2.1. Forward/reverse logistics

Many researchers have examined various practical issues to deepen the understanding of integrated forward/reverse logistics networks. For instance, Prajapati et al. [12] developed a sustainable framework for the transition of the e-commerce industry towards a circular economy in a multi-echelon, multi-product closed-loop supply chain. Their aim was to reduce the total cost associated with the forward and reverse flow of goods in the closed-loop supply chain while maximising revenue, with a focus on sustainability. Zhou et al. [13] designed a multi-echelon reverse logistics network by considering the economic and environmental costs in the recycling process of express packaging waste. This network addressed the capacity-location-routing problem in reverse logistics networks combined with customer incentive mechanisms. Yu et al. [14] proposed an integrated forward/reverse logistics network with consideration of a heterogeneous fleet with different loading capacities and transportation costs. The aim of considering different loading capacities was to deliver the demands of suppliers to customers, while the aim of considering different transportation costs was to return unsold products from customers to suppliers. Yu and Sun [15] presented an integrated and digital architecture for the design of uncertain reverse logistics networks. They also established a fuzzy optimisation model to identify potential network configurations under different demand satisfaction and capacity constraints. Li and Alumur [16] explored the effective management of water resources in the hydraulic fracturing process of natural gas extraction. In particular, they adopted a multi-period reverse logistics network design perspective and realistic modelling methods to determine the optimal locations and capacities of reservoirs, treatment facilities, and disposal sites. These efforts reduced fixed and operational costs, and decreased freshwater usage by 12.8%. Ma et al. [17] proposed a three-level interactive decision-making framework with consideration of product design. Specifically, they established an integrated crowdsourced contracting reverse logistics platform, proposed a three-level mixed 0-1 planning model, and using Stackelberg game theory to handle the complex decision-making process.

However, there is a lack of general research on integrated forward/reverse logistics networks in previous studies.

2.2. Carbon emissions

Talaei et al. [18] built a multi-objective low-carbon closed-loop green logistics network fuzzy-planning model for carbon emission reduction. In addition, a few scholars have investigated the interconnection between logistics-related carbon emissions and their influencing factors, such as the mass of transport vehicles and the distances these vehicle travel. For instance, Elhedhli and Merrick [19] developed a model for facility location and distribution to examine the link between carbon emissions and the weight of transportation vehicles. Accorsi et al. [20] created a facility-siting analytical tool for assessing the interplay between carbon emissions and overall expenditure (production and logistics expenses) due to various factors. These factors included product transportation distance and the agro-ecosystem in a food distribution network. He et al. [21] optimised a multi-level reverse logistics network for waste batteries from a circular economy perspective by introducing an expected weighting factor to balance the dual objectives of cost minimisation and carbon emission reduction. They also emphasised the importance of selecting different technology types for facilities. Chen et al. [22] integrated the offline and online trading systems for second-hand mobile phones, with consideration of multi-period, multi-product, and multi-tier characteristics; price-sensitive demand; incentives; and quality. They also proposed a novel multi-objective optimisation framework aimed at maximising profits, carbon emission reductions, and circularity in the recycling and processing process.

In the context of a logistics network, the processes of establishing or decommissioning facilities are both financially burdensome and temporally extensive. As such, the determination of locations for facilities is a strategic decision, and the locations of nodes and the paths between them affect costs and carbon emissions [23,24].

