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Research article Special Issues

Statistical inference for a competing failure model based on the Wiener process and Weibull distribution


  • Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters μ and σ2 were provided and the inverse estimation of parameters η and β were derived. The exact confidence intervals for parameters μ, σ2, and β were obtained. Furthermore, the generalized confidence interval of parameter η was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the pth percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.

    Citation: Peihua Jiang, Longmei Shi. Statistical inference for a competing failure model based on the Wiener process and Weibull distribution[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 3146-3164. doi: 10.3934/mbe.2024140

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  • Competing failure models with degradation phenomena and sudden failures are becoming more and more common and important in practice. In this study, the generalized pivotal quantity method was proposed to investigate the modeling of competing failure problems involving both degradation and sudden failures. In the competing failure model, the degradation failure was modeled through a Wiener process and the sudden failure was described as a Weibull distribution. For point estimation, the maximum likelihood estimations of parameters μ and σ2 were provided and the inverse estimation of parameters η and β were derived. The exact confidence intervals for parameters μ, σ2, and β were obtained. Furthermore, the generalized confidence interval of parameter η was obtained through constructing the generalized pivotal quantity. Using the substitution principle, the generalized confidence intervals for the reliability function, the pth percentile of lifetime, and the mean time to failure were also obtained. Simulation technique was carried out to evaluate the performance of the proposed generalized confidence intervals. The simulation results showed that the proposed generalized confidence interval was effective in terms of coverage percentage. Finally, an example was presented to illustrate the application of the proposed method.



    Scheduling models with setup times are widely used in manufacture and operational processes (see Allahverdi et al. [1] and Allahverdi [2]). Koulamas and Kyparisis [3,4] and Biskup and Herrmann [5] investigated single-machine scheduling with past-sequence-dependent setup times (~psdst). They showed that several regular objective function minimizations remain polynomially solvable. Wang [6] and Wang and Li [7] examined single-machine problems with learning effects and ~psdst. Hsu et al. [8] studied unrelated parallel machine scheduling problems with learning effects and ~psdst. They proved that the total completion time minimization remains polynomially solvable. Cheng et al. [9] investigated scheduling problems with ~psdst and deterioration effects in a single machine. Huang et al. [10] and Wang and Wang [11] studied scheduling jobs with ~psdst, learning and deterioration effects. They showed that the single-machine makespan and the sum of the αth (α>0) power of job completion times minimizations remain polynomially solvable. Wang et al. [12] dealt with scheduling with ~psdst and deterioration effects. Under job rejection, they showed that the the sum of scheduling cost and rejection cost minimization can be solved in polynomial time.

    In the real production scheduling, the jobs often have due dates (see Gordon et al. [13,14] and the recent survey papers Rolim and Nagano [15], and Sterna [16]). Recently, Wang [17] and Wang et al. [18] studied single-machine scheduling problems with ~psdst and due-date assignment. Under common, slack and different due-date assignment methods, Wang [17] proved that the linear weighted sum of earliness-tardiness, number of early and delayed jobs, and due date penalty minimization can be solved in polynomial time. Under common and slack due date assignment methods, Wang et al. [18] showed that the weighted sum of earliness, tardiness and due date minimization can be solved in polynomial time, where the weights are position-dependent weights. The real application of the position-dependent weights can be found in production services and resource utilization (see Brucker [19], Liu et al. [20] and Jiang et al. [21]). Hence, it would be interesting to investigate due date assignment scheduling with ~psdst and position-dependent weights. The purpose of this article is to determine the optimal due dates and job sequence to minimize the weight sum of generalized earliness-tardiness penalties, where the weights are position-dependent weights. The contributions of this study are given as follows:

    We focus on the due date assignment single-machine scheduling problems with ~psdst and position-dependent weights;

    We provide an analysis for the non-regular objective function (including earliness, tardiness, number of early and delayed jobs, and due date cost);

    We derive the structural properties of the position-dependent weights and show that three due date assignments can be solved in polynomial time, respectively.

    The problem formulation is described in Section 2. Three due-date assignments are discussed in Section 3. An example is presented in Section 4. In Section 5, the conclusions are given.

    The symbols used throughout the article are introduced in Table 1.

