On stochastic machine scheduling with general distributional assumptions

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

ELSEVIER

European Journal of Operational Research 105 (1998) 525-536

Theory and Methodology

On stochastic machine scheduling with general distributional assumptions Wei Li a.,, Kevin D. Glazebrook h Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100080, PR China b Department of Mathematics and Statistics, UniL'ersity of Newcastle upon Tyne, Newcastle upon Tyne, NEI 7RU, UK

Received 1 November 1995; accepted 1 February 1997

Abstract A set of stochastic jobs is to be processed on a single machine which is subject to breakdowns and repairs. Under quite general conditions on the job's processing times, the uptimes and the repairtimes of the machine, we find a simple optimal permutation policy for the weighted sum of an increasing function of the completion times. Some optimal policies for two special cases, which have not been investigated in the literature, are given. Moreover, an upper bound is established on the loss incurred when a processing policy is adopted under a simplifying assumption of exponential processing requirements. 9 1998 Elsevier Science B.V. Ke)~vords: Stochastic scheduling; Breakdowns; General uptimes and downtimes

1. I n t r o d u c t i o n The problem of scheduling jobs on machines is usually discussed under the assumptions that the processing times and due dates are known in advance and that the machines are continuously available. However, in practice, some of these assumptions are unrealistic. Machines may well be subject to lengthy and unpredictable breakdowns and repairs, whereas processing times a n d / o r due dates may also be of a stochastic nature. Single-machine stochastic scheduling models incorporating breakdowns and repairs were first studied by Glazebrook in 1984. Subsequently, many papers have investigated stochastic problems on unreliable or repairable machines, see [1-21]. However, in the existing literature, almost all of the papers have one of the following assumptions: I. the uptimes of the machines are exponentially (or geometrically) distributed; 2. job processing times are exponentially (or geometrically) distributed; 3. job processing times are deterministic. Under the first assumption, by extending the processing time to a total processing time that a job occupies the machine (i.e. the actual processing time plus the delay time because of machine failure), some stochastic

" Corresponding author. Tel.: +86-10-6495-3269; fax: +86-10-6254-1689; e-mail: [email protected]. 0377-2217/98/S19.00 9 1998 Elsevier Science B.V. All rights reserved. PII S0377-2217(97)00088-X

526

W. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998)524-536

scheduling problems on an unreliable machine can be transformed into corresponding scheduling problems with a reliable machine. See Gl~zebrook [12], Pinedo and Rammouz [19], Allahverdi and Mittenthal [3], and Li and Cao [16], etc. Based on the second assumption but with general uptime, there are few results, and these concern simple policies and are usually based on some stronger conditions. See Pinedo and Rammouz [19], Chang et al. [7], Du and Pinedo [8], and Li and Cao [15], etc. By utilising the third assumption, Allahverdi and Mittenthal [4,5], and Birge [6] obtained some powerful results. In general, when we consider stochastic scheduling problems with no specific distributional assumptions made about processing requirements, uptimes and downtimes of the machines, choices made by optimal processing strategies could depend upon both the time elapsed since the last repair was completed and the elapsed processing of the job currently being processed. Structured results of this kind have yet to appear in the literature. Frenk [10] has presented a great framework for stochastic one-machine scheduling problems which allows for the incorporation of machine breakdowns and job due dates. In this paper, we investigate a single machine scheduling model in which the machine is subject to breakdowns with general uptime and general repairtimes. Each job's processing time is also a random variable with general distribution. In developing a schedule for processing, we specialise to the class of simple recourse strategies. Under a simple recourse strategy, the sequence of job completion times in a schedule is fixed, but the completion times may be postponed as a result of the machine breakdowns (see Allahverdi and Mittenthal [4,5] or Frenk [10], etc.). This choice is motivated in part by mathematical tractability, although some of the results obtained for simple recourse schedules also hold in the larger class of general recourse schedules. Further assumptions and notations are developed as follows: A set of stochastic jobs J = {1,2 . . . . . N} is to be processed on a single machine which is subject to breakdown and repair. Each job j has a random processing requirement Xj (which can be continuous, discrete or mixed) with distribution function Fj. If job j is completed at time t, a cost oJjf(t) is incurred, where f ( t ) is an increasing bounded nonnegative function and toj is the weight associated with job j. The machine is available for processing from time 0 until the first breakdown occurs at time U I. The machine then takes time D~ to be repaired, during which time no processing takes place. The repair having been completed, the machine is again available for processing from time UI + D~ until time 0"1 + D1 -I- U2, and so on. The machine up times U~,U2 . . . . . are independent and identically distributed random variables with distribution function G, as are the repair times D~,D 2 . . . . . with distribution function H. After each repair, the job processing must resume, i.e. the work done on a job before a breakdown is not lost. Once a job is chosen for processing, it must be processed through to completion before any other job can be considered. These characteristics yield the so-called nonpreemptive resume model in the existing literatures (see [10,11,21], etc.). All the above random variables are assumed to be mutually independent. Our first goal is to schedule the jobs in such a way that the objective EY~k t%c~)f(C=(k)) is minimized, where C,~(k) denotes the completion time of the job in the kth position under policy 7r. The second goal is to evaluate the effect on the objective E E k t%(k)J(C,~tk )) of departures from convenient assumptions which yield Smith's rule as optimal (see [6,18], etc.).

