On three-machine flow shops with random job processing times

European Journal of Operational Research 125 (2000) 440±449

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Theory and Methodology

On three-machine ¯ow shops with random job processing times Jerzy Kamburowski

*

Department of Information Systems and Operations Management, The University of Toledo, Toledo, OH 43606-3390, USA Received 1 May 1998; accepted 1 February 1999

Abstract The paper deals with scheduling n jobs on three machines in JohnsonÕs ¯ow shops. The job processing times are assumed to be independent random variables, and the problem is to minimize stochastically the makespan. Using a convenient makespan representation, we present sucient conditions on the job processing time distributions which imply that the makespan becomes stochastically smaller when two adjacent jobs in a given job sequence are interchanged. They lead, in particular, to an extension of TalwarÕs rule known for two-machine stochastic ¯ow shops with exponential job processing times. Extensions of other results are also included. Ó 2000 Elsevier Science B.V. All rights reserved. Keywords: Stochastic scheduling; Flow shop; Makespan; Stochastic order relation

1. Introduction Consider a shop which has m ˆ 3 machines and must process n jobs, f1; 2; . . . ; ng, available at time zero. Each job has to be processed ®rst on machine A, next on B, and last on C. Moreover, no machine can process more than one job at a time, no preemption is allowed, all setup times are included into the job processing times, and there is unlimited intermediate storage in between the machines. The problem, F3|prmu|Cmax , is to determine a job sequence (permutation) that minimizes the com-

*

Tel.: +1-419-530-4361; fax: +1-419-530-7744. E-mail address: [email protected] (J. Kamburowski).

pletion of the last job, the so-called makespan. Recall that for m ˆ 2 and m ˆ 3 permutation schedules have been proven to be optimal among all schedules, but the shortest schedule may have di€erent job orderings on di€erent machines for m > 3 [1,2]. Recall also that the problem of minimizing the makespan in permutation ¯ow shops is strongly NP-hard for m P 3 [3]. Assume that Ak , Bk and Ck are independent random variables representing the processing times of job k on machines A, B and C, respectively. The stochastic counterpart of F3|prmu|Cmax is to determine a job sequence which minimizes the makespan in a certain stochastic sense, for example, in the sense of the expected value or the stochastic order. (A random variable X is smaller than another random variable Y in the sense of the

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 2 2 2 - 2

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stochastic order, written X 6 st Y , if P …X > t† 6 P …Y > t† for every t.) To follow pairwise job interchange arguments for two adjacent jobs i and j, let p1 ˆ …q; i; j; x† and p2 ˆ …q; j; i; x† be two job sequences, where q and x are subsequences of jobs excluding i and j, and assume that M1 and M2 denote the makespans of p1 and p2 , respectively. Ku and Niu [4] presented the following condition in the case of F2|prmu|Cmax :

can be regarded as a generalization of (1), and its proof clearly indicates how to determine an analogical condition for Fm|prmu|Cmax . We show, in particular, that if TalwarÕs rule applied to A and B, and B and C yields the same job sequence, then this sequence stochastically minimizes the makespan. Simple proofs and extensions of the results of [10,11] are also included.

if; for all a and b in the supports of Ai ‡ Aj and

2. The makespan representations

Bi ‡ Bj ; ‰min…Aj ; Bi †jAi ‡ Aj ˆ a; Bi ‡ Bj ˆ b† P st ‰min…Ai ; Bj †jAi ‡ Aj ˆ a; Bi ‡ Bj ˆ bŠ; then M1 6 st M2 :

…1†

A simple and complete proof of this condition was given in [5]. Using (1), Ku and Niu proved the optimality of TalwarÕs [6±8] scheduling rule for exponentially distributed job processing times, which, similarly as JohnsonÕs rule, states that job i precedes job j if E…min…Aj ; Bi †† > E…min…Ai ; Bj ††:

…2†

As noted in [4,6±8], if ak and bk are the shape parameters of Ak and Bk , that means E…Ak † ˆ 1=ak and E…Bk † ˆ 1=bk ; then the rule (2) is equivalent to job i precedes job j if ai ÿ bi > aj ÿ bj :

