Inequalities for stochastic flow shops and job shops

APPLIED STOCHASTIC MODELS AND DATA ANALYSIS, VOL. 2, 61-69 (1986)

Production, scheduling, routing and inventory Queueing

INEQUALITIES FOR STOCHASTIC FLOW SHOPS AND JOB SHOPS

MICHAEL PINED0 AND SUNG-HWAN WIE Department of Industrial Engineering and Operations Research, Columbia University, New York. NY 10027, U.S.A.

SUMMARY

Consider an m-machine flow shop with n jobs. The processing time of job j , j = 1, . . . , n , on each one of the m machines is equal to the random variable XI and is distributed according to 4. We show that, under certain conditions, more homogeneous distributions F l , . . . , F, result in a smaller expected makespan. We also study the effect of the variability of distribution FJ on the expected waiting costs of the n jobs and on the job sequencing which minimizes this total expected waiting cost. We show that the smallest (largest) variance first rule minimizes the total expected waiting cost on a single machine when the waiting cost function is increasing convex (concave). We also show that the smallest variance first rule minimizes, under given conditions, the total expected waiting cost in an m machine flow shop when the waiting cost function is increasing convex. Similar results are also obtained for the two-machine job shop. Similar results cannot be obtained when the processing times of job j on the various machines are i.i.d. and distributed according to F,.

KEY WORDS Stochastic scheduling Makespan Flow Time Variability ordering Majorization

1 . INTRODUCTION

Consider a shop model with m machines and n jobs. Job j , j = 1, . . . , n, has to be processed on each one of the m machines for the same random time Xj, which is distributed according to Fj. The way in which a job is routed through the system of machines depends on the par- ticular model under consideration. We consider here two models, namely:

(i) Flow shops with m machines. In a flow shop the n jobs are to be processed on the m machines with the order of processing on the different machines being the same for all jobs: each job has to be processed first on machine 1 , then on machine 2, etc. Pre- emptions are not allowed, and a job may not ‘pass’ another job while they are waiting for a machine. At the start of the process, the n jobs have to be put in a sequence, say j , , . . . , j,, which determines the order in which they go through the system.

(ii) Job shops with two machines. The set of n jobs consists of two subsets which are called set A and set B, respectively. The jobs of set A (B) have to be processed first on machine 1 (2) and after that on machine 2(1). In what follows, only policies are considered which require all jobs of set A (B) to be completed on machine 1 (2) before the initiation of a job of set B (A) on this machine.

These types of models are also used for the performance analysis of communication channels. In communication channels, the length of a message is usually random but the message does not change in length when being transmitted from one station to the next. The flow shop with m machines represents a communication channel with m transmitters in series, where each message has to follow the same route. The results obtained in this paper for the two-machine

8755-0024/86/020061-09$05 .OO 01986 by John Wiley & Sons, Ltd.

Received 23 July 1985 Revised February 1986

62 M. PINED0 AND S-H. WIE

job shop give an indication of the difficulties that arise when messages follow different routes. For past research on models where jobs have the same processing times at the different machines, see references 1-5.

We use the following notation. The random variable Cj denotes the time at which job j leaves the system. The random variable Cmax = max (Cl, . . . , C,,), i.e. the time the last job leaves the system is known as the makespan. The random variable Cg(Cj ) represents the total waiting cost; the function g is called the waiting cost function. We shall assume that this function is either increasing concave or increasing convex.

Pinedo and Weber4 have shown that when the processing time of job j on each one of the m machines is equal to Xj, the makespan in the m-machine flow shop equals XI + XZ + . . . + X n + (m - 1) max (XI, . . . , Xn). This makespan is independent of the sequence in which the jobs go through the system. Pinedo and Weber also considered the two-machine job shop where the processing time of job j on each one of the two machines is Xj. They assumed that jobs 1,. . . , p belong to set A, whereas jobs p + 1,. . . , n belong to set B and showed that the makespan under any policy from the class of policies described above equals max(Xl,X2, . . . ,Xp,Xp+l + XP+z + . . . + Xn) + rnax(Xl+ XZ + . . . + X,, &,+I, XP+z, . . . , Xn). This makespan is independent of the order in which the jobs of sets A and B go through the system.

