Scheduling two agents with controllable processing times

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    Imprecise computation

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    pression cost subject to deadline constraints (the imprecise computation model). All problems are to

    ment. We provide NP-hardness proofs for the more general problems and polynomial-time algorithms for

    heduliagentre maB hasr of jo

    paj 2 paj ; paj may decrease the job completion time, but entails anadditional cost caj x

    aj , where x

    aj paj paj is the amount of compres-

    sion of job j 2 N1 and caj is the compression cost per unit time. Thetotal compression cost is represented by a linear functionP

    j2N1caj x

    aj . Given a schedule r, the completion time of job j of agent

    A (B) is denoted by Caj r Cbj r. If there is no ambiguity, we omit

    If the processing restrictions and constraints of agent As jobsare different from the processing restrictions and constraints ofagent Bs jobs, we refer to the two-agent problem as

    ajba : bbjca : cb:For example, 1jraj ; pmtna : rbj jca : cb refers to two sets of jobs to

    be scheduled on a single machine with objectives ca and cb, respec-tively. The rst set of jobs are released at different times, i.e., ra,

    * Corresponding author.

    European Journal of Operational Research 205 (2010) 528539

    Contents lists availab


    .eE-mail address: (G. Wan).processing time, release date and due date of job j 2 N1 N2 aredenoted by paj ; r

    aj and d

    aj (p

    bj ; r

    bj and d

    bj ), respectively. In classical

    deterministic scheduling models, all processing times are xedand are known in advance. In the models considered here, thejob processing times are controllable and can be chosen by thedecision maker. Furthermore, due to symmetry of the two agents(see Agnetis et al., 2004), we assume that processing times of agentAs jobs are controllable, while processing times of agent Bs jobsare not. Formally, for each job j 2 N1, there is a maximum valueof the processing time paj which can be compressed to a minimumvalue paj paj 6 paj . Compressing paj to some actual processing time

    let ajbjc. Agnetis et al. (2004) extend this notation for the two-agent problem to ajbjca : cb. Their optimization problems can bedescribed as follows: Given that agent B keeps the value of itsobjective function cb less than or equal to Q, agent A has to mini-mize the value of its objective function ca. In this paper, we mayassume that either one set or both sets of jobs have different re-lease dates and that either one set or both sets of jobs are subjectto preemption. If the jobs of both agents are subject to the sameprocessing restrictions and constraints (as in Agnetis et al.(2004)), then the two-agent problem will be referred to as

    ajbjca : cb:Total completion timeMaximum tardinessMaximum lateness

    1. Introduction

    We consider several two-agent scsets of jobs N1 and N2 (belonging tohave to be processed on one or moschedule the n1 jobs of N1 and agentof N2. Let n denote the total numbe0377-2217/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.01.005several special cases of the problems. 2010 Elsevier B.V. All rights reserved.

    ng problems where twos A and B, respectively)chines. Agent A has toto schedule the n2 jobsbs, i.e., n n1 n2. The

    r and use Caj Cbj . The two agents may have either the same objec-tive function or two different objective functions. We consider theoptimization problem in which the value of the objective functionof agent A has to be minimized, while the value of the objectivefunction of agent B must be kept at less than or equal to a xed va-lue Q.

    The classical notation for machine scheduling is based on a trip-Controllable processing timesAvailability constraints minimize the objective function of agent A subject to a given upper bound on the objective function of

    agent B. These problems have various applications in computer systems as well as in operations manage-Discrete Optimization

    Scheduling two agents with controllable

    Guohua Wan a,*, Sudheer R. Vakati b, Joseph Y.-T. LeuaAntai College of Economics and Management, Shanghai Jiao Tong University, 535 FahubDepartment of Computer Science, New Jersey Institute of Technology, Newark, NJ 0710c Stern School of Business, New York University, 44 West Fourth Street, New York, NY 10

    a r t i c l e i n f o

    Article history:Received 3 December 2008Accepted 5 January 2010Available online 1 February 2010

    Keywords:Agent scheduling

    a b s t r a c t

    We consider several two-agents A and B have to shaing their jobs. The processitive function for agent B isfunctions are considered foimum tardiness plus comp

    European Journal of

    journal homepage: wwwll rights reserved.ocessing timesb, Michael Pinedo c

    n Road, Shanghai 200052, ChinaSA, USA

    nt scheduling problems with controllable job processing times, whereither a single machine or two identical machines in parallel while process-imes of the jobs of agent A are compressible at additional cost. The objec-ays the same, namely a regular function fmax. Several different objectiveent A, including the total completion time plus compression cost, the max-ion cost, the maximum lateness plus compression cost and the total com-

    le at ScienceDirect

    perational Research

    lsevier .com/locate /e jorj

  • constraints can often be modeled as a two-agent problem.This paper is organized as follows. The description of the prob-

    peraand are subject to preemption; the second set of jobs are alsoreleased at different times but are not subject to preemption. Weput ctrl in the second eld when the processing times arecontrollable.

