Sustainable Value Recovery of NdFeB Magnets: A Multi-Objective

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Sustainable Value Recovery of NdFeB Magnets: A Multi-Objective Network Design and Genetic Algorithm Hongyue Jin, Byung Duk Song, Yuehwern Yih, and John W. Sutherland ACS Sustainable Chem. Eng., Just Accepted Manuscript • DOI: 10.1021/ acssuschemeng.7b03933 • Publication Date (Web): 26 Feb 2018 Downloaded from http://pubs.acs.org on February 28, 2018

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ACS Sustainable Chemistry & Engineering

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Sustainable Value Recovery of NdFeB Magnets: A Multi-Objective Network

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Design and Genetic Algorithm

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Hongyue Jin1, Byung Duk Song1*, Yuehwern Yih1, John W. Sutherland2

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1

Industrial Engineering, Purdue University, 315 N. Grant Street, West Lafayette, IN 47907, USA

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2

Environmental and Ecological Engineering, Purdue University, 500 Central Drive, West Lafayette, IN

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47907, USA

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*

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Abstract

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Neodymium-iron-boron (NdFeB) magnets are widely used in clean energy applications such as wind

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turbines and electric vehicles whose demand is escalating. However, rare earth elements (REEs) for

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manufacturing NdFeB magnets are subject to significant supply uncertainty due to Chinese near-

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monopolistic supply. To mitigate the risk, companies are actively pursuing value recovery from end-of-

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life magnets. However, the questions of how to collect used magnets and smoothly transfer them through

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the reverse supply chain require further investigation. To address this challenge, this paper designs an

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efficient NdFeB magnet recovery infrastructure by identifying the optimal processing facility locations,

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and defining the capacities and transportation flows that maximize the economic and environmental

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benefits and social support for the new business. Mathematical models and a multi-objective network

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design genetic algorithm (MONDGA) were designed to calculate solutions. When compared with the

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exact solutions implemented by CPLEX (an optimization package), MONDGA provided (near) optimal

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solutions with significantly improved computation efficiencies. As the real world model application

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required a large-scale optimization, MONDGA was superior to CPLEX, which failed to provide any

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solution. The results confirmed that our proposed model and algorithm offer a promising strategy for the

Corresponding author. Tel.: 1- 765-637-8577. E-mail address: [email protected]

1 ACS Paragon Plus Environment

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NdFeB magnet recycling industry to enhance the economic and environmental sustainability and

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maximize social support.

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Keywords: rare earth elements, rare earth magnets, recycling, supply chain optimization, genetic

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algorithm.

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Introduction

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Rare earth elements (REEs) are critical materials for various applications, especially for creating

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neodymium-iron-boron (NdFeB) magnets. Clean energy products such as wind turbines and (hybrid)

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electric vehicles require these magnets, and the demand is ever-increasing.1 However, China is

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responsible for 80-95% of global REE supply, including the production of NdFeB alloys and powder.2

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The Chinese near-monopolistic supply on REEs and NdFeB magnets represents a potential supply risk for

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renewable energy technology manufacturers. The value recovery of REEs from end-of-life (EOL)

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products can help mitigate this supply rish.3–6

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Recently, several companies have begun to enter the NdFeB magnet recovery market to

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commercialize recycling technologies that increase REE or NdFeB magnet recovery from EOL products

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(e.g., Mitsubishi, Hitachi, and Urban Mining Company). However, the reverse supply chain logistics –

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collection of used magnets, dismantling, recycling/remanufacturing, and resale – have not been

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researched thoroughly. To enable success, research should focus on identifying the optimal processing

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facility locations and defining the capacities and transportation flows of the business. The associated

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decision criteria are multifaceted: economic, environmental, and social aspects all play key roles in a

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sustainable reverse supply chain. Therefore, this research aims to design an efficient value recovery

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infrastructure that maximizes the economic and environmental benefits and social support for NdFeB

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magnet recovery from EOL products with a special emphasis on the U.S. market.


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Reverse supply chain design has been studied since the early 1990s. Most of the efforts focused

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on single objective optimization7–12, while some researchers investigated bi-objective13–17 or multi-

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objective18,19 optimization. Some existing literature focused on industrial challenges including post-sale

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services, computer waste management, and bottle recycling as shown in Table 1. However, NdFeB

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magnet recycling was only studied recently, when a bi-objective optimization model for a NdFeB magnet

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reverse supply chain was proposed.20 Here, profits and environmental benefits were maximized by

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recycling the two major sources of EOL NdFeB magnets: computer hard disk drives (HDDs) and electric

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vehicle (EV) motors.3 A mixed integer linear programming (MILP) model and an epsilon-constraint

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method were developed to obtain the global optimal non-dominated solutions. Nonetheless, computation

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efficiency was a major challenge for this large-scale problem (i.e., multi-product, multi echelon, and multi

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period), which allowed only a few candidate locations as input parameters and still required a long

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computation time.