2.3. Fuzzy programming

The inherent unpredictability of an eco-friendly comprehensive forward and reverse distribution network means that its management is challenging, but there is limited literature on this topic. Talaei et al. [18] accounted for both the overall expenses and carbon footprint within such a network, in addition to incorporating uncertainties. In addition, they formulated a multi-objective fuzzy programming framework for a cohesive forward/reverse logistics framework. They validated this framework and analysed its feasibility within the context of the photocopier manufacturing sector. Pishvaee et al. [25] formulated a dual-objective fuzzy programming framework aimed at carbon emission reduction and addressing varying capacity constraints within logistics networks. Ghanbarzadeh-Shams et al. [26] introduced a novel fuzzy multi-objective programming model with chance constraints that integrated flexibility and possibility. This model considered the carpet industry’s susceptibility to high uncertainty affecting collection rates, the availability of recyclable items, and reverse channel capacities. Thus, the model addressed the multi-product, multi-site, and multi-period production planning problem under uncertainty in integrated reverse logistics. Tosarkani et al. [27], in the context of recycling plastic beverage containers, considered economic, environmental, and social objectives and established a reverse logistics network model. They also employed a scenario-based possibility approach to handle parameter uncertainty and developed a data-driven fuzzy optimisation framework. Raad and Rajendran [28] emphasised that effective network design is key to success. They also noted that decisions regarding location and capacity must consider objective data and expert opinions to avoid costly long-term consequences. Furthermore, they combined fuzzy multi-criteria decision-making methods with fuzzy stochastic chance-constrained optimisation to form an integrated multi-objective model, which they solved using the epsilon-constraint method.

2.4. Robust optimisation

Some scholars have used robust optimisation methods [29] to solve network uncertainty problems. For example, Bertsimas et al. [30,31] proposed new robust optimisation methods based on different sets of uncertain parameters and obtained linear robust pairwise equations through formula transformation. They then applied these equations to logistics network management. Hatefi and Jolai [32] simultaneously considered uncertain parameters and facility interruptions to reduce costs. They also proposed a comprehensive forward/reverse logistics network model that employed integer linear programming and aimed to reduce interruption risk by enhancing robustness criteria. Wang et al. [33] generated a logistics network encompassing both remanufacturing and the integration of forward and reverse flow models to reduce carbon trading revenue and logistics costs under tradable carbon emission rights. They also used robust optimisation methods to deal with uncertainty in the network. However, the abovementioned studies have defined only some model parameters as random or fuzzy, while real logistics networks are complex and involve multiple types of parameters. As such, the uncertainty of parameters in network models should be diverse and defined as random and/or fuzzy. Therefore, multiple types of uncertainty should be the focus of research on logistics networks [34].

In summary, the optimisation of low-carbon forward/reverse integrated logistics networks under uncertain environments is a rather new research topic. As shown in S1. although many researchers have examined this topic, few studies have simultaneously considered carbon emissions, fuzziness, and robustness in general forward/reverse integrated logistics networks. Therefore, this paper optimises a forward/reverse integrated logistics network from the perspective of uncertainty planning and accounts for aspects such as carbon emissions, fuzziness, and robustness.

3.1. Problem definition

As depicted in Fig 1, the flow of new products begins at the manufacturing/recycling nexus and advances towards the distribution facility on the path to reaching the consumer market. Concurrently, items returned by customers are routed to the recovery/inspection hub for evaluation. There, products deemed recyclable are redirected to the production/recovery facility, while those deemed unrecyclable are sent to a disposal site. The integration of hybrid facilities – where distribution and recovery/inspection functions coexist – presents cost-efficiency benefits over separate operations. These savings are quantified within the goal function within the supply chain network’s optimisation framework, as illustrated in the figure.

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Fig 1. Logistics network structure of forward and reverse integration.

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Within the logistics network, carbon emissions primarily originate from two distinct sources: (1) the internal energy use of facilities, which encompasses electricity, fuel, and heat use (this paper focuses on electricity, as the principal form of energy consumed); and (2) the transportation phase between facilities, which is influenced by variables such as vehicle mass, travel distance, and road conditions, including gradients and traffic congestion. For simplicity, the paper confines its analysis to the influence of vehicle mass and travel distance. Furthermore, this paper uses symmetric intervals or trapezoidal fuzzy numbers to delineate the fluctuation in uncertain parameters related to network attributes, as elaborated in subsequent sections.

3.2. Model assumptions and symbols

For the convenience of modelling, the following conditions are assumed: (1) there is a single-cycle single product; (2) the network’s facilities can be categorised into three tiers of capability, and their fixed operational costs and associated carbon emissions increase in tandem with the level of a tier; (3) the same product generates the same carbon emissions during the operation of the same facility; and (4) customers are willing to use remanufactured products.

