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Research article Special Issues

Reduction of greenhouse gas emissions in an imperfect production process under breakdown consideration

  • A long-run manufacturing system can experience machine breakdown at any time for various reasons such as unskilled labor or outdated machinery technology. In an integrated green inventory model, the produced green products cannot all be perfect throughout a cycle, particularly when machines malfunction. Therefore, an inspection policy is introduced to clean the production process from unusable defect products, the correctness of which depends on the discussion of the inspected errors. The perfect products detected via the inspection process are delivered to the retailer as well as the market. To transport green products, it is essential to control the capacity of the containers and the quantities of green products transported per batch. In this study, the greenhouse gas equivalence factor of CO2 emissions is calculated for all green products' manufacturing and transportation mediums. These types of energies are used in the manufacturing process: electricity, natural gas, and coal. Whereas within transportation, four transportation modes are considered: railways, roadways, airways, and waterways. The retailer can agree to transport their inventories to the customers' house according to their requirement by requiring a third-party local agency via outsourcing criteria. The model solves the problem of CO2 emissions through production and transportation within the machine breakdown.

    Citation: Bijoy Kumar Shaw, Isha Sangal, Biswajit Sarkar. Reduction of greenhouse gas emissions in an imperfect production process under breakdown consideration[J]. AIMS Environmental Science, 2022, 9(5): 658-691. doi: 10.3934/environsci.2022038

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  • A long-run manufacturing system can experience machine breakdown at any time for various reasons such as unskilled labor or outdated machinery technology. In an integrated green inventory model, the produced green products cannot all be perfect throughout a cycle, particularly when machines malfunction. Therefore, an inspection policy is introduced to clean the production process from unusable defect products, the correctness of which depends on the discussion of the inspected errors. The perfect products detected via the inspection process are delivered to the retailer as well as the market. To transport green products, it is essential to control the capacity of the containers and the quantities of green products transported per batch. In this study, the greenhouse gas equivalence factor of CO2 emissions is calculated for all green products' manufacturing and transportation mediums. These types of energies are used in the manufacturing process: electricity, natural gas, and coal. Whereas within transportation, four transportation modes are considered: railways, roadways, airways, and waterways. The retailer can agree to transport their inventories to the customers' house according to their requirement by requiring a third-party local agency via outsourcing criteria. The model solves the problem of CO2 emissions through production and transportation within the machine breakdown.



    In this paper, we consider the Schrödinger operators

    L=+V(x),xRn,n3,

    where Δ=ni=122xi and V(x) is a nonnegative potential belonging to the reverse Hölder class RHq for some qn2. Assume that f is a nonnegative locally Lq(Rn) integrable function on Rn, then we say that f belongs to RHq (1<q) if there exists a positive constant C such that the reverse Hölder's inequality

    (1|B(x,r)|B(x,r)|f(y)|qdy)1qC|B(x,r)|B(x,r)|f(y)|dy

    holds for x in Rn, where B(x,r) denotes the ball centered at x with radius r< [1]. For example, the nonnegative polynomial VRH, in particular, |x|2RH.

    Let the potential VRHq with qn2, and the critical radius function ρ(x) is defined as

    ρ(x)=supr>0{r:1rn2B(x,r)V(y)dy1},xRn. (1.1)

    We also write ρ(x)=1mV(x),xRn. Clearly, 0<mV(x)< when V0, and mV(x)=1 when V=1. For the harmonic oscillator operator (Hermite operator) H=Δ+|x|2, we have mV(x)(1+|x|).

    Thanks to the heat diffusion semigroup etL for enough good function f, the negative powers Lα2(α>0) related to the Schrödinger operators L can be written as

    Iαf(x)=Lα2f(x)=0etLf(x)tα21dt,0<α<n. (1.2)

    Applying Lemma 3.3 in [2] for enough good function f holds that

    Iαf(x)=RnKα(x,y)f(y)dy,0<α<n,

    and the kernel Kα(x,y) satisfies the following inequality

    Kα(x,y)Ck(1+|xy|(mV(x)+mV(y)))k1|xy|nα. (1.3)

    Moreover, we have Kα(x,y)C|xy|nα,0<α<n.

