Ready-Time Scheduling with Stochastic Service Times

Ready-Time Scheduling with Stochastic Service TimesAuthor(s): Thom j. Hodgson, Russell E. King and Paul M. StanfieldSource: Operations Research, Vol. 45, No. 5 (Sep. - Oct., 1997), pp. 779-783Published by: INFORMSStable URL: http://www.jstor.org/stable/172130 .

Accessed: 09/05/2014 12:50

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.

http://www.jstor.org

This content downloaded from 195.78.109.199 on Fri, 9 May 2014 12:50:22 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=informs

http://www.jstor.org/stable/172130?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


READY-TIME SCHEDULING WITH STOCHASTIC SERVICE TIMES

THOM J. HODGSON, RUSSELL E. KING, AND PAUL M. STANFIELD North Carolina State University, Raleigh, North Carolina

(Received October 1992; revisions received October 1993, May 1994; accepted May 1995)

A frequently encountered scheduling problem is to determine simultaneously a material and job ready time and production sequence based on customer-specified due dates. Each job has a stochastic production time and a deterministic due date. The ready time is constrained in that the probability that each job will be complete by its due date must meet some minimum level of confidence. The objective in such an instance is to postpone the ready time as late as possible without violating these constraints. The steps and effort necessary to determine the maximum ready time and optimal production sequence, and cases in which this effort may be significantly reduced are presented. The resulting model is applied directly to single-facility and flow-shop production environments. Methods are shown for scheduling in a dynamic environment.

M any marketing/production processes are subject to the following scenario. As part of the sales process,

a commitment for a specific delivery date is made. This, in turn, determines a time at which raw material must be "ready" for production. If the production time is stochastic, the question is: What should the "ready" time be if the due date must be met with a given probability? The problem is complicated when there are many jobs with different due dates and confidence levels. In this case, the problem becomes one of simultaneously determining the production sequence and latest job ready times given the on-time completion confidence level constraints.

A similar situation is faced by the Naval Aviation Depot (NADEP) considered by Hodgson and McDonald (1981). The NADEP provides the Navy with its most thorough level of aircraft maintenance: major repair, modification, and overhaul. The Navy maintains a list of standard repair operations and associated workload standards. Actual aircraft repair times vary widely. Scheduling in such an environment is extremely difficult. Customers (Naval Air squadrons) desire to minimize the time that their aircraft are removed from the fleet. Thus, determination of the aircraft delivery times to the NADEP in order to meet the due dates is required, along with the production sequence.

A number of authors have addressed scheduling with stochastic service times and due dates for single machines, (Banerjee 1965, Crabill and Maxwell 1969, Blau 1973, Hodgson 1977, Glazebrook 1983, Kise et al. 1982, Katoh and Ibaraki 1983, Pinedo 1983, Forst 1984, Frenk 1991, De et al. 1991, and Huang and Weiss 1992) and flow shops (Makino 1965, Talwar 1967, Cunningham and

Dutta 1973, Pinedo 1982 and 1984, Boxma and Forst 1986, and Lee and Lin 1991.) In nearly all cases, rules or methods are presented to sequence ready jobs to opti- mize some expected or stochastic measure of shop effi- ciency. Liao et al. (1993) consider scheduling arrivals of jobs to a set of equally spaced arrival slots for processing by an Erlangian server. Total customer waiting and server idle costs are minimized, but job due dates are not considered.

In what follows, a model is presented that determines both a production sequence and a latest single induction date (if jobs are to arrive at the facility at the same time). A method is then given to assign individual job ready times that improve the average job flowtime while maintaining the optimality results from the common ready time model. (Details of the model and its development can be found in Stanfield 1995.)

1. MODEL DESCRIPTION

Consider n jobs to be serviced by a production system. Each job i has a deterministic due date (di) and a stochastic nonnegative service time, (Xi). The distribution of the service time (Fxr) is known and is independent of the service time distributions of any other job and the sequence in which the jobs are performed. The goal is to sequence jobs and to determine a common ready time, (r), so that the probability of completing each job i by its due date is greater than, or equal to, some confidence level, ai. Nota- tion used in the model is shown below (unless otherwise noted, a job is indexed according to its position in the sequence):

Subject classifications: Production/Scheduling, Stochastic sequencing probabilities: scheduling facilities with stochastic service. Probability, Stochastic models: service time data probabilistically distributed. Networks, stochastic: shop modeled as stochastic networks.

