- Email: [email protected]
On the minimization of the weighted number of tardy jobs with random processing times and deadline
0305-0548/9 I
Computers Ops Res. Vol. 18. No. 5. pp. 457-463. 195’1 Primed m Great B&am. All rights reserved
53.00 + 0.00
Copyrightcc.1991 Pergamon Press plc
ON THE MINIMIZATION OF THE WEIGHTED TARDY JOBS WITH RANDOM PROCESSING DEADLINE
NUMBER OF TIMES AND
PRABUDDHADE*, JAY B. GHosHt and CHARLESE. WELLS$§ Department of MIS and Decision Sciences. University of Dayton, Dayton, OH 45469-2130. U.S.A.
(Received
October
1989;
in revised form September
1990)
and Purpose-This paper addresses the problem of sequencing a given set of jobs with random processing times on a single machine where the jobs have a common but random deadline. The objective is to find the sequence which minimizes the expected weighted number of tardy jobs or the one which stochastically minimizes the weighted number of tardy jobs. The purpose of this work is to add to the stochastic scheduling literature by identifying simple rules that yield optimal sequences. !kope
Abstract- We consider a stochastic scheduling problem where a set of jobs with random processing times are sequenced on a single machine in order to minimize the total weighted number of jobs which linish after an exponentially distributed common deadline. A simple sequencing rule is identified which minimizes the expected weighted number of tardy jobs. SulTicient conditions are determined for the existence of a sequence which stochastically minimizes the total weighted number of tardy jobs, and a sequencing rule is derived which stochastically minimizes the total number of tardy jobs. The equivalena of static and dynamic sequencing policies is shown for a large class of processing time distributions.
I. INTRODUCTION Consider the problem of deciding how to schedule n jobs with required processing times ti, i= l,..., n, on a single machine where a job is deemed tardy and assessed a penalty Of wi. i = 1, . . . , n, if it is completed after a common deadline (due-date) T. The objective is to find a sequence that minimizes the sum of the penalties. When ti, i = 1, . . . , n, and T are certain, it is well known that the weighted number of tardy jobs (WNT) problem reduces to a knapsack problem and is NP-hard [6]. When wi = 1, i = 1, . . . , n, (WNT) becomes the number of tardy jobs (NT) problem, and can be easily solved by scheduling the jobs in shortest processing time (SPT) order. Note that WNT and NT are problems of selection, where the concern simply is to select the set of jobs that should be completed before T, the sequence in which these jobs are executed is immaterial. Ifeithertheti,i= l,..., n, T, or both are modeled as random variables, the problem is no longer the determination of the best selection of tardy jobs, but the determination of the best sequence of jobs. In this case, it is not known how many jobs will be completed before T; therefore the order of the jobs becomes critical. Furthermore, the total penalty is now a random variable because it is not known if a job will be successfully processed. To illustrate such a scenario, consider a single machine that is expected to suffer a nonrepairable breakdown in some random amount of time. The jobs that are waiting to be processed on the machine have, in effect, a random and common deadline (the time of the breakdown). The jobs
lPrabuddha De is Standard Register-Sherman Distinguished Professor of MIS at the University of Dayton. He was previously a Professor of Accounting and MIS at The Ohio State University. He received his undergraduate degree in Physics from the University of Calcutta and a Ph.D. from Carnegie-Mellon University. He has published in many journals including Computers & Operations Research. Decision Sciences. Manaycmont Science, and Operations Research. tJay B. Ghosh is an Associate Professor of MIS and Decision Sciences at the University of Dayton. He received his undergraduate degree in Electronics Engineering from the Indian Institute of Technology at Kharagpur and a Ph.D. from the University of Arkansas at Fayetteville. The journals he has published in include Computers & Operations Research. Decision Sciences. European Journal I/ Operational Research. and Mana(lement Science. :Charles E. Wells is a Professor of MIS and Decision Sciences at the University of Dayton. He received his undergraduate degree in Mathematics from Harvard University and a Ph.D. from the University of Cincinnati. He has published in Computers & Operations Research, Decision Sci&es. Management Science. Netwbrks. and various other journals. $Author to whom all correspondence should be addressed. 457
458
PRABUDDHADE er al.
