Pricing decision problem in dual-channel supply chain based on experts’ belief degrees

Abstract

This paper considers a pricing decision problem in supply chain with traditional offline channel and e-commence online channel. In such supply chains, in face of highly changeable and unpredictable markets, for the lack of historical data, channel managers usually have to rely on belief degrees given by experienced experts to make pricing decisions. However, surveys have shown that these human estimations are generally take much wider ranges than they really take. Thus, uncertain measure is developed to deal with these human belief degrees and three uncertain programming models are employed to derive how channel members should make their pricing decisions under three power structures. Besides, analytical comparisons and numerical experiments are conducted to examine the effects of the power structures and experts’ estimations on the equilibrium prices and expected profits. It is revealed that the existence of dominant power, regardless of who holds the leadership, will hurt the efficiency of the channel by decreasing the profit of the whole supply chain. However, from the viewpoint of the individual firms, the firm gains the most profit as a leader while it gains the lowest as a follower. We also find that consumers will suffer from higher prices facing uncertain environment. The supply chain members may benefit from higher uncertainty level of their own costs, whereas the other channel members will gain less profits. Some other managerial highlights are also presented in this paper.

Appendices

Appendix 1

In this section, we will recall some important concepts and theorems of uncertainty theory for modeling the pricing decision problem with uncertain factors.

Let $\Gamma$ be a nonempty set, $\mathcal{L}$ a $\sigma$-algebra over $\Gamma$, and each element $\Lambda$ in $\mathcal{L}$ is called an event. Uncertain measure $\mathcal{M}$ is defined as a function from $\mathcal{L}$ to [0, 1].

Definition 1

(Liu 2007) The set function $\mathcal{M}$ is called an uncertain measure if it satisfies:

Axiom 1

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

Axiom 2

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

Axiom 3

(Subadditivity axiom) For every countable sequence of events $\Lambda _1, \Lambda _2,$ ..., we have:

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

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

Axiom 4

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

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

where $\Lambda _k$ 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 $(\Gamma ,\mathcal{L},\mathcal{M})$ 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

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

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

Definition 4

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

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

for any real number x.

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

Definition 5

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

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

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

Lemma 1

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

$$\begin{aligned} E[\xi ]=\int _0^{+\infty } (1-\varPhi (x))\mathrm{d}x-\int _{-\infty }^0 \varPhi (x)\mathrm{d}x. \end{aligned}$$

Lemma 2

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

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

Lemma 3

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

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

(37)

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

Appendix 2

1.1 Proof of Corollary 1

Referring to Table 2, we have:

$$\begin{aligned} p_\mathrm{e}^\mathrm{MD}=p_\mathrm{e}^\mathrm{RD}= & {} p_\mathrm{e}^\mathrm{ND},\quad p_\mathrm{r}^\mathrm{MD}=p_\mathrm{r}^\mathrm{RD}, \\ p_\mathrm{r}^\mathrm{MD}-p_\mathrm{r}^\mathrm{ND}= & {} \frac{D_\mathrm{r}}{12}. \end{aligned}$$

With the positive assumption, it can be easily obtained that $D_\mathrm{r}>0$; thus, $p_\mathrm{r}^\mathrm{MD}>p_\mathrm{r}^\mathrm{ND}$. Similarly, we can attain $m_\mathrm{r}^\mathrm{RD}>m_\mathrm{r}^\mathrm{ND}>m_\mathrm{r}^\mathrm{MD}$ and $w_\mathrm{r}^\mathrm{MD}>w_\mathrm{r}^\mathrm{ND}>w_\mathrm{r}^\mathrm{RD}$.