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Table 1. Representative literature on bi- and multi-objective optimization for reverse supply chain

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network design

Author (year)

Objectives Industry

Solution approach

1

2

3

4

Du & Evans (2008)13

min cost

min total tardiness

-

-

post-sale services

heuristic

Ahluwalia & Nema (2006)14

min cost

min environmental risk

-

-

electronics (computer)

simulation

Pishvaee et al. (2010)15

min cost

max responsiveness

-

-

-

memetic algorithm

Ramezani et al. (2013)18

max profit

max customer service level

min total defects

-

-

heuristic

Amin & Zhang (2013)16

min cost

max environmental benefits

-

-

-

CPLEX

Moghaddam (2015)19

max profit

min total defects

-

simulation, goal programming

Lee et al. (2015)17

min cost

min delivery tardiness

bottle distillation & sales

hybrid genetic algorithm

min late min economic delivery risk factors -

-



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This paper develops an efficient optimization model and solution approach for NdFeB magnet

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recovery with three objectives: maximizing the economic and environmental benefits and social support.

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The economic benefit was measured by the overall profit, the environmental benefit was measured by the

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reduced carbon footprint associated with recycling in comparison to new magnet production, and the

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social support was measured by ranking each location with respect to social indices like labor supply,

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quality of life, and population. Since some objectives may conflict with one another, Pareto sets are

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generated to obtain non-dominated solutions. First, global optimal (exact) Pareto solutions are pursued,

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which process require high computation capabilities. Then, to improve the computation efficiency, a

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genetic algorithm approach is developed to find the optimal or near-optimal solutions within a reasonable

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time. These two approaches were compared for the solution quality and computation efficiency. To the

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best of authors’ knowledge, this is the first research which designs an efficient NdFeB magnet reverse

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supply chain that considers all three dimensions of sustainability: economic, environment, and social, and

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provides industry with a practical tool to strategically plan a supply chain.

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Mathematical model development

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Problem description. Four major supply chain members are included in this research – collection centers,

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EOL product dismantling facilities, NdFeB magnet recycling facilities, and sales points. Each collection

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center treats one or more types of EOL products for NdFeB magnet recycling (e.g., HDDs and EV

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motors). Some collection centers send their collected EOL products to dismantlers, while some others

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choose not to due to high economic, environmental, or social costs. At dismantling facilities, EOL

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products are disassembled into multiple subcomponent levels, which include NdFeB magnets. These

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magnets are then transported to NdFeB magnet recycling facilities and reprocessed into ‘like new’

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magnets.5 Final products are transported and sold to sales points where NdFeB magnet buyers reside. In

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this network structure, key decision variables that require optimization are: 1) the locations and capacities


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of dismantlers, 2) the location and capacities of recyclers, and 3) the transportation flows between

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collection centers and dismantlers, dismantlers and recyclers, and recyclers and sales points. Since this

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infrastructure is planned for the long term (e.g., 10 years), the future market supply and demand

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projections are also important for the accuracy of the model’s results.

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Notation. Two types of variables- systemic and decision variables- are necessary to describe a proposed

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multi-objective value recovery network design model (MOVRNDM). Systemic variables are denoted in

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lowercase letters except for those which represent sets, and decision variables are denoted in uppercase

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letters. Systemic variables I

: Set of collection points

J

: Set of candidate dismantlers for EOL products

R

: Set of candidate recyclers for NdFeB magnets

S

: Set of sales points

T

: Set of time periods

K

: Set of product types

L(k)

: Set of components (other than NdFeB magnets) in product k∈K

M

: A very large number (close to infinity)

sit,k

: Supply of EOL product k∈K at collection point i∈I at time t∈T (kg/year)

ds t

: Demand for NdFeB magnets at sales point s∈S at time t∈T (kg)

dij

: Distance between collection point i∈I and dismantler j∈J (km)

djr

: Distance between dismantler j∈J and recycler r∈R (km)

drs

: Distance between recycler r∈R and sales point s∈S (km)

t pNdFeB

: Unit sales price of NdFeB magnets at time t∈T ($/kg)

plt( k )

: Unit sales price of component l ( k ) ∈L(k) at time t∈T ($/kg)

wl ( k )

: Weight percentage of component l ( k ) in product k∈K (%)

wNdFeB , k

: Weight percentage of NdFeB magnets in product k∈K (%)

ackt

: Unit acquisition cost of product k∈K at time t∈T ($/kg)

tcijt

: Unit transportation cost from collection point i∈I to dismantler j∈J at time t∈T ($/kg for



dij km) tc tjr

: Unit transportation cost from dismantler j∈J to recycler r∈R at time t∈T ($/kg for d jr

sc j

km) : Unit transportation cost from recycler r∈R to sales point s∈S at time t∈T ($/kg for d rs km) : Set-up cost of dismantler j∈J ($ per set-up)

scr

: Set-up cost of recycler r∈R ($ per set-up)

oc tj , k

ocrt

: Unit operating cost of dismantler j∈J for product k∈K at time t∈T ($/kg of EOL product) : Unit operating cost of recycler r∈R at time t∈T ($/kg of produced magnets)

η

: Magnet recycling efficiency (i.e., recovered magnet/input magnet weight) (%)

J max

: Maximum allowable number of dismantlers

R max

: Maximum allowable number of recyclers

q max j

: Maximum allowable processing capacity of dismantler j∈J (kg/year)

qrmax

: Maximum allowable processing capacity of recycler r∈R (kg/year)

co2r

: CO2 emission saved by recycling NdFeB magnets (kg CO2/kg NdFeB magnets)

co2 ( d ij )

co2 (dChina )

