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Research Article | Open Access
Jing Zhang, Jingjing Liu, Zhong Wan, "Optimizing Transportation Network of Recovering End-of-Life Vehicles by Compromising Program in Polymorphic Uncertain Environment", Journal of Advanced Transportation, vol. 2019, Article ID 3894064, 24 pages, 2019. https://doi.org/10.1155/2019/3894064
Optimizing Transportation Network of Recovering End-of-Life Vehicles by Compromising Program in Polymorphic Uncertain Environment
With rapid development of technology and improvement of living standards, the per capita holding of automobiles greatly increases, and the amount of end-of-life vehicles (ELVs) becomes larger and larger such that it is valuable to investigate an effective strategy for recycling ELVs from the viewpoints of environmental protection and resource utilization. In this paper, an optimization model with fuzzy and stochastic parameters is built to formulate the transportation planning problems of recycling ELVs in polymorphic uncertain environment, where the unit processing and transportation costs, the selling price of reused items, and the fixed cost are all fuzzy, while the demand in secondary market and the production capacity are random owing to features underlying the practical data. For this complicated polymorphic uncertain optimization model, a unified compromising approach is proposed to hedge the uncertainty of this model such that some powerful optimization algorithms can be applied to make an optimal recycling plan. Then, an interactive algorithm is developed to find a compromising solution of the uncertain model. Numerical results show efficiency of the algorithm and a number of important managerial insights are revealed from the proposed model by scenario analysis and sensitivity analysis.
With rapid development of technology and improvement of living standards, the per capita holding of automobiles greatly increases. In China, as the largest developing country with a population of around 1.3 billion, huge amount of end-of-life vehicles (ELVs) is bringing enormous pressure on its environment and human life. Actually, the civilian car ownership in China has reached 137 million in 2013, and has been almost doubling every four years. If the average lifespan of a car is 8-9 years, then the number of ELVs will exceed 14 million in 2020 . In the world, it is estimated that the number of vehicles will rise to 1.85 billion by 2030, and the scrap generated from the ELVs will be 3.71 billion tonnes . In an era of resources shortage and environmental deterioration, recycling the ELVs can give birth to new-style industry as a typical low-carbon and sustainable production approach [3, 4].
It is well known that the ELVs contain a great amount of reusable components and materials such as steel, copper, rubber, etc. Therefore, recycling the ELVs offer considerable economic and environmental benefits . This fact has been paid great attention either by governments, by industry or by academia. Actually, the European Union (EU) has established legal regulation that manufacturers are responsible for take-back of ELVs from end-users, dismantling, shredding, and recycling of ELVs . Directive 2000/53/EC required that, no later than 1 January 2015, for all the ELVs, the reuse and recovery rate shall be increased to a minimum of 95% by an average weight per vehicle and year. Within the same time limit, the reuse and recycling rate shall be increased to a minimum of 85% by an average weight per vehicle and year. Japan had an ELV recovery rate of 85% in 2002, its attained target reached 95% by 2015 .
In China, a series of relevant policies and regulations have been issued to improve the management mechanism for the ELV recycling industry since 2001. The “automotive products recycling technology policy” was implemented in 2006, which specifies utilization rate targets of the recyclable products in China . The established China automotive material data system (CAMDS) in 2009 has played an important role in implementing the automotive products recyclable rate and the managing the ELVs. Chinese government has legislated that “yellow label cars” (heavily polluting vehicles) must be eliminated by 2017 . Automotive components remanufacturing, as an essential part in automotive life-cycle development, has become a prominent direction to promote sustainable planning of automobile industry in China . Actually, these policies are bringing a great of economic and environmental benefits to China.
1.2. Literature Review
In recent years, recovery of used products has become increasingly important owing to economic reasons and growing environmental or legislative concern . Particularly, the ELV recycling plays an important role for sustainable development. For example, each remanufactured engine could save 68-83% of the energy required to manufacture a new engine, and decrease carbon dioxide emissions by 73-87%. The coolants and batteries in ELVs can also be recycled, and it reduces emissions of greenhouse gases and gases that lead to acidification. The main ingredient of coolant is Solid CO2, and the electrolyte of lead accumulator is lead accumulator . Recycling metals from the ELVs could decrease the amounts of resources consumed building new cars. If all the vehicle materials can be recycled to produce new vehicles, about 30% of the energy consumption can be saved .
