Parametric Optimization of an Interval Sustainable Supply Chain with Green–Thermal-Linked Demand Rate

Abstract

This work aims to study the optimal policy of a sustainable supply chain problem under interval uncertainty. In this chain, the customers’ demand is fluctuated within an interval and its rate is shaped by three factors: the selling price, green level, and heat-resistance efficiency. To handle this fluctuation, all interval-valued demand parameters and cost components are presented in a parametric form. And this leads to the building of imprecise profit functions in parametric form for both the centralized and decentralized scenarios of the supply chain. This work differs from existing interval-valued models by jointly incorporating green-level and heat-resistance-dependent demand within a parametric optimization framework. Now, the core research question of this study is as follows: how can one find the optimal policy of the system under imprecise parametric settings? To answer this question, this work introduces a new optimization technique, named parametric optimization, to study the optimal policy of the imprecise parametrized profits of different scenarios of the system. Using this parametric optimization technique and game-theoretic approach, the formulae for finding optimal values of decision variables along with optimal profits in all scenarios are obtained. Then, numerical examples are provided to demonstrate the optimal policy of the proposed model. Finally, a post-optimality analysis is performed to examine the effects of variations in key system parameters on optimal policies of the system. This research offers practical insights to study how the supply chain’s partners can navigate uncertainty by meeting sustainability goals.

1. Introduction

Sustainable supply chain management (SSCM) is becoming a critical research topic due to growing environmental concerns, regulatory pressures, and consumer demand for environmentally friendly products. A major challenge in designing sustainable supply chains is tackling of uncertainties in demand, cost parameters, and environmental factors while optimizing both economic and environmental performance. Traditional deterministic models often fail to capture real-world complexities, which require robust optimization techniques under uncertainty.

Green supply chain management (GSCM) emphasizes the integration of environmental sustainability into supply chain operations. Studies such as Jaggernath and Khan [1] and Green et al. [2] highlight how GSCM practices improve firm performance and reduce ecological impact. Recent works by Pal et al. [3] and Wiredu et al. [4] further explore competitive and performance-driven aspects of GSCM, demonstrating the role of green initiatives in achieving long-term sustainability.

Research on supply chain optimization under uncertainty has evolved significantly, with various approaches that address demand fluctuations and cost variations. Govindan et al. [5] provide a comprehensive review of uncertainty modeling in supply chain networks, while Borodin et al. [6] focus on agricultural supply chains. Advanced techniques such as interval-valued fuzzy sets (Foroozesh et al. [7]) and hybrid decision-making models (Zhou et al. [8]) have been proposed to enhance flexibility in uncertain environments.

This research paper is systematically structured to address an imprecise sustainable supply chain problem. The present section introduces the topic and outlines the contributions, while Section 2 reviews the relevant literature. Section 3 covers the mathematical preliminaries for interval analysis, and Section 4 details the parametric optimization technique. The problem definition is described in Section 5 and mathematically formulated in Section 6. Section 7 provides the optimality analyses for centralized and decentralized scenarios, which are then illustrated with a numerical example in Section 8. The robustness of the model is tested through sensitivity analyses in Section 9, and the paper concludes with a summary of findings and future research directions in Section 10.

2. Literature Review

Green supply chain management integrates environmental considerations into supply chain operations to enhance sustainability. Jaggernath and Khan [1] highlight the role of GSCM in fostering eco-friendly business practices, while Green et al. [2] demonstrate its positive impact on firm performance. Badi and Murtagh [9] provide a systematic review of GSCM in construction, emphasizing the need for sustainable procurement and logistics. Gawusu et al. [10] explore the role of renewable energy in GSCM, and Saini et al. [11] discuss the transition from traditional to green supply chains. Recent studies by Pal et al. [3] and Wiredu et al. [4] examine competitive dual-channel supply chains and the influence of GSCM practices on corporate environmental performance. Hosain and Mustafi [12] further investigate the mediating role of supply chain strategies in achieving environmental sustainability.

Uncertainty is one of the intrinsic attributes of supply chain management, originating from stochastic demand, variable inventory costs, environmental constraints, and complex inter-dependencies among network components. Early investigations focused on uncertainty modeling in specific contexts, such as agricultural supply chains [6] and generalized supply chain network design problems [5]. Subsequently, uncertainty was incorporated into sustainable decision-making, including hybrid formulations for circular supplier selection [13] and flexible risk-aware optimal decision frameworks [14]. To address the inherent impreciseness of real-world data, advanced fuzzy and robust mathematical approaches have been developed, notably Pythagorean fuzzy Choquet integrals [15] and interval-valued fuzzy programming for sustainable supply chain design [7]. The accumulation and propagation of uncertainty across supply chain networks may induce nervousness and systemic instability [16,17], motivating the development of mathematically sound, multi-criteria, and resilience-oriented decision models [18,19,20,21,22,23]. Recent studies further refine these formulations using interval type-2 and Fermatean fuzzy sets to enhance green supply chain management and optimization under uncertainty [8,24,25].

Parametric approaches are effective in handling interval uncertainty by converting uncertain parameters into deterministic forms. Manna et al. [26] propose a parametric approach for perishable inventory models with prepayment policies. Yadav et al. [27] develop an interval-number-based inventory model for deteriorating items, while Rahman et al. [28,29] optimize a production-inventory model under warranty-linked demand using interval techniques. Ali et al. [30] apply an improved center–radius technique for sustainable production control under interval-valued demand. These studies highlight the effectiveness of parametric optimization in addressing supply chain uncertainties.

Despite extensive works on sustainable and green supply chains ([1,2,9]), existing studies rarely modeled supply chains where demand rate depends simultaneously on green production level and product heat-resistance efficiency. Most of the studies considered only deterministic supply chains, or very few interval uncertainty-based chains within a bi-level (manufacturer–retailer) decision structure. On the other hand, although the interval parametric approach ([26,27,28,29,30]) was used by several researchers in the area of inventory control, no one used it in supply chain systems for handling interval parameters. To address these gaps, this study develops a new bi-level sustainable supply chain model with multi-factor-linked interval-valued demand under uncertainty. Then, to optimize the parametric profit functions for both centralized and decentralized structures of the system, a hybrid parametric optimization–game-theoretic approach is proposed. Numerical illustrations and detailed post-optimality analysis are performed to justify the optimal policy and variations of system’s parameters of the chain. The main contributions of this study are highlighted in the following:

• Develop a bi-level supply chain model with inter-level demand dependent on multiple factors.
• Formulate parametric profit functions for both centralized and decentralized cases.
• Introduce a parametric optimization technique to derive optimal policies.
• Conduct a post-optimality analysis to assess the impact of key parameters.

