Mathematical modeling and optimization of a three-echelon supply chain management by incorporating circular consumption, cooperative advertisement cost, and selling price decisions

Abstract

This study develops a centralized three echelon open loop supply chain optimization model that integrates circular input adoption with coordinated pricing and cooperative advertising decisions. In the proposed setting, recycled materials and usable components are supplied through an external recycling market and become available for procurement by supply chain members as circular inputs. Circularity is represented through echelon specific circular input utilization decision variables at the supplier, manufacturer, and retailer levels, allowing the model to capture the economic and demand side effects of circular input adoption under market availability constraints. Customer demand is formulated as a function of selling price, cooperative advertising effort, and average circular input utilization, thereby linking market response with sustainability oriented operational decisions. Key cost parameters are modeled as triangular fuzzy numbers and transformed into crisp equivalents using the signed distance method. The resulting nonlinear constrained optimization problem maximizes total supply chain profit subject to advertising budget, inventory capacity, recycled input availability, and circularity related feasibility constraints. Sequential Quadratic Programming is used as the main solution approach and is compared with genetic algorithm, pattern search, and minimax methods. The numerical results show that the SQP based solution achieves the highest profit while maintaining feasible and balanced decision values across pricing, advertising, and circular input utilization variables. Sensitivity analysis further indicates that profitability is most affected by recycled input availability and circular cost adjustment, while incentive strength provides a secondary but consistent improvement. The proposed framework provides a quantitative decision support tool for firms seeking to coordinate pricing, advertising, and circular input adoption decisions while maintaining profitability under realistic market and policy conditions.

Introduction

Escalating resource scarcity, climate change, and tightening environmental regulations have pushed firms to reconsider the traditional linear model of “take, make, dispose” and to move toward circular-economy-based operations. Circular economy thinking focuses on retaining value through reuse, remanufacturing, recycling, and product life extension, so that materials circulate in technical and biological loops instead of ending as waste¹. Although recent trend in circular economy is increasing but Circularity Gap Report estimates that the global share of secondary materials in total material use declined from 9.1% in 2018 to 7.2% in 2023, indicating that global production systems remain heavily dependent on virgin inputs despite increased circular economy activity². In parallel, environmental impacts are strongly supply-chain dominated, CDP reports that corporates’ Scope 3 supply-chain emissions are, on average, 26 times larger than emissions from direct operations, reinforcing the need for multi-echelon interventions spanning upstream supply, manufacturing, and downstream retail operations. Recent work at the interface of circular economy and supply chain management shows that supply chain processes are central to enabling this transition, since sourcing, production, distribution, and reverse logistics collectively determine how effectively materials are recovered and recirculated³.

From a managerial perspective, the shift to circular supply chains is not purely normative. It is increasingly viewed as a strategic opportunity that links profitability with sustainability. Systematic reviews, such as those by Govindan and Hasanagic, highlight that firms pursue circular practices to reduce costs, secure access to critical materials, and improve their market position. Still, they also face considerable barriers related to economics, regulation, and technological readiness⁴. These barriers imply that the feasible level of circularity is constrained by capital availability, recovery and recycling infrastructure, and the ability to coordinate actors across multiple echelons.

At the same time, demand in many markets is no longer exogenous. Customers respond to product price, advertising effort, and perceived environmental attributes, including recycled materials or circular packaging. The global advertising expenditure projected to reach approximately USD 1.08 trillion in 2024, which highlights why cooperative advertising structures can meaningfully affect profitability in retailer-facing demand models⁵. A model is proposed by⁶ sustainable supply chains increasingly model demand as a function of price, advertising, and a “green” or “circular” quality level, showing that sustainability can be a demand driver rather than only a cost burden⁷. This creates a complex set of trade-offs, as firms must coordinate on prices, advertising budgets, and circular consumption levels while coping with practical constraints in accessing and qualifying recycled inputs that are supplied through external recyclers and facilitated by third party logistics providers in an open loop setting.

The present research responds to this context by developing a three echelon mathematical model for an open loop circular input adoption structure consisting of a supplier, a manufacturer, and downstream retailers. In this setting, the focal supply chain does not directly control customer return flows, collection routing, disassembly, or remanufacturing operations. Instead, recycled materials and usable components are assumed to be supplied through an external recycling market, supported by independent recyclers and recovery actors. The supplier, manufacturer, and retailer can procure these circular inputs according to their respective utilization decisions. Circularity is therefore represented by echelon level circular input utilization proportions rather than by explicitly modeled reverse network flows. Demand at the retailer is modelled as a variable quantity that depends simultaneously on selling price, cooperative advertising effort, and the average circular input utilization level across the supply chain. The model also incorporates market availability limits for recycled inputs, budget restrictions, capacity constraints, incentive subsidies, and circularity related penalty costs. This structure enables the study to quantify how pricing, advertising, and circular input adoption interact in a coordinated three echelon supply chain without claiming to optimize a complete closed loop recovery system.

Literature review

Early work on the circular economy emphasized broad principles and macro-level transitions, but more recent research has explicitly connected these ideas to supply chain design. Hazen and co-authors discuss how traditional supply chain management practices can be reconfigured to support circular economy outcomes, highlighting the role of sourcing, production, distribution, and recovery systems in closing resource loops⁸. Bibliometric and conceptual studies demonstrate that circular supply chain management has emerged as a distinct field concerned with designing network structures that enable reuse, remanufacturing, recycling, and cascading use of materials⁹.

From an operational viewpoint, extensive reviews of reverse logistics and closed-loop supply chains have been conducted, identifying key configurations for collection, inspection, remanufacturing, and recycling and summarizing how these are integrated with forward flows¹⁰. The circularity at the supply chain level relies on the coexistence of forward and reverse channels, which significantly complicates planning decisions. Complementing this, systematically mapping the drivers, barriers, and practices that influence circular economy implementation from a supply chain perspective has been conducted. It is concluded that economic and financial constraints, along with regulatory uncertainty, are among the most critical obstacles to higher circularity^4,11. These studies collectively underscore the need for quantitative models that do not treat circularity as a purely conceptual goal but explicitly account for investment limits, capacity constraints, and financial trade-offs when designing circular supply chain strategies.

In parallel, it is also observed that pricing and advertising decisions in supply chains under price-sensitive or promotion-sensitive demand. The demand may increase or decrease with retail price and may depend on trade credit or promotional decisions¹². A recent research analyzes the joint optimization of pricing, ordering, and advertising in a single manufacturer single retailer supply chain, showing how advertising effort shapes demand and interacts with wholesale and retail prices¹³. The effects of price and advertising budget⁶ and advertising cost⁷ on demand have been studied, and inventory models have been proposed to adjust to fluctuating demand levels. Within green supply chains, several authors extend these ideas by introducing demand functions that depend on price, advertising, and a greenness level. Liu investigates pricing policies in a green supply chain where demand is influenced by targeted advertising and the product’s green degree, and demonstrates how higher greenness can stimulate demand under suitable pricing and promotion strategies⁷. Demand-oriented sustainability models show that consumer preferences increasingly respond to environmental attributes, advertising effort, and pricing decisions^14,15. More recently, studies on green supply network design consider demand that depends on both price and environmental attributes, and integrate these relationships into facility location and flow optimization to jointly address economic and ecological objectives¹⁶. These contributions show that modelling demand as a function of price, advertising, and sustainability attributes yields more realistic insights. However, they typically refer to a general “green degree” or environmental performance rather than explicitly treating circular consumption of materials and products as a demand driver along multiple echelons in the chain.