: CO2 emission from transportation from collection point i∈I to dismantler j∈J (kg CO2/kg product) : CO2 emission from transportation from dismantler j∈J to recycler r∈R (kg CO2/kg NdFeB magnets) : CO2 emission from transportation from recycler r∈R to sales point s∈S (kg CO2/kg NdFeB magnets) : CO2 emission saved by avoiding import from China (kg CO2/kg NdFeB magnets)

ss j

: Level of social support at dismantler j∈J (integer value; the larger, the better)

ssr

: Level of social support at recycler r∈R (integer value; the larger, the better)

tcrst

co2 (d jr )

co2 (drs )

Decision variables t

: Binary; equal to 1 if candidate dismantler j∈J is open at time t∈T

Yj

Yr

t

: Binary; equal to 1 if recycler r∈R is open at time t∈T

t

: Binary; equal to 1 if collection point i∈I is assigned to candidate dismantler j∈J at time t∈T : Binary; equal to 1 if candidate dismantler j∈J is assigned to candidate recycler r∈R at time t∈T : Binary; equal to 1 if candidate recycler r∈R is assigned to sales point s∈S at time t∈T

Xij

Xjr

t

Xrs

t

T jrt

: Integer; transportation volume from dismantler j to recycler r at time t (kg)

Oj

: Binary; equal to 1 if candidate dismantler j∈J is setup for opening

Or

: Binary; equal to 1 if recycling facility r∈R is setup for opening 6 ACS Paragon Plus Environment

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Mathematical model. Our proposed mathematical model consists of three objective functions and

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eighteen constraints as follows. 1. Objective functions. Three objectives are pursued in this research: economic and environmental

94 95

benefits and social support. The first objective function is to maximize the overall supply chain profit as

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shown in Eq. (1). NdFeB magnet recovery incurs several costs: EOL product acquisition costs,

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transportation costs, operating costs, and set up costs. Major revenue sources include the sales of

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dismantled components and recovered NdFeB magnets. The second objective function (Eq. (2)) is

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expressed as the net environmental benefit of recycling EOL NdFeB magnets (as opposed to new

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production)21 minus the environmental impacts of transportation for product collection and distribution.

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Carbon footprint is selected as the major environmental indicator because it provides optimal solutions

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similar to the solutions derived from other environmental impact categories.20 The third objective function

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(Eq. (3)) is to maximize the social support for creating a new business opportunity at a certain location.

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We defined social support from an industrial perspective, which is the societal contribution to business

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development. Therefore, we focused on identification of societies that support business, measured by

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indices including workforce, education, and business friendliness.22

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    t − tcrs ) ⋅ d st ⋅ X rst + ∑∑∑  ∑ plt( k ) ⋅wl ( k ) − ackt − tcij − octj , k  ⋅ sit, k ⋅ X ijt  max f1 = ∑  ∑∑ ( pNdFeB t  s r j k i  l  

(1)

−∑∑∑ (tc jr + ocrt ⋅η ) ⋅ T jrt − ∑ sc j ⋅ O j − ∑ scr ⋅ Or t

r

j

j

r

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   max f 2 = ∑ ∑∑ ( co2r + co2 (dChina ) − co2 (drs ) ) ⋅ d st X rst − ∑  ∑∑ co2 (dij ) ⋅ sit, k X ijt + ∑ co2 (d jr ) ⋅ Tjrt   (2) t  s r j  i k r 

109

max f 3 = ∑∑∑∑ ss j ⋅ sit,k ⋅ X ijt + ∑∑∑ ssr ⋅ d st ⋅ X rst t

j

i

k

t

s

r


(3)


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2. Constraints. The relationships between adjacent supply chain members are modeled as follows:

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1) n:1 with n being the number of collection centers for one dismantler, 2) n:1 with n being the number of

112

dismantlers for one recycler, and 3) 1: n with n being the number of sales points for one recycler. That is,

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the collection centers send their EOL products to only one dismantler; similarly, dismantlers send their

114

NdFeB magnets to only one recycler. The selected dismantlers (and recyclers) may change in each period,

115

but the number of dismantlers per collection center (and the number of recyclers per dismantler) is always

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assumed to be one. The rationale is that EOL products are scattered around the country and a relatively

117

small number of EOL products are collected at each site, so it is not necessary to distribute one collection

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volume to multiple dismantlers. Equivalently, the points of sale (i.e., motor manufacturers that purchase

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recovered NdFeB magnets) are assumed to be also dispersed and have only one supply source (i.e.,

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NdFeB recycler) per location. X ij t ≤ Y j t

∑X

≤1

∀i ∈ I , j ∈ J , t ∈ T

(4)

∀i ∈ I , t ∈ T

(5)

X jr t ≤ Y j t

∀j ∈ J , r ∈ R , t ∈ T

(6)

X jr t ≤ Yr t

∀j ∈ J , r ∈ R , t ∈ T

(7)

t ij

j∈J

∑X

t jr

≤1

∀j ∈ J , t ∈ T

(8)

r∈R

121

Equations (4) and (5) are used to assign each collection center to an appropriate dismantler.