that are not completed (tardy) induce a penalty which might reflect an opportunity cost or the cost of obtaining the additional processing time externally. When the processing times and/or deadline are random variables, the stochastic weighted number of tardy jobs (SWNT) problem can be stated as: Determine a sequence ~7E S that optimizes g(ui, wi, i = 1, . . . , n), where ul, i = 1, . . . , n, is a random variable that assumes the value I if job i is completed after T, and 0 otherwise, g is a function of the random variables ui and the weights wi, i = 1, . . . , n, and S is the set of all sequences defined for the n jobs. in this paper we examine two distinct cases of g: (1) the distribution of total penalty from tardy jobs, W = c;= 1 WjUj; and (2) the expected total penalty from tardy jobs, E{ W} = ~~=, WjE{ Uj}. A variation of (SWNT), where the due-dates are assumed to be distinct and deterministic and the job processing times are independent and normally distributed, has been examined by Baiut [I ] and Kise and lbaraki [7]. Sarin et al. [IO] has studied this variation for the special case of a common due-date. Biau [3] has explored the more general problem where the due-dates are random and distinct. The special case where the due-dates are independent and exponentially distributed has been investigated later by Pinedo [S] and Boxma and Forst [43. Derman et al. [S] (though in a different context) and Pinedo [8] have studied an exponentially distributed common due-date problem under the assumption that the processing times are also exponentially distributed. Righter [9] has recently looked at an extension of the (NT) problem with a random common due-date. In this paper, we examine the solution of the (SWNT) problem when T is modeled as an exponential random variable and the ( ti, i = I,. . . , n) are random variables with arbitrary but known distributions. We begin by investigating optimal static decision policies; that is, policies for which the sequence of jobs must be established prior to the processing of the jobs and is not subject to revision. We extend this result by showing that for a large class of processing time distributions the optimal static policy is also optimal under preemption and therefore the optimal dynamic policy. In the next section the distribution of total weighted penalty for a given sequence is derived. A simple sequencing rule is determined which minimizes the expected weighted number of tardy jobs. In Section 3 suflicient conditions are derived under which a sequence exists which yields the stochastic minimum total penalty. These results are then extended to the determination of the schedule which stochastically minimizes the total number of tardy jobs. The equivalence of static and dynamic sequencing policies is discussed in Section 4. Section 5 provides a summary of our work and its extensibility to other situations.
2.
MODEL
DEVELOPMENT
In this section we examine the (SWNT) problem when the objective is to minimize the expected weighted number of tardy jobs. We begin by deriving the distribution and expectation of the weighted number of tardy jobs for a given sequence. Assume that T is distributed as an exponential random variable with parameter Q. Let Fi denote the distribution function of ti, i = 1,. . , , n, and assume that { T, ti, i = 1,. . ., n f are mutually independent. For notational convenience, we assume that the density function fi = F;, i = 1,. . . , n exists. Consider any sequence 0 of jobs and let the jobs be numbered according to their position in Q. Thejth job will be successfully completed (and incur no penalty) if tj G T,, where q represents the time remaining until the deadline at the beginning of thejth job. Due to the memoryless property of the exponential random variable T, the probability of incurring no penalty from thejth job can be determined by deconditioning on the value of t,: P P{ T> tj} =
exp{ -ax)rj(x)
dx
s0
where i, is the Lapiace-Stieltjes
transformation
of Fj evaluated at a. Consequently, the distribution
459
Minimization of the weighted number of tardy jobs of
W, the
total
penalty from tardy jobs in sequence 6, is given by . ..X_.(l
p{w=~kwj}=j,
-J&k=
I ,... ,n
and P{W=O}=J,
. ..f”.
The expected weighted number of tardy jobs is
E{ w>= ,t, .z
[(j$k
wj)ti,
i,-*Cl -
.f’
A))].
