Review Special Issues

Value recovery from spent lithium-ion batteries: A review on technologies, environmental impacts, economics, and supply chain

  • # These authors contributed equally
  • The demand for lithium-ion batteries (LIBs) has surged in recent years, owing to their excellent electrochemical performance and increasing adoption in electric vehicles and renewable energy storage. As a result, the expectation is that the primary supply of LIB materials (e.g., lithium, cobalt, and nickel) will be insufficient to satisfy the demand in the next five years, creating a significant supply risk. Value recovery from spent LIBs could effectively increase the critical materials supply, which will become increasingly important as the number of spent LIBs grows. This paper reviews recent studies on developing novel technologies for value recovery from spent LIBs. The existing literature focused on hydrometallurgical-, pyrometallurgical-, and direct recycling, and their advantages and disadvantages are evaluated in this paper. Techno-economic analysis and life cycle assessment have quantified the economic and environmental benefits of LIB reuse over recycling, highlighting the research gap in LIB reuse technologies. The study also revealed challenges associated with changing battery chemistry toward less valuable metals in LIB manufacturing (e.g., replacing cobalt with nickel). More specifically, direct recycling may be impractical due to rapid technology change, and the economic and environmental incentives for recycling spent LIBs will decrease. As LIB collection constitutes a major cost, optimizing the reverse logistics supply chain is essential for maximizing the economic and environmental benefits of LIB recovery. Policies that promote LIB recovery are reviewed with a focus on Europe and the United States. Policy gaps are identified and a plan for sustainable LIB life cycle management is proposed.

    Citation: Majid Alipanah, Apurba Kumar Saha, Ehsan Vahidi, Hongyue Jin. Value recovery from spent lithium-ion batteries: A review on technologies, environmental impacts, economics, and supply chain[J]. Clean Technologies and Recycling, 2021, 1(2): 152-184. doi: 10.3934/ctr.2021008

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  • The demand for lithium-ion batteries (LIBs) has surged in recent years, owing to their excellent electrochemical performance and increasing adoption in electric vehicles and renewable energy storage. As a result, the expectation is that the primary supply of LIB materials (e.g., lithium, cobalt, and nickel) will be insufficient to satisfy the demand in the next five years, creating a significant supply risk. Value recovery from spent LIBs could effectively increase the critical materials supply, which will become increasingly important as the number of spent LIBs grows. This paper reviews recent studies on developing novel technologies for value recovery from spent LIBs. The existing literature focused on hydrometallurgical-, pyrometallurgical-, and direct recycling, and their advantages and disadvantages are evaluated in this paper. Techno-economic analysis and life cycle assessment have quantified the economic and environmental benefits of LIB reuse over recycling, highlighting the research gap in LIB reuse technologies. The study also revealed challenges associated with changing battery chemistry toward less valuable metals in LIB manufacturing (e.g., replacing cobalt with nickel). More specifically, direct recycling may be impractical due to rapid technology change, and the economic and environmental incentives for recycling spent LIBs will decrease. As LIB collection constitutes a major cost, optimizing the reverse logistics supply chain is essential for maximizing the economic and environmental benefits of LIB recovery. Policies that promote LIB recovery are reviewed with a focus on Europe and the United States. Policy gaps are identified and a plan for sustainable LIB life cycle management is proposed.



    In convex function theory, the classical Hermite-Hadamard inequality is one of the most well-known inequalities with geometrical interpretation, and it has a wide range of applications, see [1,2].

    Let S:KR+ be a convex function on a convex set K and ρ,ςK with ρς. Then,

    S(ρ+ς2)1ςρςρS(ϖ)dϖS(ρ)+S(ς)2. (1)

    In [3], Fejér looked at the key extensions of HH-inequality which is known as Hermite-Hadamard-Fejér inequality (HH-Fejér inequality).

    Let S:KR+ be a convex function on a convex set K and ρ,ς K with ρς. Then,

    S(ρ+ς2)1ςρD(ϖ)dϖςρS(ϖ)D(ϖ)dϖS(ρ)+S(ς)2ςρD(ϖ)dϖ. (2)

    If D(ϖ)=1, then we obtain (1) from (2). We should remark that Hermite-Hadamard inequality is a refinement of the idea of convexity, and it can be simply deduced from Jensen's inequality. In recent years, the Hermite-Hadamard inequality for convex functions has gotten a lot of attention, and there have been a lot of improvements and generalizations examined. Sarikaya [4] proved the Hadamard type inequality for coordinated convex functions such that

    Let G:ΔR+ be a coordinate convex function on Δ=[ς,ρ]×[μ,ν]. If G is double fractional integrable, then following inequalities hold:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]+Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (3)

    If α=1, then we obtain the following Dragomir inequality [5] on coordinates:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (4)

    For more details related to inequalities, see [6,7,8,9] and reference therein.

    Interval analysis, on the other hand, is a well-known example of set-valued analysis, which is the study of sets in the context of mathematical analysis and general topology. It was created as a way of dealing with the interval uncertainty that can be found in many mathematical or computer models of deterministic real-world phenomena. Archimede's method, which is used to calculate the circumference of a circle, is an old example of an interval enclosure. Moore [10], who is credited with being the first user of intervals in computational mathematics, published the first book on interval analysis in 1966. Following the publication of his book, a number of scientists began to research the theory and applications of interval arithmetic. Interval analysis is now a helpful technique in a variety of fields that are interested in ambiguous data because of its applicability. Computer graphics, experimental and computational physics, error analysis, robotics, and many more fields have applications.

    Furthermore, in recent years, numerous major inequalities (Hermite-Hadamard, Ostrowski and others) have been addressed for interval-valued functions. Chalco-Cano et al. used the Hukuhara derivative for interval-valued functions to construct Ostrowski type inequalities for interval-valued functions in [11,12,13,14]. For interval-valued functions, Román-Flores et al. developed Minkowski and Beckenbach's inequality in [15]. For fuzzy interval-valued function, Khan et al. [16,17,18] derived some new versions of Hermite-Hadamard type inequalities and proved their validity with the help of non-trivial examples. Moreover, Khan et al. [19,20] discussed some novel types of Hermite-Hadamard type inequalities in fuzzy-interval fractional calculus and proved that many classical versions are special cases of these inequalities. Recently, Khan et al. [21] introduced the new class of convexity in fuzzy-interval calculus which is known as coordinated convex fuzzy-interval-valued functions and with the support of these classes, some Hermite-Hadamard type inequalities are obtained via newly defined fuzzy-interval double integrals. We encourage readers to [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] for other related results.

    The following is an overview of the paper's structure. Section 2 recalls some preliminary notions and definitions. Moreover, some properties of introduced coordinated LR-convex IVF are also discussed. Section 3 presents some Hermite-Hadamard type inequalities for coordinated LR-convex IVF. With the help of this class, some fractional integral inequalities are also derived for the coordinated LR-convex IVF and for the product of two coordinated LR-convex IVFs. The fourth section, Conclusions and Future Work, brings us to a close.

    Let R be the set of real numbers and RI be the space of all closed and bounded intervals of R, such that URI is defined by

    U=[U,U]={yR|UyU},(U,UR). (5)

    If U=U, then U is said to be degenerate. If U0, then [U,U] is called positive interval. The set of all positive interval is denoted by R+I and defined as R+I={[U,U]:[U,U]RIandU0}.

    Let ϱR and ϱU be defined by

    ϱ.U={[ϱU,ϱU]ifϱ>0,{0}ifϱ=0,[ϱU,ϱU]ifϱ<0. (6)

    Then, the Minkowski difference DU, addition U+D and U×D for U,DRI are defined by

    [D,D][U,U]=[DU,DU],[D,D]+[U,U]=[D+U,D+U], (7)

    and

    [D,D]×[U,U]=[min{DU,DU,DU,DU},max{DU,DU,DU,DU}].