Defining symbols:

1. Sets: I contains potential locations for production and recovery operations, J contains potential locations for distribution hubs, K contains established locations within the customer region, Set L contains potential locations for recycling and inspection centres, M contains potential locations for disposal centres, N contains the facility’s capacity levels, and E contains potential integration points between recycling/inspection facilities and distribution nodes,
2. Parameters: is the fuzzy demand of customer area k; is the fuzzy recovery rate of customer area k, a constrained symmetric stochastic variable; is the average processing rate, a confined symmetric stochastic quantity; is the fixed operational expense associated with production/recovery centre i at capacity level n; is the inherent cost of the distribution center j under level n; is the constant expense associated with the recovery/inspection centre l at capacity level n; is the predetermined expense for waste disposal centre m at capacity level n; are the fixed cost savings of the joint point e of the distribution centre at capacity level n and recovery/inspection center at capacity level n^′; are respectively fuzzy unit transportation costs between facilities; is the fuzzy forward capability of production/recovery centre i at capacity level n; is the fuzzy capability of distribution centre j at capacity level n; is the fuzzy capability of recovery/inspection centre l at capacity level n; is the fuzzy capability of disposal centre m at capacity level n; is the reverse capability of production/recovery centre i at capacity level n; β is a substantially large value; is the fuzzy CO₂ emission factor, ; are the distances between facilities, respectively, km; is the carrying capacity of a vehicle; are the fixed CO₂ emission factors of initiating a manufacturing/recovery facility i, distribution center j, recycling/inspection center l and disposal center m, respectively, g/centre; and are the fuzzy CO₂ emission coefficients of unit products processed within the establishment of manufacturing/recovery facilities, fulfillment hubs, recycling/evaluation centres, and waste disposal facilities, g/ piece.
3. Decision variables: are transport volumes between facilities; is a 0-1 variable that takes a value of 1 if the manufacturing and recycling facility i with capacity tier n is selected, and 0 otherwise; is a 0-1 contingent factor that takes a value of 1 if distribution centre j with a capacity tier n is selected, and 0 otherwise; is a 0-1 variable that takes a value of 1 if the recovery/inspection centre l with the ability level n is selected, and 0 otherwise; is a 0-1 variable that takes a value of 1 if a disposal center m with ability level n is selected, and 0 otherwise.

3.3. Objective functions and constraints of the optimisation model

1. 1. The first objective function aggregates the fixed establishment expenses and deducts the incurred savings by integrating facilities and transportation costs. Therefore, Objective 1 (fixed opening and logistics cost minimisation) can be defined as follows:

(1)

1. 2. According to the problem description, Objective 2 (carbon emission minimisation) can be defined as follows:

(2)

According to the assumptions adopted, the decision variables must meet the following constraints:

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

Constraints (3) and (4) ensure that customer demand is met and that a certain proportion of the products is recycled, respectively. Constraints (5) to (7) indicate the flow balance. Constraints (8) to (15) indicate the facility capacity. Constraints (16) to (19) indicate that a certain capacity level should be selected for each open facility. Constraint (20) indicates the integrality of binary variables, and Constraint (21) indicates the integrality of non-negative variables.

As the nonlinearity is caused by , a supplementary variable is added into the model, as follows:

(22)

(23)

(24)

(25)

The treatment of , , and is performed in the same manner, and supplementary variables , , and are added into the model. Subsequently, Objective 2 is redefined as follows:

(26)

Furthermore, the term in Objective 1 is non-linear. To circumvent the intricacies of mixed-integer non-linear programming, a new variable is introduced, and the objective function is linearised as follows:

(27)

(28)

The following constraint set is added into the updated model formulation:

(29)

Formula (29) indicates that when and/or are both equal to 0 or at least one of them is 0, then the value of is 0. When both terms are equal to 1, the value of is 1.