In this paper, we establish inequalities for the expected makespan and for the total expected waiting cost. Section 2 deals with the effect of the homogeneity of the distributions F1,. . . , Fn on the expected makespan. Section 3 deals with the effect of the variability of F j on the total expected waiting cost with the waiting cost function g being either increasing concave or increasing convex. In this section, we also show which sequences minimize the total expected waiting cost in the m-machine flow shop and which policies minimize the total expected waiting cost in the two machine job shop.

The assumption that the processing times of job j on each one of the m machines is equal to Xj is crucial. If this assumption is changed in such a way that the processing times of job j on the various machines are i.i.d. according to Fj, then the process becomes considerably more complicated and results similar to those presented in what follows do not hold.

2. INEQUALITIES FOR THE EXPECTED MAKESPAN

We need the following preliminaries. Suppose a and B are n-dimensional vectors with elements al , . . . , an and PI, . . . , fin. Let a ( ~ ) , . . . a(n) and &I), . . . f i c n ) denote the rearrangement of the elements of a and 8 in decreasing order. A vector 6 is said to majorize a vector 8, ti > m p , when

k k C a ( i ) 2 C f i ( i ) , for k = 1,. . . , t~ - 1 i = l i = 1

and

A random variable X i s said to be stochastically smaller than a random variable Y, X 5 St Y, if

P ( X r t ) 5 P ( Y 2 f), for all t

Veinott extended this ordering to n-dimensional random vectors by calling a random vector

STOCHASTIC FLOW SHOPS AND JOB SHOPS 63

X stochastically smaller than a random vector P, x I st 9, if f ( X ) I st f(P) for all increasing real valued functions f.

The hazard function or simply hazard, R ( t ) , of a distribution F( t ) is defined as

R ( f ) = -log (1 - F ( t ) )

The hazards Rl( t ) , Rz( f ) , . . . , Rn(t) are said to be proportional if Ri(f ) = 4iR(t) for 4i > 0, i = 1, . . . , n, for some hazard R ( t ) . Let IF'+,, 4i > 0 ) be a family of distribution functions with proportional hazards. For CE = (at, (YZ, . . . ,an) [ B = ( / 31 , /32 , . . . ,On)] let XP, X?, . . . , X : [Xf, X I , . . . , Xfll be independent random variables with order statistics XTI), X&, . . . , XTnj [X&, X & , . . . , Xfn)] where X f [Xf] has distribution Fa, [Foil. Proschan and Sethuraman showed that if CE 5 m B then (X&, X&, . . . , XTn)) I ( X ~ I , , Xfz), . . . , Xfn)). (See also Reference 8.)

Now, consider two flow shops with m machines. The processing times of job j on the m machines in the first [second] flow shop are identical and equal to the random variable Xj'[Xf] with distribution F,,[F,3,1, where Fa,[F,3,I have proportional hazards ajR(t) [@jR( t ) ] for some hazard R ( t ) . let Cm,(6) [ C m a @ ) ] denote the makespan in the first (second) flow shop.

Theorem 1

If CE I ~ B , then Cmax(6) S st Cmax@).

Pro0 f From reference 7 it follows that

( ~ 7 1 ) ~ ~ 6 ) ~ . . . XTn)) I st ( ~ f l ) , ~ f 2 ) 9 - - Xtn))

The theorem then follows from the fact that the makespan n

1-1 C m a = C Xi + (m - 1) max (Xi, . . . , x,)

is a function that is permutation invariant and increasing in XI, X2,. . . , Xn.

factors ( ( ~ 1 , . . . , a n ) are controllable within a bound U, in such a way that So if the processing time distributions have proportional hazards and the proportionality

5 a i = U i = l

then letting the distributions be as homogenous as possible, i.e. choosing U i = U/n, for i = 1,. . . , n, minimizes the makespan stochastically. Examples of such families of distributions are families of exponential distributions and families of Weibull distributions.