    Table 1.  Symbols used in this article.
    Symbol Meaning
    N number of jobs
    Jl index of job
    pl processing time of Jl
    ~psdst past-sequence-dependent setup times
    sl setup time of ~psdst of Jl
    Cl completion time of Jl
    β a normalizing constant
    dl due date of Jl
    d common due date
    q common flow allowance
    [l] lth position in a sequence
    Ll=Cldl lateness of Jl
    Ul earliness indicator viable of Jl
    Vl tardiness indicator viable of job Jl
    ζl positional-dependent weight of lateness cost
    ηl (θl) positional-dependent weight of earliness (tardiness) indicator viable
    ϑl positional-dependent weight of due date cost
    ϱ sequence of all jobs
    ~con (~slk,~dif) common (slack, different) due date

     | Show Table
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    Suppose there are N independent jobs ˜V={J1,J2,,JN} need to be processed on a single-machine. The ~psdst setup time s[l] of job J[l] is s[l]=βl1j=1p[j], where β0 is a normalizing constant, s[1]=0, and βl1j=1p[j]+p[l] is the total processing requirement of job J[l]. Let Ll=Cldl denote the lateness of job Jl, Ul (Vl) be earliness (tardiness) indicator viable of job Jl, i.e., if Cl<dl, Ul=1, otherwise, Ul=0; if Cl>dl, Vl=1, otherwise, Vl=0.

    For the common (~con) due date assignment, dl=d (l=1,2,,N) and d is a decision variable. For the slack (~slk) due date assignment, dl=sl+pl+q and q is a decision variable. For the different due date (~dif) assignment, dl is a decision variable for l=1,2,,N. The target is to determine dl and a sequence ϱ such that is minimized.

    M=Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), (1)

    where ζl0, ηl0, ηl0 and δl0 are given positional-dependent weight constants. From Pinedo [22], the problem can be defined as:

    1|~psdst,H|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), (2)

    where H{~con,~slk,~dif}. The literature review related to the scheduling problems with ~psdst and due date assignment is given in Table 2. For a given sequence ϱ=(J[1],J[2],,J[N]), from (Wang [17]), we have

    C[l]=lj=1(s[j]+p[j])=lj=1[1+β(lj)]p[j],l=1,2,,N. (3)
    Table 2.  Problems with ~psdst and due date assignment.
    Problem Complexity Reference
    1|~psdst,~con|Nl=1(˜αEl+˜δTl+˜ηlUl+˜θlVl+˜ϑd) O(N4) Wang [17]
    1|~psdst,~con|Nl=1(˜αEl+˜δTl+˜ϑd) O(NlogN) Wang [17]
    1|~psdst,~slk|Nl=1(˜αEl+˜δTl+˜ηlUl+˜θlVl+˜ϑq) O(N4) Wang [17]
    1|~psdst,~slk|Nl=1(˜αEl+˜δTl+˜ϑq) O(NlogN) Wang [17]
    1|~psdst,~dif|Nl=1(˜αEl+˜δTl+˜ηlUl+˜θlVl+˜ϑdj) O(NlogN) Wang [17]
    1|~psdst,~con|Nl=1ζl|L[l]|+˜ϑd O(NlogN) Wang et al. [18]
    1|~psdst,~slk|Nl=1ζl|L[l]|+˜ϑq O(NlogN) Wang et al. [18]
    1|~psdst,~con|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) O(N4) This paper
    1|~psdst,~con|Nl=1(ζl|L[l]|+ϑld[l]) O(NlogN) This paper
    1|~psdst,~slk|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) O(N4) This paper
    1|~psdst,~slk|Nl=1(ζl|L[l]|+ϑld[l]) O(NlogN) This paper
    1|~psdst,~dif|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) O(NlogN) This paper

     | Show Table
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    where ˜α,˜δ,˜ϑ are given constants, ˜ηl (˜θl) is the earliness (tardiness) penalty of job Jl, El=max{0,dlCl} (Tl=max{0,Cldl}) is the earliness (tardiness) of job Jl.

    Lemma 1. For 1|~psdst,H|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) (H{~con,~slk,~dif}), an optimal sequence exists such that the first job is processed at time zero and contains no machine idle time.

    Proof. The result is obvious (see Brucker [19] and Liu et al. [20]).

    Lemma 2. For any given sequence ϱ, the optimal d is equal to the completion time of some job, i.e., d=C[a], a=1,2,,N.

