P e r g a m o n

Computers ind. Engng Vol. 35, Nos 1-2, pp. 109-112, 1998 © 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain P I h S 0 3 6 0 - g 3 5 2 ( 9 8 ) 0 0 0 3 2 - 1 0360-8352/98 $19.00 + 0.00

Scheduling Identical Jobs on Uniform Parallel Machines with

Random Processing Times

Mohamed I. Dessouky, Richard L. Marcellus Industrial Engineering Department

Northern Illinois University DeKalb, IL 60115

Li Zhang Capital One Financial Corporation

Glenn Allen, Virginia 23060

ABSTRACT

For the problem of scheduing identical jobs on a set of uniform parallel machines with random processing times, methods are given for optimizing the expected sum of weighted completion times and the probability of meeting a common due date. © 1998 Elsevier Science Lid. All rights reserved.

KEYWORDS

Scheduling, identical jobs, uniform parallel machines, random processing times.

Introduction

Scheduling identical jobs on a set of uniform parallel machines (or operators) is a problem faced when a batch of identical products need to be processed by a set of machines or operators with different efficiencies. Examples are a collection of sewers all making the same garment, machines in different states of repair making the same product, and differing technologies making the same product.

When processing times are random, any chosen schedule will result in random performance. Thus, the meaning of an efficient and cost-effective schedule becomes difficult to define. When considering makespan, for example, it is not enough to optimize the expected makespan. The variability in the makespan should also be considered, as well as the probability of an extremely long makespan. This is in sharp constrast to scheduling problems with deterministic processing times, where the exact performance can be calculated.

This paper presents some initial steps in the problem of efficiently scheduling identical jobs on uniform parallel machines with random processing times. Two per- formance measures are addressed: (1) minimizing the expected sum of the weighted completion times, and (2) maximizing the probability that a set of jobs will be com- pleted before a common due date. Complications such as precedence constraints and preemption are not considered.

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Problem statement It is assumed that there is a batch of n independent jobs, with identical process- ing requirements, that must be scheduled on a set of m parallel machines. The time that machine i requires to finish a job is random with distribution Fi(t) = Pr{processing time _< t} Thus, the completion time of job j, Cj, is also random. The problem is to assign a certain number of the jobs to each machine. This creates a schedule, N 1 , . . . , Nm, where Ni is the number of jobs assigned to machine i.

Minimizing the expected sum of the weighted completion times

For deterministic processing times, the Earliest Completion Time (ECT) schedule of Dessouky, Lageweg, Lenstra, and van de Velde (1990) minimizes the sum of the completion times. This section defines the expected time version of this schedule, the Earliest Expected Completion Time (EECT) schedule. It turns out that this schedule minimizes the expected sum of the completion times and can be used to construct a schedule that minimizes the expected sum of the weighted completion times.

Finding the EECT schedule requires maintaining a list of available expected completion times for which jobs can be scheduled. The expected times are sorted from earliest to latest. The initial list contains, for each machine, the expected time required to complete one job. As soon as the list is made, a job is assigned to the machine that corresponds to the first expected time in the list. This expected time is then increased by the expected time to complete an additional job on the machine in question, aaad the list is resorted. This process is repeated till each job has been assigned to a machine. When the procedure is finished, it will have produced a sorted sequence of expected completion times: e l , . . . , en with el _<... _< en. These are the earliest possible expected completion times for a set of n jobs on the given machines. They satisfy the following minimality property, which corresponds to the minimality property for deterministic completion times in Dessouky, Lageweg, Lenstra, and van de Velde.

MINIMALITY PROPERTY. No schedule exists with expected completion times eli < ' such that • .. < e n ej < e j f o r a n y j = 1 , . . . , n .

From the minimality property, the EECT schedule minimizes E[Cj] for job j. Hence, E[Cj] is also minimized by the EECT schedule. And since

E[~-'~ Cj] is minimized by the EECT schedule. To minimize the expected weighted sum of the completion times, first note that

Thus, minimizing E [~-~ wjCj] is equivalent to minimizing ~ wjE[Cj] . This is done by arranging the jobs in order of non-increasing weights and matching them in order with the non-decreasing expected completion times e l , . . . , en.