Consider now two job shops. In both job shops jobs 1, . . . , p have to be processed first on machine 1 and then on machine 2 and jobs p + 1, . . . , n have to be processed first on machine 2 and then on machine 1. The processing time distributions in both shops have proportional hazards.

Let CE and B be p-dimensional vectors and 7 and 6 be (n - p)-dimensional vectors of positive real numbers. In the first system the processing times are XP,Xf, . . . , X g and X;+ , ,X;+2 , . . . , X i for jobs in sets A and B, respectively. In the second system they are Xf,Xg, . . . , X$ and Xi+1, X;+2, . . . , X t for jobs in sets A and B, respectively. Let Cmax (6, 7) [Cmax@,F) ] denote the makespan in the first [second] system.


Theorem 2

If h 5 ,B and 7 S m 8 then Cmm(h , q ) i St Cm,,(B,S).

and

which shows that the makespan is permutation invariant and increasing in XI, . . . , X, (Xp+lr . . ., Xn). Since an increasing function of an increasing function is increasing and any pair of 8 " , X y , X B , X 6 are independent of each other it follows that

Eg(Cmax(c%, 7)) Eg(Cmax(h,SN &!(CmaX(,z),S))

for all increasing real functions g. The theorem then follows from the definition of stochastic order for random vectors.

3. INEQUALITIES FOR THE TOTAL EXPECTED WAITING COST

In this section we assume that there exists a partial ordering according to which the distributions F1, . . . , F n can be ordered. A random variable X1 with distribution F1 is said to be more variable than a random variable X2 with distribution F 2 when

E(h (XI )) 2 E(h W2)) for all functions h which are increasing convex. We denote this ordering by XI zcv XZ. Bessler and Veinott9 showed that this ordering holds if and only if

l t [ l - A ( x ) J d x r [ l -Fz(x)] dx, f o r a l l t 2 0

Hanoch and Levy" considered a similar ordering, which has turned out to be very useful in the study of utility theory in economics. Distribution FI was said to dominate distribution FZ when

E(h (XI )I 2 E(h (Xz)) for all functions h which are increasing concave. Hanoch and Levy showed that this ordering is equivalent to

st

n t n t 3 FI(x) dx Q Fz(x) dx, for all t 2 0 0 0

We denote this ordering by XI 2 EC XZ.


For the case where ,!?(XI) = E(X2) the following three statements are equivalent:

for all t > 0 and equal for f = 0.

(ii) E(h(X1)) 2 E(h(X2))

(iii) E(h(X1)) Q E(NX2))

for all functions h which are convex (not necessarily increasing).

for all functions h which are concave (not necessarily increasing). For more details about these orderings see Reference 11. Now consider a flow shop with m = 1, that is a single machine. For a single machine the

following is true.

Theorem 3

convex [concave], then the sequence 1, . . . , n minimizes the total expected waiting cost. If XI G C V . . . <cvX,,[X1 Q c c . . . <<ccX,,] and the waiting cost function g is increasing

Pro0 f Consider an arbitrary sequence j1, . . . , j,, and perform an adjacent pairwise interchange

between the two consecutive jobs j i and ji+1. In the first sequence (say SI) job j i is followed by job j i + 1 and in the second sequence (say SZ) job ji is preceded by job j i + 1 . We may assume that in both sequences job ji-1 is completed at time t . It is clear that the waiting cost of job j i + ~ under S1 is equal to the waiting cost of job ji under S2 as their completion times are identical, i.e. t + Xj, +Xj,+l. It suffices therefore to compare the waiting cost of job ji under SI with the waiting cost of job ji+l under SZ. The total expected waiting cost under SI is larger than the total expected waiting cost under S2 when

When the function g is increasing convex [concave] this is equivalent to

Xj, 2 cv Xji+ 1 [Xjt 2 cc Xj, + 1 I

The expected waiting cost of any sequence can be improved in this way through a series of adjacent pairwise interchanges until the sequence of the theorem is obtained.