…3†

Some special cases of the stochastic F2|prmu|Cmax problem were studied in [5,7±10]. Frostig and Adiri [11] showed a stochastic counterpart of SzwarcÕs [12] eciently solvable special case for F3|prmu|Cmax . Some results pertaining to m-machine stochastic ¯ow shops with identical machines were obtained in [10]. Excellent overviews of rather limited research on stochastic ¯ow shops can be found in [13], and in Chapter 13 of [14] written by Righter. In this paper we derive convenient representations and graphical illustrations of the makespans M1 and M2 in the case of F3|prmu|Cmax . They lead to a sucient condition for M1 6 st M2 which lays a theoretical foundation to follow pairwise job interchange arguments in the stochastic F3|prmu|Cmax problem. The proposed condition

Recall that when the jobs of a given sequence go through a ¯ow shop in the reverse order and in the opposite direction (®rst on C, next on B, and last on A), then the makespans of both sequences are the same [13, p. 96]. By the reverse sequence we mean the sequence de®ned above. For given job sequences p1 ˆ …q; i; j; x† and p2 ˆ …q; i; j; x†, denote their reverse sequences by p1 and p2 . Thus, M1 and M2 stand for the makespans of p1 and p1 , and p2 and p2 , respectively. We assume that all jobs are processed as soon as possible (without unecessary idle-times) under p1 , p1 , p2 and p2 . The following theorem shows convenient representations of M1 and M2 . Theorem 1. The makespans M1 and M2 can be represented as follows: M1 ˆ Aq ‡ Cx ‡ Aij ‡ Bij ‡ Cij ÿ V1

and

M2 ˆ Aq ‡ Cx ‡ Aij ‡ Bij ‡ Cij ÿ V2 ; where V1 and V2 are the shortest path lengths in the networks of Fig. 1, and Aq ˆ

X

Ak ;

k2q

Bij ˆ Bi ‡ Bj ;

Cx ˆ

X

Ck ;

Aij ˆ Ai ‡ Aj ;

k2x

Cij ˆ Ci ‡ Cj ;

and QAB (QAC ) is the additional time after Aq necessary to complete the processing of the jobs of q on B (C) under p1 and p2 , and QCB (QCA ) the additional time after Cx necessary to complete the processing of the jobs of x on B (A) under p1 and p2 :

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J. Kamburowski / European Journal of Operational Research 125 (2000) 440±449

Fig. 1. Shortest path networks for computing V1 and V2 :

Proof. Consider the makespan M1 of p1 and p1 . It is well-known that the makespan of a given job sequence can be represented by the longest path length in an acyclic network, see e.g., [13, p. 95]. For the de®ned times Aq ,Cx , QAB , QAC , QCB and QCA , the makespan M1 is Aq ‡ R1 ‡ Cx ; where R1 is the longest path length in the network of Fig. 2a. Since the networks of Figs. 2a

and b have the same path lengths, R1 is also the longest path length in the network of Fig. 2b. It suces now to extract the term Aij ‡ Bij ‡ Cij from R1 and observe that R1 ˆ Aij ‡ Bij ‡ Cij ÿ V1 , which veri®es the representation of M1 . When we switch i and j and notice that Aji ˆ Aij ; Bji ˆ Bij and Cji ˆ Cij , one obtains the representation of M2 . 

J. Kamburowski / European Journal of Operational Research 125 (2000) 440±449

443

Fig. 2. Longest path networks for computing R1 :

Below we present some interpretations for the case of p1 and p1 ; analogous interpretations are valid for the case of p2 and p2 . Let S1 ˆ min…Aj ‡ Bj ; Bi ‡ Ci ; Aj ‡ Ci †, that is, S1 is the shortest path length in the subnetwork restricted to heavy arcs in the network representing V1 . Then Aij ‡ Bij ‡ Cij ÿ S1 is the makespan of …i; j†. S1 can be interpreted as the time savings from allowing jobs i and j to overlap on A, B and C under …i; j†. Clearly, if i and j are the only jobs in the system, and they are not allowed to overlap, then their processing time would be

