This paper studies the problem of scheduling a set of n jobs on a single machine to minimize weighted absolute deviation of completion times from a common ...
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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH European Joumal of Operational Research 96 (1997) 559-563
Theory and Methodology
A note on minimizing the weighted sum of tardy and early completion penalties in a single machine: A case of small common due date Bahram Alidaee
Irinel Dragan b
a. * .
a Department of Management and Marketing, School of Business Administration, University of Mississippi, University, MS 38677. USA b Mathematics Department, Unioersity of Texas at Arlington, Arlington, TX 76019, USA Received February 1995; revised November 1995
Abstract This paper studies the problem of scheduling a set of n jobs on a single machine to minimize weighted absolute deviation of completion times from a common due date. It is assumed that weights of jobs are proportional to their processing times. It has been shown by other researchers that the problem can be solved efficiently for a sufficiently large due date. In the present paper we solve the problem for any given due date. Keywords: Single machine scheduling; Earliness and tardiness; Proportional weights
1. Introduction Consider scheduling a set of n independent jobs on a single machine where associated with a job i there is a positive number t i as processing time and a positive number w i as importance weight. All jobs are ready at time zero. The objective is to find a sequence or of jobs and a starting time t* to minimize weighted sum of absolute deviation of completion times from a common due date given as
(P)
z ( o- ) = ~ w[j] l ctj I j=l
* Corresponding author.
dl.
In the objective function, d is a given common due date, [ j ] is the j-th job in sequence o- and CIj I is the completion time of the j-th job. Motivation for such objective function comes from the need to minimize the penalties incurred by a supplier in a Just-In-Time production system (Baker and Scudder, 1990). The objective function Z is a non-regular measure as opposed to a regular performance measure that is an increasing function of completion times (Baker, 1974). In practice the importance weights w i ( i = 1. . . . . n) are chosen by decision maker according to some internal and external considerations and limitations that can affect the future of the company. Choosing appropriate weights in practice is difficult and time consuming. In many real applications the weights are set equal to one for all jobs. Another alternative approach is to use weights that are pro-
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B. Alidaee, L Dragan / European Journal of Operational Research 96 (1997) 559-563
portional to processing times of the jobs. Arkin and Roundy noted (1991) that typically, jobs that have longer processing times are larger jobs that have higher selling prices and, consequently, a higher priority. In many systems both the processing time and weight of a job are nearly proportional to the job's dollar value, resulting in weights that are proportional to processing times. In the present paper we assume proportional weights for the jobs, i.e.,
2. Optimal solution of the problem P In the following we introduce some notations that will be used in proving the results. Let N = {I . . . . . n} be the set of jobs and C i and S i completion time and starting time of job i, respectively. For a schedule olet A( tr) = { i ~ N I C i >
d},
B(o') =N-A(o'), w i -~ "yt i
for i = 1. . . . . n.
where 3' is a given constant number. Without loss of generality we can assume 3' = 1. Scheduling researchers have dealt with the problem P in two separate categories: 1) large due date d, i.e., d/> E t i, and 2) small due date d, i.e., d is any given positive number. The problem for a large due date and w~ = 1 for all jobs was first studied by Kanet (1981). He presented an efficient algorithm for optimal solution of the problem. Improvement of Kanet's model for single machine problem have been studied (for example, see Bagchi, Sullivan and Chang, 1986, Emmons, 1986, and Hall, 1986). Generalization of Kanet's model to m machines has been studied (Alidaee and Panwalkar, 1993; Sundararaghavan and Ahmed, 1984). It has been shown that the problem for any arbitrary value of due date d and w i = 1 (i = 1. . . . . n) is NP-hard (Hall, Kubiak and Sethi, 1991; Hoogeveen and Van de Velde, 1991). Furthermore, Hall and Posner (1991) showed that the problem for arbitrary positive weights and large due date is NP-hard. Ahmed and Sundararaghavan (1987) studied the problem for a large due date and proportional weights for all jobs. They have shown that the largest processing time (LPT) order is optimal. They gave an efficient algorithm for creating an optimal solution. Other research papers that have dealt with such problems with proportional weights include Bagchi, Chang and Sullivan (1987) and Lee, Danusputro and Lin (1991). The purpose of the present study is to show that LPT order is also optimal for the problem with a small due date and proportional weights for all jobs. We also give an efficient algorithm, similar to the one given by Ahmed and Sundararaghavan, for creating an optimal solution.