    The inclusion "⊇" means that

    UD if and only if, [U,U][D,D], and if and only if

    UD,DU. (8)

    Remark 1. [36] (ⅰ) The relation "≤p" is defined on RI by

    [D,D]p[U,U]ifandonlyifDU,DU, (9)

    for all [D,D],[U,U]RI, and it is a pseudo order relation. The relation [D,D]p[U,U] coincident to [D,D][U,U] on RI when it is "≤p"

    (ⅱ) It can be easily seen that "p" looks like "left and right" on the real line R, so we call "p" is "left and right" (or "LR" order, in short).

    For [D,D],[U,U]RI, the Hausdorff-Pompeiu distance between intervals [D,D] and [U,U] is defined by

    d([D,D],[U,U])=max{|DU|,|DU|}. (10)

    It is familiar fact that (RI,d) is a complete metric space.

    Theorem 1. [10] If G:[μ,ν]RRI is an I-V-F given by (x) [G(x),G(x)], then G is Riemann integrable over [μ,ν] if and only if, G and G both are Riemann integrable over [μ,ν] such that

    (IR)νμG(x)dx=[(R)νμG(x)dx,(R)νμG(x)dx]. (11)

    The collection of all Riemann integrable real valued functions and Riemann integrable I-V-F is denoted by R[μ,ν] and TR[μ,ν], respectively.

    Definition 1. [31,33] Let G:[μ,ν]RI be interval-valued function and GTR[μ,ν]. Then interval Riemann-Liouville-type integrals of G are defined as

    Iαμ+G(y)=1Γ(α)yμ(yt)α1G(t)dt(y>μ), (12)
    IανG(y)=1Γ(α)νy(ty)α1G(t)dt(y<ν), (13)

    where α>0 and Γ is the gamma function.

    Theorem 2. [20] Let G:[ς,ρ]RI+ be a LR-convex I-V.F such that G(y)=[G(y),G(y)] for all y[ς,ρ]. If GL([ς,ρ],R+I), then

    G(ς+ρ2)pΓ(α+1)2(ρς)α[Iας+G(ρ)+IαρG(ς)]pG(ς)+G(ρ)2. (14)

    Theorem 3. [20] Let G,S:[ς,ρ]R+I be two LR-convex I-V.Fs such that G(x)=[G(x),G(x)] and S(x)=[S(x),S(x)] for all x[ς,ρ]. If G×SL([ς,ρ],R+I) is fuzzy Riemann integrable, then

    Γ(α+1)2(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    p(12α(α+1)(α+2))M(ς,ρ)+(α(α+1)(α+2))N(ς,ρ), (15)

    and

    G(ς+ρ2)×S(ς+ρ2)
    pΓ(α+1)4(ρς)α[Iας+G(ρ)×S(ρ)+IαρG(ς)×S(ς)]
    +12(12α(α+1)(α+2))M(ς,ρ)+12(α(α+1)(α+2))N(ς,ρ), (16)

    where M(ς,ρ)=G(ς)×S(ς)+G(ρ)×S(ρ), N(ς,ρ)=G(ς)×S(ρ)+G(ρ)×S(ς),

    and M(ς,ρ)=[M(ς,ρ),M(ς,ρ)] and N(ς,ρ)=[N(ς,ρ),N(ς,ρ)].

    Note that, the Theorem 1 is also true for interval double integrals. The collection of all double integrable I-V-F is denoted TOΔ, respectively.

    Theorem 4. [35] Let Δ=[ς,ρ]×[μ,ν]. If G:ΔRI is interval-valued doubl integrable (ID-integrable) on Δ. Then, we have

    (ID)ρςνμG(x,y)dydx=(IR)ρς(IR)νμG(x,y)dydx.

    Definition 2. [36] Let G:ΔR+I and GTOΔ. The interval Riemann-Liouville-type integrals Iα,βμ+,ς+,Iα,βμ+,ρ, Iα,βν,ς+,Iα,βν,ρ of G order α,β>0 are defined by

    Iα,βμ+,ς+G(x,y)=1Γ(α)Γ(β)xμyς(xt)α1(ys)β1G(t,s)dsdt(x>μ,y>ς), (17)
    Iα,βμ+,ρG(x,y)=1Γ(α)Γ(β)xμρy(xt)α1(sy)β1G(t,s)dsdt(x>μ,y<ρ), (18)
    Iα,βν,ς+G(x,y)=1Γ(α)Γ(β)νxyς(tx)α1(ys)β1G(t,s)dsdt(x<ν,y>ς), (19)
    Iα,βν,ρG(x,y)=1Γ(α)Γ(β)νxρy(tx)α1(sy)β1G(t,s)dsdt(x<ν,y<ρ). (20)

    Definition 3. [38] The I-V.F G:ΔR+I is said to be coordinated LR-convex I-V.F on Δ if

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ), (21)

    for all (μ,ν),(ς,ρ)Δ, and τ,s[0,1]. If inequality (21) is reversed, then G is called coordinate LR-concave I-V.F on Δ.

    Lemma 1. [38] Let G:ΔR+I be an coordinated I-V.F on Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if there exist two coordinated LR-convex I-V.Fs Gx:[ς,ρ]R+I, Gx(w)=G(x,w) and Gy:[μ,ν]R+I, Gy(z)=G(z,y).

    Theorem 5. [38] Let G:ΔR+I be a I-V.F on Δ such that

    G(x,ϖ)=[G(x,ϖ),G(x,ϖ)], (22)

    for all (x,ϖ)Δ. Then, G is coordinated LR-convex I-V.F on Δ, if and only if, G(x,ϖ) and G(x,ϖ) are coordinated convex functions.

    Example 1. We consider the I-V.Fs G:[0,1]×[0,1]R+I defined by,

    G(x)(σ)={σ2(6+ex)(6+eϖ),σ[0,2(6+ex)(6+eϖ)]4(6+ex)(6+eϖ)σ2(6+ex)(6+eϖ),σ(2(6+ex)(6+eϖ),4(6+ex)(6+eϖ)]0,otherwise,

    Then, for each θ[0,1], we have G(x)=[2θ(6+ex)(6+eϖ),(4+2θ)(6+ex)(6+eϖ)]. Since end point functions G((x,ϖ),θ), G((x,ϖ),θ) are coordinate concave functions for each θ[0,1]. Hence S(x,ϖ) is coordinate LR-concave I-V.F.

    From Lemma 1, we can easily note that each LR-convex I-V.F is coordinated LR-convex I-V.F. But the converse is not true.

    Remark 2. If one takes G(x,ϖ)=G(x,ϖ), then G is known as coordinated function if G satisfies the coming inequality

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid which is defined by Dragomir [5]

    Let one takes G(x,ϖ)G(x,ϖ), where G(x,ϖ) is affine function and G(x,ϖ) is a concave function. If coming inequality,

    G(τμ+(1τ)ν,sς+(1s)ρ)
    τsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    is valid, then G is named as coordinated IVF which is defined by Zhao et al. [37, Definition 2 and Example 2]

    In this section, we shall continue with the following fractional HH-inequality for coordinated LR-convex I-V.Fs, and we also give fractional HH-Fejér inequality for coordinated LR-convex I-V.F through fuzzy order relation.

    Theorem 6. Let G:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] for all (x,y)Δ. If GTOΔ, then following inequalities holds:

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (23)

    If G(x) coordinated LR-concave I-V.F, then

    G(μ+ν2,ς+ρ2)pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)8(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (24)

    Proof. Let G:[μ,ν]R+I be a coordinated LR-convex I-V.F. Then, by hypothesis, we have

    4G(μ+ν2,ς+ρ2)pG(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Theorem 5, we have

    4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ),4G(μ+ν2,ς+ρ2)G(τμ+(1τ)ν,τς+(1τ)ρ)+G((1τ)μ+τν,(1τ)ς+τρ).