Aij ‡ Bij ‡ Cij . It is interesting to observe that either S1 ˆ min…Aj ; Bi † ‡ min…Bj ; Ci † or S1 ˆ min…Aj ‡ Bj ; Bi ‡ Ci †; and min…Aj ; Bi † ‡ min…Bj ; Ci † 6 min…Aj ‡ Bj ; Bi ‡ Ci †:

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The terms min…Aj ; Bi † and min…Bj ; Ci † can be interpreted as the time savings from allowing jobs i and j to overlap on A and B, and B and C under …i; j†. On the other hand, min…Aj ‡ Bj ; Bi ‡ Ci † is the corresponding time saving when B can be regarded as a non-bottleneck machine, or equivalently, the jobs are processed on two machines A and C, while Bk denotes the time lag between the completion of job k on A and its start on C. Moreover, R1 ˆ Aij ‡ Bij ‡ Cij ÿ V1 is the maximum of the times needed to process jobs i and j under p1 and p1 . Therefore, V1 can be interpreted as the time savings from allowing jobs i and j to overlap on A, B and C under p1 and p1 . Observe that V1 6 S1 , that is, the release times Aq ‡ QAB ; Aq ‡ QAC ; Cx ‡ QCB and Cx ‡ QCA may reduce the time savings S1 which are possible when i and j are the only jobs in the system. We also have: Aq ‡ Aij ‡ Bij ÿ min…Aij ÿ QAB ; min…Aj ; Bi †† ˆ the completion time of job j on B under p1 , Cx ‡ Cij ‡ Bij ÿ min…Cij ÿ QCB ; min…Bj ; Ci †† ˆ the completion time of job i on B under p1 , Aq ‡Aij ‡Bij ‡Cij ÿmin…S1 ; Aij ÿQAB ‡min…Bj ;Ci †; Aij ‡Bij ÿQAC †ˆthe completion time of job j on C under p1 , and Cx ‡Aij ‡Bij ‡Cij ÿmin…S1 ;Cij ÿQCB ‡min…Aj ;Bi †; Cij ‡Bij ÿQCA †ˆthe completion time of job i on A under p1 : From Theorem 1 one can easily derive the following result. Corollary 1. Sufficient conditions for M1 6 M2 are (i) V1 P V2 : (ii) min…Aj ; Bi † P min…Ai ; Bj †; min…Bj ; Ci † P min…Bi ; Cj †; and S1 P S2 [15]. …Bj ; Ci † P (iii) min…Aj ; Bi † P min…Ai ; Bj †, min…Bi ; Cj †, and min…Aj ; Ci † P min…Ai ; Cj † [15]. Moreover, ((ii) () (iii)) ) (i).

3. Sucient conditions for job interchange in the stochastic case Assume that the job processing times Ak , Bk and Ck are independent random variables. For

convenience, the conditional random variable ‰X jY ˆ yŠ is often denoted by ‰X jyŠ. Theorem 2. Sufficient conditions for M1 6 st M2 are (i) ‰V1 jhŠ P st ‰V2 jhŠ for all h in the support of H ˆ …QAB ; QAC ; QCB ; QCA ; Aij ; Bij ; Cij †: (ii) ‰min…Aj ; Bi †ja; bŠ P st ‰min…Ai ; Bj †ja; bŠ, and ‰min…Bj ; Ci †jb; cŠ P st ‰min…Bi ; Cj †jb; cŠ, ‰min…Aj ; Ci †ja; cŠ P st ‰min…Ai ; Cj †ja; cŠ for all a, b and c in the supports of Aij , Bij and Cij . Moreover, Condition (ii) implies Condition (i). Proof. To prove that Condition (i) implies M1 6st M2 , note ®rst that Aq and Cx are stochastically independent, Aq and V1 , and Aq and V2 , are dependent on QAB and QAC , while Cx and V1 , and Cx and V2 , are dependent on QCB and QCA . Moreover, X Pst Y if, and only if, ÿX 6st ÿ Y , and the stochastic order is closed under convolution. Therefore, the use of Theorem 1 leads to ‰M1 jhŠ ˆ ‰Aq jqAB ; qAC Š ‡ ‰Cx jqCB ; qCA Š ‡ a ‡ b ‡ c ÿ ‰V1 jhŠ 6 st ‰Aq jqAB ; qAC Š ‡ ‰Cx jqCB ; qCA Š ‡ a ‡ b ‡ c ÿ ‰V2 jhŠ ˆ ‰M2 jhŠ; and unconditioning gives the desired result (the order `` 6 st '' is closed under mixtures). To show that Condition (ii) implies (i), observe ®rst that ‰min…Aj ; Bi †jhŠ ˆ ‰min…Aj ; Bi †ja; bŠ; ‰min…Ai ; Bj †jhŠ ˆ ‰min…Ai ; Bj †ja; bŠ; ‰min…Bj ; Ci †jhŠ ˆ ‰min…Bj ; Ci †jb; cŠ; ‰min…Bi ; Cj †jhŠ ˆ ‰min…Bi ; Cj †jb; cŠ; ‰min…Aj ; Ci †jhŠ ˆ ‰min…Aj ; Ci †ja; cŠ; ‰min…Ai ; Cj †jhŠ ˆ ‰min…Ai ; Cj †ja; cŠ: Furthermore, under condition H ˆ h, V1 and V2 can be represented as ‰V1 jhŠ ˆ ‰min…S1 ; b ‡ c ÿ qCA ; a ‡ b ÿ qAC †jhŠ; ‰V2 jhŠ ˆ ‰min…S2 ; b ‡ c ÿ q ; a ‡ b ÿ qAC †jhŠ; CA