    Proof. For any given sequence ϱ=(J[1],J[2],,J[N]), suppose that d is not equal to the completion time of some job, i.e., C[a]<d<C[a+1], 0a<n, C[0]=0, we have

    M=al=1ζl(dC[l])+Nl=a+1ζl(C[l]d)+aj=1ηl+nj=a+1θl+Nl=1dϑl.

    (i) When d=C[a], we have

    M1=al=1ζl(C[a]C[l])+Nl=a+1ζl(C[l]C[a])+a1l=1ηl+nl=a+1θl+Nl=1C[a]ϑl.

    (ii) When d=C[a+1], we have

    M2=al=1ζl(C[a+1]C[l])+Nl=a+1ζl(C[l]C[a+1])+al=1ηl+nl=a+2θl+Nl=1C[a+1]ϑl,
    MM1=al=1ζl(dC[a])Nl=a+1ζl(dC[a])+ηa+Nl=1ϑl(dC[a])=(al=1ζlNl=a+1ζl+Nl=1ϑl)(dC[a])+ηa

    and

    MM2=al=1ζl(dC[a+1])Nl=a+1ζl(dC[a+1])+θa+1+Nl=1ϑl(dC[a+1])=(al=1ζlNl=a+1ζl+Nl=1ϑl)(dC[a+1])+θa+1.

    If al=1ζlNl=a+1ζl+Nl=1ϑl0 and C[a]<d<C[a+1], then MM10; If al=1ζlNl=a+1ζl+Nl=1ϑl0 and C[a]<d<C[a+1], then MM20. Therefore, d is the completion time of some job.

    Lemma 3. For any given sequence ϱ=(J[1],J[2],,J[N]), if θl=ϑl=0 (l=1,2,N), there exists an optimal common due date d=C[a], where a is determined by

    a1l=1ζlNl=aζl+Nl=1ϑl0 (4)

    and

    al=1ζlNl=a+1ζl+Nl=1ϑl0. (5)

    Proof. From Lemma 2, when d=C[a], we have

    M=a1l=1ζl(C[a]C[l])+Nl=a+1ζl(C[l]C[a])+Nl=1C[a]ϑl.

    (i) When d reduces ε (i.e., d=C[a]ε), we have

    M=a1l=1ζl(C[a]εC[l])+Nl=aζl(C[l]C[a]+ε)+Nl=1(C[a]ε)ϑl.

    (ii) When d increases ε (i.e., d=C[a]+ε), we have

    M=al=1ζl(C[a]+εC[l])+Nl=a+1ζl(C[l]C[a]ε)+Nl=1(C[a]+ε)ϑl.

    Hence, we have

    MM=ε(a1l=1ζlNl=aζl+Nl=1ϑl)0
    MM=ε(al=1ζlNl=a+1ζl+Nl=1ϑl)0,

    i.e., a is determined by a1l=1ζlNl=aζl+Nl=1ϑl0 and al=1ζlNl=a+1ζl+Nl=1ϑl0.

    From Lemma 2, if d=C[a], the objective function is:

    M=Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+dϑl)=a1l=1ζl(C[a]C[l])+Nl=a+1ζl(C[l]C[a])+a1l=1ηl+Nl=a+1θl+Nl=1C[a]ϑl=a1l=1ζl{aj=1[1+β(aj)]p[j]lj=1[1+β(lj)]p[j]}+Nl=a+1ζl{lj=1[1+β(lj)]p[j]aj=1[1+β(aj)]p[j]}+a1l=1ηl+Nl=a+1θl+Nl=1ϑl{aj=1[1+β(aj)]p[j]}=Nl=1Ψlp[l]+a1l=1ηl+Nl=a+1θl, (6)

    where

    Ψl={β(a1)ζ1+β(a2)ζ2+β(a3)ζ3++βζa1+βζa+1+2βζa+2++β(Na)ζN+[1+β(a1)]Nj=1ϑj,l=1,(1+β(a2))ζ1+β(a2)ζ2+β(a3)ζ3++βζa1+βζa+1+2βζa+2++β(Na)ζN+[1+β(a2)]Nj=1ϑj,l=2,(1+β(a3))(ζ1+ζ2)+β(a3)ζ3+βζa1+βζa+1+2βζa+2++β(Na)ζN+[1+β(a3)]Nj=1ϑj,l=3,(1+β)(ζ1+ζ2++ζa2)+βζa1+βζa+1+2βζa+2++β(Na)ζN+(1+β)Nj=1ϑj,l=a1,ζ1+ζ2++ζa1+βζa+1+2βζa+2++β(Na)ζN+Nj=1ϑj,l=a,ζa+1+(1+β)ζa+2+(1+2β)ζa+3++(1+β(Na1))ζN,l=a+1,ζa+2+(1+β)ζa+3+(1+2β)ζa+4++(1+β(Na2))ζN,l=a+2,ζN1+(1+β)ζN,N1,ζN,N. (7)