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Equation (4) ensures the products are delivered to a dismantler that is open for business. Equation (5)

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describes a single assignment of the transportation volume, meaning that each collection center is

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assigned to at most one dismantler and if the assignment link is active, all the volume is transferred (i.e.,

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no partial allocation is allowed). This constraint helps alleviate the computation complexities attributed to

126

searching every miscellaneous change in the transportation volume for a slightly better solution. It is also

127

a more practical approach for our research scope that is to offer a strategic plan than a tactical or

128

operational one. Equations (6-8) determine the allocation of a dismantler to a remanufacturer, similar to

129

equations (4) and (5).


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∑T

≤ ∑ ∑ wNdFeB , k ⋅ sit, k ⋅ X ijt

t jr

r ∈R

(9)

∀j ∈ J , t ∈ T

k ∈K i∈I

T ≤ M ⋅ X jr t t jr

∑η ⋅ T

t jr

j∈J

∀j ∈ J , r ∈ R, t ∈T

≥ ∑ d st ⋅ X rst

(10) (11)

∀r ∈ R , t ∈ T

s∈S

130 Equations (9-11) are used to determine the processing quantities and transportation volumes

131 132

between dismantling and recycling facilities. That is, the transportation volume is positive only for the

133

selected dismantler - recycler pairs.

X rst ≤ Yr t

∑X

∀r ∈ R, s ∈ S , t ∈T

≤1

t rs

(12)

∀s ∈ S , t ∈ T

(13)

r∈R

134 Equations (12-13) ensure a single assignment of a sales point to a recycler, similar to the case of

135 136

equations (4) and (5).

O j ≥ ∑ Y j t /T

∀j ∈ J

(14)

Or ≥ ∑ Yr t /T

∀r ∈ R

(15)

t∈T

t∈T

∑O

j

≤ J max

r

≤ R max

(16)

j∈ J

∑O

(17)

r∈R

137 Equations (14) and (15) are developed to check if a facility is ever open for setup. Candidate

138 139

facilities that are open at least once are assigned a setup cost. The total number of new facilities may be

140

limited based on financial limitations or other company constraints, as shown in equations (16) and (17).

∑∑s

t i ,k

⋅ X ijt ≤ q j max

(18)

∀j ∈ J , t ∈ T

k ∈K i∈I

∑T

t jr

≤ qr max

(19)

∀r ∈ R, t ∈ T

j∈J



Page 10 of 26

141

142 143

The maximum allowable capacities are also confined for each facility as shown in equations (18) and (19). Y j t , Yr t , X ij t , X jr t , X rs t , O j , Or ∈ {0,1} for ∀i ∈ I , j ∈ J , r ∈ R , s ∈ S , t ∈ T

(20)

T jr t is nonnegative integer for ∀j ∈ J , r ∈ R , t ∈ T

(21)

144

145

Finally, equations (20) and (21) show the binary and integer decision variables of the proposed

146

mathematical model.

147

Solution approach

148

Pareto optimality. A three-objective maximization problem can be formulated as

149

Maximize f(x) = (f1(x), f2(x), f3(x))

150

Subject to x∈X

151

where x =(x1, x2, …, xn) is a vector consisting of decision variables and X is the set of feasible solutions.

152

Since some objectives may conflict with one another, Pareto-optimal sets are acquired for non-dominated

153

solutions. In a maximization problem, Pareto optimality is defined as follows.

154

Definition 1. A solution u∈X is Pareto optimal, if and only if there is no v∈X that dominates u. That is,

155

there is no v, like the one in equation (22), that is always as good as u and sometimes strictly better for at

156

least one objective (i.e., v≻u).

157

 f i ( v ) ≥ f i (u )   f j ( v ) > f j (u )

158

A Pareto optimal set is a set consisting of Pareto optimal solutions.

for ∀ i ∈ {1,2,3}

(22)

for at least one j ∈ {1,2,3}


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159

1. Determination of the exact Pareto solutions using an epsilon-constraint method. As an effort to

160

derive an exact Pareto set for our multi-objective optimization problem, an epsilon-constraint method is

161

employed. Chankong and Haimes (1983)23 proved that epsilon-constraint methods can be used to find the

162

exact Pareto front for multi-objective problems. Recently, Abounacer et al. (2014)24 developed an

163

epsilon-constraint method to derive an exact Pareto set for a multi-objective optimization problem.

164

In this paper, a modified epsilon-constraint method is proposed to find Pareto optimal solutions for

165

our proposed multi-objective mathematical model. The method solves a single objective problem by

166

constraining the other objective function values at appropriate levels (which is called epsilon, ε) and

167

iterates through the process with updated epsilon values until there is no feasible solution. That is, a

168

Pareto set is obtained by solving the epsilon constraint model (EC1) shown below iteratively, with a new

169

set of ε2 and ε3.

170

Epsilon constraint model 1 (EC1). Maximize f1

(23)

171

Subject to f2 > ε2

172

f3 ≥ ε3

173

Eqs. (4-21)

174

where f1, f2, and f3 correspond to equations (1-3). The initial ε2 and ε3 values are both set as 0, which are

175

updated at each iteration. The specific updating procedure is outlined in Algorithm 1. S stands for the set

176

of Pareto optimal solutions, and s is an element of S. Lines 9-11 are used to check Pareto optimality of

177

each new solution so that only non-dominated solutions are kept in S. In this specific algorithm, ε3 is an

178

integer so that each increment for ε3 is 1. This constraint is valid for our model since our third objective

179

function value is an integer.