J. Kamburowski / European Journal of Operational Research 125 (2000) 440±449

where ‰S1 jhŠ and ‰S2 jhŠ are the shortest path lengths in the networks shown in Fig. 3a and b, respectively, with d ˆ a ÿ qAB and e ˆ c ÿ qCB . Therefore, to prove that Condition (ii) implies Condition (i), it suces to show that ‰S1 jhŠ Pst ‰S2 jhŠ for all h in the support of H. This will be accomplished by verifying ‰S1 jhŠ Pst ‰S 0 jhŠ P st ‰S 00 jhŠ Pst ‰S2 jhŠ, where ‰S 0 jhŠ and ‰S 00 jhŠ are the shortest path lengths in the networks depicted in Fig. 3b and c, respectively. We have ‰S1 jhŠ ˆ ‰min…min…Aj ; Bi ; d† ‡ min…Ci ; e†; min…Aj ; d† ‡ Bj †jhŠ

ˆ ‰min…min…Aj ; Bi ; d† ‡ min…Ci ; e†; min…a ÿ Ai ; d† ‡ Bj †jhŠ P st ‰min…min…Ai ; Bj ; d† ‡ min…Ci ; e†; min…a ÿ Ai ; d† ‡ Bj †jhŠ ˆ ‰min…min…Ai ; Bj ; d† ‡ min…Ci ; e†; min…Aj ; d† ‡ Bj †jhŠ ˆ ‰S 0 jhŠ ˆ ‰min…min…Ci ; Aj ; d; e† ‡ Bj ; min…Ai ; d† ‡ min…Ci ; e††jhŠ ˆ ‰min…min…Ci ; Aj ; d; e† ‡ Bj ; min…Ai ; d† ‡ min…c ÿ Cj ; e††jhŠ

Fig. 3. Shortest path networks for computing ‰S1 jhŠ, ‰S 0 jhŠ; ‰S 00 jhŠ and ‰S2 jhŠ.

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E…min…X2 ; Y1 †† P E…min…X1 ; Y2 ††

P st ‰min…min…Cj ; Ai ; d; e† ‡ Bj ; min…Ai ; d† ‡ min…c ÿ Cj ; e††jhŠ

implies

ˆ ‰min…min…Cj ; Ai ; d; e† ‡ Bj ;

‰min…X2 ; Y1 †jx; yŠ P st ‰min…X1 ; Y2 †jx; yŠ

min…Ai ; d† ‡ min…Ci ; e††jhŠ;

for all x and y in the supports of X and Y.