    Shen [1] obtained Lp estimates of the Schrödinger type operators when the potential VRHq with qn2. For Schrödinger operators L=Δ+V with VRHq for some qn2, Harboure et al. [3] established the necessary and sufficient conditions to ensure that the operators Lα2(α>0) are bounded from weighted strong and weak Lp spaces into suitable weighted BMOL(w) space and Lipschitz spaces when pnα. Bongioanni Harboure and Salinas proved that the fractional integral operator Lα/2 is bounded form Lp,(w) into BMOβL(w) under suitable conditions for weighted w [4]. For more backgrounds and recent progress, we refer to [5,6,7] and references therein.

    Ramseyer, Salinas and Viviani in [8] studied the fractional integral operator and obtained the boundedness from strong and weak Lp() spaces into the suitable Lipschitz spaces under some conditions on p(). In this article, our main interest lies in considering the properties of fractional integrals operator Lα2(α>0), related to L=Δ+V with VRHq for some qn2 in variable exponential spaces.

    We now introduce some basic properties of variable exponent Lebsegue spaces, which are used frequently later on.

    Let p():Ω[1,) be a measurable function. For a measurable function f on Rn, the variable exponent Lebesgue space Lp()(Ω) is defined by

    Lp()(Ω)={f:Ω|f(x)s|p(x)dx<},

    where s is a positive constant. Then Lp()(Ω) is a Banach space equipped with the follow norm

    fLp()(Ω):=inf{s>0:Ω|f(x)s|p(x)dx1}.

    We denote

    p:=essinfxΩp(x) and p+:=esssupxΩp(x).

    Let P(Rn) denote the set of all measurable functions p on Rn that take value in [1,), such that 1<p(Rn)p()p+(Rn)<.

    Assume that p is a real value measurable function p on Rn. We say that p is locally log-Hölder continuous if there exists a constant C such that

    |p(x)p(y)|Clog(e+1/|xy|),x,yRn,

    and we say p is log-Hölder continuous at infinity if there exists a positive constant C such that

    |p(x)p()|Clog(e+|x|),xRn,

    where p():=lim|x|p(x)R.

    The notation Plog(Rn) denotes all measurable functions p in P(Rn), which states p is locally log-Hölder continuous and log-Hölder continuous at infinity. Moreover, we have that p()Plog(Rn), which implies that p()Plog(Rn).

    Definition 1.1. [8] Assume that p() is an exponent function on Rn. We say that a measurable function f belongs to Lp(),(Rn), if there exists a constant C such that for t>0,

    Rntp(x)χ{|f|>t}(x)dxC.

    It is easy to check that Lp(),(Rn) is a quasi-norm space equipped with the following quasi-norm

    fp(),=inf{s>0:supt>0Rn(ts)p(x)χ{|f|>t}(x)dx1}.

    Next, we define LipLα,p() spaces related to the nonnegative potential V.

    Definition 1.2. Let p() be an exponent function with 1<pp+< and 0<α<n. We say that a locally integrable function fLipLα,p()(Rn) if there exist constants C1,C2 such that for every ball BRn,

    1|B|αnχBp()B|f(x)mBf|dxC1, (1.4)

    and for Rρ(x),

    1|B|αnχBp()B|f(x)|dxC2, (1.5)

    where mBf=1|B|Bf. The norm of space LipLα,p()(Rn) is defined as the maximum value of two infimum of constants C1 and C2 in (1.4) and (1.5).

    Remark 1.1. LipLα,p()(Rn)Lα,p()(Rn) is introduced in [8]. In particular, when p()=C for some constant, then LipLα,p()(Rn) is the usual weighted BMO space BMOβL(w), with w=1 and β=αnp [4].

    Remark 1.2. It is easy to see that for some ball B, the inequality (1.5) leads to inequality (1.4) holding, and the average mBf in (1.4) can be replaced by a constant c in following sense

    12fLipLα,p()supBRninfcR1|B|αnχBp()B|f(x)c|dxfLipLα,p().

    In 2013, Ramseyer et al. in [8] studied the Lipschitz-type smoothness of fractional integral operators Iα on variable exponent spaces when p+>αn. Hence, when p+>αn, it will be an interesting problem to see whether or not we can establish the boundedness of fractional integral operators Lα2(α>0) related to Schrödinger operators from Lebesgue spaces Lp() into Lipschitz-type spaces with variable exponents. The main aim of this article is to answer the problem above.

    We now state our results as the following two theorems.