,~( o') = {i e N l Si> d}.
Clearly, B(o-) represents the jobs with completion times less than or equal to d, and we have A(o') = A'(o') U {k}, where k is a job being processed during the time d. Note that if a job is completed at the time d, the set {k} will be empty. Results 1-4 given below are well known. Result 1 [10,12]. I f weights are arbitrary positive numbers and due date is a sufficiently large value, then in each optimal schedule due date d will coincide with completion time of a job. Result 2 [10,12]. If weights are arbitrary positive numbers and due date is any given number, then in each optimal schedule either the first job starts at zero or due date d will coincide with completion time of a job. Result 3 [1]. Let weights of jobs be proportional to processing times and due date d be a sufficiently large value. If o-' is a sequence obtained from or by permuting jobs in A'(cr) and B(tr) among themselves, then Z(or') = Z( or ). Result 4 [1]. If weights of jobs are proportional to processing times and due date d is a sufficiently large value, then there exists an optimal sequence or such that ( ti)ieB(cr) >1 ( ti)i~A(~).
Similar to Results 3 and 4, we give Results 5 and 6 for the small due date problem.
B. Alidaee, L Dragan / European Journal of Operational Research 96 (1997) 559-563
Result 5. Let weights of jobs be proportional to processing times and let due date d be any given value. If tr' is a sequence obtained from or by permuting jobs in A'(tr) and B ( ( r ) among themselves, then Z( o" ) = Z( o" ). Proof. It is similar to the proof of Result 3 given by Ahmed and Sundararaghavan (1987). The main result of this paper is given as follows. Result 6. I f weights of jobs are proportional to processing times and due date d is any given value, then there exists an optimal sequence or such that
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Case lai: Job l is being processed during time d; such a situation is shown in Fig. lb. Now we have -
= [ x t t + y t k] - [Zt, + yt t] = [ xt I + yt k - yt I + ytt] - [ Ztk + yt t - yt k + ytk] = [ttt k + y( t k -
t,)]
- [ttt k + y( t t - tk)]
= 2 y ( t k - tt) > o, which violates the optimality of or. Then we must have t t >/t k. Case laii: Starting time of job l is after time d; such a situation is shown in Fig. lc. Now we have -
Proof. If d is sufficiently large, then the result follows from Result 4. Now let d be any given value. We will show that any optimal sequence can be transformed to a sequence with LPT order without loosing optimality. The standard adjacent job interchange rule is used to prove the result. Let tr be an optimal sequence and t* the corresponding optimal starting time. Then there are two cases to be considered: Case 1: There is a job k in o- that is being processed during time d We have a situation such as shown in Fig. la. Now, there are two subcases to be considered Case la: Suppose job l is the largest job in B(o-) such that t t < t k. Using Result 5 we can move job l to the last position in B(o'). Obtain a sequence or' by changing the position of jobs l and k while keeping the starting time at t ' . Then we have two subcases as follows:
= [ Xtl + yt k ] - [ zt, + t,( z +
t,)]
= [ X t t + y t k ] -- [ Ztk + Ytl]
=tt(x--y) +tk(y--z) = tt( x - - y ) + tkt t = tl( X-- y ) + tt( x + y ) = 2ttt k > O, which violates the optimality of o-. This proves that t t >1 t k. Case lb: Suppose job j ~ A ' ( ~ ) is the smallest job in A'(tr) such that t k < tj. Using Result 5 we can move job j to the first position of A'(o'). Obtain a sequence tr' by changing the position of jobs k and j while keeping the starting time at t*; such a situation is shown in Fig. ld. Since y+tj=z+t~, we have Z(~) - Z(~') = [ y t k + ( y + tj)tj] - [ztj + ( z + tk)tk]
d
I
~
I
k I
I
x
I
k
I I
k
I
(a)
y
= [yt, + ( y + tj)tj] - - [ Z t j + ( y + tj)t,] ~
I
i
I
(b)
I I ~ I
i
I
(=)
k
I
(~)
z
I
J
y
z
I
J
I
I j x
I z
Fig. 1.