    Let xl,r=1 if Jl is placed in rth position, and xl,r=0; otherwise. From Eq (6), the optimalsequence of 1|~psdst,~con|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) can be formulatedasthe following assignmentproblem:

    Min Nl=1Nr=1Θl,rxl,r (8)
    s.t.{Nh=1xl,r=1,r=1,2,...,N,Nr=1xl,r=1,l=1,2,...,N,xl,r=0or1, (9)

    where

    Θl,r={Ψrpl+ηr,r=1,2,...,a1,Ψrpl,r=a,Ψrpl+θr,r=a+1,a+2,...,N, (10)

    and Ψr is given by Eq (7).

    Based on the above analysis, to solve 1|~psdst,~con|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), Algorithm 1 was summarized as follows:

    Algorithm 1
    Require: β, pl,ζl,ηl,θl,ϑl for 1lN.
    Ensure: An optimal sequence ϱ, optimal common due date d.
    Step 1. For each a (a=1,2,,N), calculate Ψr (see Eq (7)) and Θl,r (see Eq (10)), to solve the assignment problem (8)–(10), a suboptimal sequence ϱ(a) and objective function value M(a) can be obtained.
    Step 2. The (global) optimal sequence (i.e., ϱ) is the one with the minimum value
    M=min{M(a)|a=1,2,,N}.
    Step 3. Set d=C[a].

     | Show Table
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    Theorem 1. The 1|~psdst,~con|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) can be solved by Algorithm 1, and time complexity was O(N4).

    Proof. The correctness of Algorithm 1 follows the above analysis. In Step 1, for each a, solving the assignment problem needs O(N3) time; Steps 2 and 3 require O(N) time; a=1,2,,N. Therefore, the total time complexity was O(N4).

    Lemma 4. (Hardy et al. [23]). "The sum of products Nl=1albl is minimized if sequence a1,a2,,aN is ordered nondecreasingly and sequence b1,b2,,bN is ordered nonincreasingly or vice versa."

    If ηl=θl=0, a can be determined by Lemma 3 (see Eqs (4) and (5)), We

    M=Nl=1(ζl|L[l]|+ϑld[l])=Nl=1Ψlp[l], (11)

    where Ωj is given by Eq (6).

    Equation (11) can be minimized by Lemma 4 in O(NlogN) time (i.e., al=Ψl,bl=pl), hence, to solve 1|~psdst,~con|Nl=1(ζl|L[l]|+ϑld[l]), the following algorithm was summarized as follows:

    Theorem 2. The 1|~psdst,~con|Nl=1(ζl|L[l]|+ϑld[l]) can be solved by Algorithm 2, and time complexity was O(NlogN).

    Algorithm 2
    Require: β, pl,ζl,ϑl for 1lN.
    Ensure: An optimal sequence ϱ, optimal common due date d.
    Step 1. Calculate a by Lemma 3 (see Eqs (4) and (5)).
    Step 2. By using Lemma 4 (let al=Ψl,bl=pl) to determine the optimal job sequence (i.e., ϱ), i.e., place the largest pl at the smallest Ψl position, place the second largest pl at the second smallest Ψl position, etc.
    Step 3. Set d=C[a].

     | Show Table
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    Similarly, we have

    Lemma 5. For any given sequence ϱ of 1|~psdst,~slk|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), an optimal sequence exists in which

    1) C[l]d[l] implies C[l1]d[l1] and C[l]d[l] implies C[l+1]d[l+1] for all l;

    2) the optimal q is equal to the completion time of some job, i.e., q=C[b1], b=1,2,,N.

    Lemma 6. For any given sequence ϱ=(J[1],J[2],,J[N]), if θl=ϑl=0 (l=1,2,N), there exists an optimal common due date q=C[b1], where b is determined by

    b1l=1ζlNl=bζl+Nl=1ϑl0 (12)

    and

    bl=1ζlNl=b+1ζl+Nl=1ϑl0. (13)

    Proof. From Lemma 5, when q=C[b1], we have

    M=b1l=1ζl(s[b]+p[b]+C[b1]C[l])+Nl=b+1ζl(C[l]s[b]p[b]C[b1])+Nl=1ϑl(s[b]+p[b]+C[b1]).