Algorithm 1. Modified epsilon constraint method 1

S := ∅, t3 = 0

2

ε 3t = 0 3



3

While 1. EC1 has a feasible solution, do

4

t2 = 0

5

ε 2t = 0

6

While 2. EC1 has a feasible solution, do

2

7

t t X* = max (EC1, ε 22 , ε 33 )

8

S = S ∪ X*

9

for ∀ s ∈ S If (X*≻s), then S = S -s

10 11

end for t2 +1 2

12

ε

13

t2 ← t2 + 1

14

end while 2

15

ε 3t +1 = ε 3t + 1

16

t3 ← t3 + 1

17

3

t t = f2 (EC1, ε 22 , ε 33 )

3

end while 1

Output: Set of Pareto optimal solutions 180

181

The proposed epsilon-constraint method enables enumeration of all exact Pareto solutions if the

182

mixed integer programming model (MILP) formulated with Eqs. (1-21) is solved with an optimization

183

package like CPLEX. However, this is computationally expensive. Through multiple experiments with 48

184

collection centers and 6 sales points, it was found that when the number of candidate locations (e.g., two

185

dismantlers and one recycler) exceeds the maximum number of allowable new dismantlers (e.g., J max = 1)

186

and recyclers (e.g. R max = 1), this approach returns ‘out of memory’ on a desktop computer with Intel

187

Core i7 processer and 8GB RAM. This is a major drawback of the exact (or global optimization)

188

approach, as it fails to solve our proposed multi-objective optimization problem.

189

2. Multi-objective genetic algorithm. To address the aforementioned challenges and obtain (near)

190

optimal solutions with improved computation efficiency, a genetic algorithm approach is proposed.


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Genetic algorithm was first developed by Holland in 197525 and mimics nature’s evolution process

192

through operations including selection, crossover, and mutation. Each individual or ‘chromosome’

193

represents a solution that is subject to the evolutionary processes. In this paper, a multi-objective network

194

design genetic algorithm (MONDGA) is proposed to generate optimal or near optimal Pareto solutions

195

for our proposed multi-objective value recovery network design model (MOVRNDM). The overall

196

procedure of MONDGA is depicted in Figure 1, where C indicates the set of all possible weight

197

combinations and c is an element of C.

198 199

Figure 1. Overall procedure of MONDGA



200

1) Chromosome structure. In the proposed MOVRNDM, the maximum number of dismantlers (i.e., Jmax)

201

and recyclers (i.e., Rmax) are limited due to the high capital investment required for constructing relevant

202

plants and infrastructure.26,27 This characteristic enables us to construct a chromosome with a smaller

203

number of genes rather than enumerating all the possible locations with a lot of empty genes. That is, the

204

length of each chromosome (or number of genes) is equal to (Jmax + Rmax) so that the proposed MONDGA

205

may construct facilities up to (Jmax + Rmax). With this setting, each chromosome consists of two parts, one

206

for storing candidate dismantlers and the other for recyclers. Let Jc and Rc be sets consisting of candidate

207

dismantler indices and recycler indices, respectively (e.g., 1 represents a facility in California and 48 for

208

Wyoming). The initial chromosome contains genes with integer values randomly generated among {0, Jc}

209

for dismantlers and {0, Rc} for recyclers where 0 indicates no facility is constructed.

210

2) Reproduction. Three operations are performed during the reproduction stage: selection, crossover, and

211

mutation. To reproduce chromosomes (also known as offspring), two initial chromosomes (or parents) are

212

selected using a roulette wheel method that gives higher probabilities for chromosomes with higher

213

fitness values (e.g., a solution with a higher profit is more likely to be selected as a parent). If a certain

214

criteria is met (e.g., a randomly generated number is smaller than the crossover probability pc; this paper

215

used 0.7), crossover is performed that combines the selected parents to generate two new offspring (see

216

Figure 2). Otherwise, parents are copied without modification to the new generation list. In the proposed

217

MONDGA, two-point crossover is performed for the selected parents. Two random integer numbers, R1

218

and R2, are generated from {0, Jc} and {0, Rc}, respectively. Parents exchange genes up to the selected

219

two points as shown in Figure 2. The last operation is mutation, which is applied to each gene of each

220

chromosome. With a small predetermined mutation probability pm (which is 0.03 in this paper), mutation

221

randomly assigns a new gene value from {0, Jc} if the gene represents a dismantler and {0, Rc} if it

222

represents a recycler.


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Page 15 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


223 224

Figure 2. A crossover example for the proposed MONDGA.