2. Optimal permutation

policy

We firstly introduce the quantity

g,( i,j) = toiP( Xi + X j > t>_Xi), for i , j ~ { 1 , 2 . . . . . N} and t > 0 , which denotes the product of the weight associated with job i and the probability that t is between the processing time of job i and the sum of processing times of jobs i and j. It emerges that this is the key quantity for the interchange argument which yields Theorem 1.

tV. Li, K.D. Glazebrook / European Journal o f Operational Research 105 (1998) 524-536

527

T h e o r e m 1. If

g,( i,j) > g,( j,i)

(I)

holds for some i,j ~ { 1,2 . . . . . N} and for all t > 0, then the permutation policy 7r I = 7r( . . . . i,j .... ) is better than policy 7r2 = 7r( . . . . .hi .... ) with respect to minimizing the objective fimction of E E k r Proof. Denote by N ( t ) (t > 0) the number of machine failures in machine time t, i.e., N(t) = sup{k >_ 0 : Uo + U I + U 2 + 9 9 9 + U k < t} (U o = 0) and by C~(k) the completion time of the kth job under police 7r. Plainly, N(X,rlt)+X~(2)+

C~(k) = (X=(,) + X=(2) + " ' " +X=(a)) +

'''

E

+X,r(i)) D,

(2)

/=0

and therefore

Ef(C=(k~)= o

+

D, dP(X,o~ + X=o~ + . . . +X,,~) <_t) i

=/(0)+f~P(X=(i)+X,~(2)+

"'" + X = ( k ) > t ) d

EI t + E D i

~0

9

O) (4)

i=0

In order to prove that 7r I = 7r( . . . . i , j , . . . ) is better than policy 7r 2 = 7r( . . . . j,i . . . . ), it is sufficient to show that

eE

co,.,] <_e E o .2
k

k

in which C=,,(k) is the completion time of kth job under policy 7rm (m = 1,2). Suppose there are n jobs processed before job i in policy 7r I (or before job j in policy w2), then from Eq. (4) we have

e

~ w~,(k)f C~,(k) -

~ w~,(t)f C~-2(k)

k=l

=fo

k=l

f~

t=I~X~(i)
d E f t+i~=oDi) 1.

Noting the condition (1) and the fact that functions P(X,~(,) + X , o ) + 9 9 9 +X~(,) _
O.

z

k

k

This concludes the proof,

El

R e m a r k 1. (1) It is easily known that a stronger sufficient hypothesis for condition (1) of Theorem 1 to hold is X~ < ,.,.Xj and oJi > roi. The following examples illustrate that this is not necessary for the condition of Theorem 1 in (1) to hold. E x a m p l e 1. Let k > 1, and r.v.s. X i and Xj are independent exponentially distributed with rates A; and Aj respectively. In this case, if Ai > kA./> 0 and roj = kwi > 0, we can readily get X i < ,.,.Xj and ~oi < o)i, but c - A i r __ e - A /

g,( i,j) = o)i)t i

e-Ad _ e-Ad

>_g,( j,i) = rojAj Aj-

Ai

A j -- A i

528

W. Li, K.D. Glazebrook/ European Journal of Operational Research 105 (1998) 524-536

E x a m p l e 2. Let 1 / 2 > a > 0, k > 1 + 1/(1 - 2a). X i and Xj are independent discrete r.v.s, satisfying:

P(Xi=O ) = 1 -2a=

1 -P(Xi=

1)

and

P(Xj=O)=I-a=I-P(Xj=I), respectively. In this case, if toi = ktoj > 0, we can readily get X i > ,.t.Xi and toi > to i, but g t ( i , j ) > gt(j,i) is still valid. In fact, now 0, a(1-2a)

t<0, to i,

g,( i,j) = 12a2toi, ~0,

0
1 < t < 2, 2
and 0,

2a(1--a)toj,

t<0, 0
I <_t < 2,

gt( j,i) = | 2a2toj ' I

~0,

2<_t.

(2) Righter [20] gives an excellent survey of stochastic scheduling results in terms of stochastic ordering properties. Some of those results can be extended to the machine breakdown case. However, as indicated in (l) above, we get in Theorem I a new result under a condition which is strictly weaker than one of stochastic ordering. (3) As we shall see from Theorem 2, Theorem 3 and Corollary I, we are able to infer useful results from Theorem I. Nonetheless, the hypothesis (I) is strong in that it imposes conditions on each pair of jobs which need to hold for all t >__0. A key motivation for the development in Section 3 is to consider how such requirements may be relaxed in the context of particular classes of useful models. By using Theorem 1 together with an argument based on pairwise interchange, we derive some simple optimal permutation policies as follows. T h e o r e m 2. If g,(i,i + I) >_ gt(i + 1,i) holds for every i E {1,2 . . . . . N - 1} and t >_O, then 1. scheduling jobs according to permutation 1,2 . . . . . N minimizes the objective fimction of E ~ k to=(k)f(C~(k)), 2. the upper bound for the objective fitnction of EY'.k to~(k)f(C~(k)) is obtained by scheduling the jobs according

to permutation N , N -

1,... ,1.

If the processing requirement of each job i has a hazard rate Pi('), we have the following simpler results: T h e o r e m 3. If toi Pi(u) > toi+ , Pi+ I( v ) holds for every i E {1,2 . . . . . N - 1} and u,v > O, then 1. scheduling jobs according to permutation 1,2 . . . . . N minimizes the objective Jimction of E ~ k to=(k) f(C,~tk)),

2. the upper bound for the objective fimction of EE k to=ck)f(C=(k)) is obtained by scheduling the jobs according to permutation N,N - 1. . . . . 1. Proof. Since the processing requirement of each job i has a hazard rate Pi('), the result is straightforward by noting Theorem 2 and that

gt(i,j)-gt(j,i) = fo [ toiPi(X) -tojpj(t-x)]P( Xi>-x)P(Xl>-t-x)dx. []

W. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998) 524-536

529

C o r o l l a r y 1. When the processing time of job i is exponentially (or geometrically) distributed with rate I~i,

then 1. scheduling jobs in decreasing order of toi Ixi minimizes the objective fimction of EEk t%(k)f(C~ck)), 2. scheduling jobs in increasing order of toi ~i achieves the least upper bound for the objective fimction of EEk o91r(k)f(C~r(k))" Proof. This is immediate from Theorem 3.

[]

3. Some special cases In this section, we shall assume f ( t ) = at + b ( a > 0) and make a variety of assumptions about uptimes, all of which include the exponential distribution as a special case. In the first case, uptimes are DFR (Decreasing Failure Rate) and in the second IFR (Increasing Failure Rate). The latter offers a natural way of modelling machine deterioration. At first, we consider the case of a DFR uptime. Here, the distribution G ( . ) of the uptimes U/ (i >_ 1) is assumed to be a positive mixture of exponential distributions; i.e., for some m = 1,2 . . . . G(x)=