    By using Lemma 1, we have

    2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ),2G(x,ς+ρ2)G(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (25)

    and

    2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y),2G(μ+ν2,y)G(τμ+(1τ)ν,y)+G((1τ)μ+tν,y). (26)

    From (25) and (26), we have

    2[G(x,ς+ρ2),G(x,ς+ρ2)]
    p[G(x,τς+(1τ)ρ),G(x,τς+(1τ)ρ)]
    +[G(x,(1τ)ς+τρ),G(x,(1τ)ς+τρ)],

    and

    2[G(μ+ν2,y),G(μ+ν2,y)]
    p[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)]
    +[G(τμ+(1τ)ν,y),G(τμ+(1τ)ν,y)],

    It follows that

    G(x,ς+ρ2)pG(x,τς+(1τ)ρ)+G(x,(1τ)ς+τρ), (27)

    and

    G(μ+ν2,y)pG(τμ+(1τ)ν,y)+G(τμ+(1τ)ν,y). (28)

    Since G(x,.) and G(.,y), both are coordinated LR-convex-IVFs, then from inequality (14), inequalities (27) and (28) we have

    Gx(ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+Gx(ρ)+IβρGx(ς)]pGx(ς)+Gx(ρ)2. (29)

    and

    Gy(μ+ν2)pΓ(α+1)2(νμ)α[Iαμ+Gy(ν)+IανGy(μ)]pGy(μ)+Gy(ν)2 (30)

    Since Gx(w)=G(x,w), the inequality (29) can be written as

    G(x,ς+ρ2)pΓ(β+1)2(ρς)β[Iας+G(x,ρ)+IαρG(x,ς)]pG(x,ς)+G(x,ρ)2. (31)

    That is

    G(x,ς+ρ2)pβ2(ρς)β[ρς(ρs)β1G(x,s)ds+ρς(sς)β1G(x,s)ds]pG(x,ς)+G(x,ρ)2.

    Multiplying double inequality (31) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pνμρς(νx)α1(ρs)β1G(x,s)dsdx+νμρς(νx)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(νx)α1G(x,ς)dx+νμ(νx)α1G(x,ρ)dx]. (32)

    Again multiplying double inequality (31) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we have

    α2(νμ)ανμG(x,ς+ρ2)(νx)α1dx
    pαβ4(νμ)α(ρς)βνμρς(xμ)α1(ρs)β1G(x,s)dsdx
    +αβ4(νμ)α(ρς)βνμρς(xμ)α1(sς)β1G(x,s)dsdx
    pα4(νμ)α[νμ(xμ)α1G(x,ς)dx+νμ(xμ)α1G(x,d)dx]. (33)

    From (32), we have

    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(ν,ς)]
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς)+Iαμ+G(ν,ρ)]. (34)

    From (33), we have

    Γ(α+1)2(νμ)α[IανG(μ,ς+ρ2)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    pΓ(α+1)4(νμ)α[IανG(μ,ς)+IανG(μ,ρ)]. (35)

    Similarly, since Gy(z)=G(z,y) then, from (34) and (35), (30) we have

    Γ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βν,ς+G(μ,ρ)]
    pΓ(β+1)4(ρς)β[Iβς+G(μ,ρ)+Iβς+G(ν,ρ)], (36)

    and

    Γ(β+1)2(ρς)α[IβρG(μ+ν2,ς)]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)+Iα,βν,ρG(μ,ς)]
    pΓ(β+1)4(ρς)β[IβρG(μ,ς)+IβρG(ν,ς)]. (37)

    After adding the inequalities (46), (35), (36) and (37), we will obtain as resultant second, third and fourth inequalities of (23).

    Now, from left part of inequality (14), we have

    G(μ+ν2,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (38)

    and

    G(μ+ν2,ς+ρ2)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]. (39)

    Summing the inequalities (38) and (39), we obtain the following inequality:

    G(μ+ν2,ς+ρ2)
    pΓ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)], (40)

    this is the first inequality of (23).

    Now, from right part of inequality (14), we have

    Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)]pG(μ,ς)+G(μ,ρ)2, (41)
    Γ(β+1)2(ρς)β[Iβς+G(ν,ρ)+IβρG(ν,ς)]pG(ν,ς)+G(ν,ρ)2, (42)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)]pG(μ,ς)+G(ν,ς)2, (43)
    Γ(α+1)2(νμ)α[Iαμ+G(ν,ρ)+IανG(μ,ρ)]pG(μ,ρ)+G(ν,ρ)2. (44)

    Summing inequalities (41), (42), (43) and (44), and then taking multiplication of the resultant with 14, we have

    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)+IανG(μ,ς)+Iαμ+G(ν,ρ)+IανG(μ,ρ)]
    +Γ(β+1)2(ρς)β[Iβς+G(μ,ρ)+IβρG(μ,ς)+Iβς+G(ν,ρ)+IβρG(ν,ς)]
    pG(μ,ς)+G(μ,ρ)+G(ν,ς)+G(ν,ρ)4. (45)

    This is last inequality of (23) and the result has been proven.

    Remark 3. If one to take α=1 and β=1, then from (23), we achieve the coming inequality, see [38]:

    G(μ+ν2,ς+ρ2)
    p12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]p1(νμ)(ρς)νμρςG(x,y)dydxp14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    pG(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (46)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then from Remark 2 and (24), we acquire the coming inequality, see [31]:

    G(μ+ν2,ς+ρ2)Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (47)

    Let one takes α=1 and β=1, G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then Remark 2 and from (24), we acquire the coming inequality, see [37]:

    G(μ+ν2,ς+ρ2)
    12[1νμνμG(x,ς+ρ2)dx+1ρςρςG(μ+ν2,y)dy]1(νμ)(ρς)νμρςG(x,y)dydx
    14(νμ)[νμG(x,ς)dx+νμG(x,ρ)dx]+14(ρς)[ρςG(μ,y)dy+ρςG(ν,y)dy]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4. (48)

    Example 2. We consider the I-V-Fs G:[0,1]×[0,1]R+I defined by,

    G(x)=[2,6](6+ex)(6+ey).

    Since end point functions G(x,y), G(x,y) are convex functions on coordinate, then G(x,y) is convex I-V-F on coordinate. Then for α=1 and β=1, we have

    G(μ+ν2,ς+ρ2)=[2(5+e12)2,6(6+e12)2],
    Γ(α+1)4(νμ)α[Iαμ+G(ν,ς+ρ2)+IανG(μ,ς+ρ2)]+Γ(β+1)4(ρς)β[Iβς+G(μ+ν2,ρ)+IβρG(μ+ν2,ς)]
    =[4(6+e12)(5+e),12(6+e12)(5+e)],
    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)+Iα,βμ+,ρG(ν,ς)+Iα,βν,ς+G(μ,ρ)+Iα,βν,ρG(μ,ς)]
    =[2(5+e)2,6(5+e)2],
    Γ(α+1)8(νμ)α[Iαμ+G(ν,ς)GIαμ+G(ν,ρ)+IανG(μ,ς)+IανG(μ,ρ)]
    +Γ(β+1)4(ρς)β[Iβς+G(μ,ρ)˜+IβρG(ν,ς)+Iβς+G(μ,ρ)+IβρG(ν,ς)]
    =[(5+e)(13+e),3(5+e)(13+e)]
    G(μ,ς)+G(ν,ς)+G(μ,ρ)+G(ν,ρ)4=[(6+e)(20+e)+492,6((6+e)(20+e)+49)2].

    That is

    [2(5+e12)2,6(6+e12)2]p[4(6+e12)(5+e),12(6+e12)(5+e)]
    p[2(5+e)2,6(5+e)2]
    p[(5+e)(13+e),3(5+e)(13+e)]
    p[(6+e)(20+e)+492,3((6+e)(20+e)+49)].

    Hence, Theorem 3.1 has been verified

    Next both results obtain Hermite-Hadamard type inequalities for the product of two coordinate LR-convex I-V.Fs

    Theorem 7. Let G,S:ΔR+I be a coordinate LR-convex I-V.Fs on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (49)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (50)

    Where

    K(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς)+G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ),
    L(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ς)˜+G(ν,ρ)×S(μ,ρ)+G(ν,ς)×S(μ,ς)+G(μ,ρ)×S(ν,ρ),
    M(μ,ν,ς,ρ)=G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ)+G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς),
    N(μ,ν,ς,ρ)=G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)+G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς).

    and K(μ,ν,ς,ρ), ˜L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are defined as follows:

    K(μ,ν,ς,ρ)=[K(μ,ν,ς,ρ),K(μ,ν,ς,ρ)],
    L(μ,ν,ς,ρ)=[L(μ,ν,ς,ρ),L(μ,ν,ς,ρ)],
    M(μ,ν,ς,ρ)=[M(μ,ν,ς,ρ),M(μ,ν,ς,ρ)],
    N(μ,ν,ς,ρ)=[N(μ,ν,ς,ρ),N(μ,ν,ς,ρ)].