225

3) Assignment and fitness evaluation. There are three assignment links to be optimized in the proposed

226

MONDGA: assignments between 1) collection centers and dismantlers, 2) dismantlers and recyclers, and

227

3) recyclers and sales points. As noted before, not all collection centers send their EOL products to the

228

value recovery system, as it may be detrimental to the three objectives – economic and environmental

229

benefits, and social support. Therefore, a collection center is assigned to a dismantler with a certain

230

probability (e.g., 0.5 for this MONDGA). The overall fitness is evaluated for each chromosome regarding

231

the selected locations and assignments, which in turn helps produce a new generation of offspring. As

232

there are three objectives (f1, f2, and f3), a weighted sum approach is adopted to derive a single fitness, f,

233

for each chromosome. First, an upper bound is determined for each objective function value (denoted as

234

f1 , f 2 , and f3 ) as shown in Eqs. (24-26). Then the overall fitness is calculated from Eq. (27), where w1,

235

w2, and w3 denote the weights of the corresponding objective functions and sum to 1. Ultimately, the

236

parent chromosomes with higher overall fitness values are more likely to be selected based on the roulette

237

wheel method.

238

  t f1 = ∑∑∑ sit,k ⋅  wNdFeB ,k ⋅ ( pNdFeB ⋅η − min r∈R (ocrt ) ) + ∑ wl ( k ) ⋅ plt( k ) − min j∈J (octj ,k ) − ackt  − sc j − scr (24) t i k l  

239

f 2 = ∑∑∑ ( co2r + co2 (dChina ) ) ⋅ sit,k ⋅ wNdFeB ,k ⋅η t

i

(25)

k



240

f 3 = ∑∑∑ sit,k ⋅ ( max j∈J ( ss j ) + max r∈R ( ssr ) ⋅ wNdFeB ,k ⋅η )

(26)

f = w1 ⋅ f1 / f1 + w2 ⋅ f 2 / f 2 + w3 ⋅ f 3 / f 3

(27)

t

241

Page 16 of 26

i

k

242

4) Pareto optimal solutions. As mentioned above, the proposed MONDGA assigns weights for three of

243

the objective functions during the fitness evaluation process. However, the proposed MOVRNDM may

244

not require specific weights, if all the objectives are equally important. In this case, to derive Pareto

245

optimal solutions, the proposed MONDGA is run for numerous combinations of weights to obtain

246

chromosomes with good fitness values. Then the resulting chromosomes are integrated and all dominated

247

solutions are deleted, similar to the process in Algorithm 1, lines 9-11. The final solutions comprise a

248

Pareto set for our proposed MOVRNDM.

249

Results

250

Multi-objective NdFeB magnet recovery network design. Our developed MOVRNDM and MONDGA

251

were applied to design a sustainable reverse supply chain for NdFeB magnets in the U.S. A full list of the

252

input parameter values are available in Supporting Information Tables S1-S3. Throughout this paper, a

253

population of 100 initial chromosomes were reproduced for 1,000 generations. Weights (w1, w2, w3) were

254

incremented by 10% whenever applicable to balance between solution quality and computation efficiency.

255

These conditions can be easily modified according to the decision maker’s own criteria. Accordingly, all

256

48 contiguous United States were investigated for EOL product collection, dismantling, and recycling, in

257

addition to 6 sales points for recovered NdFeB magnets. The construction of up to three dismantlers (Jmax

258

=3) and one recycler (Rmax =1) were allowed, due to a limited budget as well as supply and demand

259

constraints (e.g., total EOL product supply is 7-9 million kg/year while one dismantler can process up to 5

260

million kg/year).


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261

Table 2 shows the resulting Pareto solutions. The maximum profit solution (Pareto #1)

262

recommends dismantlers located in Virginia, North Carolina, and Arizona, and a recycler in Ohio; these

263

states offer low operating costs and are close to the EOL supply (California and east coast) and magnet

264

demand (near the east coast). The solution with the highest environmental benefit (in Pareto #3) identifies

265

that dismantlers should be located in Iowa, Ohio, and Michigan and a recycler should be located in

266

Wisconsin. These states are in the middle of the U.S. so that 1) EOL products may be collected from

267

major EOL sources (i.e., east and west coasts including California, Florida, and New York), and 2)

268

recovered magnets may be sold to nearby buyers (i.e., motor manufacturers in New York, Ohio, and

269

Alabama). The net environmental savings from NdFeB magnet recycling (compared to new production)

270

are far larger than transportation emissions from collection and distribution of the reverse logistics (see

271

Supporting Information Table S1). Therefore, the second objective (f2) seeks to maximize the sales

272

volume of recovered NdFeB magnets and then minimize the total traveling distance. Lastly, the optimal

273

solution for maximum social support (Pareto #28) identifies that dismantlers should be located in North

274

Carolina, Utah, and Nebraska where the human resources and business friendliness are ranked highly. For

275

example, Utah was ranked the first by Forbes22 and CNBC28 among 50 states in the U.S. based on various

276

indices including labor supply, regulatory environment, and education. All the solutions listed in Table 2

277

are Pareto optimal (i.e., non-dominated solutions). If desired, decision makers may evaluate the solutions

278

from their own perspectives (e.g., weighing different objectives by their own willingness to tradeoff) to

279

derive single best solutions. It should be noted that the selected dismantlers (j), recyclers (r), and sales

280

points (s) change over time. Table 2 shows all the selected facilities if they were utilized at least once. All

281

solution approaches were run on a desktop computer with an Intel Core i7 processer and 8GB RAM, and

282

the total computation time to generate the results in Table 2 was 9 minutes and 19 seconds.

283

Table 2. MONDGA results considering all candidate locations (i.e., 48 contiguous United States for EOL

284

product collection, dismantling, and recycling, in addition to 6 points of sale) Pareto No.