Area of review: MANUFACTURING, OPERATIONS AND SCHEDULING.

Operations Research 0030-364X/97/4605-0779 $05.00 Vol. 46, No. 5, September-October 1997 779 ?) 1997 INFORMS



780 / HODGSON, KING, AND STANFIELD

Yi random variable corresponding to the completion time of job i, e.g., in the single facility case, Yi is 14=1 Xj;

Yi(k) Yi variate given sequence k; ri latest ready time to complete jobs 1,..., i with

confidence level ai by due date di; ri(k) ready time as determined by constraint equation

i in sequence k; a-job sequence 1, 2, .. ., i, j, . . ., n; and

a' job sequence 1, 2, . . , j, i, . .., n.

Notice that r represents the latest feasible start time for the first job if the ready times differ among individual jobs. Feasibility is defined as the ability to complete each job i by di with probability of at least ai. The general n job, single-facility problem given a specific sequence of the n jobs may be expressed as follows:

Maximize r

Subject to Fy, (di - ri) a oi V1i = 1, . ,n, (1)

r r i = 1, ...,n. (2)

The maximum ri for each of the type (1) constraints occurs when the distribution function is equal to the respective ai. Thus, one only need consider where the distributions are equal to the respective air's; i.e., Fy,(di - ri) =

ai Vi = 1, . . ., n. The common ready time r is the minimum of {r1, r2,..., rn}. The equation that renders this minimum ri for a particular sequence is termed the constraining or binding equation. The job associated with the constraining equation is termed the constraining or binding job. The best sequence is that which yields the maximum r.

To determine the maximum common ready time for a set of n jobs through enumeration requires the inversion of up to n * n! probability distributions. If an optimal sequence can be determined a priori, then at most n probability distributions require inversion. Based on the due dates, confidence levels, and service time distributions, it may also be possible to reduce the number of potentially binding equations.

This paper concentrates on the reduction of the number of potentially optimal sequences and potentially binding equations in order to reduce the number of inversions required. Inversion of distributions can be difficult and time-consuming. However, when the service times can be approximated by the Normal distribution, it is straightfor- ward. A more thorough consideration of this problem can be found in Jagerman (1978) and Platzman et al. (1988).

2. SINGLE FACILITY

The purpose of single-facility analysis is to determine when and to what degree the number of potentially optimal sequences and potentially binding equations may be reduced, and as a result, to simplify the process of finding a common ready time. Prior to this analysis, one definition (Pinedo .1984) and two facts should be stated:

Definition 1. A random variable A is stochastically greater than another random variable B if FA(t) - FB(t) Vt and written A : ST B.

Fact 1. If FA(X) = FB(y) and if A is stochastically greater than B (since their distributions are monotonically increasing), x : y.

Fact 2. If random variable Yn is the sum of n nonnegative independent random variables X1,..., Xn, then the summed random variable Yn is stochastically greater than any summed combination of the individual random variables X1, ... , Xn.

2.1. Differing Due Dates

Consider scheduling n jobs with differing due dates di and common confidence level a. The constraint equations may be stated in terms of the Yi random variable, i.e., Fy,(di - ri) = a Vi = 1, . . ., n. The optimal job sequence for this situation can be determined as follows.

Theorem 1. The latest feasible common ready time for jobs with different due dates and a common confidence level is found by sequencing jobs in Earliest Due Date (EDD) order.

Proof. Consider an arbitrary ready time r < d1 Vi =

1,..., n. Notice that for a given sequence, the maximum probability of tardiness is increasing in the value of r. The sequence that minimizes the maximum probability of tardiness will maximize the value of r. Banerjee (1965) showed that the maximum probability of tardiness for a schedule with ready time r is minimized when the jobs are sequenced in EDD order. D

After finding the optimal sequence, r is found by solving each potentially constraining equation for ri and setting r to the minimum ri. For each unique due date there exists one potentially constraining equation requiring the inversion of the corresponding distribution function. The distribution to be inverted is the convolution of the service times of all jobs due on or before the considered di.

2.2. Differing Confidence Levels

Now consider scheduling n jobs with differing confidence levels ai and common due date d. The constraint equations may be generally stated in terms of the Yi random variable, i.e., Fy(d - ri) = ai Vi = 1, ... , n.