3. Preliminaries

In this section, the parametric representation of a closed and bounded interval is discussed. Then, the arithmetic operations between intervals in parametric form are presented. Finally, the parametric representation of the interval-valued function is highlighted.

Let $A=[a_L,a_U]$ be a closed interval, where $a_L$ and $a_U$ are the lower and upper bounds of A, respectively. Then, the parametric representations of A are as follows:

(i)

The increasing representation (IR) is as follows:

$$A\left(\xi \right)=a_L+\xi (a_U-a_L),\xi \in [0,1].$$

(ii)

The decreasing representation (DR) is as follows:

$$A\left(\xi \right)=a_U-\xi (a_U-a_L),\xi \in [0,1].$$

3.1. Arithmetic Operations in Parametric Form

Let $a\left(\xi \right)$ and $b\left(\xi \right)$ be the IRs or DRs of $A=[a_L,a_U]$ and $B=[b_L,b_U]$, respectively, and let $\lambda \in \mathbb{R}$. Then, parametric arithmetic operations are defined as follows:

(i)

Addition:

$$A+B=\left\{a\right(\xi )+b(\xi ):r\in [0,1\left]\right\}.$$

(ii)

Subtraction:

$$A-B=\left\{a\right(\xi )-b(\xi ):r\in [0,1\left]\right\}.$$

(iii)

Multiplication:

$$A·B=\left\{a\right(\xi )·b(\xi ):r\in [0,1\left]\right\}.$$

(iv)

Scalar multiplication:

$$\lambda A=\{\lambda ·a(\xi ):r\in [0,1\left]\right\}.$$

(v)

Division (if $b\left(\xi \right)\ne 0$):

$$\left({\displaystyle \frac{A}{B}}\right)=\{{\displaystyle \frac{a\left(\xi \right)}{b\left(\xi \right)}}:r\in [0,1]\}.$$

(vi)

Equivalence:

$$A=B\iff \forall \xi \in [0,1],a\left(\xi \right)=b\left(\xi \right).$$

3.2. Parametric Representation of Interval-Valued Function

Let $F:D\subseteq \mathbb{R}\to {\mathbb{K}}_c$ be an interval-valued function given by

$$F\left(x\right)=[f_L\left(x\right),f_U\left(x\right)],$$

where ${\mathbb{K}}_c$ is the set of all closed intervals.

Then, parametric representations of $F\left(x\right)$ can be defined as follows:

(i)

IR:

$$F(x,\xi )=f_L\left(x\right)+\xi (f_U\left(x\right)-f_L\left(x\right)),\xi \in [0,1].$$

(ii)

DR:

$$F(x,\xi )=f_U\left(x\right)-\xi (f_U\left(x\right)-f_L\left(x\right)),\xi \in [0,1].$$

Example: Let

$$F\left(x\right)=[1,4]x^2+[2,3]x-[1,3]e^x.$$

Then, using the parametric representation, $F\left(x\right)$ can be written as follows:

$$F(x,\xi )=(1+3\xi )x^2+(2+\xi )x-(1+2\xi )e^x,\xi \in [0,1].$$

4. Parametric Optimization Technique

In this section, a new optimization technique for optimizing the interval optimization problem is proposed, which is named the parametric optimization technique.

Let $H:\mathrm{\Omega }\subseteq {\mathbb{R}}^n\to {\mathcal{K}}_c$ be an interval-valued function with the parametric form $H(u,\xi ),\xi \in [0,1]$.

Definition 1.

The point $u=u^{*}\in \mathrm{\Omega }$ is called the ${\xi }^{*}$-maximizer of the interval-valued function $H\left(u\right)=\left[h_L\left(u\right),h_U\left(u\right)\right]$ if it is the maximizer of $H(u,{\xi }^{*}),\xi ={\xi }^{*}\in [0,1]$.

Definition 2.

The function $H\left(u\right)=\left[h_L\left(u\right),h_U\left(u\right)\right]$ is called ${\xi }^{*}$- concave if $H(u,{\xi }^{*}),\xi ={\xi }^{*}\in [0,1]$ is concave.

Theorem 1.

Let the point $u=u^{*}\in \mathrm{\Omega }$ be the ${\xi }^{*}$-maximizer of the interval-valued function $H\left(u\right)=\left[h_L\left(u\right),h_U\left(u\right)\right]$. Then, $\nabla H(u^{*},{\xi }^{*})=0,\xi ={\xi }^{*}\in [0,1]$, provided $H(u,{\xi }^{*})$ is a non-constant differentiable function.

Proof. 

The proof follows from Definition 1 and the optimization theory for deterministic cases. □

Theorem 2.

Let the point $u=u^{*}\in \mathrm{\Omega }$ satisfy the condition of Theorem 1. Then, $u=u^{*}\in \mathrm{\Omega }$ is the ${\xi }^{*}$-maximizer of $H\left(u\right)=\left[h_L\left(u\right),h_U\left(u\right)\right]$ if ${\nabla }^{2}H(u^{*},{\xi }^{*})$ is negative definite, provided $H(u,\xi )$ is differentiable up to the second order.

Proof. 

The proof follows from Definition 1 and the optimization theory for deterministic cases. □

Theorem 3.

Let $H\left(u\right)=\left[h_L\left(u\right),h_U\left(u\right)\right]$ be the interval-valued function and its parametrized function $H(u,\xi )$ is differentiable up to the second order. Then, $H\left(u\right)$ is ${\xi }^{*}$- concave in Ω, if ${\nabla }^{2}H(u,{\xi }^{*})$ is definitely negative for $u\in \mathrm{\Omega }$.

Proof. 

The proof follows from Definition 2 and the theory of concavity for deterministic cases. □

These results for parametric optimization are implemented to study the optimal policy of an imprecise green supply chain model under interval uncertainty. The details regarding the problem description and assumptions of the proposed supply chain model are discussed in later sections.

5. Problem Description

In real-life supply chain systems, key factors such as demand, cost factors, and customer preferences are often uncertain and imprecise, influenced by variables like price, environmental impact, and product quality. To address these factors, the present work formulates a two-level sustainable supply chain model under interval uncertainty, where customers’ demand rate is represented as an interval-valued function dependent on green level, heat resistance, and selling price. To formulate the model mathematically, essential notations and assumptions are presented in the next two subsections.

