Decision analysis and coordination in green supply chains with stochastic demand

Figures & data

Table 1. Literature comparison.

				Decisions
Literature	Green product	Green-sensitive demand	Demand uncertainty	Price	Order	Green	Coordination mechanisms
Swami and Shah (2013)	DIGP	√		√		√	two-part tariff contract
Ghosh and Shah (2012, 2015)	DIGP	√		√		√	two-part tariff contract; cost-sharing contract through bargaining
W. G. Zhu and He (2017)	MIGP; DIGP	√		√		√	cost-sharing contract
Dey et al. (2019)	MIGP; DIGP	√		√	√	√
Cohen et al. (2015)	MIGP		√	√	√		consumer subsidies
Raza (2018); Raza and Govindaluri (2019); Raza et al. (2018)*	DIGP	√	√		√		revenue-sharing contract through bargaining
C. Liu and Chen (2019)	DIGP	√	√		√	√
W. Wang et al. (2021)	DIGP	√	√	√		√	reward contract with/without target green degree
This paper	MDIGP	√	√	√	√	√	wholesale price contract through bargaining

Notes: Sustainability issues are also included in ‘green’;

‘*’: Given that the pricing and greening decisions are exogenous, the authors use a two-phase solution approach to solve the stochastic demand model.

Figure 1. The proposed supply chain.

Table 2. Notations.

Decision-makers	Decision variables	Non-decision variables
Manufacturer${\mathrm{\Pi }}_M$: manufacturer’s expected profit	$w$: wholesale price $\theta$: greenness improvement ($\theta \ge 0$)	$C$: manufacturer’s total cost $(c+v\theta )Q+\text{β}{\theta }^2$ $c$: per-unit production cost not including green-related costs $v$: unit-variable cost coefficient $\text{β}$: fixed investment cost coefficient ($\text{β}>0$)
Retailer${\mathrm{\Pi }}_R$: retailer’s expected profit	$p$: retail price $Q$: order quantity ($p>w>c+v\theta >0$)	$a$: potential deterministic market size ($a>0$) $b_p$ and $b_g$: demand sensitivity to retail price and greenness, respectively ($b_p>0$,$b_g>0$) $D$: riskless demand for green products $a-b_{p}p+b_g\theta$ $\xi$: demand shock, a price-independent and green-independent random variable with a continuous and strictly increasing distribution $F\left(\xi \right)$ and a density function $f\left(\xi \right)$ defined on the support $[A,B]$ with a mean $\mu$ and a standard deviation $\sigma$ $S$: expected sales $E[min(Q,D+\xi \left)\right]$ transformed expected sales $S=D+z-I\left(z\right)$ with leftovers $I\left(z\right)={\int }_A^{z}F\left(\xi \right)d\xi$ $c_o$ and $c_s$: per-unit holding cost and goodwill penalty cost, respectively
Centralised supply chain ${\mathrm{\Pi }}_{SC}$: expected profit of the supply chain	$\theta$: greenness improvement $p$: retail price $z$: service level	Superscripts:
		‘gs’: green products with stochastic demand
		‘gd’: green products with deterministic demand
		Subscripts:
		‘c’: centralised decision-making
		‘m’: decentralised decision-making
		‘b’: Nash bargaining setting

Table 3. Manufacturer’s optimal greenness and profit for DIGPs and MDIGPs.

Indicators	DIGPs ($v=0$)	MDIGPs
${\theta }_m^{gd}$	${\displaystyle \frac{b_g(a-b_{p}c)}{8\text{β}{b}_p-{b_g}^2}}$	${\displaystyle \frac{(b_g-v{b}_p)(a-b_{p}c)}{8\text{β}{b}_p-{(b_g-v{b}_p)}^2}}$
${\mathrm{\Pi }}_M^{gd}$	${\displaystyle \frac{\text{β}{(a-b_{p}c)}^2}{8\text{β}{b}_p-{b_g}^2}}$	${\displaystyle \frac{\text{β}{(a-b_{p}c)}^2}{8\text{β}{b}_p-{(b_g-v{b}_p)}^2}}$
${\theta }_m^{gs}$	${\theta }_m^{gd}+{\displaystyle \frac{b_g\left(z+I\left(z\right)\right)}{8\text{β}{b}_p-{b_g}^2}}$	${\theta }_m^{gd}+{\displaystyle \frac{(b_g-v{b}_p)\left(z+I\left(z\right)\right)}{8\text{β}{b}_p-{(b_g-v{b}_p)}^2}}$
${\mathrm{\Pi }}_M^{gs}$	${\displaystyle \frac{\text{β}{\left(a-b_{p}c+z+I\left(z\right)\right)}^2}{8\text{β}{b}_p-{b_g}^2}}$	${\displaystyle \frac{\text{β}{\left(a-b_{p}c+z+I\left(z\right)\right)}^2}{8\text{β}{b}_p-{(b_g-v{b}_p)}^2}}$

Table 4. Baseline parameters.

