言

2. 第三方物流服务决策模型

2.1. 模型描述与假设

d：商品的需求量

q：商品的物流服务订购量，决策变量

w：商品的单位批发价格

t：商品的配送时间，决策变量

l：商品的物流服务价格

$\theta$ ：商品积压时，第三方物流企业赔偿损失的比例

p：商品的单位销售价格

2.2. 零售商的利润模型

① 当需求量小于订购量，即 $d\le q$ 时，由于需求量不足，商品滞销出现积压。在不考虑其他费用且积压产品残值为0的情况下，企业共损失 $\left(q-d\right)\left(w+l\right)$ 。此时，第三方物流服务商按照双方商定的赔偿比例 $\theta$ 进行赔偿，赔偿总额为 $\theta \left(q-d\right)\left(w+l\right)$

② 当需求量大于订购量，即 $d>q$ 时，产品订购不足。但是订购量过少也使得第三方物流服务商失去了一部分收入，此时，第三方物流企业不赔偿损失。

${\pi }_{R}=\left\{\begin{array}{l}dp-q\left(w+l\right)+\theta \left(q-d\right)\left(w+l\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d\le q\\ qp-q\left(w+l\right)-s\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d>q\end{array}$ (1)

${\pi }_{R}=\left\{\begin{array}{l}q\left[p-\left(w+l\right)\right]-\left(q-d\right)\left[p-\theta \left(w+l\right)\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\epsilon \le q-\left(a-bt\right)\\ qp-q\left(w+l\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon >q-\left(a-bt\right)\end{array}$ (2)

$E{\pi }_{R}=q\left[p-\left(w+l\right)\right]-\left[p-\theta \left(w+l\right)\right]{\int }_{-\infty }^{q-\left(a-bt\right)}\left(q-d\right)f\left(\epsilon \right)\text{d}\epsilon$ (3)

2.3. 第三方物流企业的利润模型

① 当需求量小于订购量，即 $d\le q$ 时，由于需求量不足，商品滞销出现积压。在不考虑其他费用且积压产品残值为0的情况下，第三方物流服务商赔偿总额为 $\theta \left(q-d\right)\left(w+l\right)$

② 当需求量大于订购量，即 $d>q$ 时，产品订购不足。第三方物流企业不赔偿损失。

${\pi }_{S}=\left\{\begin{array}{l}\left(l-\alpha -\frac{\beta }{t}\right)q-\theta \left(q-d\right)\left(w+l\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d\le q\\ \left(l-\alpha -\frac{\beta }{t}\right)q\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}d>q\end{array}$ (4)

${\pi }_{S}=\left\{\begin{array}{l}\left(l-\alpha -\frac{\beta }{t}\right)q-\theta \left(q-d\right)\left(w+l\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon \le q-\left(a-bt\right)\\ \left(l-\alpha -\frac{\beta }{t}\right)q\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\epsilon >q-\left(a-bt\right)\end{array}$ (5)

$E{\pi }_{S}=\left(l-\alpha -\frac{\beta }{t}\right)q-\theta \left(w+l\right){\int }_{-\infty }^{q-\left(a-bt\right)}\left(q-d\right)f\left(\epsilon \right)\text{d}\epsilon$ (6)

2.4. 最优化分析

$G\left(q,t\right)={\int }_{-\infty }^{q-\left(a-bt\right)}\left(q-d\right)f\left(\epsilon \right)\text{d}\epsilon$$G\left(q,t\right)$ 表示需求量小于订购量时商品需求的期望值。显然 $G\left(q,t\right)\ge 0$ ，进一步，

$\begin{array}{c}G\left(q,t\right)={\int }_{-\infty }^{q-\left(a-bt\right)}\left(q-d\right)f\left(\epsilon \right)\text{d}\epsilon \\ =\left[q-\left(a-bt\right)\right]F\left[q-\left(a-bt\right)\right]-{\int }_{-\infty }^{q-\left(a-bt\right)}\epsilon f\left(\epsilon \right)\text{d}\epsilon \end{array}$ (7)

$G\left(q,t\right)$ 代入公式(3) (6)中，得到式(8)和(9)：

$E{\pi }_{R}=q\left[p-\left(w+l\right)\right]-\left[p-\theta \left(w+l\right)\right]G\left(q,t\right)$ (8)

