Introduction

A supply chain’s design is one of the important strategic decisions that influence competitiveness and economic growth [4]. The improper design of a supply chain may reduce business performance and significantly influences its competitiveness advantage [22, 33].

Generally, a supply chain is a network of organizations that produces products and services with a specific value and delivers them to consumers through different processes and activities [2]. It is also known as a forward supply chain (FSC), which focuses on filling customer needs. Three functions mainly describe the supply chain activities,supply of material, conversion of material into intermediate and finished products, and allocation of finished products to the end of the customer [5, 18]. However, the increasing population, shortage of available resources, and other factors related to environmental impact; such as pollution, shortage of lands, and environmental policies of material disposal, have forced organizations to improve the FSC to a chain that considers these factors; or so-called closed-loop supply chains (CLSC). A typical CLSC consists of two parts: forward flow and reverse flow, which focuses on the end-of-life (EOL) products. Further, managing the EOL products and turning them into value-added products by continuing their useful life through related activities is called the Reverse Logistics Network (RLN) [9]). The RLN operations include collecting returned products, shipping, sorting, recovering energy, recycling, reusing, repairing, and also landfilling activities [3, 8].

Nowadays, there is a critical need for evolving toward sustainable operations across the supply chains. The interactions between economic, environmental, and social dimensions of improvement is referred to as sustainability [7, 31]. Worldwide, the supply chain plays an important and major role in moving toward more sustainable practices to minimize negative environmental impacts in the near time and in future. This led to the emergence of the well-known “3R” concept, which aims at reducing waste and reusing resources, and recycling value-added materials and products [22, 31]. Also, this resulted in increasing interest and investment in RLNs [6, 21, 27]. The hierarchy of RLN includes: recovery processes by reusing, refurbishing, and resale, and reprocessing processes by repairing, remanufacturing, recycling, and landfilling. From operational and competitive views, returned products from industries especially retailers increased significantly [12]. Economic savings can be found when the end-of-use products can be reused or recycled [12, 14].

In Jordan, millions of metric tons of waste are generated from agricultural, municipal, and industrial sources every year [1]. The high-growing population and industrialization have led to a rapid increase in solid waste generation in the country which put increasing pressure on the existing waste management infrastructure. The existing Solid Waste Management services in Jordan provided by local governments are no longer up to par with those provided before the enormous influx of refugees, and the daily creation rate of solid waste has increased substantially. However, currently, only 5–10 percent of Jordan's solid waste is recycled, due to the lack of a large-scale and effective government-run management solid waste sorting or recycling system [24]. Although the recovering processes are not limited to recycling processes, they extend to re-use, remanufacturing, and energy recovery activities. In Jordan, the focus is only on recycling activities under pilot projects supported by private sectors. Little efforts have been directed to remanufacturing and energy recovery activities despite their benefits in three dimensions,economic, environmental, and social. Further, the activities of an RLN are associated with random, and completely uncertain events; including demand, return quantity of products, costs, and interchange rates [31]. Two-stage stochastic modeling has been reported as an efficient approach to dealing with uncertainty. It uses two types of decision variables; the first type is fixed before observing the uncertain outcome and is called the first stage decision variables. While the second type is released only after the realization of the randomness and is known as the second-stage decision variables or recourse actions [13].

In these regards, this research develops a two-stage stochastic mathematical model that considers three recovery activities; recycling, remanufacturing, and energy recovery. The model seeks to maximize profit, maximize social impact, and minimize carbon emissions to achieve sustainability under uncertainty in the collected product, quality of collected products, and selling price. The remainder of this study is structured as follows. Section Two reviews relevant studies on RLN. Section Three develops the optimization model. Section Four applies the developed model on a real case study and conducts sensitivity analyses. Finally, Section Five summarizes conclusions and future research.