Parameter	$c_s$	$c_o$	$a$	$b_p$	$b_g$	$c$	$\text{β}$	$v$	$A$	$B$
Value	0	0	2 × 10^6	10	2 × 10^5	10^5	10^10	10^3	−2.5 × 10^5	2.5 × 10^5

Table 5. Optimal solutions under different demand settings.

Demand	Deterministic		Stochastic case 1 $U(-2.5\times {10}^5,2.5\times {10}^5)$		Stochastic case 2 $N(0,8.33\times {10}^4)$		Stochastic case 3 $Exp(1.25\times {10}^5)$		Stochastic case 4 $U(0,2.5\times {10}^5)$		Stochastic case 5 $N(1.25\times {10}^5,4.17\times {10}^4)$
Decisions	gdc	gdm	gsc	gsm	gsc	gsm	gsc	gsm	gsc	gsm	gsc	gsm
$z$			−86721	−218999	−34109	−125869	56497	9322	92536	18010	111791	64545
$\theta$	0.5221	0.2487	0.4615	0.1945	0.4934	0.2180	0.5453	0.2510	0.5605	0.2533	0.5742	0.2651
$w$		152612		141140		146113		153120		153593		156078
$p$	155482	178793	149176	161517	152530	168826	157998	179520	159656	180198	161078	183850
$D$	549600	261810	600529	423728	573380	355345	529072	255022	515547	248689	504073	214521
$Q$			513808	204729	539271	229476	585569	264344	608083	266699	615864	279066
$S$			487148	203768	520366	227123	574526	264005	590957	266050	605033	277718
$I\left(z\right)$			26660	961	18905	2353	11043	339	17126	649	10831	1348
$z/I\left(z\right)$			−3.253	−227.9	−1.804	−53.49	5.116	27.50	5.403	27.75	10.32	47.88
$R$(×10^9)		6.854		4.012		4.811		6.929		6.977		7.502
$M$(×10^9)		13.09		8.002		10.04		13.36		13.59		14.88
$SC$(×10^10)	2.748	1.994	1.892	1.202	2.275	1.486	2.893	2.028	3.006	2.056	3.221	2.239
$e$(%)		72.56		63.53		65.32		70.10		68.40		69.51
$r$(%)		65.65		66.57		67.56		65.88		66.10		66.46
$d_R$(%)				−41.46		−29.81		1.09		1.79		9.45
$d_M$(%)				−38.87		−23.30		2.06		3.82		13.67
$d_{SC}$(%)			−31.15	−39.72	−17.21	−25.48	5.28	1.71	9.39	3.11	17.21	12.29
$d_{\theta }$(%)			−11.61	−21.79	−5.50	−12.34	4.44	0.92	7.35	1.85	9.98	6.59
$d_p$(%)			−4.06	−9.66	−1.90	−5.57	1.62	0.41	2.68	0.79	3.60	2.83
$d_Q$(%)			−6.51	−21.80	−1.88	−12.35	6.54	0.97	10.64	1.87	12.06	6.59

Notes: Following Corollary 2 and Corollary 3, for the centralised supply chain, the demarcation values of $z/I\left(z\right)$ are 1.010 and 1.105; they are rounded to 1 for simplicity. For the decentralised supply chain, the demarcation value for greenness and wholesale price is −1; the demarcation value for the retail price is 0.2691.

‘gdc’ and ‘gdm’: represent centralised and decentralised supply chains under deterministic demand, respectively;

‘gsc’ and ‘gsm’ represent centralised and decentralised supply chains under stochastic demand, respectively;

‘$e$’ and ‘$r$’ denote the efficiency of the supply chain ($e={\mathrm{\Pi }}_{SCm}/{\mathrm{\Pi }}_{SCc}$) and the manufacturer’s profit share ($r={\mathrm{\Pi }}_M/{\mathrm{\Pi }}_{SC}$), respectively;

‘$d_x$’ denotes the deviation rate of each variable relative to corresponding deterministic values, i.e. $d_x=(x^{gs}-x^{gd})/x^{gd}$ where $x\in \{\theta ,p,Q,R,M,SC\}$ and please note that $Q^{gd}=D^{gd}$.