5.1. Notation

To formulate the proposed model, the following notations are used:

Parameters	:	Description
$s_m$	:	Manufacturer’s selling price per unit product
$s_r$	:	Selling price per unit product of the retailers
${\eta }_g$	:	Green level of the product
$h_c$	:	Heat-resistance indicator
$\left[D_L({\eta }_g,s_r,h_c),D_U({\eta }_g,s_r,h_c)\right]$	:	Interval-valued demand rate of customers
$c_p$	:	Unit purchase cost
$\left[{\alpha }_{0L},{\alpha }_{0U}\right]$	:	Interval-valued initial demand rate
${\alpha }_0\left(\xi \right)={\alpha }_{0L}+\xi ({\alpha }_{0U}-{\alpha }_{0L}),\xi \in [0,1]$	:	Parametric form of initial demand rate
$\left[{\alpha }_{iL},{\alpha }_{iU}\right],i=1,2,3$	:	Interval-valued parameters in demand rate
${\alpha }_i\left(\xi \right)={\alpha }_{iL}+\xi ({\alpha }_{iU}-{\alpha }_{iL}),\xi \in [0,1],i=1,2,3$	:	Parametric form of parameters in demand rate
$\left[{\lambda }_L,{\lambda }_U\right]$	:	Interval-valued hospitality effort investment
$\lambda \left(\xi \right)={\lambda }_L+\xi ({\lambda }_U-{\lambda }_L),\xi \in [0,1]$	:	Parametric form of hospitality effort investment
$\left[{\beta }_L,{\beta }_U\right]$	:	Interval-valued scaling parameter of green level
$\beta \left(\xi \right)={\beta }_L+\xi ({\beta }_U-{\beta }_L),\xi \in [0,1]$	:	Parametric form of the green investment cost
$\left[{\pi }_{mL}({\eta }_g,s_m,h_c),{\pi }_{mU}({\eta }_g,s_m,h_c)\right]$	:	Interval-valued manufacturer’s profit
${\pi }_m({\eta }_g,s_m,h_c,\xi )$	:	Parametric form of interval-valued manufacturer’s profit
$\left[{\pi }_{rL}\left(s_r\right),{\pi }_{rU}\left(s_r\right)\right]$	:	Interval-valued retailer’s profit
${\pi }_r(s_r,\xi )$	:	Parametric form of interval-valued retailer’s profit
$\left[{\pi }_{scL}({\eta }_g,s_r,h_c),{\pi }_{scU}({\eta }_g,s_r,h_c)\right]$	:	Interval-valued integrated supply chain profit
${\pi }_{sc}({\eta }_g,s_r,h_c,\xi )$	:	Parametric form of interval-valued integrated profit

5.2. Assumptions

This subsection presents the key modeling assumptions underlying our sustainable supply chain analysis. Each assumption is accompanied by economic and empirical justification to ground the model in realistic supply chain behavior.

(i)

The proposed green supply chain system has two members—manufacturer and retailer. And the model of this chain has been formulated under interval uncertainty. The proposed model considers a two-member supply chain comprising one manufacturer and one retailer. This structure will capture the fundamental decision-making dynamics in supply networks, where manufacturers produce goods and retailers distribute them to consumers. The model is formulated under interval uncertainty, which provides a mathematically rigorous yet computationally tractable way to handle parameter ambiguity without requiring precise probabilistic information.

(ii)

The customers’ demand rate of this chain is an interval-valued linear function of the green level (${\eta }_g$), heat-resistance indicator of the product ($h_c$), and retailer’s selling price ($s_r$). The demand rate is defined mathematically as

$$\left[D_L({\eta }_g,s_r,h_c),D_U({\eta }_g,s_r,h_c)\right]=\left[{\alpha }_{0L},{\alpha }_{0U}\right]+\left[{\alpha }_{1L},{\alpha }_{1U}\right]{\eta }_g+\left[{\alpha }_{2L},{\alpha }_{2U}\right]h_c-\left[{\alpha }_{3L},{\alpha }_{3U}\right]s_r,$$

where, $\left[{\alpha }_{0L},{\alpha }_{0U}\right],\left[{\alpha }_{1L},{\alpha }_{1U}\right],\left[{\alpha }_{2L},{\alpha }_{2U}\right],\left[{\alpha }_{3L},{\alpha }_{3U}\right]\in K_c^{+}$.

This specification provides a first-order approximation of consumer response while enabling closed-form solutions. Linear demand forms are standard in the operations literature and capture the essential trade-offs between sustainability attributes, quality, and price.

(iii)

The manufacturer has maintained a green investment cost for producing green products, which is an interval-valued function of the green level, and its mathematical form is given by $\left[{\beta }_L,{\beta }_U\right]{\eta }_g^2,$ where $\left[{\beta }_{0L},{\beta }_{0U}\right]$ is the fixed interval-valued purchase cost of the product.

Quadratic costs reflect increasing marginal investments needed for higher sustainability levels—initial improvements use cost-effective technologies, while advanced green features require expensive R&D and premium materials. This convex structure prevents unrealistic over-investment.

(iv)

The manufacturer also bears a eco-thermal investment cost to produce eco-thermal products, which is an interval-valued function of the heat-resistance indicator, and its mathematical form is given by $\left[{\lambda }_{cL},{\lambda }_{cU}\right]h_c^2$.

Just as with green investment, achieving higher thermal efficiency exhibits diminishing returns, where basic insulation is inexpensive but advanced thermal properties involve complex materials and manufacturing processes.

6. Mathematical Formulation

In the proposed imprecise supply chain model, the manufacturer produced $\left[D_L(·),D_U(·)\right]$ units of green products and they have been sold to customers by the retailer. Based on the considered assumptions, the interval-valued profits of the manufacturer, retailer, and integrated supply chain system are, respectively, calculated as follows:

$$\left[{\pi }_{mL}({\eta }_g,s_m,h{I}_c),{\pi }_{mU}({\eta }_g,s_m,h_c)\right]=(s_m-c_p)\left[D_L({\eta }_g,s_r,h_c),D_U({\eta }_g,s_r,h_c)\right]-\left[{\beta }_L,{\beta }_U\right]{\eta }_g^2-\left[{\lambda }_L,{\lambda }_U\right]h_c^2.$$

(1)

The parametric form of the interval-valued profit of the manufacturer is calculated as

$${\pi }_m({\eta }_g,s_m,h_c,\xi )=(s_m-c_p)\left\{{\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r\right\}-\beta \left(\xi \right){\eta }_g^2-\lambda \left(\xi \right)h_c^2.$$

(2)

The interval-valued retailer’s profit of the chain is as follows:

$$\begin{array}{ccc}\hfill \left[{\pi }_{rL}\left(s_r\right),{\pi }_{rU}\left(s_r\right)\right]& =& (s_r-s_m)\left[D_L({\eta }_g,s_r,h_c),D_U({\eta }_g,s_r,h_c)\right].\hfill \end{array}$$