    Theorem 1.3. Let potential VRHq for some qn/2 and p()Plog(Rn). Assume that 1<pp+<n(αδ0)+ where δ0=min{1,2n/q}, then the fractional integral operator Iα defined in (1.2) is bounded from Lp()(Rn) into LipLα,p()(Rn).

    Theorem 1.4. Let the potential VRHq with qn/2 and p()Plog(Rn). Assume that 1<pp+<n(αδ0)+ where δ0=min{1,2n/q}. If there exists a positive number r0 such that p(x)p when |x|>r0, then the fractional integral operator Iα defined in (1.2) is bounded from Lp(),(Rn) into LipLα,p()(Rn).

    To prove Theorem 1.3, we first need to decompose Rn into the union of some disjoint ball B(xk,ρ(xk))(k1) according to the critical radius function ρ(x) defined in (1.1). According to Lemma 2.6, we establish the necessary and sufficient conditions to ensure fLipLα,p()(Rn). In order to prove Theorem 1.3, by applying Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:

    (ⅰ) For every ball B=B(x0,r) with r<ρ(x0), then

    B|Iαf(x)c|dxC|B|αnχBp()fp();

    (ⅱ) For any x0Rn, then

    B(x0,ρ(x0))Iα(|f|)(x)dxC|B(x0,ρ(x0))|αnχB(x0,ρ(x0))p()fp().

    In order to check the conditions (ⅰ) and (ⅱ) above, we need to find the accurate estimate of kernel Kα(x,y) of fractional integral operator Iα (see Lemmas 2.8 and 2.9, then use them to obtain the proof of this theorem; the proof of the Theorem 1.4 proceeds identically).

    The paper is organized as follows. In Section 2, we give some important lemmas. In Section 3, we are devoted to proving Theorems 1.3 and 1.4.

    Throughout this article, C always means a positive constant independent of the main parameters, which may not be the same in each occurrence. B(x,r)={yRn:|xy|<r}, Bk=B(x0,2kR) and χBk are the characteristic functions of the set Bk for kZ. |S| denotes the Lebesgue measure of S. fg means C1gfCg.

    In this section, we give several useful lemmas that are used frequently later on.

    Lemma 2.1. [9] Assume that the exponent function p()P(Rn). If fLp()(Rn) and gLp()(Rn), then

    Rn|f(x)g(x)|dxrpfLp()(Rn)gLp()(Rn),

    where rp=1+1/p1/p+.

    Lemma 2.2. [8] Assume that p()Plog(Rn) and 1<pp+<, and p(x)p() when |x|>r0>1. For every ball B and fLp(), we have

    B|f(x)|dxCfLp(),χBLp(),

    where the constant C only depends on r0.

    Fo the following lemma see Corollary 4.5.9 in [10].

    Lemma 2.3. Let p()Plog(Rn), then for every ball BRn we have

    χBp()|B|1p(x),if|B|2n,xB,

    and

    χBp()|B|1p(),if|B|1.

    Lemma 2.4. Assume that p()Plog(Rn), then for all balls B and all measurable subsets S:=B(x0,r0)B:=B(x1,r1) we have

    χSp()χBp()C(|S||B|)11p,   χBp()χSp()C(|B||S|)11p+. (2.1)

    Proof. We only prove the first inequality in (2.1), and the second inequality in (2.1) proceeds identically. We consider three cases below by applying Lemma 2.3, and it holds that

    1) if |S|<1<|B|, then χSp()χBp()|S|1p(xS)|B|1p()(|S||B|)1(p)+=(|S||B|)11p;

    2) if 1|S|<|B|, then χSp()χBp()|S|1p()|B|1p()(|S||B|)1(p)+=(|S||B|)11p;

    3) if |S|<|B|<1, then χSp()χBp()|S|1p(xS)|B|1p(xS)|B|1p(xS)1p(xB)C(|S||B|)1(p)+=C(|S||B|)11p, where xSS and xBB.

    Indeed, since |xBxS|2r1, by using the local-Hölder continuity of p(x) we have

    |1p(xS)1p(xB)|log1r1log1r1log(e+1|xSxB|)log1r1log(e+12r1)C.

    We end the proof of this lemma.

    Remark 2.1. Thanks to the second inequality in (2.1), it is easy to prove that

    χ2Bp()CχBp().