Let 0’ be the sequence generated from cr by exchanging the order of jobs i and i + 1, and let W’ be the random variable which represents the total penalty incurred from sequence u’. Schedule u is judged better than 6’ if E{ W} - E{ W’)
=(Wi+l
-
Wi)[l
- j-,
+ Wi[l -wi+I[i
-_?I
-f*
*..J+l]
. . . I] **. i,-,I+,]
GO or wi+lji+ll(t
-I+*)<
wijil(l -ij)*
The following result is immediate: Theorem I The schedule that minimizes the expected weighted number of-tardy jo_bs can be obtained by ordering the jobs from largest to smallest in terms of the ratio wJ/( 1 - /i). Interestingly, Boxma and Forst [4], derived a similar sequencing rule in the context of distinct due-dates. Corollary
I. I
The schedule that minimizes the expected number of tardy jobs can be obtained by ordering the jobs from largest to smallest in terms of the 1. Proof. The result follows from Theorem 1 by letting wi = I, i = I, . . . , n. This is a direct generalization of the SPT optimal ordering to minimize the number of tardy jobs when T and tj, j= I,..., n, are known.
3. STOCHASTIC
OPTIMIZATION
In the previous section we sought a sequence which minimized the expected weighted number of tardy jobs. In this section we are interested in determining the sequence, if any, which stochastically minimizes the total penalty incurred from tardy jobs. Definition Given two random variables X and Y, X is said to be stochastically Y(X G Y) if for all real z,
less than or equal to
P{X < z} 2 P{ Y s z}. Let W(o) be the random variable representing the total penalty due to the tardy jobs for any crE S. We wish to determine the sequence CT*E S such that W(a*) G W(a) for all G c S.
PRAMJDDHADE et al.
460
Theorem
2
A st~hasti~a~ly optimal sequence O* exists if and only if the following two conditions Assume that the jobs have been renumbered according to their positions in u*.
hold.
jr>fi+,,i=
I,...,
n-
1;
(1)
wia wi+,,i=
l,...,
n-
1.
(2)
Proof, The theorem can be established by an exchange argument similar to the one used to prove Theorem 1. The following corollary gives conditions for determining the sequence that stochastically minimizes the total number of tardy jobs.
Corollary 2. I The sequence constructed by ordering the jobs from largest to smalkst according to the value of A stochastically minimizes the total number of tardy jobs. Proof. The proof follows from Theorem 2 by letting wi = 1 for i = 1, . . , , n. Note that this corollary further generalizes the result found in Corollary 1.1. coro~~~~ 2.2 WhenTandt,,i= I,..., n are mutually independent exponential random variables, the sequence that stochastically minimizes the total number of tardy jobs is in shortest expected processing time (SEPT) order. Prooj. Let f, have an exponential distribution with parameter tli, i = 1,. . . , a. By Corollary 2.1 we know that the sequence which stochastic&y minims the total number of tardy jobs is given by ordering the jobs from largest to smallest in terms of ij = ai/(ai + a), i = 1. . . . . n. However, x 2 i+,
implies that ai > aI+ ,, 8*= f,...,n-
1,
or l/a, < l/a,+,,
i = l,...,n
- 1.
Therefore the jobs are in shortest expected processing time order. This corollary may be extended to any set of processing time distributions for which /.4~~~“0.&& where pi = E { ti }, and therefore generalizes the result for deterministic processing times and due-date in which the optimal ordering is in SPT. An example
To illustrate the result of Theorem 2, consider a two job problem where t, and t, are distributed as Erlang random variables both with shape parameter 2 and scale parameters 2 and 3, respectively. T is assumed to have an exponential distribution with parameter I. Then J/$ = (2/( 2 + 1))’ = 4/9, and & = (3/(3 + 1))’ = 9/16. We consider two cases. Case I: H’~ = IO, wg = 5
The two conditions of Theorem 2 do not hold and therefore there does not exist a stochastically optimal schedule. Figure 1 shows the graph of Pf W > w ] for the schedules (A, B) and (B, A).
461
Minimization of the weighted number of tardy jobs
8
l-7 ! I--------
8
I
+________:
A
Fig. 1. P( W>
w) for Case I. -,
Schedule (A, B); ---.
0
Fig.2.P{ I+'> W)
for Case 2. -,
Case 2: wA = 5, n’g =
5
schedule (B, A); -O-O-O-.
10
Schedule (A, B). ---.
schedule (A, B) and (B, A)
15
schedule (B, A); -O-O-O-,
schedule (A, B) and (B. A).
10
The result of Theorem 2 now holds since wA,< wa and f, < f; is stochastically optimal as shown in Fig. 2.