(3)

The parametric form of the interval-valued retailer’s profit of the chain is

$$\begin{array}{cc}\hfill {\pi }_r(s_r,\xi )& =(s_r-s_m)\left\{{\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r\right\}.\hfill \end{array}$$

(4)

The interval-valued joint (integrated) profit of the chain is calculated as follows:

$$\begin{array}{cc}\hfill \left[{\pi }_{scL}({\eta }_g,s_m,h{I}_c),{\pi }_{scU}({\eta }_g,s_m,h_c)\right]& =\left[{\pi }_{mL}(·),{\pi }_{mU}(·)\right]+\left[{\pi }_{rL}(·),{\pi }_{rU}(·)\right]\hfill \\ \hfill & =(s_r-c_p)\left[D_L(·),D_U(·)\right]-\{\left[{\beta }_L,{\beta }_U\right]{\eta }_g^2+\left[{\lambda }_L,{\lambda }_U\right]h_c^2\}\hfill \end{array}$$

(5)

The parametric form of the interval-valued joint profit of the chain is as follows:

$${\pi }_{sc}({\eta }_g,s_r,I_c,\xi )=\{s_r-c_p\}\left\{{\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r\right\}-\beta \left(\xi \right){\eta }_g^2-\lambda \left(\xi \right)h_c^2.$$

(6)

7. Optimality Analyses

In this section, we have optimized the interval-valued profit functions of the retailer, manufacturer, and integrated supply chain system for centralized and contract-sharing scenarios using the parametric optimization technique.

7.1. Centralized Scenario

In this scenario, the center of the interval-valued profit of the integrated supply chain system is maximized by finding the optimal values of ${\eta }_p$, $I_c$ and $s_r$. The maximization problem is defined as

$$\underset{s_r>0,{\eta }_g>0,h_c>0}{\mathrm{Maximize}}{\pi }_{sc}(s_r,{\eta }_g,h_c,\xi )=(s_r-c_p)\left\{{\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r\right\}-\beta \left(\xi \right){\eta }_g^2-\lambda \left(\xi \right)h_c^2.$$

(7)

According to the parametric optimization technique, to optimize the interval-valued integrated profit (5), it is sufficient to optimize the parmetric form of the manufacturer’s profit (7) for a fixed value of $\xi$.

Proposition 1.

The interval-valued profit function of the integrated system given in (7) is ξ- concave if the conditions ${\alpha }_2\left(\xi \right)<2\sqrt{\{}{\alpha }_3\left(\xi \right)\lambda \left(\xi \right)\}$ and $4\lambda \left(\xi \right)\beta \left(\xi \right){\alpha }_3\left(\xi \right)>{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)+{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)$ are satisfied for a fixed value of ξ.

Proof. 

First-order partial derivatives of ${\pi }_{sc}(s_r,{\eta }_g,h_c,\xi )$ with respect to ${\eta }_g$, $h_c$, and $s_r$ are, respectively, as follows.

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\pi }_{sc}}{\partial {\eta }_g}}& =(s_r-c_p){\alpha }_1\left(\xi \right)-2\beta \left(\xi \right){\eta }_g\hfill \\ \hfill {\displaystyle \frac{\partial {\pi }_{sc}}{\partial h_c}}& =(s_r-c_p){\alpha }_2\left(\xi \right)-2\lambda \left(\xi \right)h_c\hfill \\ \hfill {\displaystyle \frac{\partial {\pi }_{sc}}{\partial s_r}}& =-(s_r-c_p){\alpha }_3\left(\xi \right)+\left({\alpha }_0\left(\xi \right)\right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r).\hfill \end{array}$$

Second-order derivatives of ${\pi }_{sc}(s_r,{\eta }_g,h_c,\xi )$ with respect to ${\eta }_g$, $h_c$, and $s_r$ are, respectively, as follows.

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g^2}}& =-2\beta \left(\xi \right)<0\hfill \\ \hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c^2}}& =-2\lambda \left(\xi \right)0\hfill \\ \hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r^2}}& =-2{\alpha }_3\left(\xi \right)<0\hfill \\ \hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g\partial s_r}}& ={\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r\partial {\eta }_g}}={\alpha }_1\left(\xi \right)>0\hfill \\ \hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g\partial h_c}}& ={\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c\partial {\eta }_g}}=0\hfill \\ \hfill {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r\partial h_c}}& ={\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c\partial s_r}}={\alpha }_2\left(\xi \right)>0.\hfill \end{array}$$

The Hessian matrix corresponding to ${\pi }_{sc}$ is calculated as follows:

$$H\left(\xi \right)=\left[\begin{array}{ccc}{\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c^2}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r\partial h_c}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g\partial h_c}}\\ {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c\partial s_r}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r^2}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g\partial s_r}}\\ {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial h_c\partial {\eta }_g}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial s_r\partial {\eta }_g}}& {\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g^2}}\end{array}\right]=\left[\begin{array}{ccc}-2\lambda \left(\xi \right)& {\alpha }_2\left(\xi \right)& 0\\ {\alpha }_2\left(\xi \right)& -2{\alpha }_3\left(\xi \right)& {\alpha }_1\left(\xi \right)\\ 0& {\alpha }_1\left(\xi \right)& -2\beta \left(\xi \right)\end{array}\right].$$

Thus, ${\pi }_{sc}$ will be $\xi$-concave if and only if the first-order and third-order principal minors of $H\left(\xi \right)$ are negative and that of the second-order principal minor is positive.

Clearly, the first-order principal minor of $H\left(\xi \right)$ is ${\displaystyle \frac{{\partial }^2{\pi }_{sc}}{\partial {\eta }_g^2}}=-2\lambda \left(\xi \right)<0$. The second-order principal minor of $H\left(\xi \right)$ will be positive if

$$\begin{array}{cc}\hfill & \left|\begin{array}{cc}-2\lambda \left(\xi \right)& {\alpha }_2\left(\xi \right)\\ {\alpha }_2\left(\xi \right)& -2{\alpha }_3\left(\xi \right)\end{array}\right|>0\hfill \\ \hfill \Rightarrow & 4\lambda \left(\xi \right){\alpha }_3\left(\xi \right)-{\left({\alpha }_2\left(\xi \right)\right)}^2>0\hfill \\ \hfill \Rightarrow & {\alpha }_2\left(\xi \right)<2\sqrt{(}{\alpha }_3\left(\xi \right)\lambda \left(\xi \right)).\hfill \end{array}$$