    Proof. Let G and S both are coordinated LR-convex I-V.Fs on [μ,ν]×[ς,ρ]. Then

    G(τμ+(1τ)ν,sς+(1s)ρ)
    pτsG(μ,ς)+τ(1s)G(μ,ρ)+(1τ)sG(ν,ς)+(1τ)(1s)G(ν,ρ),

    and

    S(τμ+(1τ)ν,sς+(1s)ρ)
    pτsS(μ,ς)+τ(1s)S(μ,ρ)+(1τ)sS(ν,ς)+(1τ)(1s)S(ν,ρ).

    Since G and S both are coordinated LR-convex I-V.Fs, then by Lemma 1, there exist

    Gx:[ς,ρ]R+I,Gx(y)=G(x,y),Sx:[ς,ρ]R+I,Sx(y)=S(x,y),

    Since Gx, and Sx are I-V.Fs, then by inequality (15), we have

    Γ(β+1)2(ρς)β[Iβς+Gx(ρ)×Sx(ρ)+IβρGx(ς)×Sx(ς)]
    p(12β(β+1)(β+2))(Gx(ς)×Sx(ς)+Gx(ρ)×Sx(ρ))
    +(β(β+1)(β+2))(Gx(ς)×Sx(ρ)+Gx(ρ)×Sx(ς)).

    That is

    β2(ρς)β[ρς(ρy)β1G(x,y)×S(x,y)ρy+ρς(yς)β1G(x,y)×S(x,y)ρy]
    p(12β(β+1)(β+2))(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))
    +(β(β+1)(β+2))(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς)). (51)

    Multiplying double inequality (51) by α(νx)α12(νμ)α and integrating with respect to x over [μ,ν], we get

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (52)

    Again, multiplying double inequality (51) by α(xμ)α12(νμ)α and integrating with respect to x over [μ,ν], we gain

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (53)

    Summing (52) and (53), we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    pΓ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    +Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    +Γ(α+1)2(νμ)αβ(β+1)(β+2)(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)). (54)

    Now, again with the help of integral inequality (15) for first two integrals on the right-hand side of (54), we have the following relation

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)). (55)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)). (56)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ))
    p(12α(α+1)(α+2))(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))
    +(α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)). (57)

    And

    Γ(α+1)2(νμ)α(Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς))
    p(12α(α+1)(α+2))(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))
    +(α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)). (58)

    From (55)–(58), inequality (54) we have

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    p(12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ).

    Hence, the result has been proven.

    Remark 4. If one to take α=1 and β=1, then from (49), we achieve the coming inequality, see [38]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    p19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (59)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [36]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (60)

    Let one takes G(x,y) is an affine function and G(x,y) is concave function. If G(x,y)G(x,y), then by Remark 2 and (50), we acquire the coming inequality, see [37]:

    1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx
    19K(μ,ν,ς,ρ)+118[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+136N(μ,ν,ς,ρ). (61)

    If G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (49), we acquire the coming inequality, see [39]:

    Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]
    (12α(α+1)(α+2))(12β(β+1)(β+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)(12β(β+1)(β+2))L(μ,ν,ς,ρ)
    +(12α(α+1)(α+2))β(β+1)(β+2)M(μ,ν,ς,ρ)+β(β+1)(β+2)α(α+1)(α+2)N(μ,ν,ς,ρ). (62)

    Theorem 8. Let G,S:ΔR+I be a coordinate LR-convex I-V.F on Δ such that G(x,y)=[G(x,y),G(x,y)] and S(x,y)=[S(x,y),S(x,y)] for all (x,y)Δ. If G×STOΔ, then following inequalities holds:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)
    +[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)
    +[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (63)

    If G and S both are coordinate LR-concave I-V.Fs on Δ, then above inequality can be written as

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)
    +[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (64)

    Where K(μ,ν,ς,ρ), L(μ,ν,ς,ρ), M(μ,ν,ς,ρ) and N(μ,ν,ς,ρ) are given in Theorem 7.

    Proof. Since G,S:ΔR+I be two LR-convex I-V.Fs, then from inequality (16), we have

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+(α(α+1)(α+2))(G(μ,ς+ρ2)×S(μ,ς+ρ2)+G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(G(μ,ς+ρ2)×S(ν,ς+ρ2)+G(ν,ς+ρ2)×S(μ,ς+ρ2)), (65)

    and

    2G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pβ2(ρς)β[ρς(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ς)+G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(G(μ+ν2,ς)×S(μ+ν2,ρ)+G(μ+ν2,ρ)×S(μ+ν2,ς)), (66)

    Adding (73) and (74), and then taking the multiplication of the resultant one by 2, we obtain

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pα2(νμ)α[νμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx+νμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dx]+β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy+ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]+(α(α+1)(α+2))(2G(μ,ς+ρ2)×S(μ,ς+ρ2)+2G(ν,ς+ρ2)×S(ν,ς+ρ2))+(12α(α+1)(α+2))(2G(μ,ς+ρ2)×S(ν,ς+ρ2)+2G(ν,ς+ρ2)×S(μ,ς+ρ2))+(β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ς)+2G(μ+ν2,ρ)×S(μ+ν2,ρ))+(12β(β+1)(β+2))(2G(μ+ν2,ς)×S(μ+ν2,ρ)+2G(μ+ν2,ρ)×S(μ+ν2,ς)). (67)

    Again, with the help of integral inequality (16) and Lemma 1 for each integral on the right-hand side of (67), we have

    α2(νμ)ανμ2(νx)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(νx)α1(ρy)β1G(x,y)dydx+νμρς(νx)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(νx)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς)). (68)
    α2(νμ)ανμ2(xμ)α1G(x,ς+ρ2)×S(x,ς+ρ2)dxpαβ4(νμ)α(ρς)β[νμρς(xμ)α1(ρy)β1G(x,y)dydx+νμρς(xμ)α1(yς)β1G(x,y)dydx]+β(β+1)(β+2)α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ς)+G(x,ρ)×S(x,ρ))dx+(12β(β+1)(β+2))α2(νμ)ανμ(xμ)α1(G(x,ς)×S(x,ρ)+G(x,ρ)×S(x,ς))dx,=Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+Γ(α+1)2(νμ)α(β(β+1)(β+2))(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))+Γ(α+1)2(νμ)α(12β(β+1)(β+2))(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς)). (69)
    β2(ρς)β[ρς2(ρy)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(ν,ρ)). (70)
    β2(ρς)β[ρς2(yς)β1G(μ+ν2,y)×S(μ+ν2,y)dy]
    pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ρG(ν,ς)×S(ν,ς)]+Γ(β+1)2(ρς)β(α(α+1)(α+2))(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))+Γ(β+1)2(ρς)β(12α(α+1)(α+2))(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(ν,ς)). (71)