3.4. Defuzzified model formulation

Objective functions (1) and (2) and Constraints (3,4,6)–(14) of the model parameters outlined in the preceding section incorporate elements of fuzziness. Thus, to facilitate calculation, these objective functions and constraints are converted into their equivalent deterministic forms. According to the literature [18], the necessity transformation can be employed within the framework of probabilistically constrained fuzzy programming to reformulate a target function and particular limitations within an optimisation framework, particularly when dealing with parameters characterised by uncertainty. The optimality of the necessity transformation can be defined as the necessity optimal solution based on randomness theory. In simple terms, the necessity optimal solution is optimal only if all possible solutions are solvable, which is meaningful for meeting chance constraints. Suppose that the linear optimisation model is as follows:

In addition, suppose that the vector f is the deterministic parameter, and that the vector sum and coefficient matrix is the indeterminate parameter of the problem. In this case, the objective function is reformulated deterministically by employing the ‘expected value’ approach, while the constraints are translated into their deterministic fuzzy chance equivalents by application of the necessity transformation. Thus, the aforementioned model is reformulated in the following manner:

Herein, the minimum degree of constraint satisfaction is defined, with its value bounded by the interval [0, 1]. Parameters exhibiting uncertainty are characterised by trapezoidal imprecise numerical values, and the corresponding deterministic model is delineated as follows:

Therefore, the equations in the model containing trapezoidal fuzzy parameters can be converted into their corresponding deterministic equivalents, as follows:

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

Equations (33)–(39) are inherently non-linear and thus must be linearised to mitigate the escalation of computational complexity within the optimisation model. Therefore, Equation (33) becomes

(41)

Similarly, let , , , . Consequently, the linear variations of Equations (34)–(39) can be derived based on Equation (41). The robust transformation can be applied for Constraints (4,6), and (7), based on concepts reported in the literature [29–33], as described below.

In Constraint (4), the recovery rate is subject to uncertainty and is treated as a constrained symmetric stochastic variable with a designated value , where, is the expected value, and is the upper limit of deviation from this expected value. For each constraint k, this paper introduces a new parameter value at , where represents the quantity of uncertainties in the recovery rate in the first constraint. represents the estimated amount of recovery uncertainty used to adjust the conservatism of the model, where recovery uncertainty is related to the decision maker’s estimation of the customer’s willingness and the quality of the recovered product. In practice, all recovery rates cannot be uncertain. Thus for each customer, the recovery rate of the product from the various distribution centres varies within the interval and within and does not specify which change. Therefore, the robust correspondence for Constraint (4) can be written as follows:

(42)

where is the variable added to the restricted value .

Given , if ream

(43)

then is equivalent to the objective of the following inquiry:

(44)

where is the variable.

Let , so. Then, according to the duality theory, the robust correspondence of Constraint (4) is transformed into:

(45)

where, and are the variables added in accordance with duality theory.

Similarly, consider that s in Constraints (6) and (7) is symmetric random variable with values of . For each constraint, this paper introduces new parameters and with values of . and represent the estimated amount of the processing rate uncertainty, which is related to the decision maker’s estimation of the quality of the recovered product and is used to adjust the conservatism of the model.

Following the same methodology, the robust correspondence of Constraint (6) is converted into

(46)

Similarly, the robust correspondence of Constraint (7) is converted into

(47)

In summary, a fuzzy robust programming model is obtained using Equations (30) and (31) as objective functions and using the constraints represented by Equations (5), (15)–(22), (29), (32), (40), (41), (45)–(47).

3.5. Transformation of the objective functions

As has been described in previous studies [35,36], there are three principal approaches for addressing multi-objective optimisation dilemmas: the a priori approach, the a posteriori approach, and the interactive approach. In the a priori approach, the decision maker articulates a preference, or weighting, for each objective prior to the optimisation model’s resolution, thereby converting the multi-objective optimisation challenge into a singular, unified objective. In the a posteriori approach, all objective functions are simultaneously optimised to yield an ensemble of efficient solutions (Pareto-efficient solutions). Subsequently, upon the conclusion of the search process, the decision maker identifies and selects the most appropriate solution from this ensemble. In the interactive method, the decision maker’s preferences for each objective are conveyed throughout the participatory decision-making process. The calculation process is interchangeable, and after several iterations, a suitable scheme is selected. In this paper, a compromise solution approach, namely the a posteriori method, is selected for model resolution. Considering that the span of each objective function encompasses the extremes (i.e., the ceiling and floor values), the target function is delineated as follows:

(48)

where

The importance of objectives is generally set by decision-makers based on their experience. This paper employs the approach that has been outlined in previous studies [37,38] to ascertain the viability of the dual objectives. Table 1 presents the initial objective function for this approach and its optimal solution.