4. EQUIVALENCE
OF
STATIC
AND
DYNAMIC
In this case, the schedule (B, A)
POLICIES
The theorems and corollaries of Sections 2 and 3 were derived assuming a static sequencing policy. We now show that for the class of new-better-than used (NBU) processing time distributions [2 3, these sequencing policies remain optimal even if preemption is allowed at any time. Let F(.) = 1 - F(a). Definition
A life distribution
F( .) is NBU if for all x, y > 0. F(x + y) < F(x)F(y).
NBU distributions include all distributions with increasing failure rates (IFR) and increasing failure rate average (IFRA). Examples of NBU distributions include all exponential distributions, gamma and two-parameter Weibull distributions with shape parameter greater than or equal to 1, and truncated normal distributions. Lemma 3
If F(.) is NBU, then f’ >, f, where f’=
= exp{ -zx}f(xlt)
dx
s0 and f(xlt) is the density function of the remaining processing time for a job which has already been processed for t > 0 time units. ProoJ. By the definition of NBU. F(x + r)/F(t)
- F(x) < 0.
Consequently, exp( -zxj[(F(x
+ 1)/f(r))
- F(x)] dx < 0.
462
PRABUDDHA DE et al.
Upon integrating by parts and simplifying, (I/a)(l
-
f”,
< (ll~I(l-9,
or
Theorem 4 If the processing time distributions are NBU, then the optimal sequences derived in (a) Theorem 1 and (b) Theorem 2 and Corollaries 1.1 and 2.1 remain optima1 under preemption and, therefore,
under a dynamic sequencing policy. Proof. For the proof of part (a), consider the preemption of job j after it has been processed for t > 0 time units. It follows from Theorem 1 that to determine the optimal sequence, it is necessary to order the jobs from largest to smallest according to the ratios
Note that job j had a larger ratio than the remaining jobs since it was sequenced first. By Lemma 3, at any time r > 0, the ratio
of the remaining processing time of job j is larger than the original ratio for job 1. Therefore
the remaining portion of job j should be scheduled first; or, in other words, job j should not be preempted. The proof for part (b) of the theorem follows in a similar way.
5. SUMMARY
AND
CONCLUSlONS
This paper has examined a stochastic scheduling problem on a single machine where the processing times of the jobs and the common deadline are modeled as random variables. A simple sequencing rule has been derived to determine the optimal schedule that minimizes the expected weighted number of tardy jobs and also st~hastically minimizes the number of tardy jobs. For a large class of job processing time distributions, it has been shown that an optima1 schedule exists that does not involve preemption of a job. This implies that the optima1 schedule under a static sequencing policy is also optimal under a dynamic sequencing policy. Although we have discussed the (SWNT) problem in a machine scheduling context, the results of this paper have application in a variety of other scenarios where a common deadline can be modeled as a random variable. For example, consider the problem of funding a number of projects during a fiscal year. If the exact amount of available funds is uncertain, as may often be the case in the government sector, then this amount serves as a random common deadline. As another example, consider scheduling any set of activities which can only be performed during some “window whose duration is random. To illustrate, during an unusual astronomical of opportunity” phenomenon, scientists would like to perform a series of observations. The unknown duration of the phenomenon induces a random common deadline.
REFERENCES I. S. J. Balut. Scheduling to minimize the number of late jobs when set-up and processing times are uncertain. Mgmr Sri. 19. 1283-1288 (1973). 2. R. E. Barlow and F. Proschan. Srarisrical ?%twry qf Reljabi/ir~ and Lr$e Testing. Holt, Rinehart & Winston, New York (1975). 3. R. A. Blau. N-Job, one machine sequencing problems under uncertainty. Mgmt Sri. 20, IOl- 109 ( 1973). 4. 0. J. Boxma and F. G. Forst, Minimizing the expected weighted number of tardy jobs in stochastic flow shops. Ops Rex Lrrt. 5. I I9- I26 ( 1986). 5. C. Derman, G. Lieberman and S. Ross, A renewal decision problem. Mgmt Sci. 24, 554-561 (1978). 6. R. M. Karp. Reducibility among combinatorial problems. fn Cr)rn~~~.~ir~of Computer Com~~rurio~s (Edited by R. E. Miller and J. W. Thatcher). pp. 85 103. New York ( 1972). 7. H. Kise and 1. Ibaraki, On Balut’s algorithm and NP-completeness for a chance-constrained scheduling problem. Mgmt Sci. 29. 384-388 (1983).