From this theorem it follows that if the n jobs have equal means and are variability ordered, the sequence which orders the jobs according to the smallest (largest) variance first rule minimizes the total expected waiting cost, when the waiting cost function is increasing convex (concave).

Now consider the case where all n jobs have a common random due date D with distribution H and where the sequence which minimizes the expected number of tardy jobs has to be determined.

66 STOCHASTIC FLOW SHOPS AND JOB SHOPS

Corollary 1

the sequence 1, . . . , n minimizes the expected number of tardy jobs. If the distribution function H i s concave over the positive reals and XI <cc . . . cCc Xn then

Pro0 f The probability that job j with completion time Cj is tardy is equivalent to the probability

that the due date occurs before the completion time of the job, i.e. H(Cj). So the distribution function H plays the same role as the waiting cost g in Theorem 3.

The class of distribution functions which are concave, which is equivalent to the class of density functions which are decreasing, includes the class of distribution functions with a decreasing hazard rate (DHR). (A distribution F with density f is DHR when the hazard rate f(t)/[ 1 - F ( t ) ] is decreasing in t.) It also includes some increasing hazard rate (IHR) distributions, like, for example, the uniform distribution with a zero lower bound.

Now, consider two systems, both with a single machine and n jobs. In both systems, the jobs are processed according to the sequence j 1 , . . . , jn. In both systems there is a waiting cost function g which is increasing convex (concave). We use priines and double primes to distinguish between quantities of interest in the two systems, e.g. C’,Xj;+,.

Theorem 4

[convex] then If Xj GCcXj’ [Xj dcvXyl for j = 1, . . . , n and the waiting cost function g is concave

wcj 1 dccCg(C:’) [Cg(Cj QcvCg(Cj’)I

Pro0 f

that it is true for n = k. Now First we prove g(Cj ) 5 Cc [ s cv ] g(C; ) by induction on n. It is clearly true for n = 1. Assume

E[h(g(Ck+1))1 = E[h(g(Ck + XL+l))l Q E[h(g(Cl+ Xl+lNl = E[h(g(Cl+1))1

for all increasing concave [convex] functions h. Hence g(Ck+1) GCc g(Ck!+l)[(g(Ck+l) Qcv g(Cl+l)] when g is concave [convex]. The theorem then follows from the fact that g(Cj)[g(C;’)] is an increasing concave [convex] function of Xi, . . ., Xi [Xf, . . . ,X,”] and hence Cg(cj) [Cg(Cj’)] is an increasing concave [convex] function of xi , . . . , x; [ Xf , . . ., x:: ] .

So we may conclude from this theorem that when the waiting cost is increasing concave (convex) and E(Xj ) = E(Xj”), j = 1, . . . , n, the total expected waiting cost is larger when the processing times are less (more) variable.

If all n jobs in the two systems have a common due date D with distribution H and if N ’ (N” ) denotes the number of jobs completed before the due date D in the first (second) system, the following result holds.

Corollary 2

E(” ) 2 E ( N ” ) . If Xj QccXj’, for j = 1, . . . , n, and the distribution function H is concave, then

M. PINED0 AND S-H. WIE 67

This implies that if the distribution function of the due date is concave and if E(XJ ) = E(XJ), j = 1, . . . , n, the expected number of tardy jobs decreases when the processing times variability increases.

Consider now a flow shop with m machines, with the processing times of job j on the m machines being identical and equal to the random variable Xj.