Summarily, the ELVs contain a great quantity of reusable components and materials such as steel, copper, rubber, plastics, etc. which can be reused or remanufactured. Thus, the remanufacturing industry of ELVs necessarily has strategic significance as it better utilizes resources and creates higher values. Especially, recycling ELVs can play an important role in realizing the country’s sustainable development goals. In 2003, China has introduced extended producer responsibility (EPR), which requires that any manufacturer should participate in ELV take-back, dismantling, remanufacturing, and so on . The “automotive product recycling technology policy” in China requires carmakers to improve the design of vehicles, spare parts, and raw materials, as well as reduce the use of lead and other environmentally hazardous substances. Actually, this policy is an encouragement to the carmakers such that more recycled materials from the ELVs are used .
It is easy to see that recycling the ELVs depends on establishment of an efficient ELV recycling network, which not only can reduce the impact on the environment during the recycling process, but also can facilitate the effective reuse of recycled resources . Furthermore, construction of optimization models for production planning problems of recycling the ELVs is helpful to provide the decision-makers an optimal plan for the practical operation of the recycling system .
In this connection, Cruz-Rivera et al. developed a reverse logistics network design for collection of the ELVs in Mexico . Demirel et al. proposed a deterministic mixed integer linear programming (MILP) model for the ELV recycling network design, where all of the end-users, collection centers, dismantlers, shredders, landfills, recycling facilities, and secondary markets are included . On the basis of , Demirel et al. in 2017 developed a closed-loop supply chain for the ELVs recycling, where some reusable components after processing are sold to suppliers for remanufacturing . Finally, new vehicles with the remanufactured components will flow to consumers. Ene et al. considered refurbishment in ELVs recycling network; the reusable parts must be refurbished before they could be sold to secondary markets . Phuc et al. designed inspection centers in the ELVs recycling system; those ELVs passing inspection is repaired in the repair centers and then is sold to the used vehicle markets . Additionally, in , municipal solid waste incinerator and advanced thermal treatment measures are applied to dispose autoshredder residue (ASR) besides land-filling.
In some of the existing models for recycling the ELVs, various kinds of uncertainties have also been considered. It suggests in  that uncertainty seems to be the key factor influencing the management of ELVs. Özkır et al. stated that the selling price of products can be described by trapezoidal fuzzy sets such that both seller’s and buyer’s satisfaction levels are reflected . In , the fixed cost, the transportation cost and the processing cost were also regarded as trapezoidal fuzzy sets in a reverse ELV recovery network. In , the capacities of sorting entities for recycling the ELVs were observed as random parameters, while the procurement cost, the transportation cost, the processing cost, and the storage cost were assumed to be interval parameters. In a global supply chain management model proposed by Wan et al., the demand of products for retailers is assumed to be stochastic and depends on the price of products .
Very recently, a polymorphic uncertain equilibrium model (PUEM) was developed by Wan et al. for the problem of decentralized supply chain management, where the demand of consumers was regarded as a continuous random variable, and the holding cost of the retailer and the transaction cost between the manufacturer and retailer were described by fuzzy sets . Then, for the PUEM, a deterministic equivalent formulation (DEF) was first derived by compromise programming approach such that the existing powerful algorithms in the standard smooth optimization were employed to find an approximate equilibrium point for the uncertain problem. It is also noted that there are many different approaches to removing uncertainty in the uncertain model. For example, expectation method was applied in  to deal with the fuzzy objective such that the fuzzy objective can be converted into a deterministic one. Chance-constrained programming method was adopted in  to deal with the random constraints.
However, in the existing results for optimizing the system of recycling the ELVs, there are still some deficiencies, which can be briefly summarized as follows.
Polymorphic uncertainty is rarely considered in this special reverse logistics network. Especially, randomness of the demand in the secondary markets of reusable components has not been taken account into design of an optimal ELV recovery network.
