Short Communication
Single machine scheduling problems with controllable processing times and total absolute differences penalties

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Abstract

In this paper, we consider single machine scheduling problem in which job processing times are controllable variables with linear costs. We concentrate on two goals separately, namely, minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing a cost function containing total waiting time, total absolute differences in waiting times and total compression cost. The problem is modelled as an assignment problem, and thus can be solved with the well-known algorithms. For the case where all the jobs have a common difference between normal and crash processing time and an equal unit compression penalty, we present an O(n log n) algorithm to obtain the optimal solution.

Introduction

Most of the scheduling literature examines regular measures of the performance, which are nondecreasing functions of job completion times. One of the most commonly occurring regular measures is the minimization of mean completion times. Its attractiveness is perhaps due to its equivalence to mean waiting time, mean lateness, and average in-process inventory. Yet in certain situations one is more interested in reducing the variability in the completion times, resulting in performance measures that are nonregular. For instance, in a service-oriented environment, one might be interested in providing as much uniform quality of service as possible based on the customers’ waiting times in system.

Another example where one might be interested in a variability measure was given by Merten and Muller (1972) in the context of organization of computer databases. They noted that, in an organization of computer files in the large databases, it is desirable to provide uniform response time to users. The objective then is to determine the arrangement that minimizes variation of access time to different records in the file.

As measures of variation, Merten and Muller (1972) considered completion time variance (CTV) and waiting time variance (WTV). These measures have also been used by Schrage, 1975, Eilon and Chowdhury, 1977, Vani and Raghavachari, 1987. Although several properties of the optimal schedules for these measures have been established, no efficient procedure exists for solving these problems. Kanet (1981) proposed using total absolute differences in completion times (TADC) as an alternative measure of completion time variation, and presented an efficient algorithm for minimizing this measure. While Bagchi (1989) proposed using total absolute differences in waiting times (TADW) as an alternative measure of waiting time variation, and presented an efficient algorithm for minimizing this measure.

In this paper, we consider the case in which job processing times are to be reduced, up to a limit, and with costs proportional to the amount of reduction in processing times. These costs will be offset by savings incurred due to the early completion. This type of problem occurs frequently in project planning, see Elmaghraby (1977). Its motive in the field of scheduling is of the same nature, that is, the assumption of controllable processing times is justified in the situations where jobs can be accomplished in shorter or longer durations caused by the increasing or decreasing additional resources. For example, in service systems, if there are too many customers, it may be important to provide customers with identical or similar service quality as well as to decrease everyone’s waiting time by increasing additional resources (otherwise the customers who wait too long time will leave). We concentrate on two goals: minimizing a cost function containing total completion time, total absolute differences in completion times and total compression cost; minimizing total waiting time, total absolute differences in waiting times and total compression cost.

Works in the scheduling problem with controllable processing times and linear cost functions are surveyed by Nowicki and Zdrzalka (1990). Vickson (1980a), who probably wrote one of the first papers on controllable processing time scheduling problems, considered the objective of minimizing the total flow time and the total processing cost incurred due to the job processing time compression. Vickson (1980b) considered the single machine scheduling of minimizing the total flow and resource costs under the assumption that the job flow costs are identical. Van Wassenhove and Baker (1982) considered single machine scheduling problems in which the objective function is to minimize the maximum completion penalty. They gave a bicriterion approach to sequencing with time/cost trade-offs which produces an efficient frontier of the possible schedules. Nowicki and Zdrzalka (1988) considered a two-machine flow shop scheduling problem with controllable job processing times. They assumed that the cost of performing a job is a linear function of its processing time, and the schedule cost to be minimized is the total processing cost plus maximum completion time cost. They showed that the problem is NP-complete, and proposed two heuristic methods for solving the problem. Daniels and Sarin (1989) considered single machine scheduling problem of joint sequencing and resource allocation when the criteria is the number of tardy jobs. Zdrzalka (1991) considered single machine scheduling problem in which each job has a release date, a delivery time and a controllable processing time. He gave an approximation algorithm for minimizing the overall schedule cost. Panwalkar and Rajagopalan (1992) considered the common due date assignment and single machine scheduling problem in which the objective is the sum of penalties based on earliness, tardiness and processing time compressions. They reduced the problem to an assignment problem. Alidaee and Ahmadian (1993) extended the results of Panwalkar and Rajagopalan (1992) to the parallel machine scheduling case. Cheng and Janiak (1994) further generalized the result to the case where the cost of compression is a general convex function of the amount of compression. Cheng et al. (1996) considered a due date assignment and single machine scheduling in which a penalty for due dates is added to the objective function which includes the penalties for earliness, tardiness and processing time compressions. Alidaee and Kochenberger (1996) considered single and parallel machine scheduling problems in which job processing time of a job was assumed to depend on the position of the job in the schedule and is a function of units of resource applied for its processing. The processing time and the processing cost functions are allowed to be nonlinear. For the single machine problem, the objective was minimization of total compression costs plus a scheduling measure. The scheduling measures included makespan, total flow time, total differences in completion times (TADC), total differences in waiting times (TADW), and total earliness and tardiness with a common due date for all jobs. Except for the total earliness and tardiness measure, they solved all variations efficiently. Under an assumption that is typically satisfied in a JIT environment, they proved that the problem with total earliness and tardiness measure was also solved efficiently. For a large class of processing time functions, they proved that the parallel machine problem with total flow time, and total earliness and tardiness measure was solved efficiently. They reduced the problems of all cases to a transportation problem which is known to be polynomially solvable. Biskup and Cheng (1999) considered a due date assignment and single machine scheduling in which a penalty for completion times is added to the objective function which includes the penalties for earliness, tardiness and processing time compressions. Biskup and Jahnke (2001) considered the problem of assigning a common due date to a set of jobs and scheduling them on a single machine with jointly reducible processing times. Besides considering due date assignment costs the first goal is to minimize the sum of earliness and tardiness penalties while the second one is to minimize the number of late jobs. For both cases polynomially solvable algorithms have been given. Hoogeveen and Woeginger (2002) combined the resource allocation and the weighted flow time costs to a single objective and proved that this problem is NP-hard. Ng et al. (2003) considered the single machine problem with a variable common due date. They presented polynomial time algorithms for minimizing a linear combination of scheduling, due date assignment and resource consumption costs. Shabtay and Kaspi (2004) considered a single machine scheduling problem with the minimum total weighted completion time criterion where the model of operations is assumed to be a specific convex function of the amount of resource consumed. They presented and analyzed some special cases that are solvable by using polynomial time algorithms. They also gave some heuristic algorithms for the general case. Ng et al. (2004) considered the single machine batch scheduling with jointly compressible setup and processing times. They presented polynomial time algorithms to find an optimal batch sequence and optimal amounts of resource consumption such that either total job completion time is minimized, subject to an upper bound on total weighted resource consumption, or total weighted resource consumption is minimized, subject to an upper bound on total job completion time.