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Maximizing the probability of meeting a common due date

Suppose a schedule has been produced that assigns N~ jobs to machine i, and there is a common due date (d) for all jobs. Let ci be the time at which machine i finishes its assigned jobs. Let Pi(d, N~) be the probability that machine i will finish its assigned jobs before the due date. That is, Pi(d, Ni) = Pr{ci < d}. Since c~ is a sum of independent random variables, this probability must be calculated by forming an N~-fold convolution. Specifically, Pi(d,N) = F*N~(d) where F *N~ is the N~-fold convolution of Fi with itself. In general, F *N~ (d) is difficult to calculate and may have to be simulated. However, for some special cases such as the normal, uniform, and exponential, analytic formulas are available.

The entire batch of jobs will be completed at time max~ ci. Thus, the due date will be met if maxi ci _< d: each and every machine must complete its assigned jobs before the due date. Since the jobs are independent, the probability that all machines will meet the due date is the product of the corresponding probabilities for the individual machines. That is,

P(d) = Pr{maxci <_ d} = H P~(d,N~). (3)

A schedule to maximize P(d) can be found by dynamic programming. Let ~-* (X, i) be the maximum probability obtainable when a batch of X jobs is sched- uled on machines 1 through i. Then, by the principle of optimality,

.~*( X, i) = max {.~* (X - k,, i - 1)Pi(d, ki) } k~

This recursive relation, together with the boundary condition

(4)

.~*(X, 1)= PI(d,X) (5)

can be used to find br*(n, m), the maximal probability of meeting the due date for n jobs and m machines.

This dynamic program has the abstract form of an allocation problem, where certain amounts of a resource are apportioned among different activities and each activity uses its portion to produce a local contribution to an overall performance measure. For the scheduling problem at hand, the batch of jobs takes the place of the resource, each machine represents an activity, and the local contribution of each machine is the probability that it will complete all its jobs before the due date. The overall performance measure is the probability that all machines will complete their jobs before the due date. This probability is the product of the individual probabilities.

Example

Suppose there are 12 jobs and 5 machines (n = 12 and m = 5). Each machine has a uniformly distributed processing time with the same expected value (2.5), but the variances differ from machine to machine. Specifically:

machine 1 has a processing time uniformly distributed from 2.0 to 3.0;

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machine 2 has a processing time uniformly distributed from 1.5 to 3.5; machine 3 has a processing time uniformly distributed from 1.0 to 4.0; machine 4 has a processing time uniformly distributed from 0.5 to 4.5; machine 5 has a processing time uniformly distributed from 0.0 to 5.0. For an EECT schedule, three of the machines are chosen to process two jobs

apiece, and the other two machines are chosen to process three jobs apiece. The resulting expected sum of completion times is 52.5. Since the machines cannot be distinguished by their expected processing times, the specific way in which the five machines are split into two groups does not matter. There are in fact 10 schedules, all of which have the same expected sum of completion times. (Ten is the number of ways that three items can be selected from a set of five items.)

However, when due dates are considered, the picture changes. For a due date of 7.5, the dynamic program produces a schedule that assigns two jobs apiece to machines 1 through 3 and three jobs apiece to machines 4 and 5. This schedule has probability 0.25 of meeting the due date. The other EECT schedules have probabilities ranging from 0.20 to 0.23 of meeting the due date.

For a due date of 8.0, the dynamic program produces a schedule that assigns three jobs apiece to machines 1 and 2 and two jobs apiece to machines 3 through 5. This schedule has probability 0.51 of meeting the due date. The other EECT schedules have probabilities ranging from 0.34 to 0.46 of meeting the due date.

It is interesting that for the tighter due date (7.5), more jobs are assigned to the machines with higher probability of low processing times (but also higher variability). For the looser due date, the schedule is more conservative, assigning more jobs to the machines with lower variability.

Conclusion

Methods have been presented for minimizing the expected sum of completion times, and maximizing the probability of meeting a common due date. An example shows that when the first of these objectives is met. the second is not necessarily satisfied.

References

Dessouky, M. I., B. J. Lageweg, J. K. Lenstra, and S. L. van de Velde (1990), Scheduling identical jobs on uniform parallel machines, Statistica NeerIandica 44(3).

Zhang, Li (1997), Scheduling identical jobs with stochastic processing times on uni- form parallel machines, Master's Thesis, Northern Illinois University, DeKalb, IL.