To hedge fuzziness of the objective function, the expectation method is often applied to transform the fuzzy objective into a deterministic one. Clearly, this method can not address the feature of variance in a fuzzy set. A more reasonable method should capture all information in a fuzzy objective function, which includes the lower and upper variances of a fuzzy set, as well as its center value.
1.3. Motivation of This Research
From the above literature review, it is necessary to build a new polymorphic uncertain optimization model for a more efficient system of the ELV recovery management. In particular, this model should simultaneously capture fuzziness and randomness of model parameters in a ELV recovery network design. Then, as done in , a unified compromising programming approach should be presented to convert the PUOM into a deterministic one such that the existing powerful optimization algorithms can be applied to find an approximately optimal strategy for recycling the ELVs.
In this paper, just like the mentioned reasons in [19, 21, 23, 24], we suppose that all of the fixed cost, the unit transportation cost, the unit processing cost, and the unit selling price of reused parts are fuzzy model parameters, and both of the capacity and the demand are regarded to be random variables. Then, our investigation proceeds along the following three subsequent steps.
Step 1. In a polymorphic uncertain environment, we construct a new optimization model to formulate the production planning problems of ELV recovery system.
Step 2. To hedge uncertainty of the model, a unified compromising programming approach will be proposed, which is associated with the following two phases: in the first phase, the original problem is converted into an auxiliary crisp multiple-objective mixed integer linear programming problem; in the second phase, a novel interactive fuzzy programming approach is proposed to find a preferred compromising solution through an interaction between the decision-maker with preference and the rational model .
Step 3. To answer what is the practical significance of the new model and the developed algorithm in this paper, we will reveal some important managerial insights from the proposed model by scenario analysis and sensitivity analysis.
The rest of the paper is organized as follows. Next section is devoted to the description of problem and construction of model. In Section 3, an interactive algorithm is developed. In Section 4, numerical results of case study are reported. In Section 5, sensitivity analysis is conducted, and some practical managerial implications are revealed from the constructed model. Some conclusions and suggestions on future research are presented in the last section.
2. Problem Description and Formulation
2.1. Problem Description
As shown in Figure 1, the network nodes basically consist of the ELV sources such as the last owners, insurance companies and abandoned vehicles, the collection centers, the dismantlers, the shredders, the recycling facilities, the secondary markets, and the landfills.
Specifically, the process flow of the recycling network can be stated as follows. All of the ELVs must be treated in formal dismantling companies. The collection centers or dismantlers first procure the ELVs from the ELV sources. Then, the ELVs in the collection centers will all be transported to the dismantlers. In the dismantlers, it is first required to remove and store separately the fuel, the motor oil, the oil from transmission system, the hydraulic oil, the cooling liquid, the liquid from the brake system, and other liquids and hazardous substances if any. Subsequently, the components or materials removed from the scrap car are considered for reuse and recycling. Reusable ferrous and nonferrous components are sold to the secondary market, while the recycling materials, such as batteries, tyres, glass, plastics, and waste oil, are sold to the recycling factories. The remaining hulks are shipped to the shredders for further recycling. In the shredders, some materials can be mechanically recycled by shredder, air suction, magnetic sorters, and eddy current sorters. Finally, the hulks are divided into ferrous material, nonferrous material, and autoshredder residue (ASR). The sorted metals will be allocated to steel mills or nonferrous smelteries for further recycling, while ASR will be directly transported to the landfill.
In the recycling network in , all of the end-users, collection centers, dismantlers, shredders, landfills, recycling facilities, and secondary markets are the nodes of this network, and it is assumed that the last owners must return their vehicles to one of the collection centers or dismantlers. Different from the network in , Figure 1 indicates that only ferrous and nonferrous components for reusing are sold to the secondary markets in our network; there is no need to build the recycling facilities for processing the battery, tyre, glass, and plastics; instead, all of them are separately sold to the existent factories which have equipment for recycling; our network is more in accordance with the suggested recycling system in China by [26, 27].