The rest of this paper is organized as follows. Notations and assumptions are given in Section 2. In Section 3, we obtain optimal compressions for any given sequence. In Section 4, we show that the problem can be formulated as an assignment problem. A special case for which there is an easy solution is presented in Section 5. In Section 6, conclusions are presented.

Section snippets

Notations and assumptions

Consider a set of n jobs J={J1,J2,,Jn} to be processed in a single machine with the following assumptions:

  • All jobs are available at time zero.

  • No job pre-emption and job splitting are allowed.

  • The machine is available at time zero and for the whole duration of time horizon.

  • The machine cannot process two or more jobs simultaneously.

  • After the process in the machine has started, no idle time can be inserted in the schedule.

The following notations will be used throughout the paper:

    s

    the sequence of

Optimal compressions

For the model (1), if we substitute, C[j]=i=1jp[i], TC=j=1nC[j], TADC=j=1n(j-1)(n-j+1)p[j] (Kanet, 1981) and x[j] = t[j]  p[j] into (1) and simplify, we havef(s,xi)=δ1j=1n(n-j+1)p[j]+δ2j=1n(j-1)(n-j+1)p[j]+δ3j=1nG[j](t[j]-p[j])=j=1n[(δ1-δ2)(n+1)+j((n+2)δ2-δ1)-j2δ2-δ3G[j]]p[j]+δ3j=1nGjtj.Letλj=(δ1-δ2)(n+1)+j((n+2)δ2-δ1)-j2δ2-δ3G[j],1jnthen λj, 1  j  n, represents the position weight of position j in the sequence s. Since δ3j=1nGjtj is a constant, for any sequence, the optimal processing

Optimal sequences

Now we discuss the determination of optimal sequences for the two models. In view of the analysis in the previous sections, where we provided the expressions for computing the optimal processing times and compressions for any given optimal sequence, the problem reduces to a pure sequencing problem. In order to obtain the optimal sequence, we formulate the models (1), (2) as an assignment problem, respectively.

For the model (1), letλij=(δ1-δ2)(n+1)+i((n+2)δ2-δ1)-i2δ2-δ3Gj,i,j=1,2,,nandpij=tj,ifλ

A special case

We now consider a special case in each model in which Gi = G and ti-ti=m, i = 1, 2,  , n. This represents the case in practice where the same means is employed to compress the processing time of each job. It can be shown that the optimal solution can be found in O(n log n) time in this case.

Theorem 2

For the models (1), (2) in which Gi = G and ti-ti=m, i = 1, 2,  , n, the optimal sequence s is the sequence obtained from matching the position weights in descending order with the normal processing times in ascending

Conclusions

The problem of scheduling n jobs with controllable processing times has been studied. The objective function is to minimize a cost function containing total completion (waiting) time, total absolute differences in completion (waiting) times and total compression cost. We have solved the problem by formulating it as an assignment problem. An O(n log n) algorithm is proved to obtain the optimal solution for a special case.

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments on an earlier version of this paper. The research was partially supposed by the foundation of Shenyang Institute of Aeronautical Engineering under Grant Number: 05YB08.

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