Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

Jun Huang; Ying Peng; Ruwen Tan; Chunxiang Guo

doi:10.3934/jimo.2020113

Article Contents

2021, Volume 17, Issue 6: 3183-3207. Doi: 10.3934/jimo.2020113

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Alliance strategy of construction and demolition waste recycling based on the modified shapley value under government regulation

1.
College of Architecture and Environment, Sichuan University, Chengdu, 610065, China
2.
Business School, Sichuan University, Chengdu, 610065, China

^* Corresponding author: Ruwen Tan
^* Corresponding author: Ruwen Tan

Received: December 31, 2019

Revised: February 29, 2020

Early access: June 2020

Published: November 2021

This work was supported by the Ministry of Education in China Project of Humanities and Social Sciences [grant numbers 17YJA630078]; Sichuan Province Cyclic Economy Research Center Project [grant numbers XHJJ-1811]; the Sichuan Science and Technology Department Soft Science Project [grant numbers 2019JDR0148]; Chengdu Science and Technology Bureau Project [grant numbers 2017-RK00-00209-ZF]; and Sichuan University [grant numbers skqy201740]

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Abstract

Construction and demolition waste (CDW) recycling enterprises have an imperfect operation system, which leads to low recovery rate and low production profits. A feasible method to improve this situation involves seeking a high-efficiency enterprise alliance strategy in the CDW recycling system composed of manufacturers, retailers and recyclers and designing a reasonable and effective coordination mechanism to enhance their enthusiasm for participa-tion. First, we constructed a Stackelberg game model of CDW recycling under government regulation and analyzed the optimal alliance strategy of CDW recycling enterprises under punishment or subsidy by the government as a game leader. In order to ensure the stable cooperation of the alliance, we used the Shapley value method to coordinate the distribution of the optimal alliance profit and improved the fairness and effectiveness of the coordination mechanism through modification of the unequal rights factor. Finally, based on the survey data of Chongqing, we further verified the conclusion through numerical simulation and an-alyzed changes in various parameters at different product costs. The results show that the alliance strategy and coordination mechanism can improve the CDW recovery rate, improve the recycling market status, and increase the production profits of enterprises.

Keywords:

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A12.

Citation:

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Figure 1. Closed-loop supply chain of building materials products

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Figure 2. Government subsidies $ (\eta<0) $ comparison between different cases when $ c $ and $ \Delta $ changes

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Figure 5. Recycler's (Manufacturer's) profit $ \pi_{C}\left(\pi_{M}\right) $ comparison between different distribution cases

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Figure 3. Retail price $ p $ (Wholesale price $ \omega $) comparison between different alliance cases

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Figure 4. Recovery rate $ \tau $ (Government utility $ \pi_{g} $) comparison between different alliance cases

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Notoations	Definitions
$ \alpha $	Potential market demand for new building products; $ (\alpha>0) $
$ \beta $	Sensitivity factor of public construction project contractors (consumers) to new building products; $ (\beta>0) $
$ D(p) $	Demand function for new building products
$ D(p, \eta) $	The demand function of new building products under government regulation
$ \eta $	Government penalizes manufacturer's products ($ \eta>0 $) or subsidies ($ \eta<0 $)
$ c_{n} $	The unit cost of producing new building products from new raw materials purchased by manufacturers; $ \left(c_{n}>0\right) $
$ c_{r} $	The unit cost of the manufacturer's use of CDW to produce new building products; $ \left(c_{r}>0\right) $
$ \omega $	The wholesale price of the new building products that the manufacturer provides to the retailer; $ (\omega>0) $
$ p $	The retail price of the new building products that the retailer manufacturer provides to the contractor; $ (p>0) $
$ c $	Collector's unit collection cost; $ (c>0) $
$ b $	The repurchase price of the CDW paid by the manufacturer to the recycler; $ (b>0) $
$ \tau $	The proportion of recycled CDW to the market demand for new building products, referred to as recovery rate; $ \tau \in[0,1] $
$ \gamma $	Environmental benefit coefficient, environmental benefits brought by unit CDW recycling
$ m $	The difficulty degree for collecting CDW; (m > 0)
$ c_{z} $	The unit cost of new building products, as new construction products include new and remanufactured products, so $ c_{z} = \tau c_{r}+ $ $ (1-\tau) c_{n} = c_{n}+\tau\left(c_{r}-c_{n}\right) = c_{n}-\Delta \tau $