The goal of this paper is to formulate the above recycling network such that the total recycling cost is minimized, which is associated with the costs of transportation, processing, disposal, and the fixed opening costs in a multistage production plan. In order to build an optimization model that is more realistic than those available in the literature, uncertainty in recycling system must be incorporated into construction of model. Actually, due to incompleteness and unavailability of desired data, it is inappropriate to assume that the fixed opening cost, the unit transportation cost, the unit processing cost, the unit selling price of reconditioned parts, the capacity, and demand are all fixed constants. An acceptable approach is to describe these parameters by uncertain mathematical concepts from the theory of stochastic or fuzzy mathematics. Especially, in this paper, we assume that the fixed opening cost, the unit transportation cost, the unit processing cost, and the unit selling price of reconditioned parts are fuzzy sets, and the capacity levels of dismantlers, shredders, landfills, and the demand of secondary markets are random variables.
For simplicity, we suppose that the membership function of the relevant fuzzy sets is subject to possibility distribution (see Figure 2). Mathematically, any trapezoidal fuzzy number is given by a membership function , where for any ,, , , and are given constants. Different values of , , , and define various fuzzy sets. From this viewpoint, any fuzzy number can be denoted by a quaternion . For fuzzy number , we call its the upper variance, call its lower variance, and the center value.
Before construction of model, we first introduce the following notations for readability.
Indexes : the labels of ELV sources, . : the labels of collection centers, . : the labels of dismantlers, . : the labels of shredders, . : the labels of secondary markets, . : the labels of steel mills, . : the labels of nonferrous smelteries, . : the labels of oil recycling factories, . : the labels of battery recycling factories, . : the labels of rubber recycling factories, . : the labels of glass recycling factories, . : the labels of plastics recycling factories, . : the labels of landfills, . : the processing periods, .
Parameters : the amount of ELVs returned from ELV source in period (ton). : the fixed opening cost for dismantler in period (yuan). : the fixed opening cost for shredder in period (yuan). : the unit cost of dismantling at dismantler in period (yuan/ton). : the unit cost of shredding at shredder in period (yuan/ton). : the unit cost of disposal at landfill in period (yuan/ton). : the unit price of selling of dismantler for ferrous components for reusing in period (yuan/ton). : the unit price of selling of dismantler for nonferrous components for reusing in period (yuan/ton). : the unit price of selling of dismantler for oil for recycling in period (yuan/ton). : the unit price of selling of dismantler for battery for recycling in period (yuan/ton). : the unit price of selling of dismantler for tyre for recycling in period (yuan/ton). : the unit price of selling of dismantler for glass for recycling in period (yuan/ton). : the unit price of selling of dismantler for plastics for recycling in period (yuan/ton). : the unit price of selling of shredder for ferrous material for recycling in period (yuan/ton). : the unit price of selling of shredder for nonferrous material for recycling in period (yuan/ton). : the unit cost of transportation between ELV source and collection center for ELV in period (yuan/tonkm). : the unit cost of transportation between ELV source and dismantler for ELV in period (yuan/tonkm). : the unit cost of transportation between collection center and dismantler for ELV in period (yuan/tonkm). : the unit cost of transportation between dismantler and shredder for hulk in period (yuan/tonkm). : the unit cost of transportation between shredder and landfill for ASR in period (yuan/tonkm). : the distance between ELV source and collection center (km). : the distance between ELV source and dismantler (km). : the distance between collection center and dismantler (km). : the distance between dismantler and shredder (km). : the distance between shredder and landfill (km). : the capacity of collection center in period (ton). : the capacity of dismantler in period (ton). : the capacity of shredder in period (ton). : the capacity of landfill in period (ton). : the demand of secondary market for reusable ferrous components in period (ton). : the demand of secondary market for reusable nonferrous components in period (ton). : the weight percentage of hulk in ELV. : the weight percentage of reusable ferrous components in ELV. : the weight percentage of reusable nonferrous components in ELV. : the weight percentage of oil in ELV. : the weight percentage of batteries in ELV. : the weight percentage of tyres in ELV. : the weight percentage of glass in ELV. : the weight percentage of plastics in ELV. : the weight percentage of ASR in hulk. : the weight percentage of ferrous material in hulk. : the weight percentage of nonferrous material in hulk.