00

ˆ ‰S jhŠ ˆ ‰min…min…Bj ; Ci ; e† ‡ min…Ai ; d†; Bj ‡ min…Cj ; e††jhŠ ˆ ‰min…min…Bj ; Ci ; e† ‡ min…Ai ; d†; bij ÿ Bi ‡ min…Cj ; e††jhŠ P st ‰min…min…Bi ; Cj ; e† ‡ min…Ai ; d†;

Proposition 1. Let the job processing times Ak , Bk and Ck be independently and exponentially distributed random variables with the shape parameters ak , bk and ck , respectively. Then a sufficient condition for M1 6st M2 is a i ÿ b i P a j ÿ bj

and

bi ÿ ci P bj ÿ cj :

…4†

bij ÿ Bi ‡ min…Cj ; e††jhŠ ˆ ‰min…min…Bi ; Cj ; e† ‡ min…Ai ; d†; Bj ‡ min…Cj ; e††jhŠ ˆ ‰S2 jhŠ:



Remark 1. Theorem 2(ii) can be regarded as a stochastic counterpart of Corollary 1(iii). The stochastic order Pst assumed in Conditions (i) and (ii) of Theorem 2 can be weakened by the increasing concave order Picv , which leads then to M1 6icx M2 , where 6icx stands for the increasing convex order; see e.g. [14] for de®nitions. Such replacement seems to be desirable when one only seeks to minimize the expected makespan, because of M1 6st M2 ) M1 6icx M2 ) E…M1 † 6 E…M2 †. The proposed proof shows rather clearly how Theorem 2 can be generalized in the case of Fm|prmu|Cmax . Namely, to secure M1 6st M2 , the conditions analogous to (1) must hold then between any pair of machines. This is not any surprise because similar conditions are necessary in the deterministic case [16].

Proof. For independently and exponentially distributed random variables X and Y, min(X,Y) is exponentially distributed with the expected value …Eÿ1 …X † ‡ Eÿ1 …Y ††ÿ1 . Therefore, (4) can be rewritten as E…min…Aj ; Bi †† P E…min…Ai ; Bj †† and E…min…Bj ; Ci †† P E…min…Bi ; Cj ††. Moreover, (4) that is, to leads to a i ÿ c i P aj ÿ c j , E…min…Aj ; Ci †† P E…min…Ai ; Cj ††. Therefore, the use Lemma 1 makes Theorem 2(ii) applicable, which completes the proof.  Corollary 2. Assume that the job processing times are independently and exponentially distributed random variables. If Talwar's rule applied to machines A and B, and B and C yields the same job sequence, then this sequence stochastically minimizes the makespan. Remark 2. Rather surprisingly, Corollary 2 includes a stronger result than could be stated in the deterministic case; compare with Corollary 1(iii). Clearly, for the exponential case, E…min…Aj ; Bi †† ˆ E…min…Ai ; Bj ††

4. Special cases The lemma below is needed to extend TalwarÕs rule (3) to the case of three machines. Lemma 1 [4]. If X1 , X2 , Y1 , and Y2 are independently and exponentially distributed random variables, X ˆ X1 + X2 and Y ˆ Y1 +Y2 , then

and E…min…Bj ; Ci †† ˆ E…min…Bi ; Cj †† imply E…min…Aj ; Ci †† ˆ E…min…Ai ; Cj ††: In the deterministic case, however,

J. Kamburowski / European Journal of Operational Research 125 (2000) 440±449

min…Aj ; Bi † ˆ min…Ai ; Bj †

bility distribution. Then from Theorem 2 and Proposition 2, one obtains the following result.

and

Corollary 3 [10]. If the processing times of all jobs are either deterministic or independent and exponential random variables, and Aj ˆ st Bj ˆ st Cj for all j, then the makespans of all job sequences are stochastically equal, that is, the jobs can be processed in any order.

min…Bj ; Ci † ˆ min…Bi ; Cj † do not imply min…Aj ; Ci † ˆ min…Ai ; Cj † [15]. Let p be a TalwarÕs (A,B)-sequence, that is a sequence satisifying TalwarÕs rule on machines A and B. A subsequence of p is said to be a segment if ai ÿ bi ˆ aj ÿ bj for all jobs i and j …i 6ˆ j† belonging to g. To check whether of the optimal sequence of Corollary 2 exists, it suces to ®nd ®rst a TalwarÕs (A; B)-sequence, and next reorder all segments according to TalwarÕs rule applied to B and C. If the resulting sequence is a TalwarÕs (B; C)-sequence, then it is optimal. If none of the sequences is both TalwarÕs (A; B)and (B; C)-sequence, we propose to schedule jobs according to the following heuristic rule:

The following result is valid under weaker assumptions than those imposed above. Proposition 2. Let the processing times of all jobs be either deterministic or independent and exponential random variables, and let either Aj ˆ st Bj or Bj ˆ st Cj for all j. If the jobs are renumbered to satisfy E…min…Aj ; Ci † P E…min…Ai ; Cj †† for all i < j, then sequence …1; 2; . . . ; n† stochastically minimizes the makespan. Proof. Let Aj ˆst Bj for all j. Then for both the deterministic and exponential cases, ‰min…Aj ; Bi †ja; bŠ ˆst ‰min…Ai ; Bj †ja; bŠ; and

job i precedes job j if

E…min…Aj ; Ci †† P E…min…Ai ; Cj ††

ai bi =…ai ‡ bi † ÿ bi ci =…bi ‡ ci † P aj bj =…aj ‡ bj † ÿ bj cj =…bj ‡ cj †:

447

…5†

The rule (5) is motivated by the well-known and widely applied JohnsonÕs heuristic rule for the deterministic case: job i precedes job j if min…Aj ‡ Bj ; Bi ‡ Ci † P min…Ai ‡ Bi ; Bj ‡ Cj †: …6† However, (5) cannot be completely regarded as a stochastic counterpart of (6), because in deriving (5) the distributions of Ak ‡ Bk and Bk ‡ Ck have been approximated by the exponential distributions with the means E…Ak † ‡ E…Bk † and E…Bk †‡ E…Ck †. As in [10], assume that the machines are identical in the sense that for each job its processing times on di€erent machines have the same proba-

implies ‰min…Aj ; Ci †ja; cŠ P st ‰min…Ai ; Cj †ja; cŠ; which is equivalent to ‰min…Bj ; Ci †jb; cŠ P st ‰min…Bi ; Cj †jj; cŠ: Therefore, Theorem 2(ii) is applicable. The case Bj ˆst Cj for all j is symmetric.  It should be pointed out here that in the general case, Ai ˆst Bi and Aj ˆst Bj only imply min…Aj ; Bi † ˆst min…Ai ; Bj †; but not ‰min…Aj ; Bi †ja; bŠ ˆst ‰min…Ai ; Bj †ja; bŠ: Szwarc [12] presented an eciently solvable special case for the deterministic three-machine problem. He showed that sequence …1; 2; . . . ; n† is

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optimal if Ai 6 Aj ; Bi ˆ Bj ; Ci P Cj for all i < j. A stochastic version of this case was proposed by Frostig and Adiri [11] who veri®ed Proposition 3 using rather lengthy arguments. Recall that a continuous random variable X with density f is said to be smaller than another continuous random variable Y with density g in the sense of the likelihood ratio order, written X 6lr Y , if f(t)/g(t) is non-increasing in t over the union of supports of X and Y. Recall also that X 6lr Y ) X 6st Y ) E…X †…Y †: Proposition 3 [11]. If the jobs can be renumbered to satisfy Ai 6 lr Aj ; Bi ˆst Bj ; Ci P lr Cj for all i < j, then sequence …1; 2; . . . ; n† stochastically minimizes the makespan. Proof. The proof is obvious in light of Theorem 2(ii) and the following result.  Lemma 2 [4]. If X and Y are independent random variables such that X 6lr Y , then ‰X jX ‡ Y ˆ zŠ 6lr ‰Y jX ‡ Y ˆ zŠ for all z: The next lemma is needed to extend the result of [11]. Lemma 3 [14, p. 384]. Let X and Y be independent random variables, and U a bivariate function U : R2 ! R such that U…x; y† ÿ U…y; x† is nondecreasing in x. Then X Pst Y implies E…U…X ; Y ††† P E…U…Y ; X ††: When we are only interested in minimizing the expected makespan, one can employ the following result. Proposition 4. If the jobs can be renumbered to satisfy Ai 6st Aj ; Bi ˆst Bj ; Ci Pst Cj for all i < j, then sequence …1; 2; . . . ; n† has the minimum expected makespan.