Finally, the third-order principal minor of H will be negative if

$$\begin{array}{cc}\hfill & \left|\begin{array}{ccc}-2\lambda \left(\xi \right)& {\alpha }_2\left(\xi \right)& 0\\ {\alpha }_2\left(\xi \right)& -2{\alpha }_3\left(\xi \right)& {\alpha }_1\left(\xi \right)\\ 0& {\alpha }_1\left(\xi \right)& -2\beta \left(\xi \right)\end{array}\right|<0\hfill \\ \hfill & \Rightarrow {\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)+{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)-4\lambda \left(\xi \right)\beta \left(\xi \right){\alpha }_3\left(\xi \right)<0\hfill \\ \hfill & \Rightarrow 4\lambda \left(\xi \right)\beta \left(\xi \right){\alpha }_3\left(\xi \right)>{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)+{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right).\hfill \end{array}$$

This completes the proof. □

Proposition 1 ensures that ${\pi }_{sc}(s_r,{\eta }_g,h_c,\xi )$ is optimal at the point ($s_r^{*}$, ${\eta }_g^{*}$, $h_c^{*}$) is satisfied. Now, equating first-order partial derivatives of ${\pi }_{sc}(s_r,{\eta }_g,h_c,\xi )$ to zero and solving, we obtain the optimal values of the decision variables $s_r$, ${\eta }_g$, $h_c$, which are as follows:

$$\begin{array}{cc}\hfill {\eta }_g\left(\xi \right)& ={\displaystyle \frac{2\{{\alpha }_0\left(\xi \right)-{\alpha }_1\left(\xi \right)c_p\}\beta \left(\xi \right)\lambda \left(\xi \right)}{4{\alpha }_3\left(\xi \right)\lambda \left(\xi \right)\beta \left(\xi \right)-{\left\{{\alpha }_1\left(\xi \right)\right\}}^2\lambda \left(\xi \right)-{\left\{{\alpha }_2\left(\xi \right)\right\}}^2\beta \left(\xi \right)}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill h_c\left(\xi \right)& ={\displaystyle \frac{2\{{\alpha }_0\left(\xi \right)-{\alpha }_2{c}_p\}\beta \left(\xi \right)\lambda \left(\xi \right)}{4{\alpha }_3\left(\xi \right)\lambda \left(\xi \right)\beta \left(\xi \right)-{\left\{{\alpha }_1\left(\xi \right)\right\}}^2\lambda \left(\xi \right)-{\left\{{\alpha }_2\left(\xi \right)\right\}}^2\beta \left(\xi \right)}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill s_r\left(\xi \right)& ={\displaystyle \frac{2\{{\alpha }_3\left(\xi \right)c_p+{\alpha }_0\left(\xi \right)\}\lambda \left(\xi \right)\beta \left(\xi \right)-c_p\{{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)+{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)\}}{4{\alpha }_3\left(\xi \right)\lambda \left(\xi \right)\beta \left(\xi \right)-{\left\{{\alpha }_1\left(\xi \right)\right\}}^2\lambda \left(\xi \right)-{\left\{{\alpha }_2\left(\xi \right)\right\}}^2\beta \left(\xi \right)}}.\hfill \end{array}$$

(10)

7.2. Decentralize Cases

In this section, interval-valued profits of the retailer and manufacturer are maximized using parametric optimization and game-theoretic approaches. The retailer’s and manufacturer’s profit functions in parametric forms ${\pi }_r(s_r,\xi )$ and ${\pi }_m({\eta }_g,s_m,h_c,\xi )$ are obtained from Equations (2) and (4), respectively.

According to the parametric optimization technique, to study the optimality of the interval-valued profits of the retailer and manufacturer, it is sufficient to study the optimality of the corresponding parametric forms of the interval-valued profits of retailer and manufacturer, keeping $\xi$ constant.

Here, at first, ${\pi }_r(s_r,\xi )$ is maximized by considering $\xi$ as a constant and the corresponding maximization problem is as follows:

$$\begin{array}{c}\hfill \underset{s_r>0}{\mathrm{Maximize}}{\pi }_r\left(s_r\xi \right)=(s_r-s_m)\{{\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_r\}.\end{array}$$

(11)

Proposition 2.

For a fixed ξ, the interval-valued retailer’s profit function is concave with respect to the retailer’s selling price ($s_r$).

Proof. 

Considering $\xi$ as a constant, the first-order partial derivatives of ${\pi }_r(s_r,\xi )$ with respect to $s_r$ is

$$\begin{array}{ccc}& & {\displaystyle \frac{d{\pi }_r(s_r,\xi )}{d{s}_r}}={\alpha }_0\left(\xi \right)+{\alpha }_3\left(\xi \right)s_m+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-2{\alpha }_3\left(\xi \right)s_r.\hfill \end{array}$$

(12)

The second-order partial derivatives of ${\pi }_r\left(s_r\xi \right)$ with respect to $s_r$ is

$$\begin{array}{ccc}& & {\displaystyle \frac{d^2{\pi }_r(s_r,\xi )}{d{s}_r^2}}=-2{\alpha }_3\left(\xi \right)<0.\hfill \end{array}$$

(13)

So, ${\pi }_r(s_r,\xi )$ is concave with respect to the retailer’s selling price ($s_r$) for a fixed $\xi$. □

From Proposition 2, ${\pi }_r(s_r,\xi )$ is optimal at the point $s_r=s_r^{*}$ such that ${\displaystyle \frac{d{\pi }_r(s_r,\xi )}{d{s}_r}}=0$ at $s_r=s_r^{*}$. Now, solving ${\displaystyle \frac{d{\pi }_r(s_r,\xi )}{d{s}_r}}=0$ , one can obtain the following:

$$\begin{array}{ccc}& & s_r={\displaystyle \frac{{\alpha }_0\left(\xi \right)+{\alpha }_3\left(\xi \right)s_m+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c}{2{\alpha }_3\left(\xi \right)}}.\hfill \end{array}$$

(14)

Now, substituting the value of $s_r$ in the manufacturer’s profit function (2), one can find

$$\begin{array}{c}\hfill {\pi }_m({\eta }_g,s_m,h_c,\xi )={\displaystyle \frac{1}{2}}(s_m-c_p)({\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_m)-\beta \left(\xi \right){\eta }_g^2.\end{array}$$

Moreover, the maximization problem related to the manufacturer’s profit ${\pi }_m(s_m,{\eta }_g,h_c,\xi )$ is defined as

$$\underset{s_m>0,{\eta }_g>0,h_c>0}{\mathrm{Maximize}}{\pi }_m(s_m,{\eta }_g,h_c,\xi )={\displaystyle \frac{1}{2}}(s_m-c_p)({\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_g+{\alpha }_2\left(\xi \right)h_c-{\alpha }_3\left(\xi \right)s_m)-\beta \left(\xi \right){\eta }_g^2.$$

(15)

Proposition 3.

The manufacturer’s interval-valued profit function is ξ-concave with respect to $s_m$, ${\eta }_g$, and $h_c$ if $2{\alpha }_3\left(\xi \right)\beta \left(\xi \right)>{\displaystyle \frac{{\left({\alpha }_1\left(\xi \right)\right)}^2}{4}}$ and $4{\alpha }_3\left(\xi \right)\beta \left(\xi \right)\lambda \left(\xi \right)>{\displaystyle \frac{1}{2}}{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)+{\displaystyle \frac{1}{2}}{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)$.

Proof. 

Considering $\xi$ as a constant, the first-order derivatives of ${\pi }_m(s_m,{\eta }_g,h_c)$ with respect to $s_m$, ${\eta }_g$, and $h_c$ are, respectively,

$$\begin{array}{ccc}& & {\displaystyle \frac{\partial {\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial s_m}}={\displaystyle \frac{1}{2}}({\alpha }_0\left(\xi \right)+{\alpha }_1\left(\xi \right){\eta }_p+{\alpha }_2\left(\xi \right)h_c-2{\alpha }_3\left(\xi \right)s_m+{\alpha }_3\left(\xi \right)c_p),\hfill \\ & & {\displaystyle \frac{\partial {\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial {\eta }_g}}={\displaystyle \frac{1}{2}}(s_m-c_p){\alpha }_1\left(\xi \right)-2\beta \left(\xi \right){\eta }_p,\hfill \\ & & and{\displaystyle \frac{\partial {\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial h_c}}={\displaystyle \frac{1}{2}}(s_m-c_p){\alpha }_2\left(\xi \right)-2\lambda \left(\xi \right)h_c.\hfill \end{array}$$

The second-order derivatives of ${\pi }_m(s_m,{\eta }_g,h_c)$ with respect to $s_m$, ${\eta }_g$, and $h_c$ are, respectively, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m^2}}=-{\alpha }_3\left(\xi \right)$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g\partial s_m}}={\displaystyle \frac{{\alpha }_1\left(\xi \right)}{2}}$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c\partial s_m}}={\displaystyle \frac{{\alpha }_2\left(\xi \right)}{2}}$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g^2}}=-2\beta \left(\xi \right)$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m\partial {\eta }_g}}={\displaystyle \frac{{\alpha }_1\left(\xi \right)}{2}}$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c\partial {\eta }_g}}=0$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c^2}}=-2\lambda \left(\xi \right)$, ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m\partial h_c}}={\displaystyle \frac{{\alpha }_2\left(\xi \right)}{2}}$, and ${\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g\partial h_c}}=0.$

$$\begin{array}{c}\hfill H_m\left(\xi \right)=\left[\begin{array}{ccc}{\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g\partial s_m}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c\partial s_m}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m\partial {\eta }_g}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g^2}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c\partial {\eta }_g}}\\ {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial s_m\partial h_c}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial {\eta }_g\partial h_c}}& {\displaystyle \frac{{\partial }^2{\pi }_m}{\partial h_c^2}}\end{array}\right]=\left[\begin{array}{ccc}-{\alpha }_3\left(\xi \right)& {\displaystyle \frac{{\alpha }_1\left(\xi \right)}{2}}& {\displaystyle \frac{{\alpha }_2\left(\xi \right)}{2}}\\ {\displaystyle \frac{{\alpha }_1\left(\xi \right)}{2}}& -2\beta \left(\xi \right)& 0\\ {\displaystyle \frac{{\alpha }_2\left(\xi \right)}{2}}& 0& -2\lambda \left(\xi \right)\end{array}\right].\end{array}$$

Here, the first principle minor of $H_m\left(\xi \right)$ is $-{\alpha }_3\left(\xi \right)$, which is negative. The second principle minor of $H_m\left(\xi \right)$ is $2{\alpha }_3\left(\xi \right)\beta \left(\xi \right)-{\displaystyle \frac{{\left({\alpha }_1\left(\xi \right)\right)}^2}{4}}$. Also, the third principle minor of $H_m\left(\xi \right)$, that is, $det\left(H_m\left(\xi \right)\right)=-4{\alpha }_3\left(\xi \right)\beta \left(\xi \right)\lambda \left(\xi \right)+{\displaystyle \frac{1}{2}}{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)+{\displaystyle \frac{1}{2}}{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)$. Now, if the Hessian matrix $H_m\left(\xi \right)$ of ${\pi }_m(s_m,{\eta }_g,h_c,\xi )$ is definitely negative, then the manufacturer’s profit function ${\pi }_m(s_m,{\eta }_g,h_c,\xi )$ is concave.

The Hessian matrix $H_m\left(\xi \right)$ of ${\pi }_m(s_m,{\eta }_g,h_c,\xi )$ is definitely negative provided that (i) the first principle minor of $H_m\left(\xi \right)$ is negative, (ii) the second principle minor of $H_m\left(\xi \right)$ is positive, and (iii) the third principle minor of $H_m\left(\xi \right)$ is negative. So, ${\pi }_m(s_m,{\eta }_g,h_c,\xi )$ is concave if $2{\alpha }_3\left(\xi \right)\beta \left(\xi \right)>{\displaystyle \frac{{\left({\alpha }_1\left(\xi \right)\right)}^2}{4}}$ and $4{\alpha }_3\left(\xi \right)\beta \lambda >{\displaystyle \frac{1}{2}}{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)+{\displaystyle \frac{1}{2}}{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)$. □

From Proposition 3, it is evident that ${\pi }_m(s_m,{\eta }_g,h_c)$ is optimal at the point ($s_m^{*}$, ${\eta }_g^{*}$, $h_c^{*}$) satisfying ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial s_m}}=0$, ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial {\eta }_g}}=0$, ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial h_c}}=0$.

Solving the equations ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial s_m}}=0$, ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial {\eta }_g}}=0$, ${\displaystyle \frac{{\pi }_m(s_m,{\eta }_g,h_c,\xi )}{\partial h_c}}=0$, one can obtain

$$\begin{array}{ccc}& & s_m={\displaystyle \frac{4\beta \left(\xi \right)\lambda \left(\xi \right)({\alpha }_0\left(\xi \right)+{\alpha }_3\left(\xi \right)c_p)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)c_p-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)c_p}{8\beta \left(\xi \right)\lambda \left(\xi \right){\alpha }_3\left(\xi \right)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & {\eta }_g={\displaystyle \frac{\lambda \left(\xi \right){\alpha }_1\left(\xi \right)({\alpha }_0\left(\xi \right)-{\alpha }_3\left(\xi \right)c_p)}{8\beta \left(\xi \right)\lambda \left(\xi \right){\alpha }_3\left(\xi \right)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)}},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & \mathrm{and}{h}_c={\displaystyle \frac{\beta \left(\xi \right){\alpha }_2\left(\xi \right)({\alpha }_0\left(\xi \right)-{\alpha }_3\left(\xi \right)c_p)}{8\beta \left(\xi \right)\lambda \left(\xi \right){\alpha }_3\left(\xi \right)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)}}.\hfill \end{array}$$

(18)

Using (16)–(18) in (14), we obtain the optimal value of $s_r$, which is

$$\begin{array}{c}\hfill s_r^{*}={\displaystyle \frac{3\{4\beta \left(\xi \right)\lambda \left(\xi \right)({\alpha }_0\left(\xi \right)+{\alpha }_3\left(\xi \right)c_p)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)c_p-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)c_p\}}{2\{8\beta \left(\xi \right)\lambda \left(\xi \right){\alpha }_3\left(\xi \right)-{\left({\alpha }_1\left(\xi \right)\right)}^2\lambda \left(\xi \right)-{\left({\alpha }_2\left(\xi \right)\right)}^2\beta \left(\xi \right)\}}}-{\displaystyle \frac{c_p}{2}}.\end{array}$$

(19)

Also, one can calculate the maximum interval-valued profits of the manufacturer and retailer by putting the optimal decision variables given in the formulae (16)–(19) in Equations (1) and (3).

Note 1.

For numerical illustration, in the optimal formulae for the central and decentralization cases, the value of the uncertainty measuring parameter can be set as $\xi =0.5$. However, one can take the other values of $\xi$ from the interval [0, 1].

Note 2.

The best way to select $\xi$ is by optimizing $\underset{\xi \in [0,1]}{Maximize}{\pi }_{sc}\left(\xi \right)$ and $\underset{\xi \in [0,1]}{Maximize}{\pi }_r\left(\xi \right)$. The function ${\pi }_{sc}\left(\xi \right)$ is obtained from (7) using the formulae (8)–(10). Similarly, ${\pi }_r\left(\xi \right)$ is obtained from (11).

8. Numerical Example

This section presents a numerical illustration to demonstrate the applicability and managerial insights of the proposed bi-level interval optimization model for a sustainable supply chain under uncertainty. The primary goals are as follows: (1) validate the model’s effectiveness, (2) compare centralized versus decentralized decision-making, and (3) analyze the economic and environmental implications of the optimal strategies. Data and modeling assumptions are partially inspired by related work in sustainable operations [31], but are adapted to the specific context of this study with interval-valued parameters.

Example 1.

Suppose a manufacturer produces a biodegradable plastic product. The unit production cost is fixed at $c_p=SAR35$. To enhance the product’s environmental appeal, the manufacturer invests a cost to increse`green level (${\eta }_g$), incurring a interval-valued cost $[{\beta }_L,{\beta }_U]·{\eta }_g^2$, where the uncertain cost coefficient lies in $[{\beta }_L,{\beta }_U]=[SAR0.3,SAR0.4]$. The green level is a normalized index (0–100), where higher values represent superior environmental performance. Simultaneously, to meet market requirements for thermal resistance (e.g., for food packaging or outdoor use), the manufacturer introduces the heat-transfer efficiency ($h_c$) in the product, with an associated quadratic cost $[{\lambda }_L,{\lambda }_U]·h_c^2$, where $[{\lambda }_L,{\lambda }_U]=[SAR0.6,SAR1.1]$. The products are sold through a retailer. Market demand is linearly affected by the green level (${\eta }_g$), heat efficiency ($h_c$), and the retail price ($s_r$). The interval parameters for demand are as follows: base market potential $[d_{0L},d_{0U}]=[1100,1200]$, green-level sensitivity $[{\alpha }_{1L},{\alpha }_{1U}]=[2,3]$, heat-efficiency sensitivity $[{\alpha }_{2L},{\alpha }_{2U}]=[4,6]$, and price sensitivity $[{\alpha }_{3L},{\alpha }_{3U}]=[17,18]$.

The manufacturer decides the green level (${\eta }_g$), heat efficiency ($h_c$), and wholesale price ($s_m$), whereas the retailer decides the retail price ($s_r$). The objective is to determine the optimal strategies for both parties under interval uncertainty.

Solution. Using the parametric approach with $\xi =0.5$ (representing a neutral decision attitude between optimistic and pessimistic bounds), the optimal decision variables and profits for both centralized and decentralized scenarios are computed. Further, applying the parameter value considered in the example in the optimal formulae of ${\eta }_g^{*}$, $s_m^{*}$, $s_r^{*}$, $h_c^{*}$, ${\xi }^{*}$, ${\pi }_m^c(s_m^{*},{\eta }_g^{*},h_c^{*})$, ${\pi }_r^c\left(s_r^{*}\right)$, ${\pi }_{sc}^c(s_r^{*},{\eta }_g^{*},h_c^{*})$ corresponding to the centralized and de-centralized policies, the numerical results are computed. The optimal results are presented in

Table 1. Optimal results of the numerical example.

Scenario	SCS Members	Profits/Decision Variables	Optimal Values
Decentralize	Manufacturer	$s_m^{*}$ (in SAR)	58.1853
		$h_c^{*}$ (in SAR)	34.0961
		${\eta }_g^{*}$ ($\in (0,100)$)	41.4024
		${\pi }_m(s_m^{*},{\eta }_g^{*},h_c^{*},\xi )$ (in SAR)	3115.5273
	Retailer	$s_r^{*}$ (in SAR)	69.7780
		${\pi }_r^c(s_r^{*},\xi )$ (in SAR)	2351.8208
	Joint Profit	${\pi }_{sc}={\pi }_r+{\pi }_m$ (in SAR)	5467.3481
Centralize		${\eta }_g^{*}$ ($\in (0,100)$)	93.4843
		$h_c^{*}$ (in SAR)	85.7856
		$s_r^{*}$ (in SAR)	82.2921
		${\pi }_{sc}(s_r^{*},{\eta }_g^{*},h_c^{*},\xi )$ (in SAR)	8303.5501

.

Result Discussion

The results summarized in Table 1 and

Table 2. Interval bounds of profit functions of different members w.r.t numerical example.

