Positive half-products and scheduling with controllable processing times

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  • uce p

    a b c,* d

    times, there are n independent and non-preemptive pj 2 0; uj, j 1; . . . ; n. A decision maker is todetermine the values of job processing timesp p1; . . . ; pn and a permutation of jobs p so asto minimize a linear combination TWC Pn

    j1 wjCj Pn

    j1 vjuj pj of the total weightedcompletion time

    Pnj1 wjCj, where Cj denotes the

    *Corresponding author.

    European Journal of Operational ReseE-mail addresses: janiak@ict.pwr.wroc.pl (A. Janiak),

    koval@newman.bas-net.by (M.Y. Kovalyov), wkubiak@mun.Adam Janiak , Mikhail Y. Kovalyov , Wieslaw Kubiak , Frank Wernera Institute of Engineering Cybernetics, Wroclaw University of Technology, Wroclaw, Poland

    b United Institute of Informatics Problems, National Academy of Sciences of Belarus, and Faculty of Economics,

    Belarus State University, 220050 Minsk, Belarusc Faculty of Business Administration, Memorial University of Newfoundland, St. Johns, Canada NFA1B 3X5

    d Otto-von-Guericke-Universitat, Magdeburg, Germany

    Received 1 November 2002; accepted 1 May 2003

    Available online 2 June 2004

    Abstract

    We study the single machine scheduling problem with controllable job processing times to minimize a linear

    combination of the total weighted job completion time and the total weighted processing time compression. We show

    that this scheduling problem is a positive half-product minimization problem. Positive half-products make up an

    interesting subclass of half-products and are introduced in this paper to provide a conceptual framework for the

    problem with controllable job processing times as well as other problems. This framework allows to readily derive in

    one fell swoop a number of results for the problem with controllable processing times from more general results ob-

    tained earlier for the half-product. We also present fast fully polynomial time approximation schemes for the problem

    with controllable processing times. The schemes apply to all positive half-products.

    2004 Elsevier B.V. All rights reserved.

    Keywords: Single machine scheduling; Controllable processing times; Pseudo-Boolean optimization; Fully polynomial time

    approximation scheme; Computational complexity

    1. Scheduling with controllable processing times

    In the problem with controllable processing

    jobs to be scheduled for processing on a single

    machine. All jobs are available for processing at

    time zero. The processing time of job j is a variablePositive half-prodwith controllablca (W. Kubiak), frank.werner@mathematik.uni-magdeburg.de

    (F. Werner).

    0377-2217/$ - see front matter 2004 Elsevier B.V. All rights reservdoi:10.1016/j.ejor.2004.04.012ts and schedulingrocessing times

    arch 165 (2005) 416422

    www.elsevier.com/locate/dswcompletion time of job j, and the total weightedprocessing time compression

    Pnj1 vjuj pj.

    ed.

  • in On2 logU=e time, where U P16 j6 n uj,2

    OperAll numerical data are positive integers. Thevalue of variable pj is a non-negative real numberin 0; uj, j 1; . . . ; n. Setting pj 0 means thateither the processing time of job j is negligible andthus it practically does not delay the completion

    times of other jobs or job j is rejected with penaltyvjuj.

    Vickson [9,10] was rst to study this problem,

    as well as a more general problem with arbitrarynon-negative lower bounds lj, lj6 uj, j 1; . . . ; n,on job processing times, more than 20 years ago.

    Other applications of the problem can be found in

    Williams [12] and Janiak [3], where the reader is

    referred to for more comprehensive references.

    For arbitrary lj and wj 1, j 1; . . . ; n, Vick-son [9] recasts the problem as an assignment

    problem. For arbitrary weights wj, Vickson [10]presents an enumerative algorithm for the prob-

    lem. Vickson [9] also shows that the search for

    optimal job processing times p p1; . . . ; pn canbe limited as follows.

    Lemma 1. There exists an optimal p p1; . . . ; pnwith pj 2 flj; ujg, j 1; . . . ; n.

    Furthermore, the shortest weighted processing

    time (SWPT) rule of Smith [8] limits the search for

    an optimal permutation p, given p, as follows.

