Possibilistic cooperative advertising and pricing games for a two-echelon supply chain

Abstract

This paper addresses coordination of pricing and cooperative advertising policies in a two-echelon supply chain under fuzziness of demand function’s parameters and manufacturing costs. Three different decentralized scenarios are introduced with regard to the players’ market power: (1) manufacturer-Stackelberg game where the manufacturer has the dominant power in the channel, (2) retailer-Stackelberg game where the manufacturer follows the strategies taken by a dominant retailer, and (3) Nash game where the manufacture and the retailer with the same market power make the decisions simultaneously. The equilibrium wholesale and retail prices, national and local advertising expenditures, and participation rate are determined using the concepts of possibilistic game theory, and the results are compared with the centralized channel scenario. A numerical example is presented to illustrate the effectiveness of the proposed modeling approach, and sensitivity analyses are carried out to measure the impact of the demand function’s parameters as well as the levels of uncertainty.

Appendix A: Preliminaries

Lemma 1

(Wang et al. 2007). Let $\xi_{i}$ be independent fuzzy variables defined on the possibility spaces ($\Theta_{i} , P\left( {\Theta_{i} } \right), {\text{Pos}}_{i} )$ with continuous membership function, $i = 1,2,\ldots,n$ and $f:X \subset R^{n} \to R$ a measurable function. If $f\left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)$ is monotonic with respect to $x_{i}$, respectively, then

• (a) $f_{\alpha }^{U} \left( \xi \right) = f\left( {\xi_{1\alpha }^{V} ,\xi_{2\alpha }^{V} , \ldots ,\xi_{n\alpha }^{V} } \right)$ where $\xi_{i\alpha }^{V} = \xi_{i\alpha }^{U}$, if $f\left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)$ is nondecreasing with respect to $x_{i} ;\xi_{i\alpha }^{V} = \xi_{i\alpha }^{L}$, otherwise,

• (b) $f_{\alpha }^{L} \left( \xi \right) = f\left( {\xi_{1\alpha }^{{\overline{V}}} ,\xi_{2\alpha }^{{\overline{V}}} , \ldots ,\xi_{n\alpha }^{{\overline{V}}} } \right)$ where $\xi_{i\alpha }^{{\overline{V}}} = \xi_{i\alpha }^{L}$, if $f\left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)$ is nondecreasing with respect to $x_{i} ;\xi_{i\alpha }^{{\overline{V}}} = \xi_{i\alpha }^{U}$, otherwise, where $f_{\alpha }^{U} \left( \xi \right)$ and $f_{\alpha }^{L} \left( \xi \right)$ denote the $\alpha$-optimistic and $\alpha$-pessimistic value of the fuzzy variable $f\left( \xi \right)$, respectively.

Definition 1

(Liu and Liu 2002). Let $\xi$ be a fuzzy variable, the expected value of $\xi$ is defined as.

$$ E\left[ \xi \right] = \mathop \int \limits_{0}^{ + \infty } Cr\left( {\left\{ {\xi \ge x} \right\}} \right)dx - \mathop \int \limits_{ - \infty }^{0} Cr\left( {\left\{ {\xi \le x} \right\}} \right)dx $$

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

Definition 2

(Liu and Liu 2002). Let $f$ be a function on $R \to R$ and $\xi$ be a fuzzy variable, then the expected value $E\left[ {f\left( \xi \right)} \right]$ is defined as

$$ E\left[ {f\left( \xi \right)} \right] = \mathop \int \limits_{0}^{ + \infty } {\text{Cr}}\left( {\left\{ {f\left( \xi \right) \ge x} \right\}} \right)dx - \mathop \int \limits_{ - \infty }^{0} {\text{Cr}}\left( {\left\{ {f\left( \xi \right) \le x} \right\}} \right)dx $$

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

Lemma 2

(Liu and Liu 2003). Let $\xi$ be a fuzzy variable with finite expected value, then

$$ E\left[ \xi \right] = \frac{1}{2}\mathop \int \limits_{0}^{1} \left( {\xi_{\alpha }^{L} + \xi_{\alpha }^{U} } \right)d\alpha $$