Minimization of the weighted number of tardy jobs
463
8. M. Pinedo, Stochastic scheduling with release dates and due dates. Ops Res. 31, 559-572 (1983). 9. R. Righter, Sequencing to stochastically maximize the number of successful job completions. Presented at the 1989 ORSA/TIMS Joint National Meeting, New York (1989). 10. S. Satin, G. Steiner and E. Erel, Sequencing jobs on a single machine with a common due date and stochastic processing times. Presented at the 1990 TIMS/ORSA Joint National Meeting, Las Vegas (1990).
Computers Ops Res. Vol. 18. No. 5. pp. 457-463. 195’1 Primed m Great B&am. All rights reserved
53.00 + 0.00
Copyrightcc.1991 Pergamon Press plc
ON THE MINIMIZATION OF THE WEIGHTED TARDY JOBS WITH RANDOM PROCESSING DEADLINE
NUMBER OF TIMES AND
PRABUDDHADE*, JAY B. GHosHt and CHARLESE. WELLS$§ Department of MIS and Decision Sciences. University of Dayton, Dayton, OH 45469-2130. U.S.A.
(Received
October
1989;
in revised form September
1990)
and Purpose-This paper addresses the problem of sequencing a given set of jobs with random processing times on a single machine where the jobs have a common but random deadline. The objective is to find the sequence which minimizes the expected weighted number of tardy jobs or the one which stochastically minimizes the weighted number of tardy jobs. The purpose of this work is to add to the stochastic scheduling literature by identifying simple rules that yield optimal sequences. !kope
Abstract- We consider a stochastic scheduling problem where a set of jobs with random processing times are sequenced on a single machine in order to minimize the total weighted number of jobs which linish after an exponentially distributed common deadline. A simple sequencing rule is identified which minimizes the expected weighted number of tardy jobs. SulTicient conditions are determined for the existence of a sequence which stochastically minimizes the total weighted number of tardy jobs, and a sequencing rule is derived which stochastically minimizes the total number of tardy jobs. The equivalena of static and dynamic sequencing policies is shown for a large class of processing time distributions.
I. INTRODUCTION Consider the problem of deciding how to schedule n jobs with required processing times ti, i= l,..., n, on a single machine where a job is deemed tardy and assessed a penalty Of wi. i = 1, . . . , n, if it is completed after a common deadline (due-date) T. The objective is to find a sequence that minimizes the sum of the penalties. When ti, i = 1, . . . , n, and T are certain, it is well known that the weighted number of tardy jobs (WNT) problem reduces to a knapsack problem and is NP-hard [6]. When wi = 1, i = 1, . . . , n, (WNT) becomes the number of tardy jobs (NT) problem, and can be easily solved by scheduling the jobs in shortest processing time (SPT) order. Note that WNT and NT are problems of selection, where the concern simply is to select the set of jobs that should be completed before T, the sequence in which these jobs are executed is immaterial. Ifeithertheti,i= l,..., n, T, or both are modeled as random variables, the problem is no longer the determination of the best selection of tardy jobs, but the determination of the best sequence of jobs. In this case, it is not known how many jobs will be completed before T; therefore the order of the jobs becomes critical. Furthermore, the total penalty is now a random variable because it is not known if a job will be successfully processed. To illustrate such a scenario, consider a single machine that is expected to suffer a nonrepairable breakdown in some random amount of time. The jobs that are waiting to be processed on the machine have, in effect, a random and common deadline (the time of the breakdown). The jobs
lPrabuddha De is Standard Register-Sherman Distinguished Professor of MIS at the University of Dayton. He was previously a Professor of Accounting and MIS at The Ohio State University. He received his undergraduate degree in Physics from the University of Calcutta and a Ph.D. from Carnegie-Mellon University. He has published in many journals including Computers & Operations Research. Decision Sciences. Manaycmont Science, and Operations Research. tJay B. Ghosh is an Associate Professor of MIS and Decision Sciences at the University of Dayton. He received his undergraduate degree in Electronics Engineering from the Indian Institute of Technology at Kharagpur and a Ph.D. from the University of Arkansas at Fayetteville. The journals he has published in include Computers & Operations Research. Decision Sciences. European Journal I/ Operational Research. and Mana(lement Science. :Charles E. Wells is a Professor of MIS and Decision Sciences at the University of Dayton. He received his undergraduate degree in Mathematics from Harvard University and a Ph.D. from the University of Cincinnati. He has published in Computers & Operations Research, Decision Sci&es. Management Science. Netwbrks. and various other journals. $Author to whom all correspondence should be addressed. 457
458
PRABUDDHADE er al.