    Lemma 2. There exists an optimal p with ppj=wpj6 ppj1=wpj1 for j 1; . . . ; n 1.

    From Lemmas 1 and 2 the following corollaryfollows immediately.

    Corollary 1. There exists an optimal solution suchthat pj 2 flj; ujg, j 1; . . . ; n, jobs with processingtimes pj lj are sequenced in the non-decreasingorder of lj=wj and jobs with processing times pj ujare sequenced in the non-decreasing order of uj=wj.

    From now on, we assume that the jobs are re-

    indexed such that u1=w16 6 un=wn, and lj 0,j 1; . . . ; n. By Corollary 1, the scheduling prob-lem with controllable job processing times, we also

    refer to it as the problem of minimizing TWC for

    convenience, reduces to deciding on a partition of

    A. Janiak et al. / European Journal ofthe set of jobs into a subset with pj 0 and athe other in On logW =e time, where W P16 j6 n wj. Finally, to briey discuss, in Section 4,

    prospects of using the FPTAS developed in this

    paper to improve eciency of the existing

    FPTASs for a special subclass of positive half-products.

    2. Scheduling with controllable processing times and

    half-product minimization

    The half-product is a pseudo-Boolean function

    of the form

    Hx Hx1; . . . ; xn D

    X

    16 i

  • OperTheorem 1. The problem of minimizing TWC andthe problem of minimizing half-products witha2=b26 6 an1=bn1, are polynomially equiva-lent.

    Proof. Let vectors w w1; . . . ;wn, u u1; . . . ;un, and v v1; . . . ; vn make up an instance ofthe problem of minimizing TWC. Dene a half-product as follows:

    TWCx Xn

    j1wjxjXj

    i1uixi

    Xn

    j1vjuj1 xj

    X

    16 i 0 andTWCx D6 TWC0; . . . ; 0 D 0. Therefore,inequality in the statement of the lemma can be

    written as

    D > f n 1TWCx:Obviously,X

    16 i6 j6 nuiwj TWC1; . . . ; 1PTWCx:

    Now, consider an instance with vj 2df n1ewj wn, for j 1; . . . ; n. We haveD 2df n 1eTWC1; . . . ; 1;for this instance and thus the lemma holds.

    Let x0 be an e-approximate solution to the

    ational Research 165 (2005) 416422problem of minimizing TWCx D. We have

  • i2P

    ci1 xi i2N

    cixi;

    Operwith all coecients standing at variables or theirproducts being non-negative. We refer to a half-

    product as a positive half-product if the constant

    DPi2P ciP 0. Thus, the positive half-productsare pseudo-Boolean functions of the form

    F x X

    16 i f nD;

    for some instances. Therefore, an e-approximatesolution to the problem of minimizing TWCxD obtained by the FPTAS of Badics and Boros [1]cannot be used to obtain an f ne-approximatesolution to the problem of minimizing TWCx forany rational function f n, a polynomial in par-ticular. Consequently, we need a dierent FPTAS

    than that of Badics and Boros. Such a FPTAS is

    presented in the following section.

    3. Positive half-products and their FPTAS

    Consider any half-product

    Hx DX

    16 i

  • eStep 1 (Initialization). Calculate d > 0 such that

    Operational Research 165 (2005) 4164221 dn 1 e. Set k 0 and X0 fKg.Step 2 (Recursive ltering). Construct set Yk

    fx0; x1jx 2 Xk1g. Calculate a1;kx andF1;kx for each x 2 Yk. If k n, then setXn Yn and go to Step 3. Otherwise, parti-tion Yk into subsets Yr;k, r 1; . . . ; sk, suchthat

    ja1;kx a1;kyj6 d minfa1;kx; a1;kyg;for any x and y in the same subset. Fromeach subset Yr;k, select a vector xr;k suchthat F1;kxr;k minfF1;kxjx 2 Yr;kg. SetXk fxr;kjr 1; . . . ; skg, k k 1 andrepeat Step 2.

    Step 3 (e-approximate solution). Select a solutionxe 2 Xn such that F xe minfF xjx 2Xng and stop.