= ( y + t j ) t j - - ( Z + t k ) t j = O. ThUS we can change the position of jobs k and j and still have optimality. Then we must have tj ~> t k. This shows that we can change the position of jobs to get tt>~t,>~t j for all l ~ B ( o ' ) , j ~ A ' ( t r ) and still have optimality.
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B. A lidaee, 1. Dragan / European Journal of Operational Research 96 (1997) 559-563 d
I
k
I
I
J
I
I
i
I
la)
k
J
(b)
x
completing job i in sequence tr at time d, and start the schedule at time ti* = ~i= l tk. Step 4. Find a schedule O'min such that Z(O'min) :
Min { Z ( ~ ) } ,
O tj > t~ for i E B(tr'). This contradicts the assumption that ti ~
t k, V i ~ B( o").
Next we present an algorithm for solving the problem.
3. C o n c l u s i o n
In this study we considered the problem of scheduling a set of n jobs on a single machine to minimize weighted sum of absolute deviation of completion times from a given common due date, where weights of jobs were proportional to processing times and the common due date was any given number. We showed that LPT order is optimal. Furthermore, we presented an efficient algorithm for generating an optimal schedule.
Algorithm
Step 1. Find a sequence o- with nonincreasing order of processing times. Step 2. Obtain a schedule tr 0 by starting the sequence or at time t* = 0. Step 3. Obtain a schedule o-i (i = 1. . . . . n) by
References Ahmed, M., and Sundararaghavan, P. (1990), "Minimizing the weighted sum of late and early completion penalties in a single machine", lie Transactions 22, 288-290.
B. Alidaee, 1. Dragan / European Journal of Operational Research 96 (1997) 559-563 Alidaee, B, and Panwalkar, S.S. (1993), "Single stage minimum absolute lateness problem with a common due date on nonidentical machines", Journal of the Operational Research Society 44, 29-36. Arkin, R., and Roundy, R. (1991), "Weighted tardiness scheduling on parallel machines with proportional weights", Operations Research 39, 64-81. Bagchi, U., Chang, Y., and Sullivan, R. (1987), "Minimizing absolute and squared deviations of completion times with different earliness and tardiness penalties and a common due date", Naval Research Logistics Quarterly 34, 739-751. Bagchi, U., Sullivan R., and Chang, Y. (1986), "Minimizing mean absolute deviation of completion times about a common due date", Naval Research Logistics Quarterly 33, 227-240. Baker, K.R. (1974), Introduction to Sequencing and Scheduling, Willey, New York. Baker, K.R., and Scudder, G. (1990), "Sequencing with earliness and tardiness penalties: A review", Operations Research 38, 22-36. Emmons, H. (1986), "Scheduling a common due date on parallel uniform processors", Naval Research Logistics Quarterly 34, 803-810. Hall, N. (1986), "Single and multiple processor models for
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minimizing completion time variance", Naval Research Logistics Quarterly 33, 49-54. Hall, N., and Posner, M. (1990, "Earliness tardiness scheduling problems 1: Weighted deviation of completion times about a common due date", 39, 836-846. Hall, N., Kubiak, W., and Sethi, S. (1991), "Earliness tardiness scheduling problems II: Deviation of completion times about a restricted common due date", 39, 847-856. Hoogeveen, J., and Van de Velde, S.L. (1991), "Scheduling around a small common due date", European Journal of Operational Research 55, 237-242. Kanet, J. (1981), "Minimizing the average deviation of job completion times about a common due date", Naval Research Logistics Quarterly 28, 643-651. Lee, C.-Y., Danusputro, S., and Lin, C.-S. (1991), "Minimizing weighted number of tardy jobs and weighted earliness-tardiness penalties about a common due date", Computers & Operations Research 18, 379-389. Sundararaghavan, P.S., and Ahmed, M. (1984), "Minimizing the sum of absolute lateness in single machine and multiple machine scheduling", Naval Research Logistics Quarterly 31, 325-333.