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Table 1. Calculation of the Shapley value of the recycler and manufacturer

S	Recycler's calculation of Shapley value		Manufacturer's calculation of Shapley value
S	$ C $	$ C \cup M $	$ M $	$ C \cup M $
$ \mathrm{v}(\mathrm{s}) $	$ {\pi_\mathrm{C}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $	$ {\pi_\mathrm{M}}^{*} $	$ {\pi_\mathrm{CM}}^{*} $
$ \mathrm{v}(\mathrm{s} / \mathrm{i}) $	0	$ {\pi_\mathrm{M}}^{*} $	0	$ {\pi_\mathrm{C}}^{*} $
$ \mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i}) $	$ \pi_{\mathrm{C}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{} $	$ \pi_{\mathrm{M}}^{*} $	$ \pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{} $
$ \|\mathrm{s}\| $	1	2	1	2
$ \omega(\|\mathrm{s}\|) $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $	$ \frac{1}{2} $
$ \omega(\|\mathrm{s}\|)[\mathrm{v}(\mathrm{s})-\mathrm{v}(\mathrm{s} / \mathrm{i})] $	$ \frac{{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}}{2} $	$ \frac{{\pi_\mathrm{M}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}}{2} $
Recycler's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{M}}^{}+\pi_{\mathrm{C}}^{*}}{2} $
Manufacturer's Shapley value distribution	$ \frac{\pi_{\mathrm{CM}}^{}-\pi_{\mathrm{C}}^{}+\pi_{\mathrm{M}}^{*}}{2} $

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Table 2. Calculation of the Shapley value of the recycler and manufacturer

Player i	Recycler	Manufacturer
$ \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}} $
$ \Delta \lambda_{i} $	$ \frac{\sqrt{t_{1}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $	$ \frac{\sqrt{t_{2}^{2}}}{\sqrt{t_{1}^{2}}+\sqrt{t_{2}^{2}}}-\frac{1}{2} $
$ \Delta \varphi_{i}(v) $	$ \left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}}^{*} $	$ \left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{C M}^{*}} $
$ \varphi_{i}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}}^{}+{\pi_\mathrm{C}}^{*}}{2} $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{C}}^{}+{\pi_\mathrm{M}}^{*}}{2} $
$ \varphi_{i}^{*}(v) $	$ \frac{{\pi_\mathrm{CM}}^{}-{\pi_\mathrm{M}^{}}+{\pi_\mathrm{C}}^{}}{2}-\left(\frac{t_{1}}{t_{1}+t_{2}}-\frac{1}{2}\right)\cdot \epsilon \cdot {\pi_{C M}}^{} $	$ \frac{\pi_{\mathrm{CM}^{}}-{\pi_\mathrm{C}^{}}+{\pi_\mathrm{M}}^{}}{2}-\left(\frac{t_{2}}{t_{1}+t_{2}}-\frac{1}{2}\right) \cdot \epsilon \cdot {\pi_{CM}}^{} $

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Table 3. The maximum profit for each alliance