    We now show that algorithm Ae producessolution xe of required relative error e. The algo-rithm complexity is shown in Theorem 3, where

    the ecient implementation of Step 2 is discussedempty word K, builds a solution to the half-product minimization. At iteration k selectedwords of length k are partitioned into subsets toensure that each subset includes only those words

    that are d-close to each other, more precisely, forany two words x and y in the same subset thealgorithm ensures

    ja1;kx a1;kyj6 dminfa1;kx; a1;kygfor some positive d dependent on e and n to bedened later. Then, F1;k is used to select a singleword x from each subset of the partition. The wordhas the smallest value F1;k among all words inthe same subset of the partition. Only the selected

    words pass to iteration k 1, where each word isextended by concatenating either 0 or 1 at its end,

    and the iteration repeats. Finally, when k reaches nthe algorithm stops selecting a word with theminimum value of F among all words thatreached iteration n. The details of the algorithmare as follows:

    Algorithm A

    420 A. Janiak et al. / European Journal ofin detail.Theorem 2. Algorithm Ae nds xe 2 Xn such thatF xe F x6 eF x.

    Proof. For an optimal x, let x0; . . . ; xn be n 1words of length n each such that

    (a) k-prex of xk is in Xk, for k 0; . . . ; n,(b) both xk and x share the same n k-sux,

    for k 0; . . . ; n,(c) k-prexes of xk1 and xk are in the same Yr;k,

    for k 1; . . . ; n.By (a) and (b), x0 x.Our proof relies on inequalities (2), (3) and (5)

    that we now prove.

    First, since all coecients in F x are non-neg-ative, we have

    a1;kxbk1;nx6 F x: 2Second, we have

    a1;kxk16 1 dk1a1;kx; k 1; . . . ; n: 3We prove this inequality by induction on k. Fork 1, (3) holds since x0 x. Assume that (3)holds for 16 k6 n 1. Let us prove that (3) holdsfor k 1. By (c), k-prexes of xk1 and xk are inthe same subset Yr;k, thus we have

    a1;kxk6 1 da1;kxk1; k 1; . . . ; n: 4Finally,

    a1;k1xk a1;kxk ak1xk16 1 da1;kxk1 ak1xk16 1 dka1;kx ak1xk16 1 dka1;k1x:

    Here, the rst equation follows from the deni-tions of Lemma 4, the rst inequality follows from

    (4), the second one follows from the inductive

    assumption, and the last one again from the de-

    nitions of Lemma 4.

    Third, we have

    F xk F xk16 d1 dk1F x: 5To prove it, we observe that by denitions of F xand xk, we have

    F xk F xk1 F1;kxk F1;kxk1 a1;kxk

    a1;kxk1bk1;nx:

  • controllable job processing times in particular.

    OperBy (c), k-prexes of xk1 and xk are in the samesubset Yr;k. Consequently, a1;kxk a1;kxk16 da1;kxk1. Moreover, the minimum value ofF1;k over all vectors in Yr;k is attained at xk, thus,F1;kxk6 F1;kxk1. Therefore, by (2) and (3),F xk F xk16 da1;kxk1bk1;nx6 d1 dk1a1;kxbk1;nx6 d1 dk1F x:

    We are now ready to prove the theorem. We have

    F xe6 F xn and x0 x. Therefore, by (5) andthe denition of d, 1 dn 1 e, we haveF xe F x6 F xn F x0

    Xn

    k1F xk F xk1

    6 dF xXn

    k11 dk1 eF x;

    which completes the proof.

    Theorem 3. Algorithm Ae can be implemented torun in On2 logA=e time, where A Pnj1 aj.

    Proof. The key to the complexity of Ae is theimplementation of set Yk partitioning in Step 2.There, we arrange the words in Yk in ascendingorder of their a1;k values, we call this order ana-order, so that 06 a1;ky16 a1;ky26 6a1;kyjYk j. Then, we assign y1; y2; . . . ; yi1 to set Y1;kuntil detecting i1 such that a1;kyi16 1 da1;ky1and a1;kyi11 > 1 da1;ky1. If such an i1 doesnot exist, then we set Y1;k Yk and stop. Next, weassign yi11; yi12; . . . ; yi2 to set Y2;k until detecting i2such that a1;kyi26 1 da1;kyi11 and a1;kyi21> 1 da1;kyi11. If such an i2 does not exist,then we set Y2;k Yk n Y1;k and stop. We continuethis partitioning until yjYk j is included in Ysk ;k, forsome sk. It is crucial for the complexity to noticehere that, if Xk1 is a-ordered, then obviously bothfx0jx 2 Xk1g and fx1jx 2 Xk1g easily inherit its a-order and their merging leads to the a-ordered setYk in linear time. Moreover, the selection of asingle vector from each set of the partitionY1;k; . . . ; Ysk ;k again inherits the a-order which