Literature review

Operational and mathematical optimization have extremely focused on the RLN design and implementation. For example, Lee and Dong [16] developed a mathematical model with a multi-period RLN with a two-stage stochastic programming. Demand was considered as an uncertain parameter with a single objective function that focused on minimizing cost. Kara and Onut [25] developed a mathematical model to determine optimal locations and the quantity of flow for a long-term strategy under uncertain events, for a paper recycling company. The proposed model was formulated by considering two methods of programming; two-stage stochastic mixed-integer and robust programming approach. Piplani and Saraswat [28] proposed a mathematical model in the form of mixed-integer linear programming which aimed to design a service network for electronic products. The model focused on minimizing the total cost with flow balance and logical constraints under uncertainty in returned products. Ayvaz et al. [9] developed a generic multi-echelon and multi-product with a capacity-constrained two-stage stochastic programming model to maximize profit for third-party waste of electrical and electronic equipment recycling companies. Capraz et al. [15] adopted a mixed-integer linear programming to determine the maximum bid price offer while determining the best operation planning strategy in a real-life case study. Fattahi and Govindan [18] developed a model to maximize profit with a multi-period integrated forward/reverse logistics network design problem under uncertain demand and quantity of returns. Ayvaz et al. [9] proposed a mixed-integer linear programming to minimize the total cost of designing EOL vehicles recycling network. The model determined the quantity of material shipped between the facilities and to decide whether to open the dismantling and shredding. The model determined the optimal number of locations and facilities, tested the impact of incremental changes of waste, and conducted a sensitivity analysis on the optimal RLN. Entezaminia et al. [17] proposed a mathematical model for a multi-product multi-period multi-site aggregate production planning problem in a green supply chain while taking into account a RLN under uncertainty. The objective function was to minimize the total cost of the supply chain including transportation cost, purchasing cost, production cost, collection cost, recycling cost, workforce cost, inventory cost, and shortage cost. Yu and Solvang [33] developed a multi-product and multi-echelon sustainable reverse logistics with a two-stage stochastic bi-objective mixed-integer programming model for the network design problem under uncertainty. A set of Pareto solutions between economic issues and environmental performance were found. Trochu et al. [31] proposed a mixed-integer linear programming model to analyze the direct impact of supply locations, the available quantity at the collection centers on RL network design. The objective functions were to minimize the total cost under flexible capacity. Baptista et al. [11] proposed a two-stage stochastic mixed 0–1 optimization model under uncertain parameters,such as, transportation cost, product demand, return volume, and returned product quality. The model sought to maximize the net present value of the expected total profit along the time horizon. Tosarkani et al. [30] developed a novel scenario-based robust possibilistic approach to optimize and configure an electronic reverse logistics network by considering the uncertainty associated with costs, the quantity of demand and return, and the quality of return products. The mathematical model was extended to multi-objective optimization by maximizing the environmental compliance of the third parties. Gholizadeh and Fazlollahtabar [19] proposed a robust optimization approach and embedded with a modified genetic algorithm in a reverse flow to maximize the total profit of a melting process under uncertainty. Reddy et al. [29] developed an Improved Benders Decomposition to solve a mixed-integer linear programming model that minimizes the total cost while comprising the carbon emission cost due to transport and production at the facilities in green RLN. They also investigated the effects of carbon emissions and the choice of the vehicle fleet on the network's structure. The proposed heuristic was applied on a set of 12 problem configurations under distinct scenarios. Balci et al. [10] designed a multi-purpose RLN that is used to create effectual management of medical waste generated in 39 districts of Istanbul. The model attempts to integrate economic, environmental, and social objectives within the sustainable development goals. It aimed to maximize the number of personnel and government earnings for the estimated MW of a megacity while minimizing the total fixed cost and the cost of carbon emissions and transportation. Sensitivity analyses showed that the amount of MW has the most significant effect on the total cost. Wang et al. [32] examined a collaborative multicenter vehicle routing problem with time windows and dynamic customer demands. They proposed a multi-objective mathematical programming model to coordinate static and dynamic customer demands while minimizing the total operating cost, the number of vehicles, and the total waiting time. A hybrid heuristic algorithm, comprising an improved clustering algorithm and a non-dominated genetic algorithm, was employed to solve the developed model. A case study of a realistic RLN in Chongqing City, China was provided for model illustration. Govindan et al. [23] presented a multi-item, multi-period, and bi-objective mixed-integer linear programming circular economy transition model for medical waste management considering the uncertainty in the amount of waste generated to design a green RLN. The queuing theory was adopted to manage waiting time of trucks carrying infectious waste in treatment centers. In addition, a stochastic scenario-based approach was employed to deal with uncertainty of the waste generated. The model efficiency was examined on a case study consisted of six hospitals, three potential collection centers, three potential treatment centers, four potential recycling centers, and three potential disposal centers in Alborz province. Gholizadeh et al. [20] used three heuristics; cross-entropy, genetic algorithm, and simulated annealing, the response surface method, and the Taguchi method to model a sustainable RLN for polystyrene disposable appliances with uncertain demand and cost of recovery of the used products. A real case study was adopted to validate the model. Karagoz et al. [26] proposed a scenario-based real-life stochastic optimization model to improve end-of-life vehicles (ELVs) supply chain network management in Istanbul the automotive industry under various uncertainties and changing parameters of the recycling process, like the number of vehicles withdrawn per year, cost items, and material composition.

