Pricing, carbon emission reduction and recycling decisions in a closed-loop supply chain under uncertain environment

Guangzhou Yan; Qinyu Song; Yaodong Ni; Xiangfeng Yang

doi:10.3934/jimo.2021181

Article Contents

2023, Volume 19, Issue 1: 224-245. Doi: 10.3934/jimo.2021181

This issue Previous Article A new data-driven robust optimization approach to multi-item newsboy problems Next Article Optimality conditions of singular controls for systems with Caputo fractional derivatives

Pricing, carbon emission reduction and recycling decisions in a closed-loop supply chain under uncertain environment

1.
College of Science, Hebei Agricultural University, Baoding 071001, China
2.
Glorious Sun School of Business and Management, Donghua University, Shanghai 200051, China
3.
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China

^* Corresponding author: Yaodong Ni
^* Corresponding author: Yaodong Ni

Received: May 31, 2021

Revised: August 31, 2021

Early access: October 2021

Published: January 2023

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Abstract

This paper studies the pricing and recycling decision problems in a closed-loop supply chain (CLSC) containing a manufacturer, a downstream retailer, and a third-party recycling left. The manufacturer is subjected to the cap-and-trade regulation and determines the wholesale price of products and carbon emission reduction rate. The retailer determines its resale price to meet customer demands. The third-party recycling left determines the collection rate of recycling and remanufacturing used products. The new product demands, total carbon emissions, and recovery of these products are characterized as uncertain variables due to lack of historical data or insufficient data collected for research. By constructing three decentralized game models, we explore the equilibrium solutions under the corresponding decision-making situation and the corresponding analytical solutions. Finally, numerical experiments are performed to show the total profit of supply chain members for each structure and some special insights are drawn.

Keywords:

Mathematics Subject Classification: Primary: 90B50; Secondary: 91A10, 91A40.

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Figure 1. The CLSC structure

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Table 1. Notations

$ \bullet $ Decision variables
$ \omega $: Unit wholesale price to retailer, the manufacturer's decision variable.
$ r $: Unit markup price of retailer, the retailer's decision variable.
$ p $: Unit retail price of retailer, where $ p=\omega+r $.
$ \tau $: Recovery rate through third-party channel,
$ \tau\in[0,1] $, the third-party's decision variable.
$ \theta $: Carbon emission reduction rate of manufacturer,
$ \theta\in[0,1] $, the manufacturer's decision variable.
$ \bullet $ Parameters
$ c_{n} $: Unit cost of manufacturing the product from raw materials.
$ c_{r} $: Unit manufacturing cost of the product from return products.
$ \widetilde{s_{r}} $: Unit sales cost of retailer, an uncertain variable.
$ \widetilde{d} $: The market base of product, an uncertain variable.
$ \widetilde{C} $: Total emissions of the manufacturer, an uncertain variable.
$ p_{c} $: Average recycling price for used product from the third-party to the manufacturer.
$ A $: Average recycling price for used products through the third-party channel.
$ e $: Initial carbon emission of unit product.
$ \Omega $: Total carbon free permits from government.
$ \rho $: Cost coefficient of emissions reduction investment.
$ \lambda $: Coefficient of carbon emissions reduction unit recovery rate.
$ b $: Carbon buying price of unit product.
$ s $: Carbon selling price of unit product.

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Table 2. Parameters for the model

Parameters	$ c_{n} $	$ c_{r} $	$ p_{c} $	$ A $	$ b $	$ s $	$ e $	$ k $	$ \lambda $	$ \rho $	$ \Omega $
Value	20	10	7	5	13	6	2.5	1200	1500	10000	5000

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Table 3. Distributions of uncertain variables

Parameters	Distribution	Expected value
$ \tilde{\beta} $	$ \mathcal{L} $(0.4, 0.8)	0.6
$ \tilde{\gamma} $	$ \mathcal{L} $(0.2, 0.4)	0.3
$ \tilde{d} $	$ \mathcal{Z} $(700,950, 1000)	900
$ \tilde{s_{r}} $	$ \mathcal{L} $(3, 5)	4

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Table 4. The optimal decisions of the three structures under buying carbon quotas

Structure	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi^{b}_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	753.7486	0.7351	224163.5012	375.3678	82851.5121	0.1856	41.3482
RS	1161.6480	0.2598	143452.4000	222.6841	15340.8000	0.0579	4.0227
VN	512.1873	0.9743	210343.7000	496.2082	145447.5000	0.2460	72.6462

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Table 5. The optimal decisions of the three structures under selling carbon quotas

Structure	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi^{b}_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	758.6526	0.3429	193461.0000	372.8177	81719.3100	0.1844	40.7821
RS	1156.2150	0.1368	107485.1000	229.5259	15617.3600	0.0572	3.9213
VN	516.9945	0.4553	174966.9000	493.6749	143955.4000	0.2448	71.9002

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Table 6. Effects of the retailer sales costs $ \tilde{s}_r $ on the optimal results under buying carbon quotas

