A scheduling problem with job values given as a power function of their completion times

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  • with some elimination properties improving its eciency) and several heuristic algorithms for the general case of the problem. 2007 Elsevier B.V. All rights reserved.

    e.g., Janiak and Li, 1994; Cheng and Kovalyov, 1995; Kovalyov and Shafransky, 1998; Ng et al., 2003). In the problems ofscheduling deteriorating jobs, the processing times of the jobs increase over time (see e.g., Kovalyov and Kubiak, 1998;

    appears here. It can be shown that the values of the computer components decrease over time as a power function andthe speed of this decrease is dierent for the particular components (see Ferrer, 1997; Voutsinas and Pappis, 2002). Obvi-ously, in the disassembling process we are interested in the component values which are determined at the moments when

    * Corresponding author. Tel.: +48 71 320 2107; fax: +48 71 321 2677.E-mail addresses: adam.janiak@pwr.wroc.pl (A. Janiak), tomasz.krysiak@pwr.wroc.pl (T. Krysiak), pappis@unipi.gr (C.P. Pappis), vutsinas@

    unipi.gr (T.G. Voutsinas).

    www.elsevier.com/locate/ejor

    Available online at www.sciencedirect.com

    European Journal of Operational Research 193 (2009) 836848Bachman and Janiak, 2000; Mosheiov, 1994, 1995, 2002).Another new kind of the scheduling problems is studied in this paper. Namely, the one in which processing times of all

    the jobs are some values which are xed in advance and constant during optimization process, but their values deteriorateover time. The precise description of the problem can be illustrated by an application example, which characterizes theutilization process of the components from some used up computers. Therefore, assume that there is given a set of someused up computers which cannot be used any more, because their further utilization is connected with high risk of a break-down, there are some applications which require faster processors, or simply some of their components are already broken.However, some of their components (e.g., monitors, oppy disks drives, network devices, power suppliers, etc.) can beutilized as spare parts in some other computers. Thus, the problem of disassembling computers into their componentsKeywords: Computational complexity; Job value; Branch and bound; Heuristic; Experimental analysis

    1. Introduction

    In order to solve the signicant real-life problems, a lot of new scheduling problems have been recently formulated. Theproblems with resource allocation and the problems with deteriorating jobs are two examples of such kinds of schedulingproblems which have been considered most often by the scientists during last few years. In the problems with resource allo-cation, the processing times, the release dates, the starting times and other parameters may depend on some resources (seeA scheduling problem with job values given as a power functionof their completion times

    Adam Janiak a,*, Tomasz Krysiak a, Costas P. Pappis b, Theodore G. Voutsinas b

    a Institute of Computer Engineering, Control and Robotics, Wroclaw University of Technology, Janiszewskiego 11/17, 50-372 Wroclaw, Polandb Department of Industrial Engineering, University of Piraeus, 80 Karaoli and Dimitriou, 185 34 Piraeus, Greece

    Available online 7 November 2007

    Abstract

    This paper deals with a problem of scheduling jobs on the identical parallel machines, where job values are given as a power functionof the job completion times. Minimization of the total loss of job values is considered as a criterion. We establish the computationalcomplexity of the problem strong NP-hardness of its general version and NP-hardness of its single machine case. Moreover, we solvesome special cases of the problem in polynomial time. Finally, we construct and experimentally test branch and bound algorithm (along0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2007.11.006

  • or dewill bincreain value. Thus, the cost represents the loss of the object value. Financial resources, which the institution has at its disposal

    problprobl

    strongly NP-hard. Moreover, unlike the problem of Voutsinas and Pappis, we prove there that the single machine versionof our problem is NP-hard. Additionally, some polynomially solvable cases of the general version of the problem are given

    there. In Sections 4 and 5 we present and experimentally test branch and bound algorithm and some heuristic algorithms,respectively, which have been constructed to solve the general version of the problem. Some concluding remarks are givenin Section 6.

    2. Problem formulation

    There are given: a set of m identical parallel machines M = {M1, . . .,Mm} and a set of n independent and non-preemp-tive jobs J = {J1, . . ., Jn} immediately available for processing at time 0. Each job Jj 2 J is characterized by its processingtime pj > 0 and a loss function lj(t) characterizing the loss of its value at time t. The loss function is dened by non-decreas-ing power function of time:

    ljt wjtaj ; 1

    where wj > 0 and aj > 0 denote proportional and exponential loss rate, respectively. Moreover, the loss function is calcu-lated only at the job completion time Cj.