Decision Variables : the amount of ELVs transported from ELV source to collection center in period . : the amount of ELVs transported from ELV source to dismantler in period . : the amount of ELVs transported from collection center to dismantler in period . : the amount of hulk transported from dismantler to shredder in period . : the amount of ASR transported from shredder to landfill in period . : the amount of ferrous components transported from dismantler to secondary market in period . : the amount of nonferrous components transported from dismantler to secondary market in period . : the amount of oil transported from dismantler to oil recycling factory in period . : the amount of batteries transported from dismantler to battery recycling factory in period . : the amount of tyres transported from dismantler to rubber recycling factory in period . : the amount of glass transported from dismantler to glass recycling factory in period . : the amount of plastics transported from dismantler to plastics recycling factory in period . : the amount of ferrous material transported from shredder to steel mill in period . : the amount of nonferrous material transported from shredder to nonferrous smeltery in period . : if dismantler is opened in period , ; otherwise, . : if shredder is opened in period , ; otherwise, .
2.3. Construction of Model for the ELV Recycling
We now formulate the objective function of the ELV recovery system in a centralized decision-making mode, being referred as to . The objective is to minimize the total cost of recycling system. It is associated with the fixed opening cost, the transportation cost, the processing cost, and the revenue from the sale income of the recycled resources. Therefore, the total cost function is written aswhere the total fixed opening cost isthe total transportation cost on each arc of the network readsthe total processing cost of dismantlers, shredders, and landfills isand the income from the sale of isolated materials is
Remark 1. Note that, in (3), (4), (5), and (6), the opening cost, the unit transportation cost, the unit sale prices of all the reused components from the ELVs, and the unit processing cost for dismantling, shredding, and disposal at landfill are all supposed to be fuzzy model parameters. The advantage of this assumption is that these fuzzy parameters can reflect uncertain information of cost or revenue which can not be precisely calculated by statistical (financial) data. Actually, it is often that depreciation degree of the production tools can only be described as higher or lower, and market reputation of the reused products can only be said to be more credible. A fuzzy set is the most appropriate mathematical concept to characterize these uncertain language evaluations such as high, low, and credible.
Next, we present some practical constraints in minimization of the total recycling expense.
The first type of constraints is on material flow balance of network. It readsThe second type of constraints is on capacities of collection centers, dismantlers, shredders, and landfills. Owing to randomness of capacities, it says that the following stochastic inequalities should be satisfied:The third type of constraints is on the limited demands in secondary markets for the used components. Since the demands are random, the following stochastic inequalities hold:The last type of constraints is on nonnegativity of decision variables and binary variables:Consequently, we obtain a polymorphic uncertain optimization model (PUOM) for the management problem of the ELV recovery system:
Remark 2. Note that in (20)-(23), the capacities for the collection centers, dismantlers, shredders, secondary markets, and landfill are supposed to be random. This assumption is based on the given probability distribution inferred from statistical data of capacity, especially after long-term practical production. In , the capacity of sorting entities for recycling the ELVs was also assumed to be random.
Remark 3. Compared with the deterministic model built in , PUOM (28) takes into account the fuzziness of cost and selling price, as well as the randomness of capacity and demand. Instead of simply describing the unit cost and selling price by interval parameters as done in , we characterize them by fuzzy sets which can contain more uncertain information, especially those perceptual evaluations of decision-makers in practice. Consequently, the proposed PUOM (28) in this paper is a polymorphic uncertain programming problem.
Remark 4. Since any uncertain optimization problem has no any optimal solution from the viewpoint of standard optimization theory, we have to make an optimal decision by a compromising programming approach, as done in . Specifically, chance-constrained and multiobjective optimization approach will be first proposed to transform the polymorphic uncertain optimization problem into a deterministic equivalent formulation in Section 3.1. Then, an interactive algorithm will be developed to find a compromising solution for the original uncertain optimization model on the basis of analytic tools and the existing powerful algorithms in the classical optimization theory.