However, little research efforts have considered the sustainable environmental and social issues in multi-objective RLN design under uncertainty. This research, therefore, aims at formulating a two-stage stochastic mixed integer quadratic programming model with multi-objectives to balance economic benefits and environmental and social measures. The developed model will consider three objective functions; maximizing profit, maximizing social impact, and minimizing environmental impact under uncertainty in the collected product quantity, quality of products, and selling price.

Model Development

The proposed RLN structure is shown in Fig. 1, which is comprised of C customer zones for P EOL products, I central collection center, M remanufacturing centers, R recycling centers, Y energy recovery centers, and L disposal for non-recyclable products. The EOL products are collected at the central collection centers, which are close to customers. Then, the locally-collected EOL products are delivered to the central inspection centers for inspection, sorting, and disassembly. The recoverable products will be sent between different facilities for energy recovery, remanufacture, and recycling centers. However, the non-recoverable products will be disposed. The following assumption are made:

  1. 1.

    Locations of customer zones, collection centers, recycling centers, remanufacturing centers, energy recovery centers, and landfills are known.

  2. 2.

    All costs including processing cost, transportation cost between entities of considered RLN, and opening cost at each facility are known in advance.

  3. 3.

    Carbon emissions associated with facility operation and the transportation of returned products are known.

Fig. 1
figure 1

The proposed RLN

Full size image

The notations used in the proposed models are listed in Table 1.

Table 1 Notations of MIQP model
Full size table

The model parameters involve: (1) Price of the products or energy generated from recovering one unit of product p at facility m, y, and r in scenario s; (2) Fixed operating costs for opening a facility at candidate locations; (3) Processing costs for treating one unit of product p at facility m, y, and r; (4) Transportation cost for shipping one unit of product p between different facilities within the reverse logistics system; (5) CO2 emissions for treating one unit of product p at facility m, y, and r; (6) CO2 emissions for landfilling one unit of product p at facility l;(7) CO2 emissions from transportation of one unit of product p between different facilities within the reverse logistics system; (8) Amount of collected p product at customer c in scenario s; (9) Capacity for dealing with product p at facility m, y, and r; (10) Required rate of utilization for treating product p at facility m, y, and r; (11) Fraction of product p suitable for remanufacturing, recycling and energy recovery; (12) Quality level of product p in scenario s; (13) Number of job opportunities created at facility m, y, and r; and (14) Average working days lost due to workplace injuries at facility m, y, and r per treated one unit of product p.

Model objectives

  1. (1)

    The first objective function aims to maximize the total profits of RLN, which is the difference between income and costs. Let Ymmps, Yyyps, Yrrps denote quantity of product p recovered at remanufacturing centers m, energy recovery center y and recycling center r, respectively. Let SPMmps, SPRrps, SPYyps denote the selling prices from recovering one tone of product p at remanufacturing center m, recycling center r, and energy recovery centers y, respectively, in scenario s. The profit is then calculated as stated in Eq. (1).

    $$Profit=Revenues-Cost$$
    (1)

    where revenues are formulated as:

    $$Revenues={\sum_{m=1}^m}{\sum_{p=1}^P}{\sum_{s=1}^S}{SPM}_{mps}{.Ym}_{mps}+{\sum_{r=1}^r}{\sum_{p=1}^P}{\sum_{s=1}^S}{SPR}_{rps}{.Yr}_{rps}+{\sum_{y=1}^y}{\sum_{p=1}^P}{\sum_{s=1}^S}{SPY}_{yps}.{Yy}_{yps}$$
    (2)

