Order allocation of logistics service supply chain with fairness concern and demand updating: model analysis and empirical examination

Abstract

Regarding a two-echelon supply chain consisting of a logistics service integrator (LSI) and several functional logistics service providers (FLSPs), this paper establishes a two-stage order allocation model considering demand updating and the FLSPs’ fairness preferences. This model is a multi-objective programming model, whose goal is to maximize profits of the LSI and the total utility of FLSPs. The ideal point method is used to obtain the optimal solution. In the numerical example, the impacts of FLSPs’ behavioral parameters and demand update parameters on the order allocation in the social services network are discussed. Besides, multi-methodological method is used to verify the theoretical perspectives through an empirical study of Tianjin SND Logistics Company. Our study obtains a few important conclusions. For example, when demand of the second stage is updated, there is an optimal updating time maximizing the supply chain performance. Increased demand of the second stage results in greater supply chain performance. When the demand during the second stage decreases, the bigger the difference of the fairness preference coefficients among FLSPs, the greater the LSI’s profits and the lower the FLSPs’ total utility will be. However, the difference of the fairness preference coefficients among FLSPs has little influence on the LSI’s profits and total utility of the FLSPs, when the demand during the second stage increases.

Appendix

Referring to Cheng (2012), the proof is listed as follows.

In this proof, we combine the Bayesian updating theory and lead time. If the lead time is $T_0$, retailer orders at $T_0$ which is the normal lead time. When the lead time is $T_n$, the retailer orders at $T_n$ which is the shortest lead time. The practical lead time point t can take values from $T_0$ to $T_n$ when demand will be updated. Sales point is 0, and the below timeline chart is shown in Fig. 20.

Fig. 20

The expressions of lead time

Bayesian demand forecast modifies the demand information through mixing the collected information samples into the existing information and updating the demand distribution. Collected information can be derived from market signal sent from pre-seasonal product (Choi et al. 2004). We assume the market demand obeys normal distribution of $\theta$, which represents the mathematical expectation of demand, so the conditional distribution of demand is $p\left( {\left. X \right| \theta } \right) \sim N\left( {\theta , \sigma ^{2}} \right)$, $P\left( {\left. X \right| \theta } \right) =\left( {2\pi \theta ^{2}} \right) ^{-n/2}\exp \left\{ {-\frac{1}{2\theta ^{2}}\mathop {\sum }\limits _{i=1}^n {\left( {x_i -\theta } \right) ^{2}} } \right\}$, in which $\sigma ^{2}$ is known. Bayesian view is that any unknown quantity can be regarded as random variables, described with probability distribution, which is called prior distribution. So we can obtain the prior distribution of $\theta$ according to the experience and data information before demand update (Choi et al. 2015).

Variance is a reflection of the prediction accuracy, so it has a certain relationship with updated lead time, namely the later the demand update and the smaller the lead time, the more fully the collected information, and the smaller the variance should be. We assume the relationship between prediction accuracy $\tau$ and t is $\tau (t)=e^{bt}$, and $\theta$ also obeys the normal distribution, namely when the lead time of demand update is t, the mean of $\theta$ is $\mu$, the variance is $\tau ^{2}(t)$, then the prior distribution of $\theta$ is $f\left( \theta \right) \sim N(\mu ,\tau ^{2}(t))$. $\pi \left( \theta \right) =\left( {2\pi \tau (T_0 )^{2}} \right) ^{-1/2}\exp \left\{ {-\frac{1}{2\tau (T_0 )^{2}}\left( {\theta -\mu } \right) ^{2}} \right\}$.

Joint distribution of X and $\theta$ can be written as

$$\begin{aligned} h\left( {X,\theta } \right) =k_1 \cdot \exp \left\{ {-\frac{1}{2} \left[ {\frac{n\theta ^{2}-2n\theta \bar{x} +\mathop {\sum }\nolimits _{i=1}^n {{x^{2}}_i } }{\sigma ^{2}}+\frac{\theta ^{2}-2\theta \mu +\mu ^{2}}{\tau (t)^{2}}} \right] } \right\} \end{aligned}$$

In which, $\bar{x} =\frac{1}{n}\mathop {\sum }\limits _{i=1}^n {x_i } ,k_1 =\left( {2\pi } \right) ^{-(n+1)/2\tau (t)^{-1}\sigma ^{-n}}$.

If $A=\frac{n}{\sigma ^{2}}+\frac{1}{\tau ^{2}(t)},B=\frac{n\bar{x} }{\sigma ^{2}}+\frac{\mu }{\tau ^{2}(t)},C=\frac{\mathop {\sum }\limits _{i=1}^n {{x^{2}}_i } }{\sigma ^{2}}+\frac{\mu ^{2}}{\tau ^{2}(t)}$ then,

$$\begin{aligned} h\left( {X,\theta } \right)= & {} k_1 \cdot \exp \left\{ {-\frac{1}{2}\left[ {A\theta ^{2}-2B\theta +C} \right] } \right\} \nonumber \\= & {} k_1 \cdot \exp \left\{ {-\frac{\left( {\theta -B/A} \right) ^{2}}{2/A}-\frac{1}{2}\left( {C-\frac{B^{2}}{A}} \right) } \right\} \end{aligned}$$

Note that A, B and C are all irrelevant with ${\uptheta }$, then the marginal density function of sample can be calculated.

$$\begin{aligned} m(X)=\int _{-\infty }^{+\infty } {h(X,\theta )d\theta } =k_1 \cdot \exp \left\{ {-\frac{1}{2}\left( {C-\frac{B^{2}}{A}} \right) } \right\} \left( {{2\pi }/A} \right) ^{1/2} \end{aligned}$$

Posteriori distribution can be obtained by Bayesian formula.

$$\begin{aligned} \pi \left( {\theta |X} \right) =\frac{h\left( {X,\theta } \right) }{m\left( X \right) }=\left( {{2\pi }/A} \right) ^{1/2}\exp \left\{ {-\frac{1}{2/A}\left( {\theta -B/A} \right) ^{2}} \right\} \end{aligned}$$

This shows that in a given sample, posteriori distribution of $\theta$ is $N\left( {B/{A,1/A}} \right)$, namely

$$\begin{aligned} \theta /M\sim N\left( {\frac{n\bar{x} \sigma ^{-2}+\mu \tau ^{-2}(t)}{n\sigma ^{-2}+\tau ^{-2}(t)},\frac{1}{n\sigma ^{-2}+\tau ^{-2}(t)}} \right) \end{aligned}$$

When the lead time of demand update is t, estimation of the demand according to the demand information collected during $T_0 \sim t$ before the second stage is $\xi$, namely $\bar{x} =\xi ,n=1$. Hence, $\mu (\xi )=\frac{\sigma ^{-2}}{\sigma ^{-2}+\tau ^{-2}(t)}\xi +\frac{\tau ^{-2}(t)}{\sigma ^{-2}+\tau ^{-2}(t)}=\frac{\sigma ^{2}\mu +\tau ^{2}(t)\xi }{\sigma ^{2}+\tau ^{2}(t)}\upsilon ^{2}(t)=\frac{1}{\sigma ^{-2}+\tau ^{-2}(t)}$.

The posteriori distribution of $\theta$ is $f\left( {\theta \left| \xi \right. } \right) \sim N(\mu (\xi ),\upsilon ^{2}(t))$. The demand distribution is $m\left( {X\left| \xi \right. } \right) \sim N\left( {\mu \left( \xi \right) ,\sigma ^{2}+\upsilon ^{2}(t)} \right)$.