3. Unified Compromising Programming Approach and an Interactive Algorithm
In this section, we intend to develop a unified compromising programming approach to treat the PUOM (28).
Our basic idea can be stated as follows. By chance-constrained programming approach, we first hedge randomness of constraints in PUOM (28) such that the obtained model is only involved with fuzzy parameters. Then, we construct an auxiliary multiple-objective optimization problem (MOOP) to convert the fuzzy model into a mixed integer linear programming model. Finally, we develop an interactive algorithm to find the compromising solution of the MOOP.
3.1. Unified Compromising Programming Approach
For simplicity of statement, we now only consider a linear stochastic constraintwhere is a vector of decision variables, is given crisp vector of coefficients, and is a random variable with probability distribution function .
Based on the idea of chance-constrained programming approach, equivalent deterministic formulation of (29) can be obtained (see, for example, [22–24, 28, 29]). Specifically, for a given confidence level (, (29) is equivalent to that the following inequality:always holds for any , where represents the probability that a stochastic inequality holds. The constraint (30) can be replaced byThus, by definition of probability distribution function, (31) can be approximated by an ordinary crisp constraint:for a given violation degree . Particularly, the random parameter is subject to normal distribution, i.e., , where and are the mean and standard deviation, respectively. Let express the standard normal distribution function. Then, (32) readsLet , , , , , and be the probability distribution functions of the random variables , , , , , and , respectively. From (32), it follows that, for a given violation degree , the constraints (20)-(25) are replaced byIf all the probability distributions are normal, then from (33), we know that (20)-(25) can be further rewritten as
We are now in a position to treat the fuzzy objective function in PUOM (28).
Since it is assumed that all the fuzzy parameters in are subject to trapezoidal membership functions, is a trapezoidal fuzzy set as a sum of trapezoidal fuzzy ones (see [25, 30, 31]). Denote . Instead of minimizing as in [19, 32, 33], we replace minimization of a fuzzy objective by minimizing an integrated crisp objective function, defined by the membership functions of fuzzified , , and (see model (57). To minimize the total cost and capture the uncertain (fuzzy) information in the total cost function, we first need to construct an auxiliary multiobjective optimization problem as follows:whereandClearly, the first objective in (46) is to minimize the total cost and the last two objectives are used to capture the uncertain information in the total cost as much as possible. In addition, if (34)-(39) in model (46) are replaced by (40)-(45), then we get a special case of model (46) under assumption of normal distribution.
Remark 5. With the proposed unified compromising programming approach, the randomness of constraints in PUOM (28) first vanishes by chance-constrained programming method. It is possible that the feasible region of PUOM (28) is empty if the choice of violation degree is very small . In other words, a suitable choice of violation degree is important to the unified compromising programming approach, which will be further addressed in Section 4. It is also noted that the constraint of PUOM (28) is involved with the inverse of distribution function. So, there does not exist an explicit expression of such a constraint in the case that it is difficult to calculate the inverse of distribution function. Consequently, many powerful algorithms in standard optimization theory can not be applied to solve PUOM (28). Under assumption of normal distribution, explicit expressions of the constraints can be obtained (see the case study in Section 4).
Remark 6. Similar to [25, 31], we construct an auxiliary multiobjective optimization model (46) to hedge the fuzziness of the objective function in (28). Intuitively, the reason why we transform a fuzzy objective function into a three-objective one is that it can push the trapezoidal possibility distribution of the fuzzy total cost to the left as far as possible (see Figure 3).
Actually, the first objective in (46) implies the total cost, especially the part with degree of membership 1, is to be minimized by minimizing the center value. The third objective in (46) can minimize the area of the region (II) in Figure 3 so as to reduce the risk of higher cost, while the second objective in (46) can maximize the area of region (I) in Figure 3 so as to increase the possibility of obtaining lower cost. Clearly, the crisp model (46) is a deterministic equivalent formulation of the PUOM (28). To the best of our knowledge, it is the first time that the management problem of ELV recovery is treated by this approach in a polymorphic uncertain environment. For example, an expectation method was recently presented to convert the fuzzy model into a crisp one in .