    Let FIi, FMm, FYy, FRr denote the fixed opening costs of collection center i, remanufacturing center m, energy recovery center y, and recycling center r, respectively. Let PCIip, PCMmp, PCYyp, PCRrp represent the processing costs of recovering one tone of product p at facility i, m, y and r, respectively. Let SCCpci denotes the shipping cost between customer zone c and collection center i. Let SCMpim, SCYpiy, SCRpir, and SCLpil denote the shipping costs between collection center i and remanufacturing center m, energy recovery center y, recycling center r and landfill site l, respectively. Let Xii, Xmm, Xyy, and Xrr represent the binary decision variables that decide whether a new facility i, m, y, and r will be opened at remanufacturing centers m, energy recovery center y and recycling center r, respectively. Let YtCpcis represents the quantity of product p shipped between customer zone c and collection center i in scenario s, Ytmpims represents the quantity of product p shipped between collection center i and remanufacturing center m in scenario s, Ytrpirs denotes the quantity of product p shipped between collection centers i and recycling center r in scenario s, Ytypiys denotes the quantity of product p shipped between collection centers i and energy recovery center y in scenario s, and Ytlpils denotes the quantity of product p shipped between collection center i and landfill site l in scenario s. Then, the cost is obtained using Eq. (3).

    $$\boldsymbol C\boldsymbol o\boldsymbol s\boldsymbol t={\sum_{i\in1}^l}{FI}_i{.Xi}_i+{\sum_{m=1}^M}{FM}_m.{Xm}_m+{\sum_{r=1}^R}{FR}_r{.Xr}_r+{\sum_{i=1}^l}{\sum_{p=1}^P}{\sum_{s=1}^S}{PCI}_{ip}{.Yi}_{ips}+{\sum_{r=1}^R}{\sum_{p=1}^P}{\sum_{s=1}^S}{PCR}_{rp}{.Yr}_{rps}+{\sum_{m=1}^M}{\sum_{p=1}^P}{\sum_{s=1}^S}{PCM}_{mp}.{Ym}_{mps}+{\sum_{y=1}^Y}{\sum_{p=1}^P}{\sum_{s=1}^S}{PCY}_{yp}{.Yy}_{yps}+{\sum_{p=1}^P}{\sum_{c=1}^C}{\sum_{i=1}^I}{\sum_{s=1}^S}{Scc}_{pci}{.Ytc}_{pcis}+{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{m=1}^M}{\sum_{s=1}^S}{Scm}_{pim}{.Ytm}_{pims}+{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{y=1}^Y}{\sum_{s=1}^S}{Scy}_{piy}{.Yty}_{piys}+{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{r=1}^R}{\sum_{s=1}^S}{Scr}_{pir}{.Ytr}_{pirs}+{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{l=1}^L}{\sum_{s=1}^S}{Scl}_{pil}{.Ytl}_{pils}$$
    (3)
  2. (2)

    The second objective function minimizes the carbon emissions, which is determined by recovering products at each facility and shipping products between facilities. Let Emmp, Errp, and Eyyp denote the CO2 emissions from recovering one tone of product p at remanufacturing center m, recycling center r, energy recovery y, respectively. Let Etcpci represents the amount of CO2 emissions from shipping product p between customer zone c and collection center i. Let Etmpim, Etrpir, Etypiy, and Etlpil denote the amount of CO2 emissions from shipping product p between collection center i and remanufacturing center m, recycling center r, energy recovery center y, and landfill site l. Then, the second objective function is formulated as:

    $$\boldsymbol M\boldsymbol i\boldsymbol n\boldsymbol\;\mathbf C\mathbf O2\;\mathbf e\mathbf m\mathbf i\mathbf s\mathbf s\mathbf i\mathbf o\mathbf n\mathbf s={\sum_{s=1}^S}Ps({\sum_{m=1}^M}{\sum_{p=1}^P}{ Em_{mp}}{.Ym_{mps}}{+}{\sum_{r=1}^R}{\sum_{p=1}^P}Er_{\mathit r\mathit p}{.Yr_{rps}}{+}{\;}{\sum_{y=1}^Y}{\sum_{p=1}^P}{ Ey_{yp}}{.Yy_{yps}}{+}{\sum_{l=1}^L}{\sum_{p=1}^P}El_{\mathit L\mathit p}{{.Yl}_{Lps}}{\sum_{p=1}^P}{\sum_{c=1}^C}{\sum_{i=1}^I}Etc_{\mathit p\mathit c\mathit i}\mathit{.YTc_{pcis}}{+}{\;}{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{m=1}^M}ETm_{\mathit p\mathit i\mathit m}\mathit{.}\mathit{ Ytm_{pims}}{+}{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{r=1}^R}Etr_{\mathit p\mathit i\mathit r}\mathit{.Ytr_{pirs}}\mathit{+}{\sum_{p=1}^P}{\sum_{i=1}^I}{\sum_{y=1}^Y}{\mathrm{Ety}}_{\mathrm{piy}}{.{\mathrm{Yty}}_{\mathrm{piys}}}{+}{\;}\sum\nolimits_{p=1}^P{\sum_{i=1}^I}{\sum_{l=1}^L}Etl_{\mathit p\mathit i\mathit l}\mathit{.Ytl_{pils}})$$
    (4)
  3. (3)