Consider the sequences o- (i -> j) and a-' (j -- i), which differ only in the order of adjacent jobs i and j with confidence levels ai : aj (all other jobs indexed by position in the sequence). If the constraining job precedes jobs i and j, switching i and j causes no change in the ready time because these jobs are not involved in the constraining equation. If the constraining job follows jobs i and j, switching i and j causes no change in the ready time because the constraining distribution is invariant in the order of its component distributions. In both cases, switching and I might cause i or]j to become the constraining job,



HODGSON. KING. AND STANFIELD / 781

and as a result, decrease the ready time. The optimal job sequence can be determined as follows.

Theorem 2. The latest feasible common ready time for jobs with a common due date and different confidence levels is found by sequencing jobs in Decreasing Confidence Level (DCL) order.

Proof. Since ai 3 cy job j will never be the constraining job in a-'. With ar it is not possible to tell which, if either, of the constraints is binding. First, consider the situation where job i is constraining for o-; i.e., Fy() (d - ri(o-)) =

Fy,(.) (d - ri(o')) = ai. Since Yi(o') :ST Yi(ur):

d - ri(a-) S d - ri(o-'); and

ri (a) 3ri (').

Thus, a- is better if job i is constraining for the sequence. Next, consider the situation where job j is constraining

for a. Since ai 3- a1, Fyj(d - rj()) Fy< (d )

and since Fy() and FY(, are equivalent:

d - r1(or) s d - ri(o-'); and

rj (a) 3ri (').

Thus, a- is better if job j is constraining for the sequence. A switch from o- to o-' can cause a reduction in the ready time. A switch from a-' to a- cannot cause any reduction in the ready time and may improve the ready time. If one were to continuously perform attractive switches, the resulting optimal sequence would be in DCL order. Eli

The number of inversions required to solve for r is the number of unique confidence levels.

2.3. Differing Due Dates and Confidence Levels

Finally, consider the case where both the due dates and confidence levels vary among jobs. It can be seen, using the arguments presented earlier, if Earliest Due Date and Decreasing Confidence Level sequencing concur, the resulting sequence yields the maximum ready time. The ordering is termed Earliest Due Date/Decreasing Confidence Level (EDD/DCL), and it should be noted both here and subsequently that an EDD/DCL ordering may not exist.

The situation where these two orders are different poses a more difficult problem. Sequences developed through pairwise comparisons may not yield the maximum r in this case. However, it may be possible to reduce significantly the number of potentially optimal sequences. Using the same argument as with adjacent jobs, it can be shown that if di - dj and ai 3 a1j, in an optimal sequence job i precedes job j.

In practice, jobs are categorized according to the neces- sity of on-time completion. Each job in a category would be assigned the same confidence level. In this case, the number of potentially optimal sequences can be reduced significantly with each category of jobs placed in EDD order within the total sequence. For example, it is common practice in the NADEP's to have two levels of priorities:

"normal" and "hot." The two-confidence level system is discussed later.

3. COMMENTS ON SINGLE FACILITY MODEL

The model exhibits intuitive behavior: jobs should be sequenced so that jobs with the most imminent due dates and the highest required due date completion probability are serviced first. Several other comments may be made:

* For the sequence with the maximum ready time, consider job i. The constraint that corresponds to this job is Fyi(di - ri) = ai. Recall that the value of ri represents the latest ready time necessary to complete jobs 1, . . ., i by due date di with confidence level ai. Let Si = (min rj) - r. Si is analogous to slackness in CPM/ PERT networks. If Si = 0, then job i or some subse- quent job is the constraining job. The individual ready time of job i may be changed from r to r + Si without violating any of the job on-time constraints. Thus, the average time in shop for the job is reduced without adverse effect on the schedule.

Theorem 3. Among all sequences that render the maximum ready time, the maximum total slack time is found by sequencing, if possible, jobs in Earliest Due Date/Decreas- ing Confidence Level EDD/DCL order.

