# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020097

## A dual-channel supply chain problem with resource-utilization penalty: Who can benefit from sales effort?

 1 School of Management, Shanghai University, Shanghai 200444, China 2 School of Economics and Management, Tongji University, Shanghai, 200092, China 3 Edward P.Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695-7906, USA

* Corresponding author: Shu-Cherng Fang

Received  October 2019 Revised  February 2020 Published  May 2020

Fund Project: This work is supported by National Natural Science Foundation of China (71502100, 71671125) and Humanities and Social Sciences Foundation of the Chinese Ministry of Education (20YJAZH135)

As manufacturers may engage in both direct sale and wholesale, the channel conflict between manufacturer and retailer becomes inevitable. This paper considers a dual-channel supply chain in which a retailer sells the product through store channel with sales effort while the manufacturer holds a direct channel and may provide an incentive measure to share the cost of sales effort. To meet social responsibility, a penalty on the total resource consumed is imposed on the manufacturer. We present a manufacturer-led decentralized model in which both members maximize individual profit, and then derive the corresponding optimal direct/store price and wholesale price. The dual-channel supply chain model without sales effort policy is also considered so as to explain the effects of sales effort policy and sharing cost measure on both parties. Special properties are presented to show (ⅰ) the influence of retailer's sales effort and manufacturer's sharing cost on the optimal strategies; (ⅱ) the resource-utilized penalty on the optimal decisions. Finally, numerical experiments are conducted to highlight the influence of various parameters on optimal solutions. We find that if the market response to retailer's sales effort is strong or the manufacturer's sharing portion of sales effort cost is increased, the retailer's profit and store selling price increase while the manufacturer's profit decreases and the direct sale and wholesale prices do not change. We also show that if the consumer's value on direct channel exceeds a threshold, the manufacturer's profit will be greater than that of the retailer. Moreover, if the market response to retailer's sales effort is strong, manufacturer's profit will be lesser than retailer's profit.