Figure 2. Profits under negative distributions.

Figure 3. Profits under nonnegative distributions.

Notes: Setting a strictly positive lower bound yields similar properties to cases with zero lower bound, but has larger differences relative to corresponding deterministic solutions. Red circles mark the profits at optimal values of service levels under stochastic demand.

Figure 4. Wholesale price and profits comparison of decentralised and coordinated cases.

Notes: ‘wb’: the wholesale price in the bargaining game;‘wm’: the wholesale price under deterministic demand with decentralised decision-making;‘UC’: unit production cost under stochastic demand with centralised decision-making;‘Rb’ and ‘Mb’: the profit of the retailer and the manufacturer in the bargaining game, respectively.

Figure 5. Impact of $v$ on optimal service level, greenness, prices, profits, and resulting ratios.

Figure 6. Impact of $b_g$ on optimal service level, greenness, prices, profits, and resulting ratios.

Swami, S., & Shah, J. (2013). Channel coordination in green supply chain management. Journal of the Operational Research Society, 64(3), 336–351. https://doi.org/10.1057/jors.2012.44

 Web of Science ®Google Scholar

Ghosh, D., & Shah, J. (2012). A comparative analysis of greening policies across supply chain structures. International Journal of Production Economics, 135(2), 568–583. https://doi.org/10.1016/j.ijpe.2011.05.027

 Web of Science ®Google Scholar

Ghosh, D., & Shah, J. (2015). Supply chain analysis under green sensitive consumer demand and cost sharing contract. International Journal of Production Economics, 164, 319–329. https://doi.org/10.1016/j.ijpe.2014.11.005

 Web of Science ®Google Scholar

Zhu, W. G., & He, Y. J. (2017). Green product design in supply chains under competition. European Journal of Operational Research, 258(1), 165–180. https://doi.org/10.1016/j.ejor.2016.08.053

 Web of Science ®Google Scholar

Dey, K., Roy, S., & Saha, S. (2019). The impact of strategic inventory and procurement strategies on green product design in a two-period supply chain. International Journal of Production Research, 57(7), 1915–1948. https://doi.org/10.1080/00207543.2018.1511071

 Web of Science ®Google Scholar

Cohen, M. C., Lobel, R., & Perakis, G. (2015). The impact of demand uncertainty on consumer subsidies for green technology adoption. Management Science, 62(5), 1235–1258. https://doi.org/10.1287/mnsc.2015.2173

 Web of Science ®Google Scholar

Raza, S. A. (2018). Supply chain coordination under a revenue-sharing contract with corporate social responsibility and partial demand information. International Journal of Production Economics, 205, 1–14. https://doi.org/10.1016/j.ijpe.2018.08.023

 Web of Science ®Google Scholar

Raza, S. A., & Govindaluri, S. M. (2019). Greening and price differentiation coordination in a supply chain with partial demand information and cannibalization. Journal of Cleaner Production, 229, 706–726. https://doi.org/10.1016/j.jclepro.2019.04.371

 Web of Science ®Google Scholar

Raza, S. A., Rathinam, S., Turiac, M., & Kerbache, L. (2018). An integrated revenue management framework for a firm’s greening, pricing and inventory decisions. International Journal of Production Economics, 195, 373–390. https://doi.org/10.1016/j.ijpe.2016.11.014

 Web of Science ®Google Scholar

Liu, C., & Chen, W. D. (2019). Decision making in green supply chains under the impact of the stochastic and multiple-variable dependent reference point. Transportation Research Part E: Logistics and Transportation Review, 128, 443–469. https://doi.org/10.1016/j.tre.2019.06.011

 Web of Science ®Google Scholar

Wang, W., Zhang, Y., Zhang, W., Gao, G., & Zhang, H. (2021). Incentive mechanisms in a green supply chain under demand uncertainty. Journal of Cleaner Production, 279, 123636. https://doi.org/10.1016/j.jclepro.2020.123636

 Web of Science ®Google Scholar

Data availability statement

Data sharing is not applicable to this article as no new data were generated in this study. The hypothetical data supporting the numerical analysis are available and analysed within the article.