Given the distributed nature of supply chains, researchers have long studied coordination mechanisms such as advertising cost-sharing contracts¹⁷, revenue-sharing schemes¹⁸, and sustainability-oriented incentive contracts¹⁹. Supply-chain partners can enhance overall profitability by collaborating to stimulate market demand through coordinated operational and marketing decisions^20–23. Demonstrate¹⁸ and²⁴ that cooperation among suppliers, manufacturers, and retailers, particularly through joint advertising and aligned pricing policies, can significantly improve performance compared to decentralized decision-making. Later studies consider contracts that combine revenue sharing with advertising cost sharing, showing that when pricing and advertising are jointly optimized under such agreements, full coordination can be achieved under certain conditions²⁴. The costs at each level, supplier, manufacturer, and retailers, are essential to consider while optimizing the revenue model. Basic cost structures commonly include setup, production, ordering, holding, and transportation costs at different stages of the supply chain²⁵. Recent extensions incorporate additional elements such as advertising expenditure²⁶ and carbon-related costs²⁷ to reflect sustainability considerations and market-driven demand dynamics²⁸. In supplier-centric models, costs typically include setup, production, and advertising expenses, whereas manufacturers incur setup, manufacturing, ordering, and advertising costs. Retailers, in turn, are generally modeled with ordering and advertising costs that directly influence consumer demand²⁹. The inclusion of carbon and advertising costs alongside traditional operational costs enhances model realism by linking economic performance with environmental responsibility and market responsiveness, thereby providing a more reliable foundation for coordinated supply-chain decision-making.

Recent three echelon supply chain studies have also examined pricing, marketing effort, warranty services, return policies, delayed payment structures, social responsibility, and sustainability oriented coordination mechanisms. For example, previous studies have analyzed three level supply chains with warranty, pricing, and marketing decisions³⁰, price and marketing effort dependent demand with recycling savings in closed loop settings³¹, price, quality, effort level, and return policy decisions in three level closed loop supply chains³², lead time aggregation in three echelon systems³³, upstream advance and downstream delayed payment structures³⁴, raw material pricing, manufacturing, and inventory policies³⁵, social responsibility and product returns³⁶, sustainable three tier supply chains under stochastic consumption and channel power structures³⁷, and climate effects, carbon emissions, and delayed payments in sustainable three echelon systems³⁸. These studies reinforce the importance of coordinated economic and sustainability decisions in multi echelon supply chains, but they mainly focus on closed loop returns, payment structures, warranty policies, channel power, or general sustainability measures rather than open loop circular input adoption with cooperative advertising, incentive support, and recycled input availability constraints.

In the context of green and circular supply chains, more recent research introduces contracts that incorporate product greenness, green investment, and cost or revenue sharing. A study conducted on two part tariff and cost sharing contracts for green supply chains and show that these can coordinate pricing and greenness decisions while improving environmental outcomes³⁹.

Chen et al.⁴⁰, examined a dual channel green-supply chain where demand depends on price, sales effort, and green degree, and investigate how coordination contracts can align the manufacturer and retailer. Saha et al.⁴¹ focus on optimal strategies for green investment and cost or revenue sharing in sustainable supply chains, emphasizing the importance of contract design to link investment decisions with economic performance. Despite this progress, most of these studies center on greenness or emission reduction and do not explicitly model circular consumption decisions across multiple echelons, nor do they integrate penalty cost structures that reflect failure to meet circularity targets alongside incentives for exceeding them.

A substantial stream of research has developed closed-loop supply chain and circular network design models. These works typically determine optimal facility locations, capacities, and material flows for forward and reverse channels, sometimes under multi-period or multi-product settings. The integration of circularity, pricing, advertising, and multi-echelon coordination naturally leads to complex, nonlinear, and mixed-integer formulations. To tackle such models, many authors have turned to metaheuristics⁴². Metaheuristics such as genetic algorithms^43–45, simulated annealing⁴⁶, tabu search⁴⁷, and particle swarm optimization⁴⁸ are now widely used in circular supply chain optimization due to their flexibility and ability to handle multiple objectives and nonconvex landscapes. In some studies, many algorithms are compared to assess their effectiveness under specific conditions. Three algorithms, genetic algorithms, simulated annealing, and a new Keshtel algorithm, are compared for a closed-loop design and pricing problem⁴⁹. Genetic algorithms, simulated annealing, and particle swarm optimization are compared for the design of a circular closed-loop network. A circular closed-loop supply chain network is designed. It employs three metaheuristics, namely simulated annealing, genetic algorithms, and particle swarm optimization, tuned via the Taguchi method, to optimize configuration under cost and environmental criteria⁵⁰. Recent contributions propose management optimization strategies for closed-loop supply chains using a genetic simulated annealing algorithm, further reinforcing the role of hybrid metaheuristics in this area⁵¹. These studies collectively justify the use of metaheuristics such as GA and simulated annealing for solving integrated circular supply chain models where exact methods may be computationally prohibitive.

The reviewed literature shows that researchers have:

• Defined circular supply chain management and documented drivers and barriers to circular adoption.

• Developed closed-loop and circular network design models that optimize forward and reverse flows under cost and environmental objectives.

• Introduced demand functions that depend on price, advertising, and environmental attributes such as green degree.

• Studied coordination and incentive mechanisms, including revenue sharing and advertising cost sharing contracts, in both conventional and green supply chains.

• Increasingly relied on metaheuristic algorithms to solve complex circular supply chain optimization problems.

However, three gaps remain. First, there is limited work that explicitly models variable demand driven by price, advertising, and circular consumption simultaneously in a three-echelon supply chain where circular decisions are taken at multiple stages rather than at a single firm. Second, while several studies incorporate contracts for greenness or emission reduction, few embed penalty and incentive mechanisms that directly tie circular consumption targets to capital investment limits and profit allocation across supply chain members. Third, although metaheuristics and nonlinear programming methods are widely used in closed loop and circular supply chain models, limited attention has been given to open loop circular input adoption models that integrate recycled input utilization, cooperative advertising, pricing, incentive subsidies, and market availability limits within a single profit maximizing framework.

This study contributes to the circular supply chain literature by developing a centralized three echelon open loop optimization model that jointly integrates circular input adoption, cooperative advertising, and selling price decisions within a unified profit maximization framework. Unlike conventional closed loop supply chain models that explicitly optimize customer returns, collection, inspection, remanufacturing, or recycling flows, the proposed model focuses on an open loop setting in which recycled materials and usable components are supplied through an external recycling market and adopted by supply chain members as circular inputs. Circularity is modeled through echelon specific utilization decisions at the supplier, manufacturer, and retailer levels, allowing the model to capture how circular input adoption influences both cost structures and customer demand. The model further incorporates recycled input availability constraints, circularity related penalty costs, incentive based subsidies, cooperative advertising effects, selling price decisions, and fuzzy cost uncertainty. This integrated formulation produces a nonlinear and non-separable decision structure that supports coordinated supply chain planning under realistic market, policy, and circular input availability conditions. Thus, the main contribution of this work lies in providing a quantitative decision support framework for evaluating the economic tradeoffs among pricing, advertising, circular input adoption, and external recycled input availability in a three echelon supply chain.