1.2 Proof of Corollary 2

The expected demands in the manufacturer-dominant case are as follows:

$$\begin{aligned} E[q_\mathrm{r}^\mathrm{MD}]= & {} E[\tilde{d}_\mathrm{r}]-E[\tilde{\beta }]p_\mathrm{r}^\mathrm{MD}+E[\tilde{\gamma }]p_\mathrm{e}^\mathrm{MD} =E[\tilde{d}_\mathrm{r}]-E[\tilde{\beta }]\frac{4E[\tilde{d}_\mathrm{r}]-D_\mathrm{r}}{4E[\tilde{\beta }]}+E[\tilde{\beta }]\frac{E[\tilde{\gamma }]}{E[\tilde{\beta }]}p_\mathrm{e}^\mathrm{MD} -E[\tilde{\gamma }]p_\mathrm{e}^\mathrm{MD}\nonumber \\= & {} \frac{E[\tilde{d}_\mathrm{r}]-(E[\tilde{\beta }^{1-\alpha }\tilde{s}_\mathrm{r}^{1-\alpha }]+E[\tilde{\beta }^{1-\alpha }\tilde{c}^{1-\alpha }])+(E[\tilde{\gamma }^{\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\gamma }^{\alpha }\tilde{s}_\mathrm{e}^{1-\alpha }])}{4};=\frac{D_\mathrm{r}}{4},\nonumber \\ E[q_\mathrm{e}^\mathrm{MD}]= & {} E[\tilde{d}_\mathrm{e}]-E[\tilde{\beta }]p_\mathrm{e}^\mathrm{MD}+E[\tilde{\gamma }]p_\mathrm{r}^\mathrm{MD} =E[\tilde{d}_\mathrm{e}]-\frac{E[\tilde{\beta }]^2-E[\tilde{\gamma }]^2}{E[\tilde{\beta }]}p_\mathrm{e}^\mathrm{MD}-E[\tilde{\gamma }]\frac{4E[\tilde{d}_\mathrm{r}]-D_\mathrm{r}}{4E[\tilde{\beta }]}\nonumber \\= & {} \frac{2E[\tilde{\beta }]D_\mathrm{e}+E[\tilde{\gamma }]D_\mathrm{r}+2(E[\tilde{\gamma }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\beta }^{1-\alpha }]-E[\tilde{\beta }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\gamma }^{\alpha }]}{4E[\tilde{\beta }]}. \end{aligned}$$

(38)

Since the sales prices of the two channels in the RD case and MD case are the same, the expected demands in the RD model can be attained similarly.

$$\begin{aligned}&E[q_\mathrm{r}^\mathrm{RD}]=\frac{D_\mathrm{r}}{4},E[q_\mathrm{e}^\mathrm{RD}]\nonumber \\&\quad =\frac{2E[\tilde{\beta }]D_\mathrm{e}+E[\tilde{\gamma }]D_\mathrm{r}+2(E[\tilde{\gamma }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\beta }^{1-\alpha }]-E[\tilde{\beta }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\gamma }^{\alpha }]}{4E[\tilde{\beta }]}.\nonumber \\ \end{aligned}$$

(39)

Similarly, we can get the expected ordering quantities in the ND model.

$$\begin{aligned} E[q_\mathrm{r}^\mathrm{ND}]= & {} E[\tilde{d}_\mathrm{r}]-E[\tilde{\beta }]p_\mathrm{r}^\mathrm{ND}+E[\tilde{\gamma }]p_\mathrm{e}^\mathrm{ND} =E[\tilde{d}_\mathrm{r}]-E[\tilde{\beta }]\frac{3E[\tilde{d}_\mathrm{r}]-D_\mathrm{r}+3E[\tilde{\gamma }]p_\mathrm{e}^{*}}{3E[\tilde{\beta }]}+E[\tilde{\beta }]\frac{E[\tilde{\gamma }]}{E[\tilde{\beta }]}p_\mathrm{e}^\mathrm{ND}-E[\tilde{\gamma }]p_\mathrm{e}^\mathrm{ND}\nonumber \\= & {} \frac{E[\tilde{d}_\mathrm{r}]-(E[\tilde{\beta }^{1-\alpha }\tilde{s}_\mathrm{r}^{1-\alpha }]+E[\tilde{\beta }^{1-\alpha }\tilde{c}^{1-\alpha }])+(E[\tilde{\gamma }^{\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\gamma }^{\alpha }\tilde{s}_\mathrm{e}^{1-\alpha }])}{3}=\frac{D_\mathrm{r}}{3},\nonumber \\ E[q_\mathrm{e}^\mathrm{ND}]= & {} E[\tilde{d}_\mathrm{e}]-E[\tilde{\beta }]p_\mathrm{e}^\mathrm{ND}+E[\tilde{\gamma }]p_\mathrm{r}^\mathrm{ND} =E[\tilde{d}_\mathrm{e}]-\frac{E[\tilde{\beta }]^2-E[\tilde{\gamma }]^2}{E[\tilde{\beta }]}p_\mathrm{e}^\mathrm{ND}-E[\tilde{\gamma }]\frac{3E[\tilde{d}_\mathrm{r}]-D_\mathrm{r}}{3E[\tilde{\beta }]}\nonumber \\= & {} \frac{3E[\tilde{\beta }]D_\mathrm{e}+ E[\tilde{\gamma }]D_\mathrm{r}+3(E[\tilde{\gamma }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\beta }^{1-\alpha }] -E[\tilde{\beta }]E[\tilde{s}_\mathrm{r}^{1-\alpha }\tilde{\gamma }^{\alpha }])}{6E[\tilde{\beta }]}. \end{aligned}$$