    Solution of the problem is represented by a set of permutations P = {p1, . . ., pm}, where pi is a sequence of jobs assignedto execute on a machine Mi, i = 1, . . ., m. Let ni denote a number of jobs processed on Mi (then

    Pmi1ni n). The objective

    is to nd such a solution P, for which the sum of losses of job values, calculated at their completion times Cpij (if a job isprocessed in the jth position of pi), is minimal, i.e.,

    TLVP Xmi1

    Xnij1

    lpijCpij Xmi1

    Xnij1

    wpijCapijpij ! min: 2

    Ferrer (1997) showed that the decrease of the computer component values can be modelled by the following time function:v(t) = xta, where the values of x and a were established experimentally for dierent kinds of the components (processors,RAM modules, hard disks, monitors, etc.). Therefore, the values of the computer components can be described by non-increasing convex power functions of time (i.e., the values of the computer components decrease fast at the beginning, whenthe components are quite new, and the values decrease slowly when the components are quite old). Thus, such situation canbe also modelled by the scheduling problem considered in the present paper, as follows. The job is to recover a given com-decreasing power model of loss of job value, and formulate a problem of minimization of the total job value loss which canbe used to solve the problem of establishing an order of object renovation. Moreover, based on the above model, we intro-duce a new model of job value, dened as a dierence between initial job value and non-decreasing power function of loss.Then, we formulate a problem of maximization of the total job value, which can be used to solve the process of computerremanufacturing. Additionally, we assume a set of identical parallel machines is available to execute the jobs. The preciseformulation of our problem is given in the next section. In Section 3, we show that the general version of our problem isponenexpreem with maximization of the total job value. They constructed some heuristic algorithms solving the single machineem, however the computational complexity status of this problem is still open. In this paper, we introduce a new non-are constrained, both in amount and time. Therefore, here appears the problem how to nd the order, in which the objects(jobs) will be renovated (executed) so that the sum of the costs (the total loss of job values) is minimized.

    The aim of this paper is to describe and solve a scheduling problem, where job values (or losses of job values) changeduring their execution and are described as a power function of the job completion times. This work was motivated byresults presented by Ferrer (1997) and Voutsinas and Pappis (2002). Ferrer presented many aspects of PC remanufacturing.Based on them, Voutsinas and Pappis introduced a non-increasing power model of job value and formulated a schedulingpartment) has to determine the order in which some objects (e.g., buildings, roads, bridges, historic monuments, etc.)e renovated. Renovation of an object is a job which has to be executed. The cost required to renovate each objectses with respect to time from its last renovation, since such objects become ruined without renovation and they lossthey are available for utilization. The component is ready to be reused after it is completely removed from the computerand its proper functionality is conrmed. Thus, the order in which the computers will be disassembled has a signicantinuence on the total prot, i.e., the sum of the component values. Therefore, maximization of the total component valuesis considered as an optimization criterion. The process of disassembling other highly technologically advanced products(such as cars, airplanes, etc.) into their components can be similarly analyzed.

    Another application example describes a renovation process of some objects. Assume that some institution (e.g., oce

    A. Janiak et al. / European Journal of Operational Research 193 (2009) 836848 837t from some used up computer. The value vj(t) of job Jj decreases over time and can be described by the followingssion:

  • into account the appropriate values of x and a. In such a case, we can also assume that the job value vj(t) is positive at any

    Notehavejobs,

    ItdierJohns

    We shthe vpj xj; wj xj; aj 1; j 1; . . . ; q; pe B; we 1=4; ae 2: PPARTITION: Given a set X = {x1, x2, . . ., xq} of q positive integers for which j1xj 2B; does there exist a partition ofthe set X into two disjoint subsets X1 and X2 such that

    Pxj2X 1xj

    Pxj2X 2xj B?

    Theorem 1. The problem 1kPwjCajj is NP-hard, even for two different values of the exponential loss rates.Proof. Given an instance of PARTITION, construct the following instance of the scheduling problem. There are n = q + 1jobs. Among them, there are q partition jobs and a single extra job Je, with the following parameters:can be also proved that the single machine version of our problem remains NP-hard, even for the case with only twoent values of the exponential loss rates. We derive this results using the NP-complete PARTITION PROBLEM (Garey andon, 1979). Pqeven for two machines (Bruno et al., 1974; Lenstra et al., 1977). Its single machine version is polynomially solvable bySWPT rule (Smith, 1956). For aj = 2 we obtain strongly NP-hard problem P jj

    PwjC

    2j (Cheng and Liu, 2004). However,

    the computational complexity of its single machine version remains open, despite the extensive research (see e.g., Town-send, 1978; Szwarc et al., 1988; Della Croce et al., 1995).

    3. Computational complexity

    The computational complexity status of the general version of our problem, PkPwjCajj , may be deduced directly fromthe considerations given above.