Theorem 5

1, . . . , n minimizes the total expected waiting cost. If X I QcV . . . GcvXn and the waiting cost function g is increasing convex, then the sequence

Proof. Consider again an arbitrary sequence j 1 , . . . , j n and perform an adjacent pairwise inter-

change between the two consecutive jobs ji and ji+1. In the first sequence, say S1, job ji is followed by job ji+ 1 and in the second sequence, say S2, job ji is preceded by job j i+ I .

Let i - 1

I = I A = 2 Xji

and

B = max(Xj,, Xj,, . . . , Xj1-J

So the departure epoch of job j i- 1 from machine k, k = 1, . . . , m, is equal to A + (k - 1)B. This implies that the time at which the last job, among the set of jobs { j l , j 2 , . . . , j i- 1 1 , leaves machine k(k= 1, . . . ,m), does not depend on the sequence in which the jobs of set ( j l , j 2 , . . . , ji-11 go through the system. This has as a consequence that the time at which job j i+l leaves the system under S1 is equal to the time at which job ji leaves the system under SZ as both of these times are equal to

Therefore, it suffices to compare the waiting cost of job j i under S1 with the waiting cost of job jj+l under SZ. Let Cji[C;+,] denote the completion time of job j i [ j i + l ] under S l [ S z ] . Now

Cjl = A + (m - l)max(B, Xjl) + Xji

and

C;+, = A + (m - l)max(B, Xil+,) + X,,, The total expected waiting cost under S1 is therefore smaller than the total expected waiting cost under SZ when

s ; g [ A + (m - l)max(B, x ) + XI W ( x ) C g [ A + (m - l)max(B, x ) + XI dFj,+,(x) s, The above inequality holds when

Xji < cv Xjt+,

as max(B, x ) is increasing convex in x and the function g is increasing convex. The total


expected waiting cost of any sequence can now be improved through a series of adjacent pairwise interchanges until the sequence of the theorem is obtained.

From this theorem, it follows that if the n jobs have equal means and are variability ordered, the sequence which orders the jobs according to the smallest variance first rule minimizes the total expected waiting cost, when the waiting cost is increasing convex.

Consider now a two-machine job shop. Again the processing times of job j on the two machines are identical and equal to the random variable Xj which is distributed according to Fj. As stated before, only policies are considered which require machine 1 (2) to complete all jobs which have to be processed first on machine 1 (2) before starting with jobs which have to be processed first on machine 2 (1). Under these policies the expected completion time of the last job that leaves the system is minimized. Let X I , . . . , X, denote the processing times of the jobs which have to go first through machine 1 and afterwards through machine 2 (the jobs of set A) and let X,+ I , . . . , Xn denote the processing times of the jobs which have to go first through machine 2 and afterwards through machine 1 (the jobs of set B).

Theorem 6

If XI Gcv . . . QcvXp, X,+l QcV . . . GCvXn and the waiting cost function g is increasing convex, then, whenever machine 1 (2) is freed, the job with the lowest index among the remaining jobs of set A (B) has to be put on the machine; if all jobs of set A (B) have completed their processing on machine 1 (2). the job with the lowest index among the remaining jobs of set B (A) has to be put on machine 1 (2), provided it has already completed its processing on machine 2 (1).

4. ADDITIONAL REMARKS

Some remarks are in order. (i) We assumed also that there is an infinite intermediate storage between two successive

machines. This implies that blocking cannot occur. If we had assumed that there is no intermediate storage between two successive machines, which may cause blocking, a result similar to Theorem 5 could not have been proven (a counterexample can be found easily).

(ii) Pinedo’* considered the effect of the variability of the processing time on the expected makespan and the total expected waiting cost when the machines are set up in parallel. These results indicate that if the variability in the processing times increases the expected makespan tends to increase and the total expected waiting cost tends to decrease.

ACKNOWLEDGEMENT

We are grateful to the referee for his careful scrutiny and his many helpful remarks.

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Inequalities for stochastic flow shops and job shops