Proof. Suppose Aj Pst Ai ; Bi ˆst Bj , and Ci Pst Cj , and i and j are adjacent. It suces to show that E…V1 † P E…V2 †. Note here that X 6st Y does not imply ‰X jX ‡ Y ˆ zŠ 6st ‰Y jX ‡ Y ˆ zŠ even if X and Y are independent. Therefore, Theorem 2 is not applicable and there is a need for a di€erent argument which demonstrates another usefulness of Theorem 1; compare with [9] and [14, p. 395]. Let us switch Aj and Ai in the network of Fig. 1 representing V1 , and denote the resulting shortest path length by V 0 . If V1 is regarded as a function U…Aj ; Ai †, then V 0 ˆ U…Ai ; Aj †. Since U…Aj ; Ai †ÿ U…Ai ; Aj † is non-decreasing in Aj , Lemma 3 is applicable and E…V1 † P E…V 0 †: Let us switch Bi and Bj in V 0 , and denote the resulting shortest path length by V 00 . Since Bi ˆst Bj , we have V 0 ˆst V 00 , so E…V 0 † ˆ E…V 00 †. Moreover, we observe that V 00 and V2 di€er only in the lengths of heavy arc (3,4) which are Ci and Cj , respectively. If V 00 is regarded as a function V 00 ˆ U…Ci ; Cj †, then V2 ˆ U…Cj ; Ci †, and Lemma 3 is again applicable. This gives E…V 00 † P …V2 †, and the proof is complete.  References [1] S.M. Johnson, Optimal two- and three-stage production schedules with setup times included, Naval Research Logistics Quarterly 1 (1954) 61±68. [2] S.S. Panwalkar, M.L. Smith, C.R. Woolam, Counterexamples to optimal permutation schedules for certain ¯owshop problems, Naval Research Logistics Quarterly 28 (1981) 339±340. [3] M.R.D. Garey, D.S. Johnson, R. Sethi, The complexity of ¯owshop and job shop scheduling, Mathematics of Operations Research 1 (1976) 117±129. [4] P.S. Ku, S.C. Niu, On JohnsonÕs two machine ¯ow shop with random processing times, Operations Research 34 (1986) 130±136. [5] J. Kamburowski, Stochastically minimizing the makespan in two-machine ¯ow shops without blocking, European Journal of Operational Research 112 (1999) 304±309. [6] P.P. Talwar, A note on sequencing problems with uncertain job times, Journal of Operations Research Society of Japan 9 (1967) 93±97. [7] A.A. Cunningham, S.K. Dutta, Scheduling jobs with exponentially distributed processing times on two machines of a ¯ow shop, Naval Research Logistics Quarterly 16 (1973) 69±81. [8] V.R. Prasad, N  2 ¯owshop sequencing problem with random processing times, Opsearch 18 (1981) 1±14.

J. Kamburowski / European Journal of Operational Research 125 (2000) 440±449 [9] C.S. Chang, D.D. Yao, Rearrangement: Majorization and stochastic scheduling, Mathematics of Operations Research 18 (1993) 658±684. [10] M. Pinedo, Minimizing the expected makespan in stochastic ¯ow shops, Operations Research 30 (1982) 148±162. [11] E. Frostig, I. Adiri, Three-machine ¯owshop stochastic scheduling to minimize distribution of schedule length, Naval Research Logistics Quarterly 32 (1985) 179±183. [12] W. Szwarc, Mathematical aspects of the 3  n jobshop sequencing problem, Naval Research Logistics Quarterly 21 (1974) 145±153.

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[13] M. Pinedo, Scheduling: Theory Algorithms and Systems, Prentice-Hall, Englewood Cli€s, NJ, 1995. [14] M. Shaked, J.G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Boston, MA, 1994. [15] F. Burns, J. Rooker, JohnsonÕs three-machine ¯ow-shop conjecture, Operations Research 24 (1976) 578±580. [16] C.L. Monma, A.H.G. Rinnooy Kan, A concise survey of eciently solvable special cases of the permutation ¯owshop problem, RAIRO Recherche Operationelle 17 (1983) 105±119.