    Corollary 1. The problem PkPwjCajj is strongly NP-hard as a more general than the strongly NP-hard problem PkPwjC2j ,Cheng and Liu (2004).that certain classical scheduling problems in fact are some special cases of the problem formulated above, but theynever been considered as the problems of minimization of the total loss of job values. Namely, for aj = 1 for all theour problem becomes the problem of the total weighted completion time minimization, P jjPwjCj, which is NP-hardmoment t of the optimization process. Thus, the following condition should be satised vjt v0j wjtaj > 0 fort 2 0;Pnj1pj. The problem is to nd such a solution P that the sum of job values calculated at their completion timesis maximal, i.e.,

    Xmi1

    Xnij1

    vpijCpij Xmi1

    Xnij1

    v0pij Xmi1

    Xnij1

    lpijCpij !

    ! max :

    SincePm

    i1Pni

    j1v0pij

    Pnj1v

    0j is constant, thus, maximization of the sum of job values is equivalent to the minimization of

    the total loss of jobs values (see Expression (2)).Consider now the process of renovation of the buildings and other objects. The cost required to renovate a given object

    increases over time (if the object is not renovated). What is more, this cost increases slowly at the beginning and faster andfaster as time passes. Thus, it should be modelled by non-decreasing and convex power function of time t, given by Expres-sion (1) with ajP 1.

    Therefore, it is enough (from the point of view of the presented practical applications) to analyze the problem of min-imization of the total loss of job values with non-decreasing power models of job value losses (formulated at the beginningof this section) for both aj 2 (0,1) and ajP 1. Using the three-eld notation a|b|c for scheduling problems, Graham et al.(1979), the considered problem is denoted by PkPwjCajj :

    For the single machine case, i.e., 1kPwjCajj , the solution is described by a single permutation p, and the objective func-tion value can be calculated as follows

    TLVp Xnj1

    lpjCpj Xnj1

    wpjCapjpj ! min :where v0j > 0 is its initial value (since lj(0) = 0) and aj 2 (0,1) (the function vj(t) is convex for the concave function lj(t) givenby Expression (1)). The specic values of v0j , wj, and aj for particular components can be established experimentally, takingvjt v0j ljt v0j wjtaj ; 3838 A. Janiak et al. / European Journal of Operational Research 193 (2009) 836848ow that PARTITION has a solution if and only if there exists a solution to the constructed instance of 1k wjCajj withalue

    PwjC

    ajj 6 y A 2B2, where A

    P16i6j6qxixj.

  • xj2Xfromsolutiarbitrany s

    Since

    O(nlo

    li li l1 gi1 l1 li1l1

    ppl ppk gi1 l1

    ppl ppk li1

    ppl l1

    ppl ppk li1

    ppl pe

    Xk1gj1

    Xi1l1

    ppl ppk Xj1li1

    ppl pe Xglj1

    ppl

    !a:

    Since pp(i) > pp(k), thus, the difference TLV(p) TLV(p 0) is positive. Thus, the global SPT rule is optimal for the jobs withthe common exponential loss rate. The position of job Je can be found in O(n) steps by checking all possible positions in the

    schedl1

    ppl ppi li1

    ppl pe gj1 l1

    ppl ppi li1

    ppl pe lj1

    ppl

    Xi1 !a Xj1 Xi1 Xg !a Xi1 Xj1 !ae

    Xi1 Xj1 !ae Xk1 Xi1 Xj1 Xg !aTLVp TLVp0 Xk1

    Cpl Xk1

    Cp0l Xi1

    ppl ppi !a

    Xj1 Xi1

    ppl ppi Xg

    ppl

    !aProof. It follows from Corollary 2 that the jobs with the common loss rate a are scheduled according to SPT rule. We willshow that it holds if there is one job Je with ae5 a. Assume that Je is placed in the jth position in some optimal permu-tation p and, contrary to the thesis of the property, pp(i) > pp(k)(i < j < k) for some jobs p(i) and p(k). Let a permutation p 0

    be obtained from p by interchanging the jobs p(i) and p(k). The difference between the objective function values obtainedfor p and p 0 is given belowgn) time, only job Je can be deferred from its position in the SPT sequence.SinceP

    xj2X 1xj P

    xj2X 2xj B; thus TLV(p) = A + 2B2 = y, as required.If. Assume now that any partition of the set X into two disjoint subsets X1 and X2 will give

    Pxj2X 1xj

    Pxj2X 2xj:

    Without loss of generality we can assume thatP

    xj2X 1xj B k andP

    xj2X 2xj B k; where k > 0. Note that expression(4) holds for any partition of the set X into X1 and X2 (even if one of them is empty). By (4), we have

    TLVp A 14B B k2 BB k y 1

    4k2 > y:

    Thus, it is shown that the decision version of our scheduling problem has a solution...

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