Selected j

Selected r

Selected s


f1 (x108)

f2 (x108)

f3 (x109)


285 286

1 VA, NC, AZ OH NY, OH, AL, MI 4.41 2 TN, LA, ND AR NY, OH, AL, MI 4.37 3 IA, OH, MI WI NY, OH, AL 4.30 4 NC, NE, OK NC NY, OH, AL 4.29 5 NE, GA, VA AR NY, OH, AL 4.28 6 GA, OH, NC IN NY, OH, AL, MI 4.25 7 UT, TX, NE MI NY, OH, AL 4.24 8 NC, TX, NE MS NY, OH, AL 4.23 9 GA, NC, CO AR NY, OH, AL, MI 4.21 10 NE, TX, TN SD NY, OH, AL, MI 4.21 11 VA, TX, NC SC NY, OH, AL 4.20 12 TX, UT, NC PA NY, OH, AL, MI 4.19 13 NC, MD, NE MS NY, OH, AL, MI 4.17 14 GA, NC, NE AL NY, OH, AL 4.17 15 NC, OH, NH MA NY, OH, AL, MI 4.16 16 NE, NC, ND NJ NY, OH, AL 4.15 17 NC, UT, CO MO NY, OH, AL 4.15 18 NE, OH, NC NE NY, OH, AL, MI 4.15 19 VA, NC, UT NE NY, OH, AL 4.14 20 VA, NC, AZ OH NY, OH, AL, MI 4.14 21 UT, NC, VA IL NY, OH, AL 4.10 22 TX, NE, NC NJ NY, OH, AL, MI 4.08 23 NC, TX, UT VA NY, OH, AL 4.05 24 NC, NE, TX NH NY, OH, AL 4.05 25 TX, UT, NE NE NY, OH, AL 4.02 26 NC, UT, TX VA NY, OH, AL 4.00 27 NC, UT, NE MA NY, OH, AL 3.96 28 NC, UT, NE MN NY, OH 3.95 29 NC, TN VA NY, OH, AL 3.75 The unit of f1 is $ earned in 10 years, and that of f2 is kg of CO2 savings in 10 years.

Page 18 of 26

1.06 0.92 1.13 0.95 0.95 0.98 0.96 0.82 0.87 0.97 0.92 0.92 1.04 1.04 1.04 1.00 0.80 0.95 0.87 1.04 0.79 0.90 0.84 1.03 0.71 0.75 0.79 0.65 1.07

2.22 2.23 2.29 3.15 3.13 2.90 3.16 3.35 3.26 3.07 3.16 3.32 2.76 2.94 2.66 3.17 3.42 3.19 3.36 2.74 3.49 3.33 3.43 3.08 3.51 3.51 3.44 3.58 2.41

To estimate the solution quality of Table 2, our proposed model (MOVRNDM) was solved in

287

CPLEX with an epsilon-constraint method for the exact Pareto solutions. However, the system ran out of

288

memory after 9 hours 54 minutes and 26 seconds of computation. The only solution it returned was the

289

maximum profit without considering the other two objectives (i.e., environmental benefits and social

290

support), which was $501,949,929 over 10 years (the entire planning horizon). Our maximum profit from

291

MONDGA, as shown in Table 2, is $441,390,258, which is 88% of the upper bound profit (i.e., the

292

maximum profit from CPLEX without considering other objectives, $501,949,929). Because of the

293

computational limitations of CPLEX and near-optimality of MONDGA, it is worth investigating various

294

other scenarios within our proposed MOVRNDM, therefore, the results of the two solutions approaches

295

are compared.


Page 19 of 26 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


296

Comparison between MONDGA and CPLEX. To understand the tradeoffs between computation

297

efficiency and solution quality, MONDGA was compared with CPLEX. Our analysis started with a

298

small-scale problem and gradually increased the problem complexities until CPLEX ran out of memory

299

(Table 3). Throughout this process, we adopted higher epsilon values for our CPLEX approach so that

300

some Pareto solutions with minor differences with each other were skipped. This is a practical approach

301

because taking our third objective as an example, Algorithm 1 suggests ε3 incremented by 1 at each

302

iteration, while the optimal f3 values are in the units of 109. Therefore, exhaustively searching for all

303

feasible solution space is computationally expensive and is likely to generate many dominated solutions.

304

Similarly, MONDGA adopted a weight (w1, w2, w3) increment of 10% in such a way that matches with

305

the total number of Pareto solutions offered by CPLEX. Instead of investigating all 48 states that was

306

infeasible for CPLEX to solve, we selected the top 5 and 10 collection centers that have the most

307

abundant EOL NdFeB magnets (|I|=5 and 10). They are California, Texas, Florida, New York,

308

Pennsylvania, Illinois, Ohio, Georgia, North Carolina, and Michigan in a decreasing order of EOL supply

309

abundance. When there are two candidate locations for dismantling or recycling, Nevada and Texas were

310

considered according to our communication with industry. The locations were expanded to include

311

Delaware when three candidates were available.