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Table 1. Game table.

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The minimum and maximum bounds for the optimisation and maximisation problems, respectively, are established by analysing the matrix in Table 2 and applying Equations (49) and (50).

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Table 2. Summary of the literature review.

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and

(49)

and

(50)

4.1. Input data generation

To demonstrate the real-world application of the discussed analytical technique, this paper performs a case analysis of an automobile manufacturer, referred to as Ws company. Ws company relies on the existing engine consumption area of its parent company to recover and repair used engines. Thus, it is necessary to build a network for production/recovery, distribution, recycling/inspection, and disposal. Hence, a reliable strategic plan is needed as a guide. In this case analysis, the facility’s capacity is categorised into three distinct levels, with the cost and carbon emissions measured in yuan and grams (g), respectively. The average processing rate is set to 0.2, while the joint points of the distribution centre and the recycling/inspection centre are J₂ and L₂、J₄ and L₄、J₅ and L₅. The weight δ is 0.5, and the capacity of the transportation vehicle is set to five pieces. The other data required for the proposed optimisation model are specified in Tables 3–11.

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Table 3. Pertinent information of the customer region.

https://doi.org/10.1371/journal.pone.0316197.t003

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Table 4. Distribution centre-related data.

https://doi.org/10.1371/journal.pone.0316197.t004

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Table 5. Recycling/inspection centre-related data.

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Table 6. Production/recovery centre-related data.

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Table 7. Disposal centre-related data.

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Table 8. Freight-related data.

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Table 9. Cost savings at joint points with a distribution centre at capacity level 1.

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4.2. Sensitivity analysis and managerial insights

To study the results of the forward/reverse integrated logistics network, let = 0.04 and = 0.01. To test the stability of the robust fuzzy model, three cases are discussed that have different levels of conservatism: (1) the certain case: = 0, ==0; (2) the intermediate case: =  2.5, =  =  5; and (3) the most conservative case: = 5, ==10. First, the robust fuzzy model is solved by using CPLEX12.5.1 with Equations (30) and (31) as targets in the three cases. Tables 12–14 present the analysis results.

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Table 10. Cost savings at joint points with a distribution centre at capacity level 2.

https://doi.org/10.1371/journal.pone.0316197.t010

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Table 11. Cost savings at joint points with a distribution centre at capacity level 3.

https://doi.org/10.1371/journal.pone.0316197.t011

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Table 12. Game table of the certain case.

https://doi.org/10.1371/journal.pone.0316197.t012

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Table 13. Game table of the intermediate case.

https://doi.org/10.1371/journal.pone.0316197.t013

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Table 14. Game table of the most conservative case.

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Based on the results in Tables 12–14, objective function ranges are obtained for the three cases and are presented in Table 15.

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Table 15. Range of objectives.

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Subsequently, CPLEX 12.5.1 was used to solve the established robust fuzzy programming model and the results are presented in Tables 16–18. In Table 17, 00 indicates that this a place is not selected, and 1, 2, 3 represents the level of at which a place choosing this place and the centre is selected). The results are used to construct the graphs and line segments in Figs 2–4, which have the same meaning and are made according to the data (the meanings represented by graphs and line segments in the figure are the same as those in Fig 1). In all three cases, the level of constraint satisfaction in all three cases is set to 0.5.

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Table 16. Optimisation results of the model.

https://doi.org/10.1371/journal.pone.0316197.t016

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Table 17. Site selection for each centre.

https://doi.org/10.1371/journal.pone.0316197.t017

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Fig 2. Forward/reverse integrated logistics network diagram of the certain case.

https://doi.org/10.1371/journal.pone.0316197.g002

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Fig 3. Forward/reverse integrated logistics network diagram of the intermediate case.

https://doi.org/10.1371/journal.pone.0316197.g003

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Fig 4. Forward/reverse integrated logistics network diagram of the most conservative case.

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Within the fuzzy parameters defined for this scenario, the level of constraint satisfaction is set to 0.5, signifying that there is a 50% likelihood that the aforementioned three cases will satisfy the constraint requirements. This setting reflects the fact that corporate executives are keen to enhance the precision of decision-making processes, which is equivalent to increasing the degree of constraint adherence.

