Dual-channel supply chain competition with channel preference and sales effort under uncertain environment

Abstract

In this paper, we investigate a dual-channel supply chain under uncertain environment. Channel preference and sales effort are taken into account to explore their effects on supply chain members’ profits with uncertain information. Then we analyze the dual-channel supply chain in centralized and decentralized cases, and give closed-form expressions for equilibriums in the two cases. A series of numerical experiments are implemented to examine the impacts of uncertainty distributions of parameters on supply chain profits. We conclude that the total profit of the supply chain in the centralized case is invariably higher than that in the decentralized case under different uncertainty degrees of these parameters. It is shown that the supplier’s profit first decreases then increases as the expected value of customers’ preference to the direct channel increases, and the retailer’s profit decreases with the increase of the expected value of customers’ preference to the direct channel. In addition, the results indicate that the total profits in the centralized and decentralized cases, the supplier’s profit and the retailer’s profit all increase when the expected value of the retailer’s sales effort elasticity increases.

Appendix

Uncertainty theory, founded by Liu (2007), is a branch of axiomatic mathematics based on normality, duality, subadditivity and product measure axioms.

Definition 1

(Liu 2007) Let $\Gamma$ be a nonempty set and $\mathcal {L}$ a $\sigma$-algebra over $\Gamma$. The set function M is called an uncertain measure if it satisfies:

Axiom 1

(Normality Axiom) ${M}\{\Gamma \}=1$ for the universal set $\Gamma$.

Axiom 2

(Duality Axiom) ${M}\{\Lambda \}+{M}\{\Lambda ^{c}\}=1$ for any event $\Lambda$.

Axiom 3

(Subadditivity Axiom) For every countable sequence of events $\{\Lambda _{i}\}$, $i = 1,2,\ldots$, we have

$$\begin{aligned} {M}\left\{ \bigcup _{i=1}^{\infty }\Lambda _{i}\right\} \le \sum _{i=1}^{\infty }{M}\{\Lambda _{i}\}. \end{aligned}$$

Besides, the product uncertain measure on the product $\sigma$-algebra $\mathcal {L}$ was defined by Liu (2009) as follows:

Axiom 4

(Product Axiom) Let $(\Gamma _k,\mathcal {L}_{k},{M}_{k})$ be uncertainty spaces for $k=1,2,\ldots$ The product uncertain measure M is an uncertain measure satisfying

$$\begin{aligned} {M}\left\{ \prod \limits _{k=1}^\infty A_{k}\right\} =\bigwedge _{k=1}^\infty {M}_{k}\{A_{k}\} \end{aligned}$$

where $A_{k}$ are arbitrarily chosen events from $\mathcal {L}_{k}$ for $k=1,2,\ldots$, respectively.

Definition 2

(Liu 2007) An uncertain variable is a measurable function $\xi$ from an uncertainty space $(\Gamma ,\mathcal {L},{M})$ to the set of real numbers, i.e., for any Borel set B of real numbers, the set

$$\begin{aligned} \{\xi \in B\}=\{\gamma \in \Gamma \ \big |\ \xi (\gamma )\in B\} \end{aligned}$$

is an event.

Definition 3

(Liu 2009) The uncertain variables $\xi _1,\xi _2,\ldots ,\xi _n$ are said to be independent if

$$\begin{aligned} {M}\left\{ \bigcap _{i=1}^n(\xi _i\in B_i)\right\} =\bigwedge _{i=1}^n {M}\{\xi _i\in B_i\} \end{aligned}$$

for any Borel sets $B_1, B_2,\ldots , B_n$.

Definition 4

(Liu 2007) The uncertainty distribution $\Phi$ of an uncertain variable $\xi$ is defined by

$$\begin{aligned} \Phi (x)={M}\{\xi \le x\} \end{aligned}$$

for any real number x.

An uncertainty distribution $\Phi$ is referred to be regular if its inverse function $\Phi ^{-1}(\alpha )$ exists and is unique for each $\alpha \in [0,1]$

Lemma 1

(Liu 2010) Let $\xi _1, \xi _2, \ldots , \xi _n$ be independent uncertain variables with regular uncertainty distributions $\Phi _1, \Phi _2, \ldots , \Phi _n$, respectively. If the function $f(x_1,x_2,\ldots ,x_n)$ is strictly increasing with respect to $x_1,x_2,\ldots , x_m$ and strictly decreasing with respect to $x_{m+1},x_{m+2},\ldots ,x_n$, then

$$\begin{aligned} \xi =f(\xi _1, \xi _2, \ldots , \xi _n) \end{aligned}$$

is an uncertain variable with inverse uncertainty distribution

$$\begin{aligned} \Phi ^{-1}(\alpha )=f(\Phi _1^{-1}(\alpha ),\ldots ,\Phi _m^{-1}(\alpha ),\Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _n^{-1}(1-\alpha )). \end{aligned}$$

Definition 5

(Liu 2007) Let $\xi$ be an uncertain variable. The expected value of $\xi$ is defined by

$$\begin{aligned} E[\xi ]=\int _0^{+\infty }{M}\{\xi \ge r\}d r-\int _{-\infty }^0{M}\{\xi \le r\} d r \end{aligned}$$

provided that at least one of the above two integrals is finite.

Lemma 2

(Liu 2010) Let $\xi$ be an uncertain variable with regular uncertainty distribution $\Phi$. If the expected value exists, then

$$\begin{aligned} E[\xi ]=\int _0^1 \Phi ^{-1}(\alpha )d \alpha . \end{aligned}$$

Lemma 3

(Liu and Ha 2010) Let $\xi _{1}, \xi _{2}, \ldots ,\xi _{n}$ be independent uncertain variables with regular uncertainty distributions $\Phi _{1},\Phi _{2}, \ldots ,\Phi _{n}$, respectively. A function $f(x_{1},x_{2},\ldots ,x_{n})$ is strictly increasing with respect to $x_{1},x_{2},\ldots ,x_{m}$ and strictly decreasing with respect to $x_{m+1},x_{m+2},\ldots ,x_{n}$. Then the expected value of $\xi =f(\xi _{1},\xi _{2},\ldots , \xi _{n})$ is

$$\begin{aligned} \text {E}[\xi ]=\int _{0}^{1} f(\Phi _{1}^{-1}(\alpha ),\ldots ,\Phi _{m}^{-1}(\alpha ),\Phi _{m+1}^{-1}(1-\alpha ),\ldots ,\Phi _{n}^{-1}(1-\alpha )))\mathrm {d}\alpha \end{aligned}$$

provided that the expected value $E[\xi ]$ exists.