Proof. Consider the sequences o-(i -> j) and o-'(j > i), which differ only in the order of adjacent jobs i and j with confidence levels ai : aj and due dates di - dj. a-' yields the maximum common ready time. It has been shown that a switch from a-' to a- would not have an adverse effect on the common ready time. In order to justify a switch from a- to 0r' to maximize total slack time, the following must hold:

TS(o>) TS(a);

Ek = Jmin, = k, . ,nrl (a) -r

E n = rmin =, . ri (o') - r];

min = i, .. ,n r1(a) + min, = ,,.., In r, (or)

A,:,min=j,....,nri(o') + min1=i,. ,nr1(af);

min[ri (a), rj (o)] + rj (a)

:min[rj(o-'), ri(o-')] + ri (o'); and

min[ri (o),rj (o)] + rj (a)

> ri (-') + ri (o').

It was shown in the proof of Theorem 2 that ri(o-) > ri(o-') and rj(o-) : ri(o-'); therefore, the total slack time can only be improved by a switch from a' to a-. Continuously per- forming such switches yields an EDD/DCL schedule. C]i

* If the service times are approximately Normally distributed, inversion of the distribution functions can be performed as simple table hookups.



782 / HODGSON, KING, AND STANFIELD

4. FLOW SHOPS

Jobs are performed in the same sequence on each machine. Let the random variable Xik represent the service time for job i on machine k. As before, the constraining equations for this problem are Fy,(di - ri) = ai Vi =

1,..., n. The added complexity of the flow shop arises because the distributions of Y2, ..., Yn are much more difficult to find and invert than in the single-machine problem.

4.1. Different Service Distributions (Two Machines)

Let the service time distributions be different for each job on each of two machines. If the jobs have a common due date and confidence level, a modified version of Johnson's rule is applicable. Ku and Niu (1986) cite sufficient conditions to stochastically minimize the makespan. Specifically, job i precedes job j if:

(min[Xi , Xj2]jXi + Xj1 = a, Xi2 + Xj2 = b)

SST(min[Xjl, Xi2 ]Xil + Xjl == a, Xi2 + Xj2 = b),

for all nonnegative a and b for which the random variable conditional distributions are well defined. This relationship is implied if either of the following two simpler relation- ships hold:

1. If two jobs i and j can be found such that Xi, is larger in likelihood ratio than Xi, and Xi2 is larger in likelihood ratio than Xj2, then an optimal sequence exists where job i precedes job j. Random variable A is larger in likelihood ratio than random variable B if:

fA(X) <A (Y)

fB(X) f() V X Y y.

2. If two jobs i and j can be found such that Xi, and Xi2 are nonoverlappingly larger than Xi1 or Xj2, then an optimal sequence exists where job i precedes job j. Random variable A is nonoverlappingly larger than random variable B if P[B - A] = 1.

4.2. Same Service Time Distributions

Finally, for m machines, let either, for a given machine, the service time distributions be the same for all n jobs, or the service time distributions for each job be the same for all m machines.

Theorem 4. The maximum common ready time for this situation is found by sequencing, if possible, jobs in Earliest Due Date/Decreasing Confidence Level EDD/DCL order.

Proof. Consider the sequences o- (i j-> ) and u' (j -i), which differ only in the sequence of jobs i and j with due dates di - dj confidence levels a1i >o aj. Boxma and Forst (1986) showed that:

Yl(o-) and Y,(o-') are identically distributed for 1 =

1,..., n and 1 i,j, Y)(o-) and Yi(o-') are identically distributed, and

Using these three facts, the optimality of the EDD/DCL schedule may be shown in the same manner as the proof of Theorem 2. C]I

5. DYNAMIC SCHEDULING (SINGLE MACHINE)

In realistic problems, the scheduling task is typically dynamic in that the sequence is specified using the available jobs, and rescheduling is done as additional jobs arrive. Preemption is not allowed. This section addresses the dynamic scheduling problem and presents a method that can be used to construct an optimal schedule quickly for either the static or dynamic single-facility problem.