In practical cases, numerous parameters are inherently uncertain, meaning it is unfeasible to attempt to meet all established constraints. Therefore, when parameters are fuzzy numbers, a satisfaction level of 0.5 is realistic. Consequently, decision-making has inherent risks, particularly within an ambiguous context, i.e., one in which it is challenging for corporate executives to assess the validity of a strategy. The model’s incorporation of fuzzy programming techniques adeptly tackles this challenge. Therefore, in this scenario, the conservatism level of the manager’s control to meet the constraint conditions is more than 50%, that is, the manager can judge that the decision obtained in the three cases is the optimal result with a probability of 50%.

In Table 16, the target value fluctuates in response to variations in , and . The increase in the squared values of and is attributable to the proliferation of disturbances in the recovery and average processing rates, and to the expansion of the uncertain sets they encompass, which increases amplifying the target value.

Table 17 shows that the quantity of central facility selections remains consistent across the three cases, with each selecting a level 1 distribution centre and a level 1 disposal centre. However, the types of other central facilities selected vary among the cases. The intermediate case selects a level 3 production/recovery centre 1, a level 2 distribution centre 2, a level 2 recycling/inspection centre 1, and level 1 recycling/‘inspection centre 2. In contrast, the definite case selects a level 2 production/recovery centre 1, a level 1 distribution centre 2, a level 1 recycling/inspection centre 1, and a level 2 recycling/inspection centre 2. The most conservative case and the certain case both select a level 1 distribution centre 2 and a level 1 recycling inspection centre 1. However, the most conservative case selects a level 3 production/recovery centre 1 and a level 1 recycling/inspection centre 2, while the certain case selects a level 2 production/recovery centre 1 and a level 2 recycling inspection centre 2. The intermediate case and the most conservative case both select a level 3 production/recovery centre 1 and a level 1 recycling/inspection centre 2. However, the most conservative case selects a level 1 distribution centre 2 and a level 1 recycling/inspection centre 1, while the intermediate case selects a level 2 distribution centre 2 and a level 2 recycling/inspection centre 1.

In Table 18, joint point 2 between the recycling/inspection centre and the distribution centre is selected for the certain and intermediate cases, while no joint point is selected for the most conservative case.

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Table 18. Selection of joint points.

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Finally, Figs 2–4 show that the logistics networks in the three cases have many identical routes. However, they also have a considerable number of different routes. The routes in Figs 2 and 4 are most concentrated on the nodes of recycling/inspection centre 2, while the routes in Fig 3 are most concentrated on the nodes of recycling/inspection centres 1 and 2. These results demonstrate that external disturbances have a significant impact on the flow network structure.

By setting the value of δ to 0 ~ 1 (with a scale of 0.1), the optimal values of the two objective functions are obtained in the deterministic, intermediate, and most conservative cases, as shown in Table 19 ~ 21. According to Tables 19–21, generate Figs 5–7 respectively. From Figs 5–7, it can be seen that in all three cases, when the value of δ is 0.5, both objectives can reach fairly low values, indicating that the current two target values belong to satisfactory values that satisfy both target requirements simultaneously.

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Table 19. Two target values in the Certain case.

https://doi.org/10.1371/journal.pone.0316197.t019

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Table 20. Two target values in the intermediate case.

https://doi.org/10.1371/journal.pone.0316197.t020

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Table 21. Two target values in the most conservative case.

https://doi.org/10.1371/journal.pone.0316197.t021

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Fig 5. Change curves of two target values in the certain case.

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Fig 6. Change curves of two target values in the intermediate case.

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Fig 7. Change curves of two target values in the most conservative case.

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In conclusion, the inherent randomness and fuzziness of parameters influence the positioning of facilities, their interconnecting routes, and the volumes of traffic they accommodate. Therefore, enterprise decision-makers should fully consider their conditions and look forward to future development, grasp the trend of the market, and make appropriate strategic decisions. Hence, the model proposed in this paper has certain applicability, as the obtained solution has superiority and provides data support for enterprise decision-making.