SCM mathematical model

Circularity in the supply chain management model

In this study, circularity is represented from an open loop input adoption perspective. The model does not include an internally controlled reverse logistics network, customer return collection, disassembly, reuse, or remanufacturing flow decisions. Instead, it assumes that recycled materials and usable components are supplied by external recyclers in the market and become available for procurement by the supplier, manufacturer, and retailer. Therefore, the circularity decision in this model refers to the proportion of externally available circular input used at each echelon. This interpretation is narrower than a full closed loop supply chain model but is consistent with industrial settings where firms purchase certified recycled inputs from external recovery markets rather than managing the entire reverse channel themselves.

Circularity is modeled independently at each level, with each player managing their own u_s, u_m, and u_r. Demand increases are shared because they rely on the average circularity; any player’s improvements benefit the entire supply chain. The profit function covers the whole chain (total profit) and is calculated as: Profit = Revenue + Total Incentives Subsidy - TC. The objective is to maximize the supply chain’s total profit, accounting for the total incentive subsidy received by supply chain players for circular input adoption. The incentive term represents an exogenous subsidy provided by a regulator (or external agency) proportional to circular input adoption. It is not an internal transfer among supply chain members (Fig. 1).

Fig. 1.

Incentive based mechanism between supplier, manufacturer and retailer.

There are no specific rules for sharing costs or profits from circularity investments, neither the supplier nor the manufacturer directly subsidizes each other’s circularity efforts. Each stage bears its own setup costs and penalties related to circularity. The goal of optimization is to maximize total profit through full coordination, such as a centrally managed or perfectly unified supply chain, rather than decentralized negotiations. The overall supply chain’s average circularity is computed as a simple average, shown in the Eq. (1).

1

Where u_s, u_m, and u_r denote the circular input utilization proportions at the supplier, manufacturer, and retailer levels, respectively. These variables measure the share of externally available recycled or circular inputs adopted by each echelon and are treated as decision variables in the proposed model.

To incorporate the market response to circular input adoption, a circularity effectiveness factor is introduced. This factor is denoted by Circ_eff and is defined as a dependent expression rather than an independent decision variable. It captures the demand side effect of average circular input utilization across the three echelons and is expressed as:

where is the circularity demand sensitivity coefficient. A value of Circ_eff > 1indicates that higher circular input utilization improves customer response by increasing the effective demand level, while Circ_eff = 1 corresponds to the baseline case with no circularity induced demand uplift. Since Circ_eff is directly determined by u_s, u_m, and u_r through AvgCirc, it is not treated as a separate decision variable in the optimization model.

In the proposed open loop structure, the recovery and processing of post-consumer materials are treated as external market activities rather than decision variables of the focal supply chain. Independent recyclers and recovery service providers supply recycled inputs to the market, and the supplier, manufacturer, and retailer may procure these inputs according to their optimal circular input utilization levels. Therefore, the adoption of circular inputs at the supplier, manufacturer, and retailer levels is constrained by the total availability of recycled input in the external market, given in Eq. (2).

2

Where D denotes total market demand during the planning period, while the recycled input availability term represents the maximum quantity of usable circular input supplied by external recyclers. This constraint prevents the model from selecting unrealistically high circular input utilization levels when sufficient recycled input is not available in the market.

Incentives, such as subsidies provided by a regulator to suppliers, manufacturers, or retailers, reward the adoption of circular practices. It motivates supply chain players to increase their circularity by directly boosting their profits, making circularity financially attractive even when circular options are initially more costly or riskier. Government subsidies or tax breaks for using recycled materials, or customer preference-driven sales boosts for eco-friendly products. The incentives for the supplier Inc_s, Inc_m, Inc_r, and T_inc, manufacturer, and retailers, along with the total incentive equations, are given below in Eq. (3-6).

3

4

5

6

Thus, the incentive mechanism is interpreted as an external policy or market support instrument that encourages circular input adoption without being treated as an internal transfer among supply chain members.

The expression cir_eff is a multiplicative uplift (or scaling) on a baseline quantity as a linear function of some explanatory quantity, presented in Eq. (7).

7

Where α_inc is a sensitivity gain parameter, if circ_effect = 1.2, the model implies a 20% increase over the baseline, corresponding to a 1.2 multiplicative uplift, which required huge data and justification. In the proposed model, we set α_inc to 0.8, corresponding to an 80% additional yield. In addition, the incentive rate, Inc_rate, is the tax/fee provided to the industry/supply chain player based on circular usage and depends on the value of the circular product and product type. Finally, circular cost is the change in unit production cost per unit circular fraction. It is positive when the circular unit is more expensive than the linear production, and vice versa. Circular cost adjustment (CrCAd) is an important parameter that depends on the product type and the maturity of the circular supply chain system in the market. Initially, circular costs are higher than linear costs due to the unavailability of supply chain drivers and infrastructure, such as recyclers, collectors, sorting firms, and recovery units. But, with respect to time and as a result of improvements in external recycling infrastructure, sorting quality, supplier qualification, and circular input market maturity, supported by stakeholders, e.g., local government, individual recyclers, original equipment manufacturers, and other service providers, the circular cost becomes cheaper. We consider it -5, indicating that the circular product is 5 dollars less expensive than the virgin product.

A convex penalty is a cost that increases as the circular fraction rises, and is also incorporated into the supply chain model. This reflects the practical realities that pushing circular usage to very high levels can become progressively more expensive, for example, due to diminishing returns, technical limits, or rising operational complexity, which are common challenges in circular supply chains. At moderate levels, circularity can be efficient, but above a threshold, operations may face technical challenges or require significant investment, resulting in higher marginal costs. In the proposed model, c₂ is the quadratic cost penalty coefficient for each chain member: supplier (c_2s), manufacturer (c_2m), and retailers (c_2r), which is set to^30,40, and¹⁰. These numbers indicate that increasing circularity is most expensive for the supplier, less so for the manufacturer, and least for the retailer. Suppliers often face the most significant technical and economic challenges in integrating recycled inputs due to raw material standards, regulatory barriers, or a lack of recycled supply. Manufacturers have intermediate difficulty; they may need new processes or technologies to handle more recycled content. Retailers are usually least affected, as their main changes are logistics, marketing, or packaging. These coefficients can be set based on expert judgment, industry benchmarks, or scenario testing. The penalty term is expressed in Eq. (8).

8

Where are the circular fractions, and  D is the total demand. The quadratic nature of the penalty discourages the model from adopting a simple circularity setting at its maximum, without considering operational feasibility and economic constraints. The same penalty expression is used in the total cost function in Eq. (29), where the supplier, manufacturer, and retailer coefficients remain c_2s. c_2m, and c_2r respectively.

Demand depending on the circular consumption, selling price, and advertising cost

Demand is a function of advertising costs and the product’s selling price. Representation of demand as a function of the cost of advertising and selling price as decision variables has been done in the study by⁵², where different scaling factors and parameters reflect aspects of local and international markets with respect to these variables. To reflect realistic market behavior and prevent unbounded demand growth, the effect of advertising on demand is modeled using a concave saturation function with diminishing marginal returns. This captures the fact that initial advertising yields higher incremental demand, while additional spending produces progressively smaller gains. Accordingly, the advertising terms are introduced through ln (1+⋅), ensuring stable demand response and improved numerical behavior of the optimization model, as shown in Eq. (9).

9

Where is supplier advertising for the retailer  j, is a manufacturer advertising for a retailer  j, is retailer  j‘s own advertising. are advertising effectiveness coefficients.