~pk[1--exp(--akx)], k=l

where 0 < a I < a 2 < 9 9 9 < am, E~= I p~ = 1. Note that we can approximate a wide range of DFR distributions by such mixtures. For this case, if m = 1, i.e. the uptime is exponential, the optimal permutation policy is to schedule the jobs in decreasing order of toi//EX i, which has been obtained by many authors (see [6,11,16,19], etc.). For the general case m > 1, we have the following result. T h e o r e m 4. Suppose condition

vjtoi+iEexp(-vjXi+, ) 1-Eexp(-vjXi+,)

<

vjtoiEexp(-vjXi) 1-eexp(-vjXi)

holds for i = I . . . . . N - 1, j = 0,1 . . . . . m - 1, where v o = 0 (in this case, the condition means the limiting case as v o ~ 0), and - vj ( j = 1,2 . . . . . m - 1) is the sequence of all real-valued negative roots of the equations ~

Pk

k=l

~ -I- a k

O.

Then the optimal permutation policy is to schedule the jobs in the order 1,2 . . . . . N. Proof. Under the condition that the distribution of Ui, i >_ 1, is a positive mixture of exponential distributions, we know from [6] or [9] that the renewal function equals

EN(t) = - -

m - I Ak

t

+ E --(1 -exp(-vkt)), EU k= i v~

t>0;

where

Ak .

.

. iffil

. (Oli-- lYk)

2

>0,

k = l,2 . . . . . m - 1 .

(5)

IV. Li, K.D. Glazebrook/ European Journal of Operational Research 105 (1998)524-536

530

Substituting Eq. (5) into Eq. (3) for any permutation policy 7r, we have

(ED)* EC=(k)= I + - ~

m-I E EX=(i)+(EO) ~-,

j=l

j=l

AY[ 1 -Ee-V'(x""'+---+x,,,)]. vy

(6)

Denote by 7r o = (1,2 . . . . . N ) the policy described in Theorem 4, and let 7ri be the same policy as 7r o except for an interchange between i and i + 1. In order to prove that zr 0 is optimal, it is enough ([22]) to show that E[E~= ~OJ=0(k)f(C=o(k))] < E[IE~r ~w,~,(k)f(C=,(k))]. According to Eq. (6), we can easily get

a e

~'~o~)f(C~o(,))- E I

'~

kffil

= =

1 + -~

EXi+I

Ex, ex,+, + ( c o ) E - - [ e e -~'(x'+ '

~Ai]

j=l

b'j

+x,_,)]

~i+-tEe--~'x'-----+' wiEe-~'x' ]

•

[ 1 -

Ee-

~jx,+,

The result then follows by the condition stated in the theorem.

1 - Ee-

~jx,

9

[]

R e m a r k 2. (1) In the case j = 0 and i = 1,2 . . . . . N - 1, the condition of Theorem 4 becomes ("Oi+ I

EXi+ l

s -

EXi"

This has a double significance. Firstly, it is made use of in the calculation contained in the proof. Secondly, it demonstrates that Theorem 4 includes the known result in the exponential case m = 1. (2) See Remark 1(3). A necessary condition for the hypothesis of Theorem 1 to hold (in the case j = i + 1) is that

twi+ I E e x p ( 1 -

-

tXi+ i)

Eexp(-tS,+,)

<

-

twiEexp( - tXi) l - Eexp(-tSi)

(7)

holds for any t >__0. In the special case considered in Theorem 4, this is relaxed to the requirement that (7) holds only when t = v j , 1 < j < m - 1 , and in the limit as t goes to zero. Note also that if Y / ~ e x p ( v j ) are independent of the processing requirements, then the hypothesis of Theorem 4 can be re-expressed as:

oj,o,+,P(X,+, <_r3 < vj,,,,P(x,<_~) 1 - P( Xi+ , < Yj)

1 - P( Xi <_Yj) "

This implies that the optimal permutation favours jobs with high weights w~ and short processing times as indicated by large values of P(X i <_Yj). This accords with intuition. Next, we investigate a case of IFR uptime. Now, the distribution of the uptimes Ui, i > 1, is chosen to be a positive mixture of an exponential distribution and an Erlang distribution, i.e., the density of Ui is given by

g( x) =pae -~x + (1 - p ) a2xe -'~x (where a > 0 and 0 < p < 1). In this case, we get the following theorem.