that are not completed (tardy) induce a penalty which might reflect an opportunity cost or the cost of obtaining the additional processing time externally. When the processing times and/or deadline are random variables, the stochastic weighted number of tardy jobs (SWNT) problem can be stated as: Determine a sequence ~7E S that optimizes g(ui, wi, i = 1, . . . , n), where ul, i = 1, . . . , n, is a random variable that assumes the value I if job i is completed after T, and 0 otherwise, g is a function of the random variables ui and the weights wi, i = 1, . . . , n, and S is the set of all sequences defined for the n jobs. in this paper we examine two distinct cases of g: (1) the distribution of total penalty from tardy jobs, W = c;= 1 WjUj; and (2) the expected total penalty from tardy jobs, E{ W} = ~~=, WjE{ Uj}. A variation of (SWNT), where the due-dates are assumed to be distinct and deterministic and the job processing times are independent and normally distributed, has been examined by Baiut [I ] and Kise and lbaraki [7]. Sarin et al. [IO] has studied this variation for the special case of a common due-date. Biau [3] has explored the more general problem where the due-dates are random and distinct. The special case where the due-dates are independent and exponentially distributed has been investigated later by Pinedo [S] and Boxma and Forst [43. Derman et al. [S] (though in a different context) and Pinedo [8] have studied an exponentially distributed common due-date problem under the assumption that the processing times are also exponentially distributed. Righter [9] has recently looked at an extension of the (NT) problem with a random common due-date. In this paper, we examine the solution of the (SWNT) problem when T is modeled as an exponential random variable and the ( ti, i = I,. . . , n) are random variables with arbitrary but known distributions. We begin by investigating optimal static decision policies; that is, policies for which the sequence of jobs must be established prior to the processing of the jobs and is not subject to revision. We extend this result by showing that for a large class of processing time distributions the optimal static policy is also optimal under preemption and therefore the optimal dynamic policy. In the next section the distribution of total weighted penalty for a given sequence is derived. A simple sequencing rule is determined which minimizes the expected weighted number of tardy jobs. In Section 3 suflicient conditions are derived under which a sequence exists which yields the stochastic minimum total penalty. These results are then extended to the determination of the schedule which stochastically minimizes the total number of tardy jobs. The equivalence of static and dynamic sequencing policies is discussed in Section 4. Section 5 provides a summary of our work and its extensibility to other situations.
2.
MODEL
DEVELOPMENT
In this section we examine the (SWNT) problem when the objective is to minimize the expected weighted number of tardy jobs. We begin by deriving the distribution and expectation of the weighted number of tardy jobs for a given sequence. Assume that T is distributed as an exponential random variable with parameter Q. Let Fi denote the distribution function of ti, i = 1,. . , , n, and assume that { T, ti, i = 1,. . ., n f are mutually independent. For notational convenience, we assume that the density function fi = F;, i = 1,. . . , n exists. Consider any sequence 0 of jobs and let the jobs be numbered according to their position in Q. Thejth job will be successfully completed (and incur no penalty) if tj G T,, where q represents the time remaining until the deadline at the beginning of thejth job. Due to the memoryless property of the exponential random variable T, the probability of incurring no penalty from thejth job can be determined by deconditioning on the value of t,: P P{ T> tj} =
exp{ -ax)rj(x)
dx
s0
where i, is the Lapiace-Stieltjes
transformation
of Fj evaluated at a. Consequently, the distribution
459
Minimization of the weighted number of tardy jobs of
W, the
total
penalty from tardy jobs in sequence 6, is given by . ..X_.(l
p{w=~kwj}=j,
-J&k=
I ,... ,n
and P{W=O}=J,
. ..f”.
The expected weighted number of tardy jobs is
E{ w>= ,t, .z
[(j$k
wj)ti,
i,-*Cl -
.f’
A))].