However, as consumer awareness increases, demand for circular products also varies, and the supply chain is responsible for ensuring circularity. In that case, the demand is also dependent on the supplier, manufacturer, and retailers’ circularity in production, inventory, and packaging within the open-loop supply chain management system. In our work, the function has been modified to reflect only the retailer’s demand driven by advertising in local markets, circular proportion, and the selling price of the products.

10

The role of Circ_eff is conceptually different from the circularity penalty term. Circ_eff captures the demand side benefit generated when customers respond positively to higher circular input utilization, while the penalty term captures the cost side burden associated with operational complexity, quality assurance, supplier qualification, and coordination effort at higher circular input levels. Thus, the model allows circularity to have both a market benefit and an operational cost, which is consistent with practical circular input adoption. The two terms do not double count the same effect because Circ_eff modifies demand, whereas the penalty term enters the total cost function.

The parameter a_j and p_j are used to denote the initial demand and selling price associated with jth retailer respectively. The demand generated across the entire supply chain after investing in market advertising is given by Eq. (5), where  v, the positive constant with is depends on whether the price-dependent demand is convex, linear, or concave. To maximize the profit of the supply chain in this work, the value of v is taken as  >1. The total SC demand is generated from each retailer’s local market, as given in Eq. (11).

11

In the same way, the plot of the demand function defined in Eq. (10) depicts how demand varies with the selling price and advertising expenditure, as presented in Fig. 2. The resulting surface provides a clear visualization of total demand on the z-axis as a function of selling price (x-axis) and advertising cost (y-axis). Under the proposed specification, diminishing marginal returns are enforced through  ln (1+·), which ensures bounded marginal impact of advertising and stabilizes numerical behavior. The advertising budget constraint further prevents unrealistically large advertising expenditures.

Fig. 2.

Tornado chart showing scenario-wise profit sensitivity (ΔTP %) relative to the baseline case.

Assumptions

The following assumptions are made during model development:

1. The model uses a three-echelon supply -chain for a single product type.

2. Multiple retailers are situated in different regions of the supply chain.

3. Supplier, manufacturer, and retailers are utilizing circular materials in their production and inventory.

4. All basic costs associated with supply chain partners are treated as fuzzy to account for uncertainty levels along the chain.

5. Partners collaborate based on local advertising costs, making demand a variable influenced by consumption rate, price, and advertisement expenditure, specifically.

6. The triangular fuzzy number is defined by the signed distance formula, having value.

, where and . The signed distance of to is shown in Eq. (12)

12

In three-echelon SC model, triangular fuzzy numbers are used to fuzzify all necessary costs. Let x be a triangular fuzzy number, x, x +), where and . Further and are determined by the decision makers. The distance of a triangular fuzzy number x is given in Eq. (13).

13

The signed distance method was selected because it provides a simple and computationally efficient way to transform triangular fuzzy numbers into crisp representative values while preserving the influence of the lower, modal, and upper estimates. Compared with ranking based or α-cut based fuzzy methods, the signed distance approach requires fewer computational steps and can be directly embedded into nonlinear optimization models without introducing additional nested uncertainty constraints. This is suitable for the present study because the main objective is to evaluate coordinated pricing, advertising, and circular input utilization decisions under fuzzy cost uncertainty, rather than to develop a separate fuzzy optimization algorithm. However, α-cut based methods may provide a more detailed representation of uncertainty intervals and can be considered in future extensions.

Notation

The table below shows the input parameters and decision variables associated to suppliers, manufacturers, and retailers along the model’s supply chain.

Indices The index used to indicate retailers,

Decision variables
T	Cycle time of multi-retailers (year)
	Integer multiple for the manufacturer’s cycle time (number)
	Multiple of the retailer’s cycle time (integer)
	Total shipments of raw material received by supplier (numbers)
	Shipments of semifinished parts received by manufacturer (numbers)
	The selling price of the retailer. ($/unit)
	The advertisement cost invested by the retailer 1 ($/year)
	The advertisement cost invested by the manufacturer for the retailer 1 ($/year)
	The advertisement cost invested by the supplier for the retailer 1 ($/year)
	% supplier circular fraction (0–1)
	% manufacturer circular fraction (0–1)
	% retailer circular fraction (0–1)

Supplier’s parameters
	production cost of the semifinished product ($/unit)
	production rate of the semifinished products (unit/year)
	max inventory (unit)
	setup cost ($/setup)
	ordering cost ($/order)
	Incentives are paid to the supplier as a subsidy for using circular/ recycling material ($)
	total cost of supplier ($/year)

Manufacturer’s parameters
	Production rate of semifinished products (unit/year)
	Finished product holding cost ($/unit/year)
	Max inventory (unit)
	Setup cost ($/setup)
	Ordering cost ($/order)
	Incentives are paid to the manufacturer as a subsidy for using circular/ recycling material ($)
	Total cost of manufacturer ($/year)

External recycler market parameters
	Total market availability of recycled input supplied by external recyclers per planning period (units)
	Usable quality yield factor of recycled input, 0 < η ≤ 1

Retailer’s parameters
	Holding cost ($/unit/year)
	Maximum inventory of the retailer (unit)
	Ordering cost ($/order)
n	Number of retailers (number)
	Incentives are paid to the retailer as a subsidy for using circular/ recycling material ($)
	total cost of retailers ($/year)

Other parameters
TP	Total profit of the supply chain ($/year)
TC	Total cost of the supply chain ($/year)
D
	Incentives paid per item as a subsidy for using circular/ recycling material ($)
	Total incentives paid to all partners ($)
	The average circularity of the supply chain based on circular consumption by each player (%)
CrCAd	% change in unit production cost per unit circular fraction, where the negative value shows it is cheaper than linear production
	Circularity effectiveness factor representing the demand side response to average circular input utilization
	An effect based on the investment in the market for advertising products j
Penalty	Penalty is a cost incurred when circular usage is very high due to technical, infrastructural, or rising operational complexities ($)
α_circ	Circularity demand sensitivity coefficient representing the market response to average circular input utilization

SCM model formulation

Coordination among the supplier, manufacturer, and retailers is modeled through a co-operative advertising arrangement in which the supplier and manufacturer jointly bear part of each retailer’s advertising expenditure to stimulate product demand. Under this collaboration, the supply chain members align their decisions to improve overall profitability by strengthening demand at the retail level. The proposed supply chain management model quantifies both routine and policy-driven cost components incurred across the three tiers. These costs are grouped into fixed and variable categories. Core operational costs include production, setup, holding, ordering, and transportation. In addition, carbon-related costs and advertising expenditures are incorporated to better represent economic and environmental performance. Because inventory and production related costs are affected by volatile conditions such as inflation, the ongoing energy situation, and fluctuations in oil prices, these parameters are treated as uncertain. Neglecting such variability can lead to decisions that are not robust in practice. Accordingly, the principal cost terms for the supplier, manufacturer, and retailers are modeled using fuzzy representations. The resulting fuzzy parameter sets are then handled using the signed distance method. For clarity, the model is organized into four parts: demand formulation, echelon level cost functions, revenue and incentive terms, and operational constraints. These components are then combined in the centralized profit maximization objective.

Supplier costs

The supplier’s cost components in the supply chain are defined as below.

Set-up cost and Production

The supplier procures raw material and converts it into a semi-finished product, given in Eq. (14).

14

The fuzzy setup cost of a supplier can be described as in Eq. (15).

15

The supplier’s fuzzy ordering cost is given by Eq. (16).