Let each job have one of two confidence levels (a < f3). This situation is common in repair operations where jobs are characterized as "normal" or "hot." Consider adjacent jobs i and j in the sequence where ai = a and aj = P3. Each job a has due date da and Normally distributed service time with mean ga and variance o-a. In this case, the maximum ready time problem is essentially reduced to determining the optimal sequence. Let F71 (A, [k, &) be the inverse of a Normal distribution function with mean g and variance 2 evaluated at confidence level A. The case of interest occurs when EDD and DCL conflict, i.e., di <

dj. Based on the ready time maximization objective, job j should come first if the minimum of ri and rj is greater when j precedes i, i.e.,

- I ~ ~ ~~k k

min di - F 71 a, ptE + E_ ,, i2 + E2 1=1 ~~1=1

k k

dj - F-1 3, Ai + Aj + 11,i2Cj+ 2 < 1=1 1=1

- ~ ~ ~ ~~~ X k .

mind - FK1( 13, -N j + T J+ El 1=11=

di- F1((a, pj + +Zi + i, Lj + +E 2

where k is the job immediately preceding i and j. Follow- ing from Facts 1 and 2, if 0.5 - a - 1.0, b - c, and d - e, then FJl(a, b, d) - Fkl(a, c, e). As a result, the inequality reduces to:

di - d 4-(D ) _ q-'(ae) >

4( 1 = 1 1 ) + + ?2

where 1D-1 ( ) is the inverse of the standard normal distribution. In practice, this rule implies that when consid- ering a large number of jobs, the initial part of the sequence tends to be in EDD order, and the latter part of the sequence tends to be in DCL order. In addition, in the limit, as service time variance tends towards 0, EDD dominates. Conversely, as service time variance increases, DCL dominates. Also, the mean service time (pti) does not influ- ence the sequencing decision.

Consider a set of "hot" jobs H that immediately follow a "normal" job i in an optimal sequence. H need not contain



HODGSON, KING, AND STANFIELD / 783

all hot jobs immediately following i. Let h be the job in H with the latest due date. As arriving jobs are inserted before i, i and H will switch positions if the total variance of the jobs through h exceeds:

h [ dh di 12

If two jobs are in DCL order in an optimal sequence, then the optimal order will not change as new jobs arrive. A new job is inserted at position i determined by

= fmax= 1 . nida > dj= if a, = a minj= ,. nlda<dj if aa =13.

Hot jobs in positions i, . .. , n are then evaluated to determine if they should be moved up in the sequence. In the worst case, job insertion is (O(n log(n)), though in practice, insertion typically is accomplished much more effi- ciently. Based upon these observations it can be seen that the preceding rule can be used to dynamically add jobs to an existing sequence and, thus, dispatch jobs to the resources.

ACKNOWLEDGMENT

This research was funded, in part, by the Office of Naval Research (Grant N00014-90-J-1045).

REFERENCES

BAKER, K. R. 1974. Introduction to Sequencing and Scheduling. John Wiley and Sons, New York.

BANERJEE, B. P. 1965. Single Facility Sequencing With Ran- dom Execution Times. Opns. Res. 13, 358-364.

BLAU, R. G. 1973. N Job, One Machine Sequencing Problems Under Uncertainty. Mgmt. Sci. 20, 101-109.

BOXMA, 0. J. AND F. G. FORST. 1986. Minimizing the Ex- pected Weighted Number of Tardy Jobs in Stochastic Flow Shops. 0. R. Letts. 5, 119-126.

CUNNINGHAM, A. A. AND S. K. DUlTA. 1973. Scheduling Jobs with Exponentially Distributed Processing Times on Two Machines of a Flow Shop. Naval Res. Logist. 20, 69-81.

CRABILL, T. B. AND W. L. MAXWELL. 1969. Single Machine Sequencing with Random Processing Times and Random Due-Dates. Naval Res. Logist. 16, 549-554.

DE, P., J. B. GHOSH, AND C. E. WELLS. 1991. On the Minimi- zation of the Weighted Number of Tardy Jobs with Ran- dom Processing Times and Deadline. Comp. and Opns. Res. 18, 457-463.

FORST, F. G. 1984. A Review of the Static, Stochastic Sched- uling Job Sequencing Literature. Opsearch 21, 127-144.

FRENK, J. B. G. 1991. A General Framework for Stochastic One-Minute Scheduling Problems with Zero Release Times and No Partial Ordering. Prob. in Eng. and Inf. Sci. 5, 297-315.

GLAZEBROOK, K. D. 1983. On Stochastic Scheduling Problems with Due Dates. Int. J. Sys. Sci. 14, 1259-1271.

HODGSON, T. J. 1977. A Note on Single Machine Sequencing with Random Processing Times. Mgmt. Sci. 23, 1144-1146.

HODGSON, T. J. AND G. W. MCDONALD. 1981. Interactive Scheduling of a Generalized Fellowshop. Interfaces, 11, 2-4, 35-47 