    The third objective function seeks to maximize the social impact of reverse logistics activities, which is evaluated by number of job opportunities created, average working days lost due to workplace injuries, and service level. Let Jmm, Jrr,and Jyy represent the number of job opportunities created at facility m, r, and y, respectively. Let Wmmp, Wrrp, and Wyyp be the average working days lost due to workplace injuries at at facility m, r, and y, respectively, per treated one tone of product p. Let Ytcpci represents the amount of product collected from customer zones c to collection center i (Service Level). The third objective function is then expressed as given in Eq. (5).

    $$\boldsymbol M\boldsymbol a\boldsymbol x\boldsymbol\;\mathbf s\mathbf o\mathbf c\mathbf i\mathbf a\mathbf l\boldsymbol\;\mathbf i\mathbf m\mathbf p\mathbf a\mathbf c\mathbf t=\left({\sum_{m=1}^M}{Jm}_m{Locm}_m+{\sum_{r=1}^R}{Jr}_r{Locr}_r+{\sum_{y=1}^Y}{Jy}_y{Locy}_y\right)+\sum\nolimits_{s\in S}{\sum_{p=1}^P}{\sum_{c=1}^C}{\sum_{i=1}^I}{Ps\;Ytc}_{pci}-\sum\nolimits_{s\in S}Ps\left({\sum_{m=1}^M}{\sum_{p=1}^P}{Wm}_{mp}{Qnm}_{mps}+{\sum_{r=1}^R}{\sum_{p=1}^P}{Wr}_{rp}{Qnr}_{rps}+{\sum_{y=1}^Y}{\sum_{p=1}^P}{Wy}_{yp}{Qny}_{yps}\right)$$
    (5)

Model constraints

The objective functions are subjected to the following constraints:

  • Demand satisfaction: Let dpcps denotes the amount of collected product p at customer zones c in scenario s. Constraint (4) guarantees that the quantity of the collected product are shipped to the inspection and meet the customer demand.

    $$\begin{array}{cc}\sum_{i=1}^IYtc_{\;pcis}=dp_{\;cps}&\forall S\;\in S\;d\in D,\;p\in P\end{array}$$
    (6)

  • Flow Balance: Constraints (7–12) guarantee the balance of flows at each facility and each route.

    $$\sum_{c=1}^C\begin{array}{cc}Ytc_{\;pci}=YI_{pis}&\forall S\in Si\in I,\;p\in P\end{array}$$
    (7)
    $$\begin{array}{cc}\sum_{i=1}^IYtr_{\;pims}=Ym_{pms}&\forall S\in S\;m\in M,\;p\in P\end{array}$$
    (8)
    $$\begin{array}{cc}\sum_{i=1}^IYtr_{\;pirs}=Yr_{prs}&\forall S\in S\;r\in R,\;p\in P\end{array}$$
    (9)
    $$\begin{array}{cc}\sum_{i=1}^IYty_{\;piys}=Yy_{pys}&\forall S\in S\;y\in Y,\;p\in P\end{array}$$
    (10)
    $$\begin{array}{cc}\sum_{i=1}^IYtl_{\;pils}=YI_{pis}&\forall S\in S\;l\in L,\;p\in P\end{array}$$
    (11)
    $$\begin{array}{cc}YI_{\;Ips}={\sum_{m=1}^M}Ytm_{\;pims}+{\sum_{r=1}^R}Ytr_{pirs}+{\sum_{y=1}^Y}{ Y}{ t}{{\scriptstyle y}_{piys}}{+}{\sum_{l=1}^L}{ Y}{ t}{{\scriptstyle l}_{pils}}\,&\forall s\in S,\;i\in I,\;p\in P\end{array}$$
    (12)

  • Capacity constraints: Let Cmpm, Crpr, Cipi,, and Cypy denote the capacity for product p at remanufacturing center m, recycling center r, collection centers i and energy recovery center y, respectively. Inequalities (13–15) guarantee the capacity for each facility with respect to each type of product will not be exceeded at scenario s.