    (i) When q reduces ε (i.e., q=C[b1]ε), we have

    M=b1l=1ζl(s[b]+p[b]+C[b1]εC[l])+Nl=bζl(C[l]s[b]p[b]C[b1]+ε)+Nl=1(s[b]+p[b]+C[b1]ε)ϑl.

    (ii) When q increases ε (i.e., q=C[b1]+ε), we have

    M=bl=1ζl(s[b]+p[b]+C[b1]+εC[l])+Nl=b+1ζl(C[l]s[b]p[b]C[b1]ε)+Nl=1(s[b]+p[b]+C[b1]+ε)ϑl.

    Hence, we have

    MM=ε(b1l=1ζlNl=bζl+Nl=1ϑl)0
    MM=ε(bl=1ζlNl=b+1ζl+Nl=1ϑl)0,

    i.e., b is determined by b1l=1ζlNl=bζl+Nl=1ϑl0 and bl=1ζlNl=b+1ζl+Nl=1ϑl0.

    From Lemma 5, if q=C[b1] (i.e., d[l]=s[l]+p[l]+C[b1]), the objective function is:

    M=Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l])=b1l=1ζl(s[l]+p[l]+C[b1]C[l])+Nl=b+1ζl(C[l]s[l]p[l]C[b1])+b1l=1ηl+Nl=b+1θl+Nl=1(s[l]+p[l]+C[b1])ϑl=b1l=1ζl(C[b1]C[l1])+Nl=b+1ζl(C[l1]C[b1])+b1l=1ηl+Nl=b+1θl+Nl=1(s[l]+p[l])ϑl+Nl=1C[b1]ϑl=b1l=1ζl{b1j=1[1+β(b1j)]p[j]l1j=1[1+β(l1j)]p[j]}+Nl=b+1ζl{l1j=1[1+β(l1j)]p[j]b1j=1[1+β(b1j)]p[j]}+b1l=1ηl+Nl=b+1θl+Nl=1(βl1j=1p[j]+p[l])ϑl+Nl=1ϑl{b1j=1[1+β(b1j)]p[j]}=Nl=1Φlp[l]+b1l=1ηl+Nl=b+1θl, (14)

    where

    Φl={(1+β(b2))ζ1+β(b2)ζ2+β(b3)ζ3++βζb1+βζb+1+2βζb+2++β(Nb)ζN+[1+β(b2)]Nj=1ϑj+ϑ1+βNj=2ϑj,l=1,(1+β(b3))(ζ1+ζ2)+β(b3)ζ3+β(b4)ζ4++βζb1+βζb+1+2βζb+2++β(Nb)ζN+[1+β(b3)]Nj=1ϑj+ϑ2+βNj=3ϑj,l=2,(1+β(b4))(ζ1+ζ2+ζ3)+β(b4)ζ4++βζb1+βζb+1+2βζb+2++β(Nb)ζN+[1+β(b4)]Nj=1ϑj+ϑ3+βNj=4ϑj,l=3,(1+β)(ζ1+ζ2++ζb2)+βζb1+βζb+1+2βζb+2++β(Nb)ζN+(1+β)Nj=1ϑj+ϑb2+βNj=b1ϑj,l=b2,ζ1+ζ2++ζb1+βζb+1+2βζb+2++β(Nb)ζN+Nj=1ϑj+ϑb1+βNj=bϑj,l=b1,ζb+1+(1+β)ζb+2+(1+2β)ζb+3++(1+β(Nb1))ζN+ϑb+βNj=b+1ϑj,l=b,ζb+2+(1+β)ζb+3+(1+2β)ζb+4++(1+β(Nb2))ζN+ϑb+1+βNj=b+2ϑj,l=b+1,ζN+ϑN1+βϑN,N1,ϑN,N. (15)

    Similarly, from Eq (14), the optimal sequence of 1|~psdst,~slk|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) can be obtained as follows:

    Min Nl=1Nr=1Ξl,rxl,r (16)
    s.t.{Nh=1xl,r=1,r=1,2,...,N,Nr=1xl,r=1,l=1,2,...,N,xl,r=0or1, (17)

    where

    Ξl,r={Φrpl+ηr,r=1,2,...,b1,Φrpl,r=b,Φrpl+θr,r=b+1,b+2,...,N, (18)

    and Φr is given by (15).