Scenario	SCS Members	Interval-Valued Profit	Interval Uncertainty of Optimal Values
Decentralized policy	Manufacturer	$[{\pi }_{mL}(·),{\pi }_{mU}(·)]$(in SAR)	[2656.5104, 3659.9078]
	Retailer	$[{\pi }_{rL}(·),{\pi }_{rU}(·)]$ (in SAR)	[1881.6949, 2968.4045]
	Joint Profit	(in SAR)	[4538.2053, 6628.3123]
Centralized		$[{\pi }_{sc}(·),{\pi }_{sc}(·)]$ (in SAR)	[6853.0357, 8686.6879]

highlight the performance differences between centralized and decentralized decision-making within the proposed uncertain sustainable supply chain model. The results highlight several key managerial insights:

• Superiority of Centralized Coordination: The centralized model yields a significantly higher system profit (SAR 8303.55) compared to the decentralized outcome (SAR 5467.35), demonstrating a 52% improvement. This coordination gap arises because the centralized planner internalizes the positive demand effects of sustainability investments (${\eta }_g$ and $h_c$), leading to higher optimal levels (e.g., ${\eta }_g^{*}$ increases from 41.40 to 93.48). The higher retail price in the centralized case (SAR 82.29) reflects consumers’ willingness to pay for enhanced product attributes.
• Interpretation of Interval Profits: The profit intervals displayed in Table 2 represent the range of possible outcomes due to the parameters’ uncertainty. For practitioners, the lower bound (${\pi }_L$) indicates a conservative or worst-case scenario profit, while the upper bound (${\pi }_U$) represents an optimistic scenario. The choice of the parameter $\xi \in [0,1]$ allows a decision-maker to select a strategy anywhere between these extremes. Setting $\xi =0.5$, as in Table 1, represents a balanced or neutral risk attitude, using the midpoint values of all interval-valued parameters.
• Scale of Green Level: The green level (${\eta }_g$) is defined as a normalized index ranging from 0 to 100, where 100 represents the maximum feasible environmental performance given current technology. The optimal value of 93.48 in the centralized model indicates a near-maximum level, justified by the high demand sensitivity to green features and the quadratic cost structure, which makes high levels economically viable only under coordinated, system-wide optimization.

9. Sensitivity Analyses

A sensitivity analysis was conducted by varying the uncertainty parameter $\xi$ from 0 (pessimistic) to 1 (optimistic); to examine the the feasibility of the optimal policy, we have performed sensitivity analyses. Here, we have illustrated the impact of system parameters on the center joint profit of the system.

The sensitivity analyses have been carried out with the help of the data considered in the numerical example, and simulated results are presented in Figure 1, Figure 2, Figure 3 and Figure 4.

Sensitivity analyses were performed to evaluate the stability of the optimal solutions across a range of values of the uncertainty parameter $\xi \in [0,1]$. Figure 1, Figure 2, Figure 3 and Figure 4 show how the optimal decision variables and profits respond to increasing levels of uncertainty.

• Figure 1 reveals that the green level (${\eta }_g$) increases with rising $\xi$, suggesting that as the system becomes more optimistic (higher upper bounds of uncertain parameters), greater investment in sustainability becomes optimal. Similarly, Figure 2 shows that the optimal heat-resistance efficiency ($h_c$) also increases with $\xi$, reinforcing the importance of risk mitigation under favorable uncertainty. The retailer’s selling price ($s_r$), as illustrated in Figure 3, shows a mild upward trend, consistent with the rising demand potential and increased willingness to pay.
• Finally, Figure 4 demonstrates that the centralized policy consistently outperforms the decentralized one in terms of profitability across all values of $\xi$. This consistent advantage underlines the value of coordination in managing both economic and environmental objectives, even in the face of uncertain system parameters.

Practical Rationale for $\xi$: The parameter $\xi$ can be interpreted as a managerial “optimism index.” A risk-averse manager (e.g., in a volatile market) might choose a $\xi$ close to 0, leading to conservative investments and lower, more guaranteed profits. A risk-tolerant or optimistic manager (e.g., in a growing green market) might choose a $\xi$ close to 1, pursuing aggressive strategies to capture higher potential profits. The model thus provides a flexible framework for decision-making under varying risk appetites.

10. Conclusions

In this research, a non-deterministic sustainable supply chain problem with green technology–thermal effect-linked customer demand has been modeled using a parametric approach under interval uncertainty. The core finding of this study is that centralized decision-making is more economically viable than that of the decentralized cases by generating higher joint profits, superior environmental outcomes, and greater robustness when facing uncertain system parameters. A central contribution of the model is the explicit treatment of the uncertainty parameter $\xi$, which controls the decision-maker’s stance from pessimistic to optimistic within the interval framework. The analysis reveals that $\xi$ is not merely a theoretical device but a pivotal factor that directly shapes optimal investment levels, pricing strategies, and profit intervals. The centralized model exhibits a lower sensitivity to variations in $\xi$, providing more stable and preferable interval outcomes across its entire range. This underscores the value of coordination not only for expected performance but also for managing risk and variability.

This model provides a clear strategic argument for supply chain coordination, but its application in the real world has several challenges. The main hurdle is obtaining the data needed to define the model’s uncertainty ranges (e.g., for costs or demand). Companies would need to rely on estimates, historical data, or expert forecasts, which can be subjective. Choosing the right level of optimism or pessimism $\xi$ also depends on the specific risk appetite of a company. Perhaps the biggest challenge is organizational. The centralized approach requires a high degree of trust, transparency, and shared goals between separate companies (e.g., a manufacturer and a supplier). Implementing joint investments and profit-sharing often means creating new contracts and governance structures. Therefore, this model is likely the easiest to adopt within a single vertically integrated company or in a long-term, close partnership where goals are already aligned.

To address these limitations and extend the work, future research should move beyond the current static, single-echelon setting. Substantive directions include the following:

• Dynamic, Multi-Period Modeling: Incorporating temporal dynamics would allow for the analysis of phased investment strategies, learning effects, and the evolution of uncertainty over time.
• Policy-Aware Modeling: A fruitful avenue is to explicitly model the impact of specific regulatory policies (e.g., carbon taxes, cap-and-trade, subsidies) on the interval-valued decisions, providing guidance for both firms and policymakers.
• Empirical and Case-Study Validation: Applying the system to real-world case studies or calibrating it with industry data, one can evaluate its practicality by improving parameter estimation.

In summary, the proposed chain has provided a structured framework for navigating sustainable investments under uncertainty. It highlighted the unequivocal benefits of coordination while also charting a path for future research to enhance its realism, applicability, and integration with the complex realities of supply chain management and environmental policy.