    $$\begin{array}{cc}Yi_{pis}\leq Cl_{pi}.XI_i&\forall S\in S,\;i\in I,\;p\in P\end{array}$$
    (13)
    $$\begin{array}{cc}Ym_{pms}\leq CM_{pm}.XM_m&\forall S\in S,\;m\in M,\;p\in P\end{array}$$
    (14)
    $$\begin{array}{cc}Yr_{prs}\leq CR_{pr}.XR_r&\forall S\in S,\;r\in R,\;p\in P\end{array}$$
    (15)
    $$\begin{array}{cc}Yy_{py}\leq CY_{py}.XY_y&\forall S\in S,\;yg\in Y,\;p\in P\end{array}$$
    (16)

  • Utilization: Let Uiip, UMmp, URrp, UYyp denote the utilization rates for recovering product p at collection centers i, remanufacturing center m, recycling center r, and energy recovery centers y, respectively. Inequalities (17–20) guarantee a minimum levels of utilization at the facilities to avoid inefficient use of facilities.

    $$\begin{array}{cc}Yi_{ips}\geq Ui_{pi}.Ci_{pi}.Xi_i&\forall S\in S,\;i\in I,\;p\in P\end{array}$$
    (17)
    $$\begin{array}{cc}Ym_{pms}\geq Um_{pm}.Cm_{pm}.Xm_m&\forall S\in S,\;i\in I,\;p\in P\end{array}$$
    (18)
    $$\begin{array}{cc}Yr_{prs}\geq Ur_{pr}.Cr_{pr}.Xr_r&\forall S\in S,\;i\in I,\;p\in P\end{array}$$
    (19)
    $$\begin{array}{cc}Yy_{pys}\geq Uy_{py}.Cy_{py}.Xy_y&\forall S\in S,\;i\in I,\;p\in P\end{array}$$
    (20)

  • Conversion: Let Frmpm, Frrpr, and Frypy represent the fractions of product p appropriate for remanufacturing center m, recycling center r, and energy recovery center y, respectively. Constraints (21–24) guarantee that the percentage of end-of-use products delivered for recycling r, energy recovery y, remanufacturing center m and landfill site l comply with the quality and proportion requirements.

    $$\begin{array}{cc}\sum_{i=1}^IYtm_{\;pims}\leq QI_{sp}.Frm_{pm}.YI_{ips}&\forall s\in S\;i\in I,\;p\in P\end{array}$$
    (21)
    $$\begin{array}{cc}\sum_{i=1}^IYty_{\;piys}\leq QI_{sp}.Fry_{py}.YI_{ips}&\forall s\in S\;i\in I,\;p\in P\end{array}$$
    (22)
    $$\begin{array}{cc}\sum_{i=1}^IYtr_{\;pirs}\leq QI_{sp}.Frr_{pr}.YI_{ips}&\forall s\in S\;i\in I,\;p\in P\end{array}$$
    (23)
    $$\begin{array}{cc}{\sum_{i=1}^I}Ytl_{\;pils}\leq\left(1-QI_{sp}\right).YI_{ips}&\forall s\in S\;i\in I,\;p\in P\end{array}$$
    (24)

Constraints (25) defines decision variables of opening the facilities as binary variables each has only two possible values 0 or 1.

$$\begin{array}{cc}Xr_r,Xi_i,Xm_m,Xy_y\;\in\left\{0,1\right\}&\forall m\in M,\;i\in I,\;r\in R,\;y\in Y\end{array}$$
(25)

Constraints (26) guarantee the non-negativity variables.

$$\begin{array}{ccc}Yi_{ips},Ym_{mps},Yr_{rp},Y_{yps},Yr_{rps},Ytc_{pcis},Ytm_{pims},Ytrr_{pirs},Yty_{piys}\;\geq\;0&\forall s\;m\in M,\;i\in I,\;r&\in R,\;y\in Y,\;s\in S\end{array}$$
(26)

By solving the model, the obtained result would support decision-makers in determining which facility has to be opened at maximum profit, maximum social impact, and minimum CO2 emissions. The solution of the model is classified into two types of decision variables: decision variables of stage 0 and decision variables of the second stage under all scenarios regarding the possible values of the random variables. In stage 0, a decision of establishing the facilities is taken. In other words, the values of \(X_{rr},Xi_i,Xm_m,Xy_y\) are determined. In the following stage, the quantity of the products at each facility Yiips, Ymmps, Yrrps, Yyyps and the quantity of product shipped between facilities Ytcpcis, Ytmpims, Ytrpirs, Ytypiys, and Ytlpils are determined under each scenario.