	$R U C$	$R \cup M$
$\eta^{*}$	$\frac{(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{((c-\Delta)(c-4 \gamma-\Delta)+4 m \mu)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-4 m\right)+\beta^{2}\left(c^{2}+4 \gamma \Delta+\Delta^{2}-2 c(2 \gamma+\Delta)+4 m \mu\right) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-6 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(1+\beta \mu)) c_{n}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$	$\frac{2 \alpha\left(\beta(c-\Delta)^{2}-2 m\right)+\beta(\beta(c-\Delta)(c-4 \gamma-\Delta)+4 m(-1+\beta \mu)) c_{n}}{\beta(\zeta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$	$\frac{2(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu)}$
	$C \cup M$	$R \cup C \cup M$
$\eta^{*}$	$\frac{(2 \beta \gamma(-c+\Delta)+m(2+\beta \mu))\left(\alpha-\beta c_{n}\right)}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{(2 c \gamma-2 \gamma \Delta-m \mu)\left(-\alpha+\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\omega^{*}$	$\frac{\alpha\left(-2 m+\beta(c-\Delta)^{2}\right)+\beta^{2}(-2 c \gamma+2 \gamma \Delta+m \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	---
$b^{*}$	$\frac{c+\Delta}{2}$	$\frac{c+\Delta}{2}$
$p^{*}$	$\frac{\alpha\left(-3 m+\beta(c-\Delta)^{2}\right)+\beta(m+2 \beta \gamma(-c+\Delta)+m \beta \mu) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$	$\frac{\alpha\left(-m+\beta(c-\Delta)^{2}\right)+\beta(2 \beta \gamma(-c+\Delta)+m(-1+\beta \mu)) c_{n}}{\beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))}$
$\tau^{*}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$	$\frac{(c-\Delta)\left(\alpha-\beta c_{n}\right)}{\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu)}$
$\pi_{\mathrm{rc}}^{*}$	$\frac{2 m\left(8 m+\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rm}}^{*}$	$\frac{4 m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{\beta(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{cm}}^{*}$	$\frac{m\left(4 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$
$\pi_{\mathrm{rcm}}$	$\frac{m\left(2 m-\beta(c-\Delta)^{2}\right)\left(\alpha-\beta c_{n}\right)^{2}}{2 \beta(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}}$

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Table 4. Calculation of the Shapley value of the retailer

s	$ R $	$ R \cup C $	$ R \cup M $	$ R \cup C \cup M $
$ \mathbf{v}(s) $	$ \pi_{r 3}^{*} $	$ \pi_{r 3}^{}+\pi_{c 3}^{} $	$ \pi_{\mathrm{rm} 4}^{*} $	$ \pi_{\mathrm{rcm} 7}^{*} $
$ \mathbf{v}(s/i) $	0	$ \pi_{\mathrm{c} 3}^{*} $	$ \pi_{\mathrm{m} 3}^{*} $	$ \pi_{\mathrm{cm} 5}^{*} $
$ \mathbf{v}(s)-\mathbf{v}(s/i) $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{r} 3}^{*} $	$ \pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{} $	$ \pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{} $
$ \|s\| $	1	2	2	3
$ \omega (\|s\|) $	$ \frac{1}{3} $	$ \frac{1}{6} $	$ \frac{1}{6} $	$ \frac{1}{3} $
$ \omega(\|s\|)[v(s)-v(s)i)] $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{3} $	$ \frac{\pi_{\mathrm{r} 3}^{*}}{6} $	$ \frac{\pi_{\mathrm{rm} 4}^{}-\pi_{\mathrm{m} 3}^{}}{6} $	$ \frac{\pi_{\mathrm{rcm} 7}^{}-\pi_{\mathrm{cm} 5}^{}}{3} $
Retailer 's Shapley value distribution	$ \frac{m^{2}(c-\Delta)(c-4 \gamma-\Delta)(\beta(7 c-12 \gamma-7 \Delta)(c-\Delta)+8 m(-2+\beta \mu))\left(\alpha-\beta c_{n}\right)^{2}}{3(\beta(c-\Delta)(c-2 \gamma-\Delta)+m(-2+\beta \mu))^{2}(\beta(3 c-4 \gamma-3 \Delta)(c-\Delta)+4 m(-2+\beta \mu))^{2}} $

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