3.2. Interactive Algorithm
Step 1. Choose an acceptable violation degree for the random constraints. Solve the following three mixed integer linear programming problems (MILP): Denote , , and by the optimal solutions of (50), (51), and (52), respectively. The corresponding optimal values of the three objective functions are referred to as , , and , respectively. ComputeClearly, and are the upper and lower bounds of the -th objective function, .
Step 2. With and , formulate a trapezoidal membership function for the objective function . Specifically, we fuzzify and and by specifying their membership functions (also see Figures 4(a) and 4(b)):andrespectively.
Step 3. The decision-maker decides a compensation coefficient and a weight for the -th objective function, . Then, an integrated model is defined bywhere is an auxiliary variable.
Step 4. Select a violation degree as small as possible to ensure solution existence of the deterministic model (57).
Step 5. Solve model (57). Denote and the optimal solution of (57). Clearly, corresponding to , the satisfaction degree of the three objective functions at least attains .
Step 6. If the decision-maker is satisfied with this current compromising solution, go to Step 7. Otherwise, update the value of compensation coefficient by , where . Return to Step 5.
Step 7. Choose a smaller . If the feasible region of (57) is empty for , then the algorithm stops and the decision-maker determines the most satisfactory , and amongst all of the previous numerical results. Otherwise, return to Step 5.
(a) Membership function for
(b) Membership function for
Remark 7. In Step 2, the three membership functions defined by (54), (55), and (56) are degenerate trapezoidal functions. Since our aim is to minimize , smaller values of than are always preferable. Thus, it is nature to make the degree of membership equal to 1 for any value of less than . In this case, the left side of the membership function is degenerate. The similar reason can be used to explain the definitions of and .
Remark 8. In Step 3, as done in , a fuzzy programming approach is employed to convert a multiobjective optimization problem into a single-objective problem such that an interactive algorithm is developed to find a compromising solution of the PUOM (28). Clearly, by maximizing the integrated degree of membership in model (57), we obtain a compromising solution such that both the total cost and its variance (the decision-making risk) become as smaller as possible. Actually, the first term in the objective function of model (57) describes the lower bound of the membership degree of all the three fuzzy objectives; greater lower bound means higher degree of overall satisfaction, with which the three objectives are simultaneously improved. The second term in the objective function of model (57) makes a trade-off between the overall satisfaction and the individual one.
Remark 9. In , an interactive algorithm was developed to solve a problem of purchasing, production, and distribution, which is involved in multiple suppliers, one manufacturer, and multiple distribution centers. It is shown that this algorithm always generates unbalanced and balanced solutions based on the decision-maker’s preferences. Similar to , we develop Algorithm 1 to find a compromising solution of the polymorphic uncertain recycling problems of ELVs in this paper. Numerical experiments in Sections 4 and 5 will show that Algorithm 1 can reveal more valuable managerial insights from the proposed model (2) than the solution methods available in the literature. Especially, by our algorithm, we can show how to choose an optimal compensation coefficient and a violation degree (see Section 4).
4. Case Study with Different Algorithmic Parameters
4.1. Case Description
According to , China has built a number of ELV recycling enterprises. In this case study, we attempt to deal with the end-of-life car recovery management problem of the five key cities (Changsha, Zhuzhou, Xiangtan, Hengyang, and Shaoyang) in Hunan (see Figure 6), considering that the data can be collected completely.
(a) The map of Hunan province
(b) Recovery network of five key cities
For convenience, the centers of the five cities are regarded as the ELV sources, and all of the relevant collecting enters, dismantlers, shredders, landfills, secondary markets, and recycling factories are distributed in these cities with given locations (see Figure 6(b) and Tables 1–6). It is noted that, for some cities, there are more than one dismantler, secondary market, oil, or glass factory, or there is no any landfill, secondary market, rubber, or plastics factory in practice.
A part of available data on unit costs of transportation and processing, selling price and material weight percentages of ELVs are deregistered from  (see Tables 7–11). For simplifying this case study, we only consider a single-period recycling problem. Referring to the government data on scrapped vehicles , the amount of ELVs from the five sources are estimated byThe aim of this case study is to answer the following questions: