Optimal pricing decision for supply chains with risk sensitivity and human estimation

Abstract

In reality, consumption markets are changeable and uncertain in terms of consumer demands, product manufacturing costs, sales costs and so on. When facing these markets, the small and medium manufacturers are usually weak and risk sensitive vis-à-vis their counterpart power retailers, like Walmart, Carrefour, and TESCO. Therefore, human estimations on uncertain market information given by experienced experts and risk sensitivity of the weak manufacturers are of great significance for the optimal pricing strategies. Accordingly, this paper investigates the pricing decision problem in an uncertain supply chain where two risk-sensitive manufacturers distribute substitutable products into the same market through a common dominant and risk-neutral retailer. Uncertainty theory is employed to deal with human-estimation information and chance-constrained programming models are proposed to formulate the pricing decision problem with risk sensitivity. Moreover, numerical experiments are provided to validate the effectiveness of the proposed model. Interestingly, we show that the results critically depend on the comparison between how much the demands and manufacturing costs of two competing products change with the manufacturers’ risk-sensitivity levels. Correspondingly, based on the estimated uncertainty distributions of uncertain variables by experts, the weak manufacturers are suggested to adjust their risk-sensitivity levels strategically for their profitability.

Appendix

To model pricing decision problem with uncertain variables, we introduce some important concepts and theorems on uncertain variable. Let $\Gamma$ be a nonempty set and ${\mathcal {L}}$ a $\sigma$-algebra over $\Gamma$. Each element $\Lambda$ in ${\mathcal {L}}$ is called an event.

Definition 1

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

Axiom 1

(Normality Axiom) ${\mathcal {M}}\{\Gamma \}=1$.

Axiom 2

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

Axiom 3

(Subadditivity Axiom) For every countable sequence of events $\{\Lambda _{i}\}$, $i = 1,2,\ldots$, 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 [27] as follows:

Axiom 4

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

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

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

Definition 2

([26]) 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

([27]) 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

([26]) The uncertainty distribution $\Phi$ of an uncertain variable $\xi$ is defined by

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

for any real number x.

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

Lemma 1

([28]) 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

([26]) Let $\xi$ be an uncertain variable. The expected value of $\xi$ is defined by

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

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

Lemma 2

([28]) 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

([30]) 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

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

(25)

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