312

Table 3. Solution performance of CPLEX for solving small size problems

313 314

f1 (x108) f2 (x107) f3 (x109) Case No. of CPU |I| |J| |R| Jmax Rmax No. time Min Max Min Max Min Max Pareto 5 2 2 1 1 1.97 2.03 5.51 5.87 1.41 1.48 10 30" 1 5 3 3 1 1 1.97 2.03 5.51 5.87 1.41 1.48 11 66" 2 1.78 2.05 5.51 1.04 1.48 3 5 2 2 2 2 6.89 20 34" 5 3 3 2 2 1.78 2.05 5.51 6.94 0.66 1.48 25 138" 4 10 2 2 1 1 2.65 2.81 6.93 7.79 1.78 2.05 10 124" 5 3 3 1 1 2.65 2.81 6.93 7.79 1.78 2.05 13 286" 6 10 2 2 2 2 ≤2.47 2.85 ≤6.99 8.79 ≤1.42 2.06 N/A 9,398" 7 10 3 3 2 2 N/A 7,293" 8 10 2 2 1 1 ≤3.06 3.22 ≤8.04 8.98 ≤2.02 2.29 N/A 5,644" 9 15 3 3 1 1 ≤3.20 3.22 ≤8.04 ≤2.23 2.29 N/A 7,706" 10 15 |I| represents the number of collection centers; |J| is the number of candidate dismantlers; |R| is the number of candidate remanufacturers; N/A is for the out of memory cases.



Page 20 of 26

With this background, Pareto solutions of MONDGA and CPLEX were compared at ten different

315 316

conditions as shown in Table 4. Table 4 shows that MONDGA effectively generates solutions that are

317

within 88-100% of the global optimal solutions for all three objectives. When the problem size is small

318

(e.g., |I| =5, |J| =2, |R|=2, Jmax=1, Rmax=1), CPLEX was faster than MONDGA (e.g., 30 seconds vs. 46

319

seconds). As the problem size increases (e.g., |I| =10, |J| =3, |R|=3, Jmax=1, Rmax=1), CPLEX gets slower

320

than MONDGA (e.g., 4 minutes and 46 seconds for CPLEX to obtain 13 Pareto solutions vs. 58 seconds

321

for MONDGA for 26 Pareto solutions). When the problem becomes more complex (e.g., |I| =10, |J| =3,

322

|R|=3, Jmax=2, Rmax=2), CPLEX ran out of memory (e.g., in 2 hours 2 minutes). By contrast, MONDGA

323

generated solutions within a short period of time (e.g., 29 Pareto solutions in less than 2 minutes). In real

324

life, the supply chain design of NdFeB magnet recycling is likely to require more decision variables and

325

thus would exceed the computation capabilities of CPLEX. It renders MONDGA a promising tool for

326

(near) optimal solutions.

327

Table 4. Solution performance of MONDGA and comparison with Table 3 Case No. 1 2 3 4 5 6 7 8 9 10

f1 (x108) Min Max 1.98 2.01 1.98 2.01 1.98 2.01 1.98 2.01 2.67 2.77 2.68 2.76 2.73 2.78 2.71 2.77 3.08 3.16 3.09 3.17

f2 (x107) Min Max 5.48 5.80 5.48 5.80 5.48 6.42 5.48 6.43 6.86 7.58 6.90 7.55 6.72 7.77 6.99 7.74 7.99 8.61 7.97 8.59

f3 (x109) Min Max 1.42 1.48 1.42 1.48 1.18 1.48 1.11 1.48 1.84 2.05 1.84 2.03 1.71 2.06 1.10 2.04 2.09 2.26 2.12 2.27

No. of Pareto 11 9 17 17 22 26 25 29 25 30

CPU time 46" 39" 75" 80" 55" 58" 107" 116" 64" 84"

Gap with CPLEX (%) f1 f2 f3 99 99 100 99 99 100 98 93 100 98 93 100 98 97 100 98 97 99 97 88 100 N/A N/A N/A 98 96 99 98 N/A 99

328

329

The comparison with CPLEX confirmed that our proposed MONDGA effectively solves

330

MOVRNDM with (near) optimal solutions, while significantly reducing the computation time and

331

identifying solutions which CPLEX was not able to obtain. These results suggest that MONDGA is

332

suitable for optimizing the economic, environmental, and social aspects of NdFeB magnet reverse supply

333

chain. It is confirmed that our mathematical model (MOVRNDM) and solution approach (MONDGA)


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334

offer a helpful tool for the industry to design the reverse supply network for the optimal locations,

335

capacities, and transportation flows for circular economy.



336

Supporting Information

337

Input parameter values, assumptions, and data sources; EOL product supply, operating costs, and social

338

support ranking data by state; future market projections; demand for recovered NdFeB magnets (Word

339

document).

340

Acknowledgements

341

This research is supported by the Critical Materials Institute (an Energy Innovation Hub funded by the

342

U.S. Department of Energy, Advanced Manufacturing Office). Hongyue Jin gratefully acknowledges

343

support from the Environmental Research & Education Foundation Scholarship. The authors would like

344

to thank Dr. Gamini Mendis for his editorial assistance and Mr. Justin S. Richter for his insights on social

345

sustainability.

346

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Synopsis: An efficient NdFeB magnet recovery network was designed to maximize the economic and

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environmental sustainability and social support using mathematical models.

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Sustainable Value Recovery of NdFeB Magnets: A Multi-Objective

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