    Lemma 2.5. [1] Suppose that the potential VBq with qn/2, then there exists positive constants C and k0 such that

    1) ρ(x)ρ(y) when |xy|Cρ(x);

    2) C1ρ(x)(1+|xy|ρ(x))k0ρ(y)Cρ(x)(1+|xy|ρ(x))k0/(k0+1).

    Lemma 2.6. [11] There exists a sequence of points {xk}k=1 in Rn such that Bk:=B(xk,ρ(xk)) satisfies

    1) Rn=kBk,

    2) For every k1, then there exists N1 such that card {j:4Bj4Bk}N.

    Lemma 2.7. Assume that p()P(Rn) and 0<α<n. Let sequence {xk}k=1 satisfy the propositions of Lemma 2.6. Then a function fLipLα,p()(Rn) if and only if f satisfies (1.4) for every ball, and

    1|B(xk,ρ(xk))|αnχB(xk,ρ(xk))p()B(xk,ρ(xk))|f(x)|dxC,forallk1. (2.2)

    Proof. Let B:=B(x,R) denote a ball with center x and radius R>ρ(x). Noting that f satisfies (1.4), and thanks to Lemma 2.6 we obtain that the set G={k:BBk} is finite.

    Applying Lemma 2.5, if zBkB, we get

    ρ(xk)Cρ(z)(1+|xkz|ρ(xk))k0C2k0ρ(z)C2k0ρ(x)(1+|xz|ρ(x))k0k0+1C2k0ρ(x)(1+Rρ(x))C2k0R.

    Thus, for every kG, we have BkCB.

    Thanks to Lemmas 2.4 and 2.6, it holds that

    B|f(x)|dx=BkBk|f(x)|dx=kG(BBk)|f(x)|dxkGBBk|f(x)|dxkGBk|f(x)|dxCkG|Bk|αnχBkp()C|B|αnχBp().

    The proof of this lemma is completed.

    Corollary 1. Assume that p()P(Rn) and 0<α<n, then a measurable function fLipLα,p() if and only if f satisfies (1.4) for every ball B(x,R) with radius R<ρ(x) and

    1|B(x,ρ(x))|αnχB(x,ρ(x))p()B(x,ρ(x))|f(x)|dxC. (2.3)

    Let kt(x,y) denote the kernel of heat semigroup etL associated to L, and Kα(x,y) be the kernel of fractional integral operator Iα, then it holds that

    Kα(x,y)=0kt(x,y)tα2dt. (2.4)

    Some estimates of kt are presented below.

    Lemma 2.8. [12] There exists a constant C such that for N>0,

    kt(x,y)Ctn/2e|xy|2Ct(1+tρ(x)+tρ(y))N,x,yRn.

    Lemma 2.9. [13] Let 0<δ<min(1,2nq). If |xx0|<t, then for N>0 the kernel kt(x,y) defined in (2.4) satisfies

    |kt(x,y)kt(x0,y)|C(|xx0|t)δtn/2e|xy|2Ct(1+tρ(x)+tρ(y))N,

    for all x,y and x0 in Rn.

    In this section, we are devoted to the proof of Theorems 1.3 and 1.4. To prove Theorem 1.3, thanks to Corollary 1 and Remark 1.2, we only need to prove that the following two conditions hold:

    (ⅰ) For every ball B=B(x0,r) with r<ρ(x0), then

    B|Iαf(x)c|dxC|B|αnχBp()fp();

    (ⅱ) For any x0Rn, then

    B(x0,ρ(x0))Iα(|f|)(x)dxC|B(x0,ρ(x0))|αnχB(x0,ρ(x0))p()fp().

    We now begin to check that these conditions hold. First, we prove (ⅱ).

    Assume that B=B(x0,R) and R=ρ(x0). We write f=f1+f2, where f1=fχ2B and f2=fχRn2B. Hence, by the inequality (1.3), we have

    BIα(|f1|)(x)dx=BIα(|fχ2B|)(x)dxCB2B|f(y)||xy|nαdydx.

    Applying Tonelli theorem, Lemma 2.1 and Remark 1.2, we get the following estimate

    BIα(|f1|)(x)dxC2B|f(y)|Bdx|xy|nαdyCRα2B|f(y)|dyC|B|αnχBp()fp(). (3.1)

    To deal with f2, let xB and we split Iαf2 as follows:

    Iαf2(x)=R20etLf2(x)tα21dt+R2etLf2(x)tα21dt:=I1+I2.