IV. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998)524-536

531

T h e o r e m 5. The optimal permutation policy is to schedule the jobs in order 1,2 . . . . . N, provided that the compatibility condition wi+l

wi

EXi+ , -

wiEexp[-(2-p)aXi]

EX i

1-

Eexp[

-

wi+lEexp[-(2-p)aXi+l]

(2-p)aXi]

< l-Eexp[-(2-p)~X~+,]

holds for i = 1,2 . . . . . N - 1. Proof. Under the condition that the distribution of Ui, i > 1, is a positive mixture of an exponential distribution and an Erlang distribution, we can easily show that the renewal function equals

EN(t)

1-e-"(2-P~'],

EU

t>__O,

(8)

where E U = (2 - p ) / a . Substituting Eq. (8) into Eq. (3) yields ED F. EX,,(~) EC~c~~ = 1 + -i-d ~_ ,

- (ED)

t~ 1

[1 -

Ee- o(2-p•

... + x . , o ] .

Therefore, for the same reason as in the proof of Theorem 4, we have a E

t~

)--' w~,,(k)f(C~M,)

[_k=l

=

k=l

(

1 +-i-J

~-/+,

exi

ex, ex,+, + ( e D )

x[1 -Ee-a(2-')x'][1 - e e -~

ee-~215

~'Ee-~

1 - Ee -~(2-p)x' -

~

1 - Ee -a(2-p)x'*'

"

(9)

Noting the compatibility condition in the Theorem 5. by using the interchange argument and the Eq. (9). we conclude the proof. [] R e m a r k 3. (1) It is valuable to point out that if Y denotes an exponentially distributed random variable with rate ( 2 - p ) a , then the compatibility condition in Theorem 5 can be re-expressed as: ~oi+ , -

-

~o, <

- -

, , i + , P ( X , + , _< Y) ==:.

EXi+ ' - EX i

~o,P(X, _< Y) >

1-P(Xi+,

<-Y ) -

.

l_P(Xi<_y)

(2) The compatibility condition in Theorem 5 is rather strong and it might be that the permutation policy 7r 0 = {1,2 . . . . . N} which schedules the jobs in order of decreasing ooi/EX i performs well even when it does not hold, We shall develop a suboptimality hound for 7r o expressed as a natural measure of the extent to which the compatibility condition fails, The desired result is given as Corollary 2, We write _ pj-

max

i
1 - Eexp[-(2 -p)~X,]

wiEexp[-(2-p)aXj] 1-Eexp[-(2-p)aXj]

}) + '

W. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998)524-536

532

noting that ~j will be zero if jobs {1,2 . . . . . j} meet the compatibility requirements. The proof of Corollary 2 utilizes the calculation in ihe proof of Theorem 5 together with an argument which constructs an arbitrary permutation from 7ro by successive pairwise interchanges.

Corollary 2. (S,tboptimality bound for 1FR mixture model.) E Y'~ oJ,~o(k)f(C,~o(k) ) -- min~ E ~ o~(k)f(C~(k) ) < (ED)

~bk .

k=l

k=l

Note that Corollary 2 holds for 0 _< p < 1. Substituting p = 1 into the above bound yields the optimality of go in the exponential uptime case.

4. T h e loss i n c u r r e d u p o n m a k i n g the simplifying exponential a s s u m p t i o n In stochastic scheduling, an assumption that processing times are exponential is frequently made in the interests of simplicity. In this section, we assess how much might be lost by making such an assumption in the context of a general breakdown process. We seek results in the spirit of Corollary 2 - namely suboptimality bounds expressed in terms of natural measures of the extent to which the exponential assumption fails. A natural approach is via hazard rates. Suppose that pj(t) is the hazard rate for processing time of job j, assumed continuous. The parameter ej = sup pj(t) -- inf p./(t) t~O

t~=O

is a simple measure of the extent to which X./ fails to be exponential. We also require p T = inft> o pj(t) be assumed positive for all j ( j = 1. . . . . N). Our goal will be to derive bounds on the quantity

and the proportionate version:

E

tO=oCk)f(C=o~k ) -- min,~ E

t%cof(C,~cO

in terms of the discrepancy parameters s i, j = 1,2 . . . . . N. In the above, 7r o will be the permutation which processes the jobs in decreasing order of wi/EX i, i.e. Smith's rule. This will indeed be the chosen policy when the exponential assumption is made and so A and /~ measure the damage caused by making such a simplification. Suppose that w i ,tt i > wi+ l/-ti+ t (i = 1,2 . . . . . N -- t). In the statement of Theorem 6, X ~ ~ exp(p.j) and X ~ ~ F(2,/_t~) is the sum of two independent copies of X ~ The notation Ef(Ck,YI,Y 2 . . . . . YN) is used to denote Ef(C~) under an assumption that Yj is the processing time for job j ( j = 1,2 . . . . . N).

IV. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998) 524-536

533

Theorem 6. N

o<_a<_ E ( o , ~

~..... x ~

N

N

E

E o ~ , - - e s ( c , , x , ~. . . . . x ~ 1 7 6 1 7 6

k-I

k=l

~k

i=k N

..... x ~

]'s N

~k

o ..... ,
I , X k o. -,X~

I," 9 9 ,x o)

~s

where ~ok~ - to k 1-I~= ,(( ~, + 6t)/Ix~), ~o; ~ to k 1-I ~_ ,( g , / ( Ixt + 6,)). Proof. Let {X~, i = 1 . . . . . N} be a sequence of independent exponential random variables each with rate P-'i - sup,:, o Pi(t) = I-ti + ei. Then by the comment before the theorem and the definition of to'i and P-'i, and by simple calculation, we have t

t

t

t

toi P.i < toi+ I txi+ l,

(10)

which holds for i = 1,2 . . . . . N dP

l. Since /.t/< p j ( x ) < tzi + ej, X~ < x

< dP

E Xj < x

j=l

<

Of 1=1 ~

j=l

]s

]1

E X~ < x . j=l

By using Corollary 1, Eq. (3) and the result in Eq. (10), and if we let K ( t ) = E f < (t + ,-.i=ruO)no v i , ~, then it can be shown that N

m i n . E r~

. . . . . X,~(u)]

k-I

N >_ minT, Y'. to~,o)

k

k- I

~,~(1) + ~tt) s

)]J:

K(t) dP(X;o)+

... +X.o)
)

N

= min~. ~

to,~(~)Ef[C~(k),X,~tl ' ;" ' ),

"'" ' X : (] N - - ) ,i

k=l N t t E o~,ef[c,.x, .....

X I

.l

k-1

and N

E o~:,<,>e:[c:,,,,,.xo,.> . . . . . x,.>] k=l <

k~--=I w~,~,#<,t, [~=I(g'"(O+#'G'"
=

g

o.L.,e.r[c:,,<~,.xL<,,

k-I N

= E o.~176 k=l

..... x ~

.....

x=~

X.,,(i)

+

" " 9 +

X~

,

IV. Li, K.D. Glazebrook / European Journal of Operational Research 105 (1998)524-536

534

It is valuable to note that the above two inequalities can also be straightforwardly derived by noting that the system with the 0 superscritSt has larger weights and stochastically larger completion times than the original system, and the system with the ' superscript has smaller weights and stochastically smaller completion times than the original system. Therefore, by using the result in Corollary 1 again, we have N

0 < A = E W~o(oEf[ C~o(k),X~,,O) . . . . . X~o(u)] - min~ Y'~ t%(k)Ef[ C,,o),X,~(, ) . . . . . X~(N) ] k=l

~"

N

N

t <- E o ~0 , e y [ q , x , 0 . . . . . x ~ - E o ~t e y [ q , x , t . . . . . xN]

k=l

k=l N

N

= g

(o, ~ - o~'~)e : [ c ~ , x ~ . . . . . x ~ ] + E , 4 [ ~ r

~ . . . . . x g ) - ~ y ( q , x ' , . . . . . x',,)]

k=l

k=l

-0,+0

(11)

2.