Application of the proposed model

The model was implemented in the redesign of reverse logistics for a recycling service company in Jordan. The current reverse logistics is composed of 10 customer zones to collect end-of-use products, two collection centers to inspect, sort, and disassemble, and four recycling centers. The disassembled products with high quality will be moved to one of the recycling centers and the products with low quality will be sent to the landfill. Remanufacturing center deals with p3; metal, and recycling centers deal with p1 and p2; paper and plastic products, respectively. Due to the fact that millions of tons of waste are generated in Jordan yearly from different sectors, in addition to the growing industrialization and high population growth rate increases, there is a need to design sustainable RLN with more than recoverable options, such as remanufacturing, energy recovery, and recycling processes. The proposed RLN includes the current RLN with new remanufacturing and energy recovery centers as shown in Fig. 2. The proposed RLN is composed of 10 customer zones (C = 10), 2 collection centers (I = 2), 4 recycling centers (R = 4), 3 remanufacturing centers (M = 3), 2 energy recovery centers(Y = 2) and 2 landfill sites (L = 2).

Fig. 2
figure 2

RLN design

Full size image

The collected data for the existing RLN are obtained from company records, while the data regarding the remanufacturing centers and energy recovery centers are estimated from a paper presented by [33]. The parameters related to RLN are presented in Tables 2 and 3. The potential locations of facilities are presented in Fig. 3.

Table 2 parameters related RLN
Full size table
Table 3 Transportation cost and CO2 emissions
Full size table
Fig. 3
figure 3

Potential locations of RLN

Full size image

The maximum capacity of the trucks used to transfer different components between the facilities is 5 tons of fuel consumption about 0.18 Liter/Km. The price of the fuel is 1.4$/ Liter and the distance between the facility are known. The amount of CO2 emissions is obtained from the website of NTM “Network for Transportation and Environment”. Further, the average transportation cost and emissions between two facilities are listed in Table 4.

Table 4 Uncertainty in the proposed model
Full size table

In the proposed model, the selling price of used products, quantity of products collected, and quality of these products are assumed to be stochastic (uncertain) variables. Price, quality, and amount of collected product are modeled by using the normal distribution as shown in Table 4.

Lingo 19.0 optimization package was used to code and solve the proposed MIQP model on a personal computer with Intel Core i5-9300H 2.4 GHz (8 CPUs) processor and 32 GB memory under Window 10 operating system. Four main steps were conducted to set up the stochastic model in Lingo. The first step was coding the programming model and set the parameter’s value. The second step was identifying the amounts of collected products, selling price of the recoverable products, and quality of returned products which were treated as random variables. The third step was defining the decision variables among stages. The final step was determining the distribution of the random variables and determining the sample size to generate the scenarios. Because the normal distribution generates infinite possible outcomes for the random variable assigned to, the Monte Carlo sampling technique was employed to generate scenarios of the uncertain variables. The sample size was set to 10; thus, 10 samples were generated at a single stage model and thereby the total number of scenarios is 101 = 10 scenarios. Figure 4 presents the flow of searching for the global optimal solution of the proposed model.

Fig. 4
figure 4

The flow of searching for the global optimal solution

Full size image

In Lingo, the RLN problem was classified as a mixed integer quadratic program with a total number of 10 scenarios, the total number of random variables of 50, and the total number of variables of 258, from which 11 variables were integer variables. The total number of constraints was 380, while the total number of non-zeros variables was 955. The total number of iterations that were used to obtain the global optimal solution was 6151. The decision variables of stage 0 that were decided to be opened to achieve the optimal solution, including the collection centers Xi1and Xi2, the remanufacturing center Xm3, the recycling center Xr3, and the energy recovery center Xy1. The related fixed cost of establishing these facilities is 725000$ and it is expected that 80 jobs opportunities will be created. The estimated objective functions at the optimal solution are presented in Table 5.