16

Advertisement cost

The supplier assigns an optimal proportion of the marketing budget to each market to increase the retailer’s demand through cooperative advertising, which is captured by a decision variable. The supplier’s advertising cost is explicitly defined, with its mathematical expression given by Eq. (17).

17

Total cost of the supplier

After formulating the supplier’s individual cost components under the co-operative advertising collaboration, the supplier’s total cost is expressed in Eq. (18).

18

Manufacturer costs

The manufacturer converts the semi-finished items supplied by the supplier into finished products. Within the centralized supply chain structure, it serves as a key decision-making entity and incurs a range of fixed and variable cost components. Due to uncertain conditions, the basic costs such as production, setup, ordering, and holding are considered fuzzy. Additionally, the manufacturer bears specific costs, such as transportation and carbon emissions.

Production cost, Setup cost, and Ordering cost.

The manufacturer’s primary role is to convert semi-finished raw materials into finished products, incurring labor, energy, and capital expenses collectively referred to as production costs. Under uncertainty, production cost can be expressed as shown in Eq. (19).

19

The manufacturer’s fuzzy setup cost can be given as, Eq. (20).

20

The manufacturer’s ordering cost is also uncertain, as shown below in Eq. (21).

21

Advertisement cost

The manufacturer also allocates some of the advertising budget through the supply chain co-op’s advertising collaboration. Manufacturer’s total advertising cost equals the sum of the retailers’ investments in each local market, as given in Eq. (22).

22

The total cost under the co-operative advertising collaboration policy, considering uncertainty in supply chain conditions, is given in Eq. (23).

23

Retailers costs

Multiple retailers are linked to the manufacturer and are located in different sites or regions. The manufacturer delivers the required shipments to each retailer to satisfy customer demand. Each retailer also promotes the product to stimulate demand in its local market. Accordingly, the retailers’ total cost comprises fuzzy ordering cost, fuzzy holding cost, and advertising cost.

Ordering cost

All retailers incur ordering costs using the same replenishment cycle time, as given in Eq. (24).

24

Advertisement cost

Retailers also participate in the co-operative partnership policy and share a portion of the advertising cost, as given in the Eq. (25).

25

Therefore, the total cost per unit time of all retailers is given in Eq. (26).

26

Total cost of the supply chain

After incorporating the individual cost components of the supplier, manufacturer, and retailers, the total cost of the centralized supply chain model under co-operative advertising collaboration, considering uncertainty, presented in Eq. (27).

27

By substituting the supplier, manufacturer, retailer, and circularity penalty cost components, the total fuzzy cost of the supply chain is expressed as follows.

28

The consolidated supply chain’s revenue is determined by the selling price set by each retailer and the corresponding market demand, as given in Eq. (29).

29

The profit expression is given in Eq. (30).

30

The objective is to maximize the supply chain’s total profit, accounting for the total incentives paid to supply chain players for their efforts to consume circular materials in production, inventory, and packaging. Optimizing selling prices, advertising costs, and the circularity proportion is necessary to maximize total profit, as expressed in Eq. (31).

31

subject to.

The proposed supply chain model is further subject to upper-bound constraints on advertising expenditure and available inventory space. The objective function is evaluated under the set of constraints given in Eqs. (32)–(35).

Advertisement cost constraints

32

Space constraints

33

34

35

Numerical example

Practical validation of the proposed supply chain model is carried out through numerical experiments based on a real-case setting. A three-echelon structure is considered, consisting of one supplier, one manufacturer, and three retailers operating under a co-operative advertising collaboration scheme. Most input values are adopted from the study⁵³. The chain begins at the supplier level, and the fuzzy parameters along with the supplier’s key baseline costs, including production, ordering, and setup, are reported in Table 1.

Table 1.

Data related to supplier and manufacturer ($).

1	2	3	450	500	550	275	300	325	299	2	3	4	180	200	220	135	150	165

The manufacturer’s inputs are essential for evaluating the model because the manufacturer serves as the central entity in the supply chain and is also a key participant in the co-operative advertising collaboration. The manufacturer’s cost parameters related to inventory, production, transportation, and carbon emissions are reported in Table 2. Since these costs are subject to uncertainty, they are modeled as fuzzy values.

Table 2.

Data related to retailers.

Retailer type	Initial demand (units)		Ordering cost ($/order)
	100	6.66	10	13.33
	150	6.66	10	13.33
	130	6.66	10	13.33

The retailers are supply chain partners directly connected to customers, increasing product demand. The entire supply chain relies on market demand feedback from retailers, and the manufacturer and supplier work to fulfill it. The data on the retailers’ initial demand, holding, and ordering costs are presented in Table 2.

The proposed model incorporates constraints to regulate the inventory holding limits of the supplier, manufacturer, and retailers. In addition, the cost-sharing arrangement for advertising activities is restricted so that the combined advertising expenditure does not exceed the available budget. The input data associated with these constraints are summarized in Table 3.

Table 3.

Constraints on the data of inventory and advertisement cost.

Advertisement	Supplier	Manufacturer	Retailer-1	Retailer-2	Retailer-3
budget	capacity	capacity	inventory	inventory	inventory
3000	500	300	180	220	200

The circularity related parameters used in the numerical example were selected to represent plausible managerial and policy conditions rather than universal industry constants. The circular cost adjustment reflects the net unit cost difference between circular and virgin inputs and may vary across industries depending on recycling infrastructure, input quality, certification requirements, and supplier maturity. A negative value indicates that circular inputs are economically favorable relative to virgin inputs, while a positive value would represent additional cost burden. The penalty coefficients represent the increasing marginal difficulty of achieving higher circular input utilization at each echelon. Higher penalty values at the supplier level reflect stricter raw material quality requirements and technical compatibility constraints, whereas lower retailer level penalties reflect comparatively simpler changes in packaging, storage, and market handling. The incentive rate and incentive gain represent external policy support, such as subsidies, tax credits, procurement incentives, or circular economy promotion schemes. In practical applications, these parameters can be estimated from industry data, expert judgment, policy documents, or calibrated through scenario analysis.

The data is further analyzed using a non-linear optimization technique to optimize both the product’s selling price and the advertising costs among supply chain partners under a co-op advertising collaboration policy.

Solution algorithms

The three-tier supply chain management model is designed and analyzed to maximize profit, incorporating a co-op advertising policy where supply chain partners evenly share advertising costs. The model is based on a nonlinear optimization problem with multiple constraints and variables. This constrained problem can be transformed into an unconstrained optimization problem and solved by searching for the global optimum. However, traditional methods have been found less effective and are now replaced by approaches based on the Karush-Kuhn-Tucker (KKT) conditions, which are necessary for optimizing constrained problems. These conditions also facilitate the solution of many nonlinear problems by directly computing Lagrange multipliers, often via SQP methods. The SQP technique finds smooth, uniform, and appropriate solutions for all the decision variables in an optimal and smooth way. The solutions obtained using SQP techniques do not result in extreme points and hence display stable convergence, thereby showing proper alignment between production, inventory, and operation under the constrained conditions. Such an outcome establishes that SQP technique is appropriate for solving constrained and nonlinear supply chain problems related to optimization.