    Scheduling models with controllable processing times have re-ceived considerable attention in the literature since the originalstudies by Vickson (1980a,b). The scheduling problem of minimiz-ing total (weighted) completion time plus compression cost has at-tracted much attention (see, for example, van Wassenhove andBaker, 1982; Nowicki and Zdrzalka, 1995; Janiak and Kovalyov,1996; Wan et al., 2001; Hoogeveen and Woeginger, 2002; Janiaket al., 2005; Shakhlevich and Strusevich, 2005). Shabtay and Stei-ner (2007) survey relevant results up to 2007.

    Scheduling models with multiple agents have already receivedsome attention in the literature. Baker and Smith (2003) and Agne-tis et al. (2004) consider scheduling models with two agents inwhich all jobs of the two sets are released at time 0 and both setsof jobs are subject to the same processing restrictions and con-straints. The objective functions considered in their models includethe total weighted completion time (

    PwjCj, wherewj is the weight

    of job j), the number of tardy jobs (P

    Uj, where Uj 1 if Cj > dj andUj 0 otherwise) and the maximum of regular functions (fmax,where fmax maxjfjCj and fjCj is a nondecreasing function withrespect to Cj). Cheng et al. (2007) consider scheduling models withmore than two agents and each agent has as an objective the totalweighted number of tardy jobs. Cheng et al. (2008) consider sched-uling models with more than two agents and with precedence con-straints. Leung et al. (2010) consider a scheduling environmentwithmP 1 identical machines in parallel and two agents, and gen-eralize the results of Baker and Smith (2003) and Agnetis et al.(2004) by including the total tardiness objective, allowing for pre-emption, and considering jobs with different release dates.

    Scheduling models with controllable processing times are moti-vated by numerous applications in production and operationsmanagement as well as in computing systems. The main issue ofconcern is the trade-off between job completion times and thecosts of compression. Two-agent models have also importantapplications in practice. In the remaining part of this section wediscuss some of the applications of our models.

    For an application of controllable processing times in manufac-turing and production management, assume the values pj repre-sent the processing requirements under normal situations. Theseprocessing requirements can be controlled by the allocation levelof the resources (e.g., people and/or tools). When additional re-sources are allocated to job j, its regular processing time pj canbe reduced by an amount xj to some value pj 2 pj; pj at a cost ofcjxj (i.e., the cost of the additional resources), where cj is the unitcost of additional resources allocated (cf., Cheng and Janiak,1994; Cheng et al., 1998, 2001; Grabowski and Janiak, 1987; Janiak,1986, 1987a,b, 1988, 1989, 1998; Janiak and Kovalyov, 1996; Now-icki and Zdrzalka, 1984; Shakhlevich and Strusevich, 2005). In theproject management literature the compression of activities is usu-ally referred to as crashing.

    In supply chainmanagement, rms often have tomake decisionswith regard to outsourcing, i.e., they have to decidewhich part of anorder to process in-house and which part of the order to outsourceto a third party since it may be protable for a rm to process only apart of an order internally for pj time units and outsource theremainder (xj pj pj time units) to a third party (see, e.g., Chaseet al., 2004). A good strategy here is to process the order in-houseas much as possible by setting the lower bound pj (representingthe minimum amount of in-house production) close to the ordersize pj, and to minimize the payment to the third party

    Pcjxj (see

    G. Wan et al. / European Journal of Oalso Shakhlevich and Strusevich, 2005).Another application of controllable processing times occurs in

    computer programming. An iterative algorithm typically involveslems and their applications is presented in Section 2. In Section 3,we consider the problem of minimizing the compression cost, sub-ject to the constraint that agent As jobs have to meet their dead-lines. In Section 4, we consider the total completion time plusthe compression cost as the objective function of agent A. In Sec-tion 5, we consider the maximum tardiness (or lateness) plus thecompression cost as the objective function of agent A. Both Sec-tions 3 and 5 consider a single machine only, while Section 4 con-siders a single machine as well as two identical machines inparallel. We conclude in Section 6 with a discussion of future re-search opportunities.