Let 0’ be the sequence generated from cr by exchanging the order of jobs i and i + 1, and let W’ be the random variable which represents the total penalty incurred from sequence u’. Schedule u is judged better than 6’ if E{ W} - E{ W’)
=(Wi+l
-
Wi)[l
- j-,
+ Wi[l -wi+I[i
-_?I
-f*
*..J+l]
. . . I] **. i,-,I+,]
GO or wi+lji+ll(t
-I+*)<
wijil(l -ij)*
The following result is immediate: Theorem I The schedule that minimizes the expected weighted number of-tardy jo_bs can be obtained by ordering the jobs from largest to smallest in terms of the ratio wJ/( 1 - /i). Interestingly, Boxma and Forst [4], derived a similar sequencing rule in the context of distinct due-dates. Corollary
I. I
The schedule that minimizes the expected number of tardy jobs can be obtained by ordering the jobs from largest to smallest in terms of the 1. Proof. The result follows from Theorem 1 by letting wi = I, i = I, . . . , n. This is a direct generalization of the SPT optimal ordering to minimize the number of tardy jobs when T and tj, j= I,..., n, are known.
3. STOCHASTIC
OPTIMIZATION
In the previous section we sought a sequence which minimized the expected weighted number of tardy jobs. In this section we are interested in determining the sequence, if any, which stochastically minimizes the total penalty incurred from tardy jobs. Definition Given two random variables X and Y, X is said to be stochastically Y(X G Y) if for all real z,
less than or equal to
P{X < z} 2 P{ Y s z}. Let W(o) be the random variable representing the total penalty due to the tardy jobs for any crE S. We wish to determine the sequence CT*E S such that W(a*) G W(a) for all G c S.
PRAMJDDHADE et al.
460
Theorem
2
A st~hasti~a~ly optimal sequence O* exists if and only if the following two conditions Assume that the jobs have been renumbered according to their positions in u*.
hold.
jr>fi+,,i=
I,...,
n-
1;
(1)
wia wi+,,i=
l,...,
n-
1.
(2)
Proof, The theorem can be established by an exchange argument similar to the one used to prove Theorem 1. The following corollary gives conditions for determining the sequence that stochastically minimizes the total number of tardy jobs.
Corollary 2. I The sequence constructed by ordering the jobs from largest to smalkst according to the value of A stochastically minimizes the total number of tardy jobs. Proof. The proof follows from Theorem 2 by letting wi = 1 for i = 1, . . , , n. Note that this corollary further generalizes the result found in Corollary 1.1. coro~~~~ 2.2 WhenTandt,,i= I,..., n are mutually independent exponential random variables, the sequence that stochastically minimizes the total number of tardy jobs is in shortest expected processing time (SEPT) order. Prooj. Let f, have an exponential distribution with parameter tli, i = 1,. . . , a. By Corollary 2.1 we know that the sequence which stochastic&y minims the total number of tardy jobs is given by ordering the jobs from largest to smallest in terms of ij = ai/(ai + a), i = 1. . . . . n. However, x 2 i+,
implies that ai > aI+ ,, 8*= f,...,n-
1,
or l/a, < l/a,+,,
i = l,...,n
- 1.
Therefore the jobs are in shortest expected processing time order. This corollary may be extended to any set of processing time distributions for which /.4~~~“0.&& where pi = E { ti }, and therefore generalizes the result for deterministic processing times and due-date in which the optimal ordering is in SPT. An example
To illustrate the result of Theorem 2, consider a two job problem where t, and t, are distributed as Erlang random variables both with shape parameter 2 and scale parameters 2 and 3, respectively. T is assumed to have an exponential distribution with parameter I. Then J/$ = (2/( 2 + 1))’ = 4/9, and & = (3/(3 + 1))’ = 9/16. We consider two cases. Case I: H’~ = IO, wg = 5
The two conditions of Theorem 2 do not hold and therefore there does not exist a stochastically optimal schedule. Figure 1 shows the graph of Pf W > w ] for the schedules (A, B) and (B, A).
461
Minimization of the weighted number of tardy jobs
8
l-7 ! I--------
8
I
+________:
A
Fig. 1. P( W>
w) for Case I. -,
Schedule (A, B); ---.