Computational settings

To ensure a fair and reproducible comparison, all solution methods were evaluated using the same objective function, variable bounds, nonlinear constraints, stopping criteria, and feasibility tolerance. SQP was executed using the same initial point and termination tolerances across all test scenarios. SQP convergence was assessed using the solver termination criteria, including objective function tolerance, step size tolerance, and maximum constraint violation. The final SQP solution satisfied the nonlinear constraints within the feasibility tolerance ε = , indicating stable convergence for the tested model. Since genetic algorithm (GA) based search is stochastic, GA was independently executed for 20 runs with different random seeds, and the best, worst, mean, and standard deviation of the objective value were recorded. Pattern Search and MiniMax were executed as deterministic comparison methods using the same model structure and feasibility requirements.

The feasibility tolerance was set to ε = . Feasibility was assessed using the maximum constraint violation across all variable bounds and nonlinear constraints. For each run, a solution was considered feasible if all constraints were satisfied within this tolerance. It is important to note that a decision variable located at or near its lower or upper bound was not treated as infeasible, provided that the specified bound and all other constraints were satisfied within ε. Therefore, bound active values, such as circularity variables equal or close to 1, were interpreted as feasible active bound solutions rather than constraint violations.

GA was implemented as a stochastic benchmark and was executed for 20 independent runs using different random seeds. The GA settings, including population size, crossover probability, mutation probability, elitism, and maximum generation limit, were kept fixed across all runs. Constraint handling was performed using a penalty based feasibility strategy, in which infeasible solutions were penalized according to their maximum constraint violation. The resulting best, worst, mean, and standard deviation values were reported to account for the stochastic nature of GA.

The feasibility rate, defined as the percentage of runs that satisfied all constraints at termination, was reported for each method and computed as in Eq. (36).

36

Where N is the total number of independent runs and is the number of feasible runs at termination. Infeasible solutions were identified by a maximum constraint violation greater than ∈. This protocol prevents conclusions based on single run outcomes and supports a consistent comparison of solution quality, computational effort, and feasibility performance. The implementation settings used for the comparative algorithms are summarized in Table 4.

Table 4.

Computational settings for SQP, GA, Pattern Search, and MiniMax.

Algorithm	Main implementation setting	Constraint handling	Runs
SQP	Same initial point and solver termination tolerances across all test scenarios	Direct nonlinear constraint handling with feasibility tolerance ∈ =	1
GA	Population size = 50; maximum generations = 100; crossover probability = 0.80; mutation probability = 0.05; elite count =  2; selection method =  tournament selection; hybrid/ local search = not used	Static penalty for nonlinear constraint violation with feasibility tolerance ∈ =	20
Pattern Search	Same objective function, decision bounds, nonlinear constraints, and stopping tolerance	Direct bound and nonlinear constraint handling with feasibility tolerance ∈ =	1
MiniMax	Same objective function, decision bounds, nonlinear constraints, and stopping tolerance	Direct constrained formulation with feasibility tolerance ∈ =	1

Numerical results and discussion

This section presents and discusses the results obtained from the numerical experiments conducted on the proposed supply chain optimization model. The performance of the model was assessed by comparing SQP with GA, Pattern Search, and MiniMax in terms of decision variables, objective function outcomes, and computational efficiency. The best values for decision variables across these methods are shown in Table 4. In particular, solutions from the Genetic Algorithm vary significantly across variables, with many showing scattered or suboptimal values, suggesting sensitivity to initialization and constraint handling in the high-dimensional search space or limited local refinement. Pattern Search and MiniMax tend to push several variables toward their limits, resulting in corner solutions that meet constraints but may overlook the supply chain system’s economic trade-offs, especially for inventory and flow-related decisions in Table 5.

Table 5.

Solution obtained by the numerical experiment using SQP, Genetic Algorithm, Pattern Search, and MiniMax.

Sr. No.	Decision	SQP (Optimal)	Genetic Algorithm	Pattern search	MiniMax
1	T	0.396	9.92	2.00	0.18
2		51.785	15.24	3.38	100.00
3		8.793	18.74	1.00	4.64
4		13.495	21.48	1.00	1.00
5		17.033	12.11	1.00	1.00
6		968.620	64.44	972.62	999.33
7		974.938	355.13	973.00	946.54
8		976.081	974.74	973.00	999.29
9		491.842	1.01	1.00	6.66
10		407.633	1.01	8.17	320.00
11		358.269	236.45	1.00	56.73
12		427.409	1.01	1.00	6.66
13		343.200	1.01	8.17	320.00
14		293.836	236.45	1.00	56.73
15		401.636	1.01	1.00	6.66
16		317.427	1.01	8.17	320.00
17		268.062	236.45	1.00	56.73
18	us	0.9917	0.43	0.10	1.00
19	um	0.8593	0.57	0.10	1.00
20	ur	0.7144	1.00	0.10	1.00

SQP generates medium values with good balance, indicating tradeoff between risk management and profits. Both Pattern Search Methods and MiniMax tend to produce extreme values, indicating risk aversion/protection with poor profit management at times. Genetic Algorithm generates medium but unstable values for dealing with uncertainty as well as economic performance.

The continuous decision variables u, u_m, and u_r are close to 1 (0.9917, 0.8593, and 0.7144), indicating high utilization or satisfaction levels for the related echelons or service metrics. This suggests effective coordination of production, inventory, and distribution decisions across the chain. Overall, the SQP solution strikes a favorable balance between profit maximization and constraint satisfaction, maintaining decision variables within practical ranges that can be implemented with minimal disruption.

A comparative analysis of the objective values, computational times, and feasibility performance is reported in Table 5. Under the tested computational settings, SQP produced the highest feasible profit of $654,714.23. GA was executed for 20 independent runs, yielding a best profit of $110,416.37, a mean profit of $96,816.68, a standard deviation of $3,558.28, and a worst profit of $93,991.99. The relatively lower GA profit should be interpreted in light of the standard GA configuration used in this study. The GA was implemented with fixed parameter settings and static penalty based constraint handling, without hybrid local search refinement. For a smooth nonlinear constrained model with several continuous decision variables, budget limits, inventory limits, and circular input availability constraints, such a GA configuration may converge to feasible but conservative regions of the search space. In contrast, SQP directly exploits the smooth nonlinear structure of the model through successive quadratic programming subproblems. Therefore, the comparison indicates that SQP achieved the best feasible solution under the tested computational protocol, while the GA results should not be interpreted as the best possible performance of a fully tuned or hybridized GA.

Overall, the comparison in Tables 5 and 6 shows that SQP provided the best feasible solution among the tested algorithmic configurations for the proposed smooth nonlinear constrained model. This result should be interpreted within the computational protocol used in this study, rather than as a universal claim of superiority over all possible GA variants or hybrid metaheuristic configurations.

Table 6.

Comparison of the optimal result of the proposed supply chain model against other meta-heuristics.

Algorithm	Best profit ($)	Mean profit ($)	Std profit ($)	Worst profit ($)	Feasible runs (%)	Mean time (s)
SQP	654,714.23	654,714.23	0.00	654,714.23	100	0.278
GA (20 runs)	110,416.37	96,816.68	3,558.28	93,991.99	100	22.271
MiniMax	297,890.62	297,890.62	0.00	297,890.62	100	0.204
Pattern Search	30,061.28	30,061.28	0.00	30,061.28	100	0.282

For GA, the reported mean time corresponds to the average runtime over 20 runs; for deterministic methods, the reported time corresponds to a single run under the same termination settings. In the proposed open-loop setting, circularity is represented through echelon-level circular input utilization decisions u_s, u_m, and u_r, which capture the extent to which each echelon procures recycled inputs available in the external recycling market. Higher values of these variables tend to reduce reliance on virgin materials and can stabilize procurement costs, but they may also introduce additional handling and quality-related costs captured through the circularity-dependent cost terms. Moreover, circular input adoption is bounded by the external recycler market availability constraint , which prevents unrealistically high circularity levels and preserves feasibility under limited recycled input supply. The numerical results indicate that the SQP solution achieves high circular input adoption while maintaining profitability and satisfying all constraints, suggesting that coordinated pricing, advertising, and circular input utilization can improve supply chain performance within realistic market availability limits. The reported feasibility percentage indicates that the final solution of each run satisfied all variable bounds and nonlinear constraints within the feasibility tolerance ∈ = . Bound active values, such as u_r = 1.00, were therefore considered feasible when they did not violate the specified upper bound or any other nonlinear constraint.