Table 5 Estimated objective functions under scenarios
Full size table

According to Table 5, scenarios 2, 3, and 8 are the best, moderate, and worst scenarios, respectively. Hence, these scenarios will be considered for further analysis. The values of the random variables under these scenarios are presented in Table 6. The optimal quantities transported between facilities are presented in Table 7, where the large amounts of products are shipped to energy recovery centers because the center deals with all the types of products, while remanufacturing center deals with p3; metal, and recycling centers deal with p1 and p2; paper and plastic. Therefore, the model identifies only two customer zones C2 and C5 due to the capacity limitation at centers. Figure 5 depicts the profit and cost for each scenario, where it is noted that profit decreases is mainly caused by the increase in processing cost and reducing processing cost is necessary to maintain the profit at a high level.

Table 6 Random variables under selected scenarios
Full size table
Table 7 Decision variable values under scenarios
Full size table
Fig. 5
figure 5

Profit vs Cost

Full size image

Figure 6 shows the carbon emissions for each scenario, in which the carbon emission depends mainly by transportation.

Fig. 6
figure 6

CO2 emissions in RLN

Full size image

Further, a comparisons of profit, establishing cost, transportation cost, processing cost, revenue, CO2 emissions, and job created were conducted between the proposed RLN and the existing RLN as presented in Table 8 and Fig. 7.

Table 8 Computational results for existing and improved RLN
Full size table
Fig. 7
figure 7

Comparison between existing and proposed RLNs

Full size image

From Table 8 and Fig. 7, it is noted that:

  • The profit of the improved RLN is greater than the existing RLN under all three scenarios. This increasing in profit is expected since the selling price for the remanufactured products and energy recovery are more than the recycled product as in the existing case.

  • The amount of product transported from customer zones to collection for the improved RLN is larger than it corresponding of existing RLN because the former deals with two types of products, paper, plastic, whereas the latter includes remanufacturing and energy recovery centers and deals with three types of products. paper, plastic, and metal.

  • CO2 emissions are affected mainly by transportation. The amount of CO2 emissions in the improved RLN is less than the existing RLN by about 43%, because the distance between facilities in the improved RLN is shorter than its corresponding of the existing RLN.

  • The transportation cost for the improved RLN is greater than the existing RLN; that is predictable because the service level (amount of product transported from customer zones to collection centers) is greater than the existing RLN. However, if transportation cost is determined per 1000 tons, the result will show that the transportation cost in the existing case will be greater since the distance between facilities is greater.

  • The number of jobs created in improved RLN is greater than the existing RLN by about 100%, because more centers will be added in improved RLN and therefore the number of jobs increases.

Sensitivity analysis

Sensitivity analysis of the important parameters of the model was conducted to examine their impact on the objective functions as shown in Table 9. It is found that as quality level (percentage of the recoverable product p) increases, the amount of recoverable products, profit, and revenue increase. It is worth mentioning that increasing the quality by 20% results in increasing profit by 40%. Moreover, job opportunities decrease with decreasing the quality level because the number of facilities that will be opened decreases.

Table 9 Sensitivity analysis on quality level
Full size table

Further, Table 10 shows sensitivity analysis by changing the capacity. It is noticed that at when capacity at each center increases, the amount of product that will be recoverable increases due to the fact that the centers can to recover more product. Furthermore, increasing the amount of recoverable products leads to increasing profit, revenue, processing cost, and transportation cost. However, the establishing cost remains the same with the increasing in the capacity because the number of opened facilities remains the same.

Table 10 Sensitivity analysis on capacity
Full size table

Conclusions

In this research, a two-stages stochastic programming model for sustainable reverse logistics was developed with multi-objectives and multi-products under uncertain selling price of recoverable products, amount of product collected, and quality level of the product collected. The model focused on economic issue represented by profit, environmental issue by determining CO2 emissions, and social issues by the number of created job opportunities and minimizing work-days lost due to injuries. Three processes were considered in the designed RLN; recycling, energy recovery and, remanufacturing processes with three types of products; paper, plastic, and metal. The model was implemented in the redesign of RLN for a recycling service company in Jordan. The obtained results indicated the feasibility of adding two recoverable options to improve sustainable RL performance. Compared to the existing RLN, the proposed RLN improved profit increased by four times, decreased the recoverable products by 37%. Furthermore, the amount of CO2 emissions was decreased whereas the social impact was increased. In conclusion, the proposed RLN model is found valuable and can provide great support to supply chain planners in the design of effective sustainable RLN that achieves company goals under emergent internal and external issues. Future research considers the adoption of block chain technology in the management and planning of RLN.