0
Fig.2.P{ I+'> W)
for Case 2. -,
Case 2: wA = 5, n’g =
5
schedule (B, A); -O-O-O-.
10
Schedule (A, B). ---.
schedule (A, B) and (B, A)
15
schedule (B, A); -O-O-O-,
schedule (A, B) and (B. A).
10
The result of Theorem 2 now holds since wA,< wa and f, < f; is stochastically optimal as shown in Fig. 2.
4. EQUIVALENCE
OF
STATIC
AND
DYNAMIC
In this case, the schedule (B, A)
POLICIES
The theorems and corollaries of Sections 2 and 3 were derived assuming a static sequencing policy. We now show that for the class of new-better-than used (NBU) processing time distributions [2 3, these sequencing policies remain optimal even if preemption is allowed at any time. Let F(.) = 1 - F(a). Definition
A life distribution
F( .) is NBU if for all x, y > 0. F(x + y) < F(x)F(y).
NBU distributions include all distributions with increasing failure rates (IFR) and increasing failure rate average (IFRA). Examples of NBU distributions include all exponential distributions, gamma and two-parameter Weibull distributions with shape parameter greater than or equal to 1, and truncated normal distributions. Lemma 3
If F(.) is NBU, then f’ >, f, where f’=
= exp{ -zx}f(xlt)
dx
s0 and f(xlt) is the density function of the remaining processing time for a job which has already been processed for t > 0 time units. ProoJ. By the definition of NBU. F(x + r)/F(t)
- F(x) < 0.
Consequently, exp( -zxj[(F(x
+ 1)/f(r))
- F(x)] dx < 0.
462
PRABUDDHA DE et al.
Upon integrating by parts and simplifying, (I/a)(l
-
f”,
< (ll~I(l-9,
or
Theorem 4 If the processing time distributions are NBU, then the optimal sequences derived in (a) Theorem 1 and (b) Theorem 2 and Corollaries 1.1 and 2.1 remain optima1 under preemption and, therefore,
under a dynamic sequencing policy. Proof. For the proof of part (a), consider the preemption of job j after it has been processed for t > 0 time units. It follows from Theorem 1 that to determine the optimal sequence, it is necessary to order the jobs from largest to smallest according to the ratios
Note that job j had a larger ratio than the remaining jobs since it was sequenced first. By Lemma 3, at any time r > 0, the ratio
of the remaining processing time of job j is larger than the original ratio for job 1. Therefore
the remaining portion of job j should be scheduled first; or, in other words, job j should not be preempted. The proof for part (b) of the theorem follows in a similar way.
5. SUMMARY
AND
CONCLUSlONS
This paper has examined a stochastic scheduling problem on a single machine where the processing times of the jobs and the common deadline are modeled as random variables. A simple sequencing rule has been derived to determine the optimal schedule that minimizes the expected weighted number of tardy jobs and also st~hastically minimizes the number of tardy jobs. For a large class of job processing time distributions, it has been shown that an optima1 schedule exists that does not involve preemption of a job. This implies that the optima1 schedule under a static sequencing policy is also optimal under a dynamic sequencing policy. Although we have discussed the (SWNT) problem in a machine scheduling context, the results of this paper have application in a variety of other scenarios where a common deadline can be modeled as a random variable. For example, consider the problem of funding a number of projects during a fiscal year. If the exact amount of available funds is uncertain, as may often be the case in the government sector, then this amount serves as a random common deadline. As another example, consider scheduling any set of activities which can only be performed during some “window whose duration is random. To illustrate, during an unusual astronomical of opportunity” phenomenon, scientists would like to perform a series of observations. The unknown duration of the phenomenon induces a random common deadline.
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Minimization of the weighted number of tardy jobs
463
8. M. Pinedo, Stochastic scheduling with release dates and due dates. Ops Res. 31, 559-572 (1983). 9. R. Righter, Sequencing to stochastically maximize the number of successful job completions. Presented at the 1989 ORSA/TIMS Joint National Meeting, New York (1989). 10. S. Satin, G. Steiner and E. Erel, Sequencing jobs on a single machine with a common due date and stochastic processing times. Presented at the 1990 TIMS/ORSA Joint National Meeting, Las Vegas (1990).