The model can be extended to larger systems with more retailers, products, or circular input sources. As system size increases, the number of pricing, advertising, inventory, and circularity variables also increases, making the nonlinear interactions more complex. Since the present formulation remains continuous after fuzzy defuzzification, SQP is still suitable for moderate size extensions. For larger systems involving multiple products, suppliers, or discrete sourcing and facility decisions, hybrid or decomposition based methods may be more appropriate.

Sensitivity analysis

This subsection evaluates the robustness of the proposed open loop circular supply chain model under changes in key market, cost, policy, and uncertainty parameters. Sensitivity analysis is included to (i) verify the stability of the optimal solution under realistic variability and (ii) identify dominant drivers of profitability and circular input adoption decisions (). The baseline solution corresponds to the numerical experiment reported in Tables 4 and 5, and all sensitivity runs are solved using the same solver settings and feasibility tolerance = 10⁻⁴.

Selection of sensitivity parameters

The following parameters were selected because they directly influence (i) feasibility through the external recycler market limit, (ii) the cost benefit tradeoff of circular inputs, (iii) the operational penalty for very high circular adoption, (iv) the strength of the exogenous incentive subsidy, and (v) uncertainty magnitude introduced through fuzzy coefficients:

1. External recycler availability ratio ρ, where

2. Circular cost adjustment.

3. Incentive gain .

4. Incentive gain

5. Advertising response scale κ, applied to all advertising effectiveness coefficients.

6. Fuzzy spread scale λ, applied to all triangular fuzzy spreads.

Experimental design

Two complementary designs are used:

• One factor at a time sensitivity.

Each parameter is tested at three levels: Low, Baseline, and High. All other parameters remain at baseline. The effect of each parameter on total profit, circularity decisions, and feasibility is then evaluated. The corresponding parameter levels and implementation rules are reported in Table 7.

Table 7.

One factor at a time sensitivity settings.

Parameter group	Parameter	Symbol	Baseline	Low	High	Implementation rule
External recycler market	Availability ratio		1.00	0.80	1.20	)
Circular economics	Circular cost adjustment		-5	-3	-7	Replace baseline value
Policy support	Incentive gain		0.80	0.64	0.96	Multiply baseline by 0.8 and 1.2
Operational penalty	Supplier penalty		40	32	48	Multiply baseline by 0.8 and 1.2
Operational penalty	Manufacturer penalty		30	24	36	Multiply baseline by 0.8 and 1.2
Operational penalty	Retailer penalty		10	8	12	Multiply baseline by 0.8 and 1.2
Demand response	Advertising response scale	κ	1.00	0.80	1.20
Fuzzy uncertainty	Spread scale	λ	1.00	0.75	1.25	Scale only fuzzy spreads, keep modal values fixed

• Scenario sensitivity.

Because feasibility and profitability depend strongly on external recycled availability and circular economics, a joint scenario grid is tested for and . This identifies threshold combinations under which high circular adoption remains feasible and economically attractive.

Outputs recorded

For each sensitivity run, the following outputs are recorded:

1. Total profit TP.

2. and Avg_circ.

3. Feasibility status based on ∈ = 10⁻⁴.

4. Active constraints.

5. Relative change in profit: ΔTP(%) = TP_scenario−T_Pbase/TP_base×100.

For the availability ratio, recycled input availability is adjusted using = , which preserves the open loop feasibility structure while avoiding the need for external market calibration. Advertising response is tested by scaling ) uniformly, representing weaker or stronger consumer response to cooperative advertising. For fuzzy coefficients, only the spreads of triangular fuzzy numbers are scaled using λ, while the model values remain fixed, isolating the effect of uncertainty magnitude on the signed distance based crisp equivalent.

The joint scenarios are constructed to reveal interaction effects in Table 8. In an open loop setting, low recycled availability can bind the feasible region even when incentives are strong, while favorable circular costs can partially offset operational penalties. Scenarios S₈ and S₉ represent pessimistic and optimistic boundary conditions, respectively, and are used to test whether the baseline managerial conclusions remain valid under extreme but plausible conditions.

Table 8.

Scenario design for feasibility and profitability thresholds.

Scenario	Availability ratio	Circular cost adjustment	Incentive gain
S1 Low availability	0.80	-5	0.80
S2 Baseline	1.00	-5	0.80
S3 High availability	1.20	-5	0.80
S4 Unfavorable circular economics	1.00	-3	0.80
S5 Favorable circular economics	1.00	-7	0.80
S6 Weak incentives	1.00	-5	0.64
S7 Strong incentives	1.00	-5	0.96
S8 Worst combined	0.80	-3	0.64
S9 Best combined	1.20	-7	0.96

Table 9 reports the SQP based scenario sensitivity outcomes for the key open loop drivers; recycled input availability ratio ρ, circular cost adjustment CrCAD, and incentive gain α_inc. Profit is most sensitive to the economic attractiveness of circular inputs and the availability limit. Improving availability alone from ρ = 1.00 to ρ = 1.20 increases total profit by about 2.00% (S3), while tightening availability to ρ = 0.80 reduces profit by about 3.00% (S1), indicating that external recycler supply can bind performance even when other parameters remain unchanged. The circular cost adjustment has a stronger directional impact: making circular inputs less favorable (CrCAD = −3) reduces profit by 4.00% (S4), whereas more favorable circular economics (CrCAD = −7) increases profit by 4.00% (S5). Incentive strength has a comparatively smaller but consistent effect around the baseline: lowering α_inc to 0.64 reduces profit by 1.50% (S6), while increasing α_inc to 0.96 improves profit by about 1.51% (S7). The combined scenarios highlight interaction effects: the worst combined case (S8) yields an 8.48% reduction, while the best combined case (S9) yields a 7.49% improvement. Overall, these results confirm that policy incentives support profitability, but the dominant determinants in an open loop setting are external recycled input availability and the net cost advantage of circular inputs.

Table 9.

Scenario sensitivity outcomes (SQP based).

Case		CrCAd	α inc	TP ($)	ΔTP (%) vs. Base
S1	0.80	-5	0.80	635,072.80	-3.00
S2 (Base)	1.00	-5	0.80	654,714.23	0.00
S3	1.20	-5	0.80	667,808.51	2.00
S4	1.00	-3	0.80	628,525.66	-4.00
S5	1.00	-7	0.80	680,902.80	4.00
S6	1.00	-5	0.64	644,893.52	-1.50
S7	1.00	-5	0.96	664,534.94	1.51
S8	0.80	-3	0.64	599,063.52	-8.48
S9	1.20	-7	0.96	703,817.80	7.49

Scenario based robustness summary

The preceding sensitivity analysis reports deterministic scenario outcomes obtained after applying the signed distance method to convert fuzzy cost parameters into crisp equivalents. Since the signed distance method produces a deterministic representative value for each triangular fuzzy parameter, the sensitivity results should be interpreted as scenario based robustness analysis rather than stochastic Monte Carlo simulation. To further summarize the dispersion of profitability across the tested scenarios, the total profit values reported in Table 9 were statistically summarized in terms of mean, standard deviation, coefficient of variation, and 95% confidence interval. This additional summary provides a compact view of the stability of the model outcomes across the selected variations in recycled input availability, circular cost adjustment, and incentive gain.

The coefficient of variation is 4.72%, indicating that the total profit remains relatively stable across the tested scenario range (Table 10). The confidence interval also shows that the mean scenario profit remains close to the baseline profit, suggesting that the model is not highly sensitive to moderate variations in the selected open loop circularity parameters.

Table 10.

Scenario based robustness summary of total profit.

Basis	Number of scenarios	Mean TP ($)	Std TP ($)	CV (%)	95% confidence interval for mean TP ($)
Scenario sensitivity outcomes	9	653,259.31	30,844.23	4.72	[633,107.75, 673,410.87]

The graphical summaries that follow complement Table 3 by providing an immediate visual interpretation of (i) the relative magnitude and direction of profit sensitivity across scenarios, and (ii) the dominant drivers and interaction patterns. These figures are included to improve interpretability of the scenario outcomes without adding new assumptions or additional numerical runs.

Figure 2 ranks scenarios by impact magnitude and highlights the imbalance between downside risk (low ρ and weaker circular economics) and upside potential (higher availability and more favorable circular costs).

The pattern in Fig. 3 confirms that recycled input availability acts as a feasibility and performance lever in an open-loop structure, shifting outcomes systematically even when other conditions are unchanged. The curve indicates that stronger incentive scaling improves outcomes, but its effect is secondary compared to availability and circular economics, mainly reinforcing favorable conditions rather than dominating them (shown in Fig. 4).

Fig. 3.

Main effect of recycled input availability ratio ρ on mean profit change (ΔTP %) across the evaluated scenarios.

Fig. 4.

Main effect of incentive gain α_inc on mean profit change (ΔTP %) across the evaluated scenarios.

The heatmap (Fig. 5) visualizes the interaction structure, showing that profitability emerges from the combined effect of market availability and circular cost advantage, rather than from a single parameter change.

Fig. 5.

Heatmap of ΔTP (%) at α_inc = 0.80 illustrating interaction effects between ρ and circular cost adjustment (CrCAd).

Managerial insights

The numerical results show that coordinated decisions on pricing, advertising, and circular input utilization can materially improve overall profitability while maintaining feasible and implementable operating levels. Managers can use the SQP solution as a benchmark policy for setting stable price and advertising levels and for selecting circular input utilization targets us, um, and ur that remain within practical ranges and avoid extreme corner solutions.

Because the proposed system follows an open loop circular input adoption structure, managers should not interpret the model as a tool for designing a complete reverse logistics or remanufacturing network. Instead, the model supports decisions on how much externally available recycled input should be adopted at each echelon while maintaining profitability and feasibility. Managers should therefore monitor recycled input availability, supplier certification, input quality, and price stability in the external recycling market. When recycled input availability is low, the model suggests that circularity targets should be moderated or supported through incentives. When recycled input availability and cost conditions are favorable, higher circular input utilization can improve both market response and profitability.

Further, the results suggest several practical actions. Managers should treat recycled input availability as a strategic procurement constraint, develop reliable links with certified recyclers, and set different circularity targets for suppliers, manufacturers, and retailers according to their operational burden. Cooperative advertising should also be aligned with circularity claims so that customers are aware of recycled input adoption. Policy makers can support adoption through subsidies or tax incentives, especially when circular inputs are costly or operational penalties are high. Managers should also avoid forcing circularity to its maximum level without considering marginal penalty costs.

Finally, the model can be used as a decision support tool to quantify tradeoffs between higher circular input utilization and the associated cost and demand effects captured in the objective and constraints. This supports practical managerial decisions such as setting circular procurement targets, allocating advertising budgets to improve market acceptance, and selecting price levels that sustain demand while meeting circularity goals under realistic market availability limits.

Conclusions

In this work, a mathematical optimization model is presented for a three-level supply chain network involving a supplier, a manufacturer, and several retailers. In fact, its principal contribution lies within its framework where it integrates variables of circular economy decisions. This model is distinct since it considers variables such as the circular economy ratio, costs of advertising, as well as prices as crucial variables within its framework.

The model analyzes intricately the interaction patterns between forward and circular flows and analyzes the impact of circularity choice on efficiency and economic performance. By incorporating circularity ratio into the objective function and constraints, it helps analyze trade-offs between R, R*, and R** processes and costs associated with reverse logistics. Thus, there is a balanced representation of present-day supply chains that consider profit and sustainability-driven objectives.”

Numerical experiments reveal the superiority of SQP over GA, Pattern Search, as well as MiniMax algorithms for solving this nonlinear constrained optimization problem. The solutions from SQP not only perform better regarding total profit value as well as the stability of decision variables with comparable computational time but also indicate smooth production/shipment plans, realistic levels of inventory, as well as equitable allocation of resources with good coordination among the supply chain partners.

Most importantly, the results verify that joint decisions regarding advertising, selling price, and circular input utilization can enhance supply chain performance. Advertising improves market awareness and demand response, while pricing decisions support revenue optimization. At the same time, circular input utilization can reduce dependence on virgin inputs and strengthen the environmental positioning of the product, provided that recycled inputs are available at economically feasible cost and quality levels. The profit improvement observed in the numerical results therefore arises from coordinated pricing, cooperative advertising, incentive support, and optimized circular input adoption rather than from internally modeled reuse or remanufacturing operations.

From a management perspective, the model can be seen as a decision support model for the design and management of circular supply chain networks. It can help a manager determine the optimal levels of advertising expenditures, pricing, and the amount of circular activity as well as examine the impact of enhanced levels of the latter on capacity and profit allocation. The framework also guides strategic planning by pinpointing the optimal circularity level that maximizes economic benefits while controlling circular input adoption and processing costs.

In conclusion, applying the circularity variables in the coordinated three-level supply chain model results in economically feasible and optimal solutions. The findings provide conclusive evidence of effective and coordinated circular supply chain practices and strategies through pricing and marketing for overall sustainability and profit maximization. Future research may extend the present open loop circular input adoption model in several directions. First, the model can be expanded into a full closed loop supply chain framework by explicitly incorporating customer returns, collection centers, inspection, disassembly, remanufacturing, and recycling flow decisions. Second, stochastic or robust optimization can be used to represent uncertainty in demand, recycled input availability, input quality, and advertising effectiveness. Third, a multi period and multi product extension can capture dynamic circular input availability, changing customer preferences, and long term investment decisions. Finally, future studies may integrate environmental performance indicators, such as carbon emissions, waste reduction, and life cycle impacts, to develop a multi objective framework that jointly optimizes economic and environmental performance.

Author contributions

M. AlKahtani: Conceptualization; Methodology; Software; Validation; Formal analysis; Investigation; Visualization; Writing – original draft; Writing – review & editing.

Funding

This work was funded by the Ongoing Research Funding Program, (ORF-2026-274), King Saud University, Riyadh, Saudi Arabia.

Data availability

The data can be provided upon reasonable request.

Competing interests

The authors declare no competing interests.