    A. Janiak et al. / European Journal ofresults in a-ordered Xk. Consequently, the a-orderAnother scheme with time complexity

    On2 logB=e, where B Pnj1 bj, can be derived ina similar way as Ae. The scheme relies on valuesbk1;nx and Fk1;nx for its recursive ltering inStep 2, and builds word xnxn1 . . . x1 starting fromempty word K.

    4. Conclusions and further research

    We have shown that the single machine sched-

    uling problem with controllable job processingtimes is polynomially equivalent to the problem of

    maximizing a special subclass of half-products,

    namely, positive half-products. This immediately

    proves that not only is the former problem NP-

    hard but also that it can be solved in pseudo-

    polynomial time by dynamic programs proposed

    earlier for the half-product minimization, see [4].

    We have also developed a couple of fully polyno-mial time approximation schemes for the problem

    with controllable processing times. The schemes

    apply to a general class of problems called positive

    half-products that we have also introduced in this

    paper. The class includes, for instance, the two

    machine weighted completion time problem, and it

    is very likely to include many more schedulingof words is an invariant of Step 2, and therefore thestep can be implemented in OjYkj, and the wholealgorithm in OPnk1 jYkj time. Furthermore, wehave jYkj 2jXk1j 2sk1and sk 6K 1, k 1;. . . ; n, where K is an integer that satises1 dKPA. Consequently, the algorithm runs inOnK time, and it remains to estimate the value ofK. We have KP logA= log1 d. From therelationship between e and d dened in Step 1 ofthe algorithm, we have log1 d log1 e=n.Since log1 e6 e, for 0 < e6 1, then KPn logA=e. Notice that if e > 1, then a 1-approxi-mate solution can be taken as an e-approximatesolution, and we may assume 0 < e6 1 withoutloss of generality. Thus, by setting K dn logA=ee,we obtain the required complexity.

    Theorems 2 and 3 prove that Ae is a FPTAS forany positive half-product, and the problem with

    ational Research 165 (2005) 416422 421problems. The search for them seems an exciting

  • and practically important topic for further re-search since it may ultimately lead to more ecient

    approximation schemes, based on the schemes

    presented in this paper, for many scheduling

    positive half-product problems.

    Acknowledgements

    M.Y. Kovalyov was supported in part by

    INTAS under grant number 00-217. W. Kubiak

    has been supported by the Natural Sciences and

    Engineering Council of Canada Research GrantOGP0105675. The authors would like to thank

    anonymous referees for their constructive com-

    ments that resulted in an improved paper.

    References

    [1] T. Badics, E. Boros, Minimization of half-products,

    Mathematics of Operations Research 23 (1998) 649660.

    [2] M.R. Garey, D.S. Johnson, Computers and Intractability:

    G. Zapfel (Eds.), Modern Production Concepts, Springer-

    Verlag, Berlin, 1991, pp. 404415.

    [4] B. Jurisch, W. Kubiak, J. Jozefowska, Algorithms for

    minclique scheduling problems, Discrete Applied Mathe-

    matics 72 (1997) 115139.

    [5] M.Y. Kovalyov, W. Kubiak, Fully polynomial approxi-

    mation schemes for decomposable partition problems, in:

    Operations Research Proceedings 1999, Selected papers of

    the Symposium on Operations Research (SOR99), Mag-

    deburg, September 13, 1999, pp. 397401.

    [6] W. Kubiak, New results on the completion time variance

    minimization, Discrete Applied Mathematics 58 (1995)

    157168.

    [7] W. Kubiak, Minimization of ordered, symmetric half-

    products, submitted for publication.

    [8] W.E. Smith, Various optimi...

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