$E{\pi }_{S}=\left(l-\alpha -\frac{\beta }{t}\right)q-\theta \left(w+l\right)G\left(q,t\right)$ (9)

$S=E{\pi }_{R}+E{\pi }_{S}$ 代表零售商和第三方物流服务商的利润总和。

$F\left(q-a+bt\right)=\frac{q-a+bt+\lambda }{2\lambda }$ (10)

$G\left(q,t\right)=\frac{{\left(q-a+bt+\lambda \right)}^{2}}{4\lambda }$ (11)

${{G}^{\prime }}_{q}\left(q,t\right)=F\left(q-a+bt\right)=\frac{q-a+bt+\lambda }{2\lambda }$ (12)

${{G}^{\prime }}_{t}\left(q,t\right)=bF\left(q-a+bt+\lambda \right)=\frac{b\left(q-a+bt+\lambda \right)}{2\lambda }$ (13)

${q}^{*}=\frac{2\lambda \left[p-\left(w+l\right)\right]}{\left[p-\theta \left(w+l\right)\right]}+\left(a-bt-\lambda \right)$

$\begin{array}{c}\frac{\partial E{\pi }_{R}}{\partial q}=\left[p-\left(w+l\right)\right]-\left[p-\theta \left(w+l\right)\right]{{G}^{\prime }}_{q}\left(q,t\right)\\ =\left[p-\left(w+l\right)\right]-\frac{\left[p-\theta \left(w+l\right)\right]\left(q-a+bt+\lambda \right)}{2\lambda }\end{array}$

$\frac{{\partial }^{2}E{\pi }_{R}}{\partial {q}^{2}}=-\frac{\left(p-\theta \left(w+l\right)\right)}{2\lambda }<0$

${t}^{*}=\sqrt{\frac{2\beta \lambda \left(p-w-l\right)+\beta \left(a-\lambda \right)}{b\left(l-\alpha \right)}}$

$\begin{array}{c}E{\pi }_{S}=\left(l-\alpha -\frac{\beta }{t}\right){q}^{*}-\theta \left(w+l\right)G\left({q}^{*},t\right)\\ =\left(l-\alpha -\frac{\beta }{t}\right)\left\{\frac{2\lambda \left[p-\left(w+l\right)\right]}{\left[p-\theta \left(w+l\right)\right]}+\left(a-bt-\lambda \right)\right\}-\theta \left(w+l\right){\left[\frac{p-\left(w+l\right)}{p-\theta \left(w+l\right)}\right]}^{2}\end{array}$ (14)

$\frac{\partial E{\pi }_{S}}{\partial t}=\frac{\beta }{{t}^{2}}\left\{\frac{2\lambda \left[p-\left(w+l\right)\right]}{\left[p-\theta \left(w+l\right)\right]}+\left(a-bt-\lambda \right)\right\}-b\left(l-\alpha -\frac{\beta }{t}\right)$

$\frac{\partial {E}^{2}{\pi }_{S}}{\partial {t}^{2}}=-\frac{2\beta }{{t}^{3}}\left\{\frac{2\lambda \left[p-\left(w+l\right)\right]}{\left[p-\theta \left(w+l\right)\right]}+\left(a-bt-\lambda \right)\right\}-\frac{2b\beta }{{t}^{2}}<0$

${t}^{*}=\sqrt{\frac{2\beta \lambda \left(p-w-l\right)+\beta \left(a-\lambda \right)}{b\left(l-\alpha \right)}}$

${t}^{*}$ 代入到 ${q}^{*}$ 中得：

${q}^{*}=\frac{2\lambda \left(p-w-l\right)}{p-\theta \left(w+l\right)}+\left(a-\lambda \right)-\sqrt{\frac{2\beta b\lambda \left(p-w-l\right)+\beta b\left(a-\lambda \right)}{l-\alpha }}$

3. 数值算例

Table 1. Optimal decision-making results of two-level supply chain under different demand uncertainties

$p=50,l=5,w=10,a=100,b=5,\theta =2,\alpha =1,\beta =0.3,\lambda =50$

Table 2. Optimal decision-making results of two-level supply chain under different risk sharing ratios

4. 结论

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