    And

    2G(μ+ν2,ς)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ς)+IανG(μ,ς)×S(μ,ς)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ς)+G(ν,ς)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ς)+G(ν,ς)×S(μ,ς)), (72)
    2G(μ+ν2,ρ)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ρ)+IανG(μ,ρ)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ρ)×S(μ,ρ)+G(ν,ρ)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ρ)+G(ν,ρ)×S(μ,ρ)), (73)
    2G(μ+ν2,ς)×S(μ+ν2,ρ)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ς)×S(ν,ρ)+IανG(μ,ς)×S(μ,ρ)]+α(α+1)(α+2)(G(μ,ς)×S(μ,ρ)+G(ν,ς)×S(ν,ρ))+(12α(α+1)(α+2))(G(μ,ς)×S(ν,ρ)+G(ν,ς)×S(μ,ρ)), (74)
    2G(μ+ν2,ρ)×S(μ+ν2,ς)pΓ(α+1)2(νμ)α[Iαμ+G(ν,ρ)×S(ν,ς)+IανG(μ,ρ)×S(μ,ς)]
    +α(α+1)(α+2)(G(μ,ρ)×S(μ,ς)+G(ν,ρ)×S(ν,ς))+(12α(α+1)(α+2))(G(μ,ρ)×S(ν,ς)+G(ν,ρ)×S(μ,ς)), (75)
    2G(μ,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(μ,ρ)+IβρG(μ,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(μ,ς)×S(μ,ς)+G(μ,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(μ,ρ)+G(μ,ρ)×S(μ,ς)), (76)
    2G(ν,ς+ρ2)×Sϕ(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(ν,ρ)+IβρG(ν,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(ν,ς)×S(ν,ς)+G(ν,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(ν,ρ)+G(ν,ρ)×S(ν,ς)), (77)
    2G(μ,ς+ρ2)×S(ν,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(μ,ρ)×S(ν,ρ)+IβρG(μ,ρ)×S(ν,ς)]+β(β+1)(β+2)(G(μ,ς)×S(ν,ς)+G(μ,ρ)×S(ν,ρ))+(12β(β+1)(β+2))(G(μ,ς)×S(ν,ρ)+G(μ,ρ)×S(ν,ς)), (78)

    and

    2G(ν,ς+ρ2)×S(μ,ς+ρ2)pΓ(β+1)2(ρς)β[Iβς+G(ν,ρ)×S(μ,ρ)+IβρG(ν,ρ)×S(μ,ς)]+β(β+1)(β+2)(G(ν,ς)×S(μ,ς)+G(ν,ρ)×S(μ,ρ))+(12β(β+1)(β+2))(G(ν,ς)×S(μ,ρ)+G(ν,ρ)×S(μ,ς)), (79)

    From inequalities (68) to (79), inequality (67) we have

    8G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)2(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+(2α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))]+2(12α(α+1)(α+2))[Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))]+2(β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))]+2(12β(β+1)(β+2))[Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ρ)+Iαμ+G(ν,ρ)×S(ν,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))]
    +2α(α+1)(α+2)β(β+1)(β+2)K(μ,ν,ς,ρ)++(12α(α+1)(α+2))2β(β+1)(β+2)L(μ,ν,ς,ρ)
    +2α(α+1)(α+2)(12β(β+1)(β+2))M(μ,ν,ς,ρ)+2(12α(α+1)(α+2))(12β(β+1)(β+2))N(μ,ν,ς,ρ). (80)

    Again, with the help of integral inequality (15) and Lemma 1, for each integral on the right-hand side of (80), we have

    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(μ,ρ)+Iβς+G(ν,ρ)×S(ν,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(μ,ς)+IβρG(ν,ς)×S(ν,ς))p(12β(β+1)(β+2))K(μ,ν,ς,ρ)+β(β+1)(β+2)M(μ,ν,ς,ρ). (81)
    Γ(β+1)2(ρς)β(Iβς+G(μ,ρ)×S(ν,ρ)+Iβς+G(ν,ρ)×S(μ,ρ))+Γ(β+1)2(ρς)β(IβρG(μ,ς)×S(ν,ς)+IβρG(ν,ς)×S(μ,ς))p(12β(β+1)(β+2))L(μ,ν,ς,ρ)+β(β+1)(β+2)N(μ,ν,ς,ρ). (82)
    Γ(α+1)2(νμ)α(Iαμ+G(ν,ς)×S(ν,ς)+Iαμ+G(ν,ρ)×S(ν,ρ))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ς)+IανG(μ,ρ)×S(μ,ρ))p(12α(α+1)(α+2))K(μ,ν,ς,ρ)+α(α+1)(α+2)L(μ,ν,ς,ρ). (83)
    Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))+Γ(α+1)2(νμ)α(IανG(μ,ς)×S(μ,ρ)+IανG(μ,ρ)×S(μ,ς))p(12α(α+1)(α+2))M(μ,ν,ς,ρ)+α(α+1)(α+2)N(μ,ν,ς,ρ). (84)

    From (77) to (84), (80) we have

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)pΓ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (85)

    This concludes the proof of Theorem 8 result has been proven.

    Remark 5. If we take α=1 and β=1, then from (63), we achieve the coming inequality, see [38]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)p1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (86)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64), we acquire the coming inequality, see [37]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)1(νμ)(ρς)νμρςG(x,y)×S(x,y)dydx+536K(μ,ν,ς,ρ)+736[L(μ,ν,ς,ρ)+M(μ,ν,ς,ρ)]+29N(μ,ν,ς,ρ). (87)

    Let one takes G(x,y) is an affine function and G(x,y) is convex function. If G(x,y)G(x,y), then from Remark 2 and (64) we acquire the coming inequality, see [36]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (88)

    If we take G(x,y)=G(x,y) and S(x,y)=S(x,y), then from (63), we acquire the coming inequality, see [39]:

    4G(μ+ν2,ς+ρ2)×S(μ+ν2,ς+ρ2)Γ(α+1)Γ(β+1)4(νμ)α(ρς)β[Iα,βμ+,ς+G(ν,ρ)×S(ν,ρ)+Iα,βμ+,ρG(ν,ς)×S(ν,ς)+Iα,βν,ς+G(μ,ρ)×S(μ,ρ)+Iα,βν,ρG(μ,ς)×S(μ,ς)]+[α2(α+1)(α+2)+β(β+1)(β+2)(12α(α+1)(α+2))]K(μ,ν,ς,ρ)+[12(12α(α+1)(α+2))+α(α+1)(α+2)β(β+1)(β+2)]L(μ,ν,ς,ρ)+[12(12β(β+1)(β+2))+α(α+1)(α+2)β(β+1)(β+2)]M(μ,ν,ς,ρ)+[14α(α+1)(α+2)β(β+1)(β+2)]N(μ,ν,ς,ρ). (89)

    In this study, with the help of coordinated LR-convexity for interval-valued functions, several novel Hermite-Hadamard type inequalities are presented. It is also demonstrated that the conclusions reached in this study represent a possible extension of previously published equivalent results. Similar inequalities may be discovered in the future using various forms of convexities. This is a novel and intriguing topic, and future study will be able to find equivalent inequalities for various types of convexity and coordinated m-convexity by using different fractional integral operators.

    The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research. All authors read and approved the final manuscript. This work was funded by Taif University Researchers Supporting Project number (TURSP-2020/345), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no competing interests.