    Similarly, to solve 1|~psdst,~slk|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), the following algorithm can be proposed:

    Theorem 3. The 1|~psdst,~slk|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) can be solved by Algorithm 3, and time complexity was O(N4).

    Algorithm 3
    Require: β, pl,ζl,ηl,θl,ϑl for 1lN.
    Ensure: An optimal sequence ϱ, optimal common flow allowance q.
    Step 1. For each b (b=1,2,,N), calculate Φr (see Eq (15)) and Ξl,r (see Eq (18)), to solve the assignment problem (16)–(18), a suboptimal sequence ϱ(b) and objective function value M(b) can be obtained.
    Step 2. The (global) optimal sequence (i.e., ϱ) is the one with the minimum value
    M=min{M(b)|b=1,2,,N}.
    Step 3. Set q=C[b1].

     | Show Table
    DownLoad: CSV

    Similarly, if ηl=θl=0, we have

    Theorem 4. The problem 1|~psdst,~slk|Nl=1(ζl|L[l]|+ϑld[l]) can be solved in O(NlogN) time.

    Lemma 7. For a given sequence π of 1|~psdst,~dif|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]), an optimal solution exists such that d[l]C[l].

    Proof. For a given sequence ϱ, the objective function for job J[l] was:

    M[l]=ζl|C[l]d[l]|+ηlU[l]+θlV[l]+ϑld[l]. (19)

    If d[l]>C[l] (i.e., the job J[l] is an early job), it follows that

    M[l]=ζl(d[l]C[l])+ηlU[l]+ϑld[l].

    Move d[l] to the left such that d[l]=C[l], we have

    M[l]=ϑld[l]=ϑlC[l]<M[l],

    therefore, d[l]C[l].

    Lemma 8. For a given sequence ϱ, if ϑlζl, d[l]=0; otherwise d[l]=C[l] (l=1,2,,N).

    Proof. For a given sequence ϱ, from Lemma 7, we have d[l]C[l] and

    M[l]=ζl(C[l]d[l])+θlV[l]+ϑld[l]=ζlC[l]+θl+(ϑlζl)d[l]. (20)

    From Eq (20), when ϑlζl0, d[l] was equal to 0; otherwise, then d[l] was equal to C[l].

    From Lemma 8, if ϑlζl, we have d[l]=0 and

    M=Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l])=Nl=1ζlC[l]+Nl=1θl. (21)

    If ϑl<ζl, we have d[l]=C[l] and

    M=Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l])=Nl=1ϑlC[l]. (22)

    From Eqs (21) and (22), minimizing Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) is equal to minimizing the expression

    M=Nl=1min{ϑl,ζl}C[l]=Nl=1min{ϑl,ζl}lj=1[1+β(lj)]p[j]=Nl=1Υlp[l], (23)

    where

    Υl={min{ϑ1,ζ1}+(1+β)min{ϑ2,ζ2}++(1+(N1)β)min{ϑN,ζN},l=1,min{ϑ2,ζ2}+(1+β)min{ϑ3,ζ3}++(1+(N2)β)min{ϑN,ζN},l=2,min{ϑN1,ζN1}+(1+β)min{ϑN,ζN},N1,min{ϑN,ζN},N, (24)

    i.e.,

    Υl=Nj=l[1+β(jl)]min{ϑj,ζj},    l=1,2,,N. (24')

    Obviously, Eq (23) can be minimized by Lemma 4.

    Theorem 5. The 1|~psdst,~dif|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]+ϑld[l]) can be solved by Algorithm 4, and time complexity was O(NlogN).

    Algorithm 4
    Require: β, pl,ζl,ηl,θl,ϑl for 1lN.
    Ensure: An optimal sequence ϱ, optimal common due date dl.
    Step 1. By using Lemma 4 (let al=Υl,bl=pl) to determine the optimal job sequence (i.e., ϱ), i.e., place the largest pl at the smallest Υl position, place the second largest pl at the second smallest Υl position, etc.
    Step 2. If ϑlζl, d[l]=0; otherwise d[l]=C[l] (l=1,2,,N).

     | Show Table
    DownLoad: CSV

    We present an example to illustrate the calculation steps and results of the three due date assignments.