However, by noticing that

d P ( X ff < t) - d e ( X;~ < t)
and

de

EX}+X~

E

j=t

X~

1+

<

j=k+l

de

EX~

L/=1

+

jr1

E

X~

,

jfk+l

we easily deduce from Eq. (3) that

02 =

N

N

k=l

i=1

E E<[es(ci,x;, N

E k=l

y'

x k~ ~ .

,x~ .

x 'k - I ' x k~, ' ' "

,x ~

"'',"Xk-I~

, x. '~ _ , , x , , x k +o, . . . .

xo)]

e f ( c , , x ; . . . . , x'~_, ' x~,x~+, , o . . . ."

xo)]

.

.

N

E <[ef(c,,x;,

"'''

i=k

N

<

E

Ewe--

k=l i=k

Ef t +

~l~k

Di

dP

i

Exj+

+

j= I

E

x ~ <- t

j=k+ 1

N

< E k=l

E w: i=k

N

N

E

E

k=l

i= k

1 + -[l=l

E f ( C i , X ~ '''''

X k~- I ' X k~ ' X k~+ l ' ' ' "

,X ~

].3q

03i~kEf(Ci,

XO,

I~k

vO y,02 vO 9 . . ~ t L k _ 1 ~-L k ~ - ' L k + i ~ . . 9

,xO).

(12)

Thus, by appeal to (11) and (12), the first upper bound on is derived. The second follows by placing a simple bound on w~

w~ < w k

i=,

This completes the proof.

1+ []

-

i=,

1-

~

< 2

-

+de i=1

6

535

IV. Li, K.D. Glazebrook/ EuropeanJournal of OperationalResearch 105 (1998) 524-536

If we now suppose that ./(0) = 0 and that f has derivative everywhere bounded by a, then it will follows that f ( t ) < at for all t < 0. Under these mild conditions, we are able to bound A and zl by quantities which (to first

order in the sj) are simple functions of the parameters of the model, and hence easily computable. Corollary 3.

(I) 0 <- A < 2-

wkAk k=l

"1- E E O ) i k A i + k=l i=k ILk [

a

1 + ED

N

+~

~

ek

'

where 1.

j=!

(2) 0 < zl < A maxk ek + B max k

+ dT{(maxk ~k)2},

where A = A N + 2 A o and B = a ( 2 N + 1)(1 + ( E D / E U ) ) .

Proof. By using the inequality (see p. 668 in [6]) ( t / E U ) - 1 < E N ( t ) < ( t / E U ) + ( E U 2 / ( E U ) 2) - 1, from Eq. (3) we get

Ef(Ck,X ~..... X~

fo

[t+(ED)EN(t)IdP(X tED


t + E--'--~+ED

~ + "" +X ~
(EU2)] (EU)2

1

dP(X~

+X~

=A k.

Similarly it follows that 9~.o2,-,k+ v o i,- . . , X ~ E f ( C i , XO . . . . . X L i,-,k

+ - /'tk 1 +

.

Corollary 3(1) now follows directly from Theorem 6. Corollary 3(2) is a consequence of simple algebraic arguments. This concludes the proof. []

5. Conclusions We study a class of single machine stochastic scheduling problems, with no specific distributional assumptions made aboat processing requirements or the availability of the machine. Theorem 1 elucidates a general condition which facilitates comparisons based on pairwise interchanges. We can ease this condition considerably when the uptimes are drawn either from a general (DFR) mixture of exponentials or from a (IFR) mixture of an exponential and a Gamma distribution. When conditions sufficient for optimal policies of simple structure fail, it is beneficial to understand the cost implications of implementing simple policies nevertheless. Several suboptimality bounds are given in the paper, most of which assess the loss incurred when making the simplifying assumption that processing times are exponential.

536

W. Li, K.D. G!azebrook / European Journal of Operational Research 105 (1998) 524-536

Acknowledgements T h e w o r k o f W e i Li was partially supported by the U n i v e r s i t y o f N e w c a s t l e upon T y n e by m e a n s o f a S e n i o r Visiting F e l l o w s h i p and by the National Natural S c i e n c e Foundation o f China, respectively. M a n y thanks are due to the a n o n y m o u s referees for the constructive suggestions and the detailed c o m m e n t s contributing to i m p r o v e the legibility o f the paper.

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