    [1] Grand view research (2020) Lithium-ion battery market Size, Share & Trends analysis report by product (LCO, LFP, NCA, LMO, lithium nickel manganese cobalt), by application, by region, and segmented forecasts, 2020-2027.
    [2] Lithium ion battery Market, Size, Share, COVID impact analysis and forecast to 2027. ResearchAndMarket, 2021.
    [3] Boudette NE, Davenport C (2021) GM announcement Shakes Up U.S, Automakers' transition to electric cars. New York Times. Available from: https://www.nytimes.com/2021/01/29/business/general-motors-electric-cars.html.
    [4] Deng J, Bae C, Denlinger A, et al. (2020) Electric vehicles batteries: requirements and challenges. Joule 4: 511-515. doi: 10.1016/j.joule.2020.01.013
    [5] Yu A, Sumangil M (2021) Top electric vehicle markets dominate lithium-ion battery capacity growth. S & P glob mark Intell. Available from: https://www.spglobal.com/marketintelligence/en/news-insights/blog/top-electric-vehicle-markets-dominate-lithium-ion-battery-capacity-growth.
    [6] Roskill (2020) The resurgence of LFP cathodes: A safe and cost-efficient battery active material to support the EV revolution.
    [7] Yan W, Cao H, Zhang Y, et al. (2020) Rethinking Chinese supply resilience of critical metals in lithium-ion batteries. J Clean Prod 256: 120719. doi: 10.1016/j.jclepro.2020.120719
    [8] Forecasts (2021) Lithium, cobalt, graphite, and nickel. Benchmark Mineral Intelligence.
    [9] Tolomeo R, De Feo G, Adami R, et al. (2020) Application of life cycle assessment to lithium ion batteries in the automotive Sector. Sustainability 12: 4628. doi: 10.3390/su12114628
    [10] Melin HE (2019) The lithium-ion battery end-of-life market - A basesline study. Glob Batter Alliance 1-11.
    [11] Arshad F, Li L, Amin K, et al. (2020) A comprehensive review of the advancement in recycling the anode and electrolyte from spent lithium ion batteries. ACS Sustainable Chem Eng 8: 13527-13554. doi: 10.1021/acssuschemeng.0c04940
    [12] Li L, Qu W, Zhang X, et al. (2015) Succinic acid-based leaching system: A sustainable process for recovery of valuable metals from spent Li-ion batteries. J Power Sources 282: 544-551. doi: 10.1016/j.jpowsour.2015.02.073
    [13] Melin HE (2018) The lithium-ion battery end-of-life market 2018-2025. Circular Energy Storage. United Kingdom 6.
    [14] Campagnol N, Eddy J, Hagenbruch T, et al. (2018) Metal mining constraints on the electric mobility horizon, McKinsey Company.
    [15] Roskill (2021) Cathode and precursor materials: outlook to 2030, 1st ed 2021.
    [16] Merriman D, Liang L, Shang KG, et al. (2021) VW Power Day: A roadmap for e-mobility, battery manufacturing and smart grid rollout.
    [17] Gaines L, Dai Q, Vaughey JT, et al. (2021) Direct recycling R & D at the ReCell center. Recycling 6: 31. doi: 10.3390/recycling6020031
    [18] Ding Y, Cano ZP, Yu A, et al. (2019) Automotive Li-ion batteries: current status and future perspectives. Electrochem Energ Rev 2: 1-28. doi: 10.1007/s41918-018-0022-z
    [19] EFORE. Comparison of lithium-ion batteries 2020. Available from: https://www.efore.com/content/uploads/2020/12/Comparison_of_lithium_batteries_20201209.pdf.
    [20] The White House (2021) Building resilient supply chains, revitalizing american manufacturing, and fostering broad-based growth: 100-day reviews under executive order 14017.
    [21] FCAB (2021) National blueprint for lithium batteries 2021-2030. Available from: https://www.energy.gov/sites/default/files/2021-06/FCAB%20National%20Blueprint%20Lithium%20Batteries%200621_0.pdf.
    [22] NAATBatt (2021) Developing a supply chain for lithium-ion batteries in north America. NAATBatt - webinar.
    [23] Gaines L (2018) Lithium-ion battery recycling processes: research towards a sustainable course. Sustain Mater Technol 17: e00068.
    [24] Huang Y, Han G, Liu J, et al. (2016) A stepwise recovery of metals from hybrid cathodes of spent Li-ion batteries with leaching-flotation-precipitation process. J Power Sources 325: 555-564. doi: 10.1016/j.jpowsour.2016.06.072
    [25] Wang L, Wang X, Yang W (2020) Optimal design of electric vehicle battery recycling network - From the perspective of electric vehicle manufacturers. Appl Energy 275.
    [26] Vieceli N, Nogueira CA, Guimarães C, et al. (2018) Hydrometallurgical recycling of lithium-ion batteries by reductive leaching with sodium metabisulphite. Waste Manag 71: 350-361. doi: 10.1016/j.wasman.2017.09.032
    [27] Zhang X, Xie Y, Cao H, et al. (2014) A novel process for recycling and resynthesizing LiNi1/3Co1/3Mn1/3O2 from the cathode scraps intended for lithium-ion batteries. Waste Manag 34: 1715-1724. doi: 10.1016/j.wasman.2014.05.023
    [28] Rothermel S, Evertz M, Kasnatscheew J, et al. (2016) Graphite recycling from spent lithium-ion batteries. ChemSusChem 9: 3473-3484. doi: 10.1002/cssc.201601062
    [29] Ahmadi L, Young SB, Fowler M, et al. (2017) A cascaded life cycle: reuse of electric vehicle lithium-ion battery packs in energy storage systems. Int J Life Cycle Assess 22: 111-124. doi: 10.1007/s11367-015-0959-7
    [30] Pesaran A (2021) Supporting the Li-ion battery supply chain via PEV battery reuse, developing a supply chain for lithium-ion batteries in north America. NAATBatt - webinar.
    [31] Neubauer J, Smith K, Wood E, et al. (2015) Identifying and Overcoming Critical Barriers to Widespread Second Use of PEV Batteries. Natl Renew Energy Lab 23-62.
    [32] Heymans C, Walker SB, Young SB, et al. (2014) Economic analysis of second use electric vehicle batteries for residential energy storage and load-levelling. Energy Policy 71: 22-30. doi: 10.1016/j.enpol.2014.04.016
    [33] Dai Q, Spangenberger J, Ahmed S, et al. (2019) EverBatt: A closed-loop battery recycling cost and environmental impacts model. Argonne Natl Lab 1-88.
    [34] Alfaro-Algaba M, Ramirez FJ (2020) Techno-economic and environmental disassembly planning of lithium-ion electric vehicle battery packs for remanufacturing. Resour Conserv Recycl 154.
    [35] Gentilini L, Mossali E, Angius A, et al. (2020) A safety oriented decision support tool for the remanufacturing and recycling of post-use H & EVs lithium-ion batteries. Procedia CIRP 90: 73-78. doi: 10.1016/j.procir.2020.01.090
    [36] Rallo H, Benveniste G, Gestoso I, et al. (2020) Economic analysis of the disassembling activities to the reuse of electric vehicles Li-ion batteries. Resour Conserv Recycl 159.
    [37] Wegener K, Andrew S, Raatz A, et al. (2014) Disassembly of electric vehicle batteries using the example of the Audi Q5 hybrid system. Procedia CIRP 23: 155-160. doi: 10.1016/j.procir.2014.10.098
    [38] Herrmann C, Raatz A, Mennenga M, et al. (2012) Assessment of automation potentials for the disassembly of automotive lithium ion battery systems. In: Leveraging Technolnology for a sustainable world, Springer, Berlin, Heidelberg, 149-154.
    [39] McIntyre TJ, Harter JJ, Roberts TA (2019) Development and Operation of a High Throughput Computer Hard Drive Recycling Enterprise, 1-18.
    [40] Xia W, Jiang Y, Chen X, et al. (2021) Application of machine learning algorithms in municipal solid waste management: A mini review. Waste Manag Res 2021: 34269157.
    [41] European Parliament (2021) EU Legislation in Progress New EU regulatory framework for batteries Setting sustainability requirements.
    [42] Zheng R, Wang W, Dai Y, et al. (2017) A closed-loop process for recycling LiNixCoyMn(1-x−y)O2 from mixed cathode materials of lithium-ion batteries. Green Energy Environ 2: 42-50. doi: 10.1016/j.gee.2016.11.010