    Example 1. Consider a 6-job problem, where β=1, p1=7, p2=9, p3=4, p4=6, p5=8, p6=5, ζl,ηl,θl and ϑl are given in Table 3.

    Table 3.  Values of ζl,ηl,θl and ϑl.
    l=1 l=2 l=3 l=4 l=5 l=6
    ζl 6 8 14 3 15 7
    ηl 8 4 9 10 12 5
    θl 10 8 6 5 14 17
    ϑl 12 16 7 13 8 9

     | Show Table
    DownLoad: CSV

    From Algorithm 1, For the ~con assignment, if a=1, the values Ψ1=205,Ψ2=140,Ψ3=93,Ψ4=54,Ψ5=29,Ψ6=7, (see Eqs (7) or (7')) and Θl,r (see Eq (10)) are given in Table 4. By the assignment problems (8)–(10), the sequence is ϱ(1)=(J3,J6,J4,J1,J5,J2) and M(1)=2801. Similarly, for a=2,3,4,5,6, the results are shown in Table 5. From Table 5, the optimal sequence is ϱ=(J3,J6,J4,J1,J5,J2), M=2801 and d=C[2]=14.

    Table 4.  Values Θl,r for a=1.
    r=1 r=2 r=3 r=4 r=5 r=6
    J1 1435 988 657 383 217 66
    J2 1845 1268 843 491 275 80
    J3 820 568 378 221 130 45
    J4 1230 848 564 329 188 59
    J5 1640 1128 750 437 246 73
    J6 1025 708 471 275 159 52

     | Show Table
    DownLoad: CSV
    Table 5.  Results for ~con.
    a ϱ(a) M(a)
    1 (J3,J6,J4,J1,J5,J2) 2801
    2 (J3,J6,J4,J1,J5,J2) 3017
    3 (J3,J6,J4,J1,J5,J2) 3615
    4 (J3,J6,J4,J1,J5,J2) 5335
    5 (J3,J6,J4,J1,J5,J2) 7451
    6 (J3,J6,J4,J1,J5,J2) 11,382

     | Show Table
    DownLoad: CSV

    For the ~slk assignment, the results are shown in Table 6. From Table 6, the optimal sequence is ϱ=(J3,J6,J4,J1,J5,J2), M=2832 and q=C[0]=0.

    Table 6.  Results for ~slk.
    b ϱ(b) M(b)
    1 (J3,J6,J4,J1,J5,J2) 2832
    2 (J3,J6,J4,J1,J5,J2) 2928
    3 (J3,J6,J4,J1,J5,J2) 3286
    4 (J3,J6,J4,J1,J5,J2) 4310
    5 (J3,J6,J4,J1,J5,J2) 5934
    6 (J3,J6,J4,J1,J5,J2) 9049

     | Show Table
    DownLoad: CSV

    For the ~dif assignment, Υ1=137,Υ2=98,Υ3=65,Υ4=40,Υ5=22,Υ6=7, the optimal sequence is ϱ=(J3,J6,J4,J1,J5,J2), M=1987, d3=0, d6=0, d4=C4=28, d1=0, d5=C5=80 and d2=0.

    Under ~con, ~slk and ~dif assignments, the single-machine scheduling problem with ~psdst and position-dependent weights had been addressed. The goal was to minimize the weighted sum of lateness, number of early and delayed jobs and due date cost. Here we showed that the problem remains polynomially solvable. If the due dates are given, from Brucker [19], the problem 1|~psdst|Nl=1(ζl|L[l]|+ηlU[l]+θlV[l]) is NP-dard. For future research, we suggest some interesting topics as follows:

    1) Considering the problem 1|{\widetilde{psdst}}|\sum_{l = 1}^{N}\left(\zeta_l |L_{[l]}|+\eta_l{U_{[l]}}+\theta_l{V_{[l]}}\right) ;

    2) Investigating the problem in a flow shop setting;

    3) Studying the group technology problem with learning effects (deterioration effects) and/or resource allocation (see Wang et al. [24], Huang [25] and Liu and Xiong [26]);

    4) Investigating scenario-dependent processing times (see Wu et al. [27] and Wu et al. [28]).

    This research was supported by the National Natural Science Regional Foundation of China (71861031 and 72061029).

    The authors declare that they have no conflicts of interest.



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