    2. Problem description

    In the scheduling problems considered in this paper, each job jof agent B has a penalty function f bj Cbj , where f bj Cbj is a nonde-creasing function with respect to Cbj j 1; . . . ;n2, and the objec-tive function of agent B is simply f bmax maxf b1 Cb1; . . . ; f bn2 C


    Given that agent B keeps the value of f bmax less than or equal to Q,agent A has to minimize the value of one of the following objectivefunctions:

    (1) In the rst problem each job j of agent A must meet a dead-line daj . The goal is to determine the actual processing timesof agent As jobs so that the total compression cost is mini-mized. Using the notation introduced above, we denote theproblems by 1jctrla; raj ; daj ; pmtna : pmtnbj

    Pcaj x

    aj : f

    bmax and

    1jctrla; raj ; daj ; pmtna : bjP

    caj xaj : f

    bmax. In Section 3, we show

    that 1jctrla; raj ; daj ; pmtna : bjP

    caj xaj : f

    bmax is unary NP-hard,

    while 1jctrla; raj ; daj ; pmtna : pmtnbjP

    caj xaj : f

    bmax is solvable in

    polynomial time. We also consider the case where all jobsof both agents are released at time 0 (i.e., 1jctrla; daj ;pmtna : bjP caj xaj : f bmax), and show that this problem is solv-able in polynomial time.some initial setup that takes pj time units. After the initial setup,the algorithm typically goes through many iterations that take anadditional pj pj time units. Ideally, the program should be runfor exactly pj time units, but because of deadline constraints, itmay not be possible to run the program in its entirety. However,we can still get some useful (though not exact) result if the pro-gram is run for pj time units, where pj 6 pj 6 pj. In the computerscience community, this mode of operation is called imprecisecomputation (see Blazewicz, 1984; Blazewicz and Finke, 1987;Leung, 2004; Leung et al., 1994; Liu et al., 1991; Potts and vanWas-senhove, 1992; Shih et al., 1989, 1991). In the imprecise computa-tion model, the total weighted error is equivalent to the totalcompression cost

    Pcjxj, where xj pj pj.

    Two-agent models have various important applications in prac-tice as well. For example, consider a machine that has to undergomaintenance at regular intervals. One can imagine the mainte-nance process to be the responsibility of, say, agent B. There area number of maintenance tasks that have to be performed in giventime windows, each one specied by a release date and a due date.In order to ensure that the maintenance tasks do not deviate toomuch from the specied time windows, agent B tries to minimizean objective fmax. Agent A has the responsibility of scheduling thereal jobs and may be allowed to do some compression of thesejobs. A machine scheduling problem that is subject to availability

    tional Research 205 (2010) 528539 529(2) The second problem is to minimize the total completiontime plus job compression costs. Again, the jobs of agent Amay be preempted. We show in Section 4 that when the jobs

  • i.e., the problems 1jctrl : j Cj cj xj : fmax and

    eraproperty. In some practical situations, a job with large duedate means the processing of the job can be done later thusits compression cost should be large so that the jobs withsmall due dates can be processed earlier. In Section 5, we de-velop polynomial-time algorithms for the problem withagreeability property.

    Note that in all these problems, the objective functions for agentB are always fmax. It is regular, i.e., it is a nondecreasing functionwith respect to job completion times. To make the description ofthe algorithms more concise, we rst describe a procedure to sche-dule the jobs of agent B.ProFor

    (assumPcaj x

    aj : f bmax and 1jctrla; raj ; pmtna : bjLmax

    Pcaj x

    aj : f bmax

    are both unary NP-hard. When the jobs of agent A havethe same release date, then the computational complexitiesof the nonpreemptive and preemptive cases are identical.Although we do not know the exact status of computationalcomplexity of these two problems, we are again able to pro-vide a polynomial-time algorithm for the special case when

    da1 6 da2 6 6 dan1


    ca1 6 ca2 6 6 can1 :

    Again, we call this the agreeability property and the prob-lems with equal due dates or equal job compression costsare both special cases of the problems with agreeabilityity property. In some practical situations, a long job process-ing time means processing of the job is complicated thus it isdifcult to compress and incurs large unit job compressioncost. In Sections 4.1 and 4.2, we develop polynomial-timealgorithms for the problems of a single machine and of twoidentical machines in parallel with agreeability property,respectively.

    (3) The third and fourth problems concern the minimization ofthe maximum tardiness plus job compression cost and theminimization of the maximum lateness plus job compres-sion cost, respectively. The jobs of agent A may again be pre-empted. We show in Section 5 that if the jobs of agent Ahave arbitrary release dates, then both problems ar...


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