    [43] Liang HJ, Hou BH, Li WH, et al. (2019) Staging Na/K-ion de-/intercalation of graphite retrieved from spent Li-ion batteries: in operando X-ray diffraction studies and an advanced anode material for Na/K-ion batteries. Energy Environ Sci 12: 3575-3584. doi: 10.1039/C9EE02759A
    [44] Sloop S, Crandon L, Allen M, et al. (2020) A direct recycling case study from a lithium-ion battery recall. Sustain Mater Technol 25: e00152.
    [45] Shi Y, Chen G, Chen Z (2018) Effective regeneration of LiCoO2 from spent lithium-ion batteries: A direct approach towards high-performance active particles. Green Chem 20: 851-862. doi: 10.1039/C7GC02831H
    [46] Ji Y, Edwin E, Kpodzro CTJ, et al. (2021) Direct recycling technologies of cathode in spent lithium-ion batteries. Clean Technol Recycl 1: 124-151. doi: 10.3934/ctr.2021007
    [47] Zhang X, Bian Y, Xu S, et al. (2018) Innovative application of acid leaching to regenerate Li(Ni1/3Co1/3Mn1/3)O2 cathodes from spent lithium-ion batteries. ACS Sustainable Chem Eng 6: 5959-5968. doi: 10.1021/acssuschemeng.7b04373
    [48] Träger T, Friedrich B, Weyhe R (2015) Recovery concept of value metals from automotive lithium-ion batteries. Chem Ing Tech 87: 1550-1557. doi: 10.1002/cite.201500066
    [49] Zhou M, Li B, Li J, et al. (2021) Pyrometallurgical Technology in the Recycling of a Spent Lithium Ion Battery: Evolution and the Challenge. ACS ES & T Eng 1: 1369-1382
    [50] Velázquez-Martínez O, Valio J, Santasalo-Aarnio A, et al. (2019) A critical review of lithium-ion battery recycling processes from a circular economy perspective. Batteries 5: 5-7. doi: 10.3390/batteries5010005
    [51] Accurec Recycling GmbH (2021) Available from: https://accurec.de/lithium.
    [52] Lv W, Wang Z, Cao H, et al. (2018) A critical review and analysis on the recycling of spent lithium-ion batteries. ACS Sustainable Chem Eng 6: 1504-1521. doi: 10.1021/acssuschemeng.7b03811
    [53] Hu J, Zhang J, Li H, et al. (2017) A promising approach for the recovery of high value-added metals from spent lithium-ion batteries. J Power Sources 351: 192-199. doi: 10.1016/j.jpowsour.2017.03.093
    [54] Fan E, Li L, Zhang X, et al. (2018) Selective recovery of Li and fe from spent lithium-ion batteries by an environmentally friendly mechanochemical approach. ACS Sustainable Chem Eng 6: 11029-11035. doi: 10.1021/acssuschemeng.8b02503
    [55] Yang Y, Song S, Lei S, et al. (2019) A process for combination of recycling lithium and regenerating graphite from spent lithium-ion battery. Waste Manag 85: 529-537. doi: 10.1016/j.wasman.2019.01.008
    [56] Wang B, Lin XY, Tang Y, et al. (2019) Recycling LiCoO2 with methanesulfonic acid for regeneration of lithium-ion battery electrode materials. J Power Sources 436.
    [57] Yang Y, Xu S, He Y (2017) Lithium recycling and cathode material regeneration from acid leach liquor of spent lithium-ion battery via facile co-extraction and co-precipitation processes. Waste Manag 64: 219-227. doi: 10.1016/j.wasman.2017.03.018
    [58] Nayaka GP, Zhang Y, Dong P, et al. (2019) An environmental friendly attempt to recycle the spent Li-ion battery cathode through organic acid leaching. J Environ Chem Eng 7.
    [59] Gao W, Zhang X, Zheng X, et al. (2017) Lithium carbonate recovery from cathode scrap of spent lithium-ion battery: A closed-loop process. Environ Sci Technol 51: 1662-1669. doi: 10.1021/acs.est.6b03320
    [60] Golmohammadzadeh R, Rashchi F, Vahidi E (2017) Recovery of lithium and cobalt from spent lithium-ion batteries using organic acids: process optimization and kinetic aspects. Waste Manag 64: 244-254. doi: 10.1016/j.wasman.2017.03.037
    [61] Ghassa S, Farzanegan A, Gharabaghi M, et al. (2021) Iron scrap, a sustainable reducing agent for waste lithium ions batteries leaching: an environmentally friendly method to treating waste with waste. Resour Conserv Recycl 166: 105348. doi: 10.1016/j.resconrec.2020.105348
    [62] Wang F, Sun R, Xu J, et al. (2016) Recovery of cobalt from spent lithium ion batteries using sulphuric acid leaching followed by solid-liquid separation and solvent extraction. RSC Adv 6: 85303-85311. doi: 10.1039/C6RA16801A
    [63] Chan KH, Anawati J, Malik M, et al. (2021) Closed-loop recycling of lithium, cobalt, Nickel, and manganese from waste lithium-ion batteries of electric vehicles. ACS Sustainable Chem Eng 9: 4398-4410. doi: 10.1021/acssuschemeng.0c06869
    [64] Zeng X, Li J, Shen B (2015) Novel approach to recover cobalt and lithium from spent lithium-ion battery using oxalic acid. J Hazard Mater 295: 112-118. doi: 10.1016/j.jhazmat.2015.02.064
    [65] Li L, Fan E, Guan Y, et al. (2017) Sustainable recovery of cathode materials from spent lithium-ion batteries using lactic acid leaching system. ACS Sustainable Chem Eng 5: 5224-5233. doi: 10.1021/acssuschemeng.7b00571
    [66] Li L, Bian Y, Zhang X, et al. (2018) Economical recycling process for spent lithium-ion batteries and macro- and micro-scale mechanistic study. J Power Sources 377: 70-79. doi: 10.1016/j.jpowsour.2017.12.006
    [67] Ning P, Meng Q, Dong P, et al. (2020) Recycling of cathode material from spent lithium ion batteries using an ultrasound-assisted DL-malic acid leaching system. Waste Manag 103: 52-60. doi: 10.1016/j.wasman.2019.12.002
    [68] Tanong K, Coudert L, Mercier G, et al. (2016) Recovery of metals from a mixture of various spent batteries by a hydrometallurgical process. J Environ Manage 181: 95-107. doi: 10.1016/j.jenvman.2016.05.084
    [69] Natarajan S, Boricha AB, Bajaj HC (2018) Recovery of value-added products from cathode and anode material of spent lithium-ion batteries. Waste Manag 77: 455-465. doi: 10.1016/j.wasman.2018.04.032
    [70] Esmaeili M, Rastegar SO, Beigzadeh R, et al. (2020) Ultrasound-assisted leaching of spent lithium ion batteries by natural organic acids and H2O2. Chemosphere 254: 126670. doi: 10.1016/j.chemosphere.2020.126670
    [71] Montgomery DC (2017) Design and analysis of experiments, 8th ed., John Wiley & Sons.
    [72] Tapani VM (2012) Experimental optimization and response surfaces. Chemom Pract Appl 91-138.
    [73] Gaines L (2019) Profitable recycling of low-cobalt lithium-ion batteries will depend on new process developments. One Earth 1: 413-415. doi: 10.1016/j.oneear.2019.12.001
    [74] Zheng X, Zhu Z, Lin X, et al. (2018) A mini-review on metal recycling from spent lithium ion batteries. Engineering 4: 361-370. doi: 10.1016/j.eng.2018.05.018
    [75] GEM (2021) Available from: http://en.gem.com.cn/en/AboutTheGroup/index.html.
    [76] Brunp (2021) Available from: https://www.catl.com/en/solution/recycling/.
    [77] Putsche V, Witter E, Santhanagopalan S, et al. (2021) NAATBatt lithium-ion battery supply chain database. National Renewable Energy Laboratory. Version 1.
    [78] Larouche F, Tedjar F, Amouzegar K, et al. (2020) Progress and status of hydrometallurgical and direct recycling of Li-Ion batteries and beyond. Materials (Basel) 13.
    [79] Redwood materials (2021) Available from: https://www.redwoodmaterials.com/.
    [80] Intertek (2019) The future of battery technologies-part V environmental considerations for lithium batteries.
    [81] U.S. geological survey (2021) Mineral commodity summaries. Available from: https://pubs.usgs.gov/periodicals/mcs2021/mcs2021-cobalt.pdf.
    [82] Zeuner B (2018) An Obsolescing Bargain in a Rentier State: multinationals, artisanal miners, and cobalt in the Democratic Republic of Congo. Front Energy Res 6: 1-6. doi: 10.3389/fenrg.2018.00001