Citation: Lianxia Zhao, Jianxin You, Shu-Cherng Fang. A dual-channel supply chain problem with resource-utilization penalty: Who can benefit from sales effort?. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020097
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##### References:
Variation of profits with the change of $\delta$ for $c = 0.20, \lambda = 0.75, \beta = 0.15, a = 0.35$
Variation of profits with the change of $a$ for $c = 0.25, \lambda = 0.75, \beta = 0.08, \delta = 0.50$
Variation of profits with the change of $\lambda$ for $c = 0.15, \beta = 0.20, \delta = 0.55, a = 0.35$
Variation of profits with the change of $\beta$ for $c = 0.25, \lambda = 0.70, \delta = 0.55, a = 0.30$
Notations
 Notation Description Parameters $\theta$ Willing-to-pay of the consumer for retailer channel, $0\leq\theta\leq1$. $\delta$ Consumer's preference to select direct channel, $0\leq\delta<1$. $c$ Marginal costs incurred by the retailer for the product sold through the store channel, $0\leq c <1$. $\beta$ The coefficient of resource-utilization penalty, $\beta\geq0$. $a$ Market response to retailer's sales effort, $0\leq a\leq1$. $\lambda$ Retailer's cost-sharing proportion for sales effort, $0\leq\lambda\leq1$. $NS/S$ Without/with retailer's sales effort. $q_r, q_r^i$ Market demand for the retailer, $i\in\{NS,S\}$. $q_d, q_d^i$ Market demand for the manufacturer, $i\in\{NS,S\}$. $\Pi_r^i/\Pi_d^i$ Retailer's/Manufacturer's profit, $i\in\{NS,S\}$. $CS^i$ Consumer surplus, $i\in\{NS,S\}$. Superscript $*$ The optimal value of each decision variable. Decision variables $p_r, p_r^i$ Retail price at the retail store channel, $i\in\{NS,S\}$. $p_d, p_d^i$ Retail price at the manufacturer's online channel, $i\in\{NS,S\}$. $s$ Sales effort level provided by the retailer. $w, w^i$ Manufacturer's wholesale price to retailer, $i\in\{NS,S\}$.
 Notation Description Parameters $\theta$ Willing-to-pay of the consumer for retailer channel, $0\leq\theta\leq1$. $\delta$ Consumer's preference to select direct channel, $0\leq\delta<1$. $c$ Marginal costs incurred by the retailer for the product sold through the store channel, $0\leq c <1$. $\beta$ The coefficient of resource-utilization penalty, $\beta\geq0$. $a$ Market response to retailer's sales effort, $0\leq a\leq1$. $\lambda$ Retailer's cost-sharing proportion for sales effort, $0\leq\lambda\leq1$. $NS/S$ Without/with retailer's sales effort. $q_r, q_r^i$ Market demand for the retailer, $i\in\{NS,S\}$. $q_d, q_d^i$ Market demand for the manufacturer, $i\in\{NS,S\}$. $\Pi_r^i/\Pi_d^i$ Retailer's/Manufacturer's profit, $i\in\{NS,S\}$. $CS^i$ Consumer surplus, $i\in\{NS,S\}$. Superscript $*$ The optimal value of each decision variable. Decision variables $p_r, p_r^i$ Retail price at the retail store channel, $i\in\{NS,S\}$. $p_d, p_d^i$ Retail price at the manufacturer's online channel, $i\in\{NS,S\}$. $s$ Sales effort level provided by the retailer. $w, w^i$ Manufacturer's wholesale price to retailer, $i\in\{NS,S\}$.
Equilibrium decisions under different strategies
 Variables $i=NS$ $i=S$ $p_r^i$ $\frac{(\beta+2\delta)(1+c)+\beta\delta}{2(\beta+2\delta)}$ $\frac{\lambda(1-\delta)[(\beta+2\delta)(c+1)+\delta\beta ]-a^2[\delta(\beta+\delta)+c(\beta+2\delta)]}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]}$ $p_d^i$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $w^i$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $s_i$ N/A $\frac{a(1-c-\delta)}{2\lambda(1-\delta)-a^2}$ $q_r^i$ $\frac{1-c-\delta}{2(1-\delta)}$ $\frac{\lambda(1-c-\delta)}{2\lambda(1-\delta)-a^2}$ $q_d^i$ $\frac{2c\delta-\beta(1-c-\delta)}{2(1-\delta)(\beta+2\delta)}$ $\frac{\lambda[2c\delta-\beta(1-c-\delta)]-a^2\delta}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]}$ $\Pi_r^i$ $\frac{(1-c-\delta)^2}{4(1-\delta)}$ $\frac{\lambda(1-c-\delta)^2}{2[2\lambda(1-\delta)-a^2]}$ $\Pi_d^i$ $\frac{\delta^2}{2(\beta+2\delta)}$ $\frac{\delta^2[2\lambda(1-\delta)-a^2]^2 +a^2(\beta+2\delta)(\lambda-1)(1-c-\delta)^2}{2(\beta+2\delta)[2\lambda(1-\delta)-a^2]^2}$ NS=No sales effort, S=Sales effort
 Variables $i=NS$ $i=S$ $p_r^i$ $\frac{(\beta+2\delta)(1+c)+\beta\delta}{2(\beta+2\delta)}$ $\frac{\lambda(1-\delta)[(\beta+2\delta)(c+1)+\delta\beta ]-a^2[\delta(\beta+\delta)+c(\beta+2\delta)]}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]}$ $p_d^i$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $w^i$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $\frac{\delta(\beta+\delta)}{\beta+2\delta}$ $s_i$ N/A $\frac{a(1-c-\delta)}{2\lambda(1-\delta)-a^2}$ $q_r^i$ $\frac{1-c-\delta}{2(1-\delta)}$ $\frac{\lambda(1-c-\delta)}{2\lambda(1-\delta)-a^2}$ $q_d^i$ $\frac{2c\delta-\beta(1-c-\delta)}{2(1-\delta)(\beta+2\delta)}$ $\frac{\lambda[2c\delta-\beta(1-c-\delta)]-a^2\delta}{(\beta+2\delta)[2\lambda(1-\delta)-a^2]}$ $\Pi_r^i$ $\frac{(1-c-\delta)^2}{4(1-\delta)}$ $\frac{\lambda(1-c-\delta)^2}{2[2\lambda(1-\delta)-a^2]}$ $\Pi_d^i$ $\frac{\delta^2}{2(\beta+2\delta)}$ $\frac{\delta^2[2\lambda(1-\delta)-a^2]^2 +a^2(\beta+2\delta)(\lambda-1)(1-c-\delta)^2}{2(\beta+2\delta)[2\lambda(1-\delta)-a^2]^2}$ NS=No sales effort, S=Sales effort
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