(40)

Then, it can be easily attained that

$$\begin{aligned} E[q_\mathrm{r}^\mathrm{MD}]-E[q_\mathrm{r}^\mathrm{ND}]= & {} E[q_\mathrm{r}^\mathrm{RD}]-E[q_\mathrm{r}^\mathrm{ND}]=-\frac{D_\mathrm{r}}{12}<0,\\ E[q_\mathrm{e}^\mathrm{MD}]-E[q_\mathrm{e}^\mathrm{ND}]= & {} E[q_\mathrm{e}^\mathrm{RD}]-E[q_\mathrm{e}^\mathrm{ND}]=\frac{D_\mathrm{r}}{12E[\tilde{\beta }]}>0. \end{aligned}$$

That is

$$\begin{aligned} E[q_\mathrm{e}^\mathrm{MD}]= & {} E[q_\mathrm{e}^\mathrm{RD}]>E[q_\mathrm{e}^\mathrm{ND}],\\ E[q_\mathrm{r}^\mathrm{MD}]= & {} E[q_\mathrm{r}^\mathrm{RD}]<E[q_\mathrm{r}^\mathrm{ND}]. \end{aligned}$$

1.3 Proof of Corollary 3

Since the sales prices and ordering quantities of the two channels in the MD model and RD model are the same, we can easily obtain $E[\pi _{t}^\mathrm{MD}]=E[\pi _{t}^\mathrm{RD}]$. From the results gained from the three models, we have

$$\begin{aligned}&E[\pi _{t}^\mathrm{ND}]-E[\pi _{t}^\mathrm{RD}]=E[(p_\mathrm{e}^\mathrm{ND}-\tilde{c}-\tilde{s}_\mathrm{e})(\tilde{d}_\mathrm{e}-\tilde{\beta } p_\mathrm{e}^\mathrm{ND} +\tilde{\gamma } p_\mathrm{r}^\mathrm{ND}) \quad +(p_\mathrm{r}^\mathrm{ND}-\tilde{c}-\tilde{s}_\mathrm{r})(\tilde{d}_\mathrm{r}-\tilde{\beta } p_\mathrm{r}^\mathrm{ND}+\tilde{\gamma } p_\mathrm{e}^\mathrm{ND})]\nonumber \\&\quad -E[(p_\mathrm{e}^\mathrm{RD}-\tilde{c}-\tilde{s}_\mathrm{e})(\tilde{d}_\mathrm{e}-\tilde{\beta } p_\mathrm{e}^\mathrm{RD} +\tilde{\gamma } p_\mathrm{r}^\mathrm{RD}) +(p_\mathrm{r}^\mathrm{RD}-\tilde{c}-\tilde{s}_\mathrm{r})(\tilde{d}_\mathrm{r}-\tilde{\beta } p_\mathrm{r}^\mathrm{RD}+\tilde{\gamma } p_\mathrm{e}^\mathrm{RD})]\nonumber \\&=p_\mathrm{e}^\mathrm{ND}E[q_\mathrm{e}^\mathrm{ND}]-p_\mathrm{e}^\mathrm{RD}E[q_\mathrm{e}^\mathrm{RD}]+p_\mathrm{r}^\mathrm{ND}E[q_\mathrm{r}^\mathrm{ND}]-p_\mathrm{r}^\mathrm{RD}E[q_\mathrm{r}^\mathrm{RD}]+(E[\tilde{\beta }^{1-\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\beta }^{1-\alpha }\tilde{s}_\mathrm{r}^{1-\alpha }]-E[\tilde{\gamma }^{\alpha }\tilde{c}^{1-\alpha }]\nonumber \\&\quad +E[\tilde{\gamma }^{\alpha }\tilde{s}_\mathrm{e}^{1-\alpha }])(p_\mathrm{r}^\mathrm{ND}-p_\mathrm{r}^\mathrm{RD})=\frac{(3E[\tilde{d}_\mathrm{r}]-D_\mathrm{r})D_\mathrm{r}}{9E[\tilde{\beta }]}-\frac{(4E[\tilde{d}_\mathrm{r}]-D_\mathrm{r})D_\mathrm{r}}{16E[\tilde{\beta }]}\nonumber \\&\quad -\frac{(E[\tilde{\beta }^{1-\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\beta }^{1-\alpha }\tilde{s}_\mathrm{r}^{1-\alpha }] -E[\tilde{\gamma }^{\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\gamma }^{\alpha }\tilde{s}_\mathrm{e}^{1-\alpha }])D_\mathrm{r}}{12E[\tilde{\beta }]}\nonumber \\&=\frac{(12E[\tilde{d}_\mathrm{r}]-7D_\mathrm{r})D_\mathrm{r}-12(E[\tilde{\beta }^{1-\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\beta }^{1-\alpha }\tilde{s}_\mathrm{r}^{1-\alpha }]-E[\tilde{\gamma }^{\alpha }\tilde{c}^{1-\alpha }]+E[\tilde{\gamma }^{\alpha }\tilde{s}_\mathrm{e}^{1-\alpha }])D_\mathrm{r}}{144E[\tilde{\beta }]}\nonumber \\&=\frac{5D_\mathrm{r}^{2}}{144E[\tilde{\beta }]}>0. \end{aligned}$$

(41)

Thus, we can conclude that $E[\pi _{t}^\mathrm{ND}]>E[\pi _{t}^\mathrm{RD}]=E[\pi _{t}^\mathrm{MD}]$.

Similarly, we can attain

$$\begin{aligned} E[\pi _\mathrm{r}^\mathrm{ND}]-E[\pi _\mathrm{r}^\mathrm{MD}]= & {} \frac{7D_\mathrm{r}^{2}}{144E[\tilde{\beta }]}>0, \\ E[\pi _\mathrm{r}^\mathrm{ND}]-E[\pi _\mathrm{r}^\mathrm{RD}]= & {} \frac{D_\mathrm{r}^{2}}{72E[\tilde{\beta }]}>0,\\ E[\pi _\mathrm{m}^\mathrm{ND}]-E[\pi _\mathrm{m}^\mathrm{MD}]= & {} \frac{D_\mathrm{r}^{2}}{72E[\tilde{\beta }]}>0,\\ E[\pi _\mathrm{m}^\mathrm{ND}]-E[\pi _\mathrm{m}^\mathrm{RD}]= & {} \frac{3D_\mathrm{r}^{2}}{144E[\tilde{\beta }]}>0. \end{aligned}$$

That is,

$$\begin{aligned}&E[\pi _\mathrm{m}^\mathrm{MD}]>E[\pi _\mathrm{m}^\mathrm{ND}]>E[\pi _\mathrm{m}^\mathrm{RD}],\quad E[\pi _\mathrm{r}^\mathrm{RD}]>E[\pi _\mathrm{r}^\mathrm{ND}]\\&>E[\pi _\mathrm{r}^\mathrm{MD}],\quad E[\pi _{t}^\mathrm{MD}]=E[\pi _{t}^\mathrm{RD}]>E[\pi _{t}^\mathrm{ND}]. \end{aligned}$$