    [83] Tsurukawa N, Prakash S, Manhart A (2011) Social impacts of artisanal cobalt mining in Katanga, Democratic Republic of Congo. Ö ko-Inst EV - Inst Appl Ecol Freibg 49: 65.
    [84] Huijbregts MAJ, Steinmann ZJN, Elshout PMF, et al. (2017) ReCiPe2016: a harmonised life cycle impact assessment method at midpoint and endpoint level. Int J Life Cycle Assess 22: 138-147. doi: 10.1007/s11367-016-1246-y
    [85] Richa K, Babbitt CW, Nenadic NG, et al. (2017) Environmental trade-offs across cascading lithium-ion battery life cycles. Int J Life Cycle Assess 22: 66-81. doi: 10.1007/s11367-015-0942-3
    [86] Ciez RE, Whitacre JF (2019) Examining different recycling processes for lithium-ion batteries. Nat Sustain 2: 148-156. doi: 10.1038/s41893-019-0222-5
    [87] Wang S, Yu J (2021) A comparative life cycle assessment on lithium-ion battery: case study on electric vehicle battery in China considering battery evolution. Waste Manag Res 39: 156-164. doi: 10.1177/0734242X20966637
    [88] Richa K, Babbitt CW, Gaustad G (2017) Eco-efficiency analysis of a lithium-ion battery waste hierarchy inspired by circular economy. J Ind Ecol 21: 715-730. doi: 10.1111/jiec.12607
    [89] Jin H, Frost K, Sousa I, et al. (2020) Life cycle assessment of emerging technologies on value recovery from hard disk drives. Resour Conserv Recycl 104781.
    [90] EUR-Lex (2006) Directive 2006/66/EC of the European Parliament and of the Council of 6 September 2006 on batteries and accumulators and waste batteries and accumulators and repealing Directive 91/157/EEC. L 266.
    [91] Wang G, Zhao G, Wu W, et al, (2018) Cascade utilization and recycling of driving LiB, 2nd ed. Beijing: China Electric Power Press.
    [92] CBEA Summary of waste driving battery recycling technology and their profits 2018. Available from: http://www.cbea.com/dianchihuishou/201810/274400.html.
    [93] Green & low carbon development foundation. Research on the power battery recycling mechanism and policy for Shenzhen, 2018.
    [94] Wang X, Gaustad G, Babbitt CW, et al. (2014) Economies of scale for future lithium-ion battery recycling infrastructure. Resour Conserv Recycl 83: 53-62. doi: 10.1016/j.resconrec.2013.11.009
    [95] Electric vehicles revolution, China leads the global boom 2017. Available from: https://www.televisory.com/blogs/-/blogs/electric-vehicles-revolution-china-leads-the-global-boom.
    [96] EU (European Union) (2000) Directive 2000/53/EC of the European Parliament and of the council (2000.9.18), Brussels: European Union.
    [97] GPO (1996) US. Public Law 104-142-Mercury Containing and Rechargeable Battery Management Act.
    [98] CA Code (2006) Rechargeable battery recycling act of 2006.
    [99] New York State rechargeable battery law. New York environmental conservation law. In: Title 18. Rechargeable battery recycling, New York, 2010.
    [100] MN PCA (2015) Product stewardship for rechargeable batteries. St. Paul, MN, USA: Minnesota Pollution Control Agency.
    [101] Call2Recycle (2021) Available from: https://www.call2recycle.org/what-can-i-recycle/.
    [102] Mikolajczak C, Kahn M, White K, et al. (2012) Lithium-ion batteries hazard and use assessment. Springer Science & Business Media.
    [103] European Commission (2020) Regulation of the European Parliament and of the Council concerning batteries and waste batteries, repealing. Directive 2006/66/EC and amending Regulation (EU) No 2019/1020;0353.
    [104] Li L, Dababneh F, Zhao J (2018) Cost-effective supply chain for electric vehicle battery remanufacturing. Appl Energy 226: 277-286. doi: 10.1016/j.apenergy.2018.05.115
    [105] Hoyer C, Kieckhäfer K, Spengler TS (2015) Technology and capacity planning for the recycling of lithium-ion electric vehicle batteries in Germany. J Bus Econ 85: 505-544.
    [106] Tadaros M, Migdalas A, Samuelsson B, et al. (2020) Location of facilities and network design for reverse logistics of lithium-ion batteries in Sweden. Oper Res Int J.
    [107] Hendrickson TP, Kavvada O, Shah N, et al. (2015) Life-cycle implications and supply chain logistics of electric vehicle battery recycling in California. Environ Res Lett 10.
    [108] Zhalechian M, Torabi SA, Mohammadi M (2018) Hub-and-spoke network design under operational and disruption risks. Transp Res E 109: 20-43. doi: 10.1016/j.tre.2017.11.001
    [109] Paul SK, Sarker R, Essam D (2017) A quantitative model for disruption mitigation in a supply chain. Eur J Oper Res 257: 881-895. doi: 10.1016/j.ejor.2016.08.035
    [110] Paul SK, Sarker R, Essam D (2014) Real time disruption management for a two-stage batch production-inventory system with reliability considerations. Eur J Oper Res 237: 113-128. doi: 10.1016/j.ejor.2014.02.005
    [111] Lücker F, Seifert RW, Biçer I (2019) Roles of inventory and reserve capacity in mitigating supply chain disruption risk. Int J Prod Res 57: 1238-1249. doi: 10.1080/00207543.2018.1504173
    [112] Li Q, Zeng B, Savachkin A (2013) Reliable facility location design under disruptions. Comput Oper Res 40: 901-909. doi: 10.1016/j.cor.2012.11.012
    [113] Saha AK, Paul A, Azeem A, et al. (2020) Mitigating partial-disruption risk: A joint facility location and inventory model considering customers' preferences and the role of substitute products and backorder offers. Comput Oper Res 117.
    [114] Li-cycle. Spoke & hub technologies, mechanical & hydrometallurgical process, Li-cycle 2021. Available from: https://li-cycle.com.
    [115] Gao G, Luo X, Lou X, et al. (2019) Efficient sulfuric acid-vitamin C leaching system: Towards enhanced extraction of cobalt from spent lithium-ion batteries. J Mater Cycles Waste Manag 21: 942-949. doi: 10.1007/s10163-019-00850-4
    [116] Cheng Q, Chirdon WM, Lin M, et al. (2019) Characterization, modeling, and optimization of a single-step process for leaching metallic ions from LiNi 1/3 Co 1/3 Mn 1/3 O 2 cathodes for the recycling of spent lithium-ion batteries. Hydrometallurgy 185: 1-11. doi: 10.1016/j.hydromet.2019.01.003
    [117] Xie J, Huang K, Nie Z, et al. (2021) An effective process for the recovery of valuable metals from cathode material of lithium-ion batteries by mechanochemical reduction. Resour Conserv Recycl 168: 105261. doi: 10.1016/j.resconrec.2020.105261
    [118] Chen X, Ma H, Luo C, et al. (2017) Recovery of valuable metals from waste cathode materials of spent lithium-ion batteries using mild phosphoric acid. J Hazard Mater 326: 77-86. doi: 10.1016/j.jhazmat.2016.12.021
    [119] Musariri B, Akdogan G, Dorfling C, et al. (2019) Evaluating organic acids as alternative leaching reagents for metal recovery from lithium ion batteries. Miner Eng 137: 108-117. doi: 10.1016/j.mineng.2019.03.027
    [120] Fan E, Yang J, Huang Y, et al. (2020) Leaching mechanisms of recycling valuable metals from spent lithium-ion batteries by a malonic acid-based leaching system. ACS Appl Energy Mater 3: 8532-8542. doi: 10.1021/acsaem.0c01166
    [121] Roshanfar M, Golmohammadzadeh R, Rashchi F (2019) An environmentally friendly method for recovery of lithium and cobalt from spent lithium-ion batteries using gluconic and lactic acids. J Environ Chem Eng 7.
    [122] Sun LY, Liu BR, Wu T. et al. (2021) Hydrometallurgical recycling of valuable metals from spent lithium-ion batteries by reductive leaching with stannous chloride. Int J Miner Metall Mater 28: 991-1000
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