    For I1, if xB and yRn2B, we note that |x0y|<|x0x|+|xy|<C|xy|. By Lemma 2.8, it holds that

    I1=|R20Rn2Bkt(x,y)f(y)dytα21dt|CR20Rn2Btn2e|xy|2t|f(y)|dytα21dtCR20tn+α21Rn2B(t|xy|2)M/2|f(y)|dydtCR20tMn+α21dtRn2B|f(y)||x0y|Mdy,

    where the constant C only depends the constant M.

    Applying Lemma 2.1 to the last integral, we get

    Rn2B|f(y)||x0y|Mdy=i=12i+1B2iB|f(y)||x0y|Mdyi=1(2iR)M2i+1B|f(y)|dyCi=1(2iR)Mχ2i+1Bp()fp().

    By using Lemma 2.4, we arrive at the inequality

    Rn2B|f(y)||x0y|MdyCi=1(R)M(2i)nnp+MχBp()fp()CRMfp()χBp(). (3.2)

    Here, the series above converges when M>nnp+. Hence, for such M,

    |R20etLf2(x)tα21dt|CRMfp()χBp()R20tMn+α21dtC|B|αn1fp()χBp().

    For I2, thanks to Lemma 2.8, we may choose M as above and NM, then it holds that

    |R2etLf2(x)tα22dt|=|R2Rn2Bkt(x,y)f(y)dytα22dt|CR2Rn2BtαnN22ρ(x)Ne|xy|2t|f(y)|dydtCρ(x)NR2tαnN22Rn2B(t|xy|2)M/2|f(y)|dydt.

    As xB, thanks to Lemma 2.5, ρ(x)ρ(x0)=R. Hence we have

    |R2etLf2(x)tα21dt|CRNR2tM+αnN21dtRn2B|f(y)||x0y|Mdy.

    Since M+αnN<0, the integral above for variable t converges, and by applying inequality (3.2) we have

    |R2etLf2(x)tα21dt|C|B|αn1fp()χBp(),

    thus we have proved (ⅱ).

    We now begin to prove that the condition (ⅰ) holds. Let B=B(x0,r) and r<ρ(x0). We set f=f1+f2 with f1=fχ2B and f2=fχRn2B. We write

    cr=r2etLf2(x0)tα21dt. (3.3)

    Thanks to (3.1), it holds that

    B|Iα(f(x))cr|BIα(|f1|)(x)dx+B|Iα(f2)(x)cr|dxC|B|αn1χBp()fp()+B|Iα(f2)(x)cr|dx.

    Let xB and we split Iαf2(x) as follows:

    Iαf2(x)=r20etLf2(x)tα21dt+r2etLf2(x)tα21dt:=I3+I4.

    For I3, by the same argument it holds that

    I3=|r20etLf2(x)tα21dt|C|B|αn1fp()χBp().

    For I4, by Lemma 2.9 and (3.3), it follows that

    |r2etLf2(x)tα21dtcr|r2Rn2B|kt(x,y)kt(x0,y)||f(y)|dytα21dtCδr2Rn2B(|xx0|t)δtn/2e|xy|2Ct|f(y)|dytα21dtCδrδRn2B|f(y)|r2t(nα+δ)/2e|xy|2Ctdttdy.

    Let s=|xy|2t, then we obtain the following estimate

    |r2etLf2(x)tα21dtcr|CδrδRn2B|f(y)||xy|nα+δdy0snα+δ2esCdss.

    Notice that the integral above for variable s is finite, thus we only need to compute the integral above for variable y. Thanks to inequality (3.2), it follows that

    |r2etLf2(x)tα21dtcr|CδrδRn2B|f(y)||xy|nα+δdyCi=1Rαn(2i)αnp+δχBp()fp()C|B|αnnfp()χBp(),

    so (ⅰ) is proved.

    Remark 3.1. By the same argument as the proof of Theorem 1.3, thanks to Lemma 2.2 we immediately obtained that the conclusions of Theorem 1.4 hold.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Ping Li is partially supported by NSFC (No. 12371136). The authors would like to thank the anonymous referees for carefully reading the manuscript and providing valuable suggestions, which substantially helped in improving the quality of this paper. We also thank Professor Meng Qu for his useful discussions.

    The authors declare there are no conflicts of interest.



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