Structure	$ \tilde{s}_{r} $	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	$ \mathcal{L} $(2.5, 5.5)	753.7724	0.7351	224152.9000	375.4051	82890.7900	0.1856	41.3454
	$ \mathcal{L} $(3, 5)	753.7486	0.7351	224163.5012	375.3678	82851.5121	0.1856	41.3482
	$ \mathcal{L} $(3.5, 4.5)	753.7789	0.7351	224176.7000	375.3218	82777.9400	0.1856	41.3517
RS	$ \mathcal{L} $(2.5, 5.5)	1161.6360	0.2598	143442.6000	222.7092	15407.3100	0.0579	4.0218
	$ \mathcal{L} $(3, 5)	1161.6480	0.2598	143452.4000	222.6841	15340.8000	0.0579	4.0227
	$ \mathcal{L} $(3.5, 4.5)	1161.6620	0.2599	143464.5000	222.6533	15264.5700	0.0579	4.0239
VN	$ \mathcal{L} $(2.5, 5.5)	512.1714	0.9743	210333.8000	496.2412	145507.4000	0.2460	72.6412
	$ \mathcal{L} $(3, 5)	512.1873	0.9743	210343.7000	496.2082	145447.5000	0.2460	72.6462
	$ \mathcal{L} $(3.5, 4.5)	512.2070	0.9743	210355.9000	496.1676	145379.4000	0.2460	72.6524

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Table 7. Effects of the retailer sales costs $ \tilde{s}_r $ on the optimal results under selling carbon quotas

Structure	$ \tilde{s}_{r} $	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	$ \mathcal{L} $(2.5, 5.5)	758.6278	0.3429	193450.0000	372.8552	81758.7700	0.1840	40.7793
	$ \mathcal{L} $(3, 5)	758.6526	0.3429	193461.0000	372.8177	81719.3100	0.1844	40.7821
	$ \mathcal{L} $(3.5, 4.5)	758.6834	0.3429	193474.6000	372.7715	80964.4000	0.1844	40.7856
RS	$ \mathcal{L} $(2.5, 5.5)	1158.2020	0.1368	107475.7000	229.5509	15952.6800	0.0572	3.9204
	$ \mathcal{L} $(3, 5)	1156.2150	0.1368	107485.1000	229.5259	15617.3600	0.0572	3.9213
	$ \mathcal{L} $(3.5, 4.5)	1156.2300	0.1368	107496.8000	229.4951	15541.0800	0.0572	3.9223
VN	$ \mathcal{L} $(2.5, 5.5)	516.9780	0.4553	174957.1000	493.7081	143955.3000	0.2448	71.8953
	$ \mathcal{L} $(3, 5)	516.9945	0.4553	174966.9000	493.6749	143889.4000	0.2448	71.9002
	$ \mathcal{L} $(3.5, 4.5)	517.0149	0.4553	174979.1000	493.6339	143887.0000	0.2448	71.9062

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Table 8. Effects of the greening level elastic coefficient $ \tilde{\gamma} $ on the optimal results under buying carbon quotas

Structure	$ \tilde{\gamma} $	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	$ \mathcal{L} $(0.15, 0.45)	753.7486	0.7351	224163.5000	375.3678	82826.4400	0.1856	41.3482
	$ \mathcal{L} $(0.2, 0.4)	753.7486	0.7351	224163.5012	375.3678	82851.5121	0.1856	41.3482
	$ \mathcal{L} $(0.25, 0.35)	753.7486	0.7351	224163.5000	375.3678	82851.4700	0.1856	41.3482
RS	$ \mathcal{L} $(0.15, 0.45)	1161.6480	0.2598	143452.4000	222.6842	15340.7800	0.0579	4.0227
	$ \mathcal{L} $(0.2, 0.4)	1161.6480	0.2598	143452.4000	222.6841	15340.8000	0.0579	4.0227
	$ \mathcal{L} $(0.25, 0.35)	1161.6480	0.2598	143452.4000	222.6841	15340.7900	0.0579	4.0227
VN	$ \mathcal{L} $(0.15, 0.45)	512.1873	0.9743	210343.7000	496.2082	145447.5000	0.2460	72.6462
	$ \mathcal{L} $(0.2, 0.4)	512.1873	0.9743	210343.7000	496.2082	145447.5000	0.2460	72.6462
	$ \mathcal{L} $(0.25, 0.35)	512.1873	0.9743	210343.7000	496.2082	145447.5000	0.2460	72.6462

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Table 9. Effects of the greening level elastic coefficient $ \tilde{\gamma} $ on the optimal results under selling carbon quotas

Structure	$ \tilde{\gamma} $	$ \omega^{\ast} $	$ \theta^{\ast} $	$ \pi_{m} $	$ r^{\ast} $	$ \pi_{r} $	$ \tau^{\ast} $	$ \pi_{t} $
MS	$ \mathcal{L} $(0.15, 0.45)	758.6554	0.3429	193462.2000	372.8136	81714.9800	0.1844	40.7825
	$ \mathcal{L} $(0.2, 0.4)	758.6554	0.3429	193461.0000	372.8177	81719.3100	0.1844	40.7825
	$ \mathcal{L} $(0.25, 0.35)	758.6554	0.3429	193462.2000	372.8136	81714.9600	0.1844	40.7825
RS	$ \mathcal{L} $(0.15, 0.45)	1156.2160	0.1368	107486.2000	229.5232	15612.7900	0.0572	3.9214
	$ \mathcal{L} $(0.2, 0.4)	1156.2160	0.1368	107485.1000	229.5259	15617.3600	0.0572	3.9214
	$ \mathcal{L} $(0.25, 0.35)	1156.2160	0.1368	107486.1000	229.5232	15612.7800	0.0572	3.9214
VN	$ \mathcal{L} $(0.15, 0.45)	516.9964	0.4553	174968.0000	493.6712	143951.5000	0.2448	71.9007
	$ \mathcal{L} $(0.2, 0.4)	516.9964	0.4553	174966.9000	493.6749	143955.4000	0.2448	71.9007
	$ \mathcal{L} $(0.25, 0.35)	516.9964	0.4553	174968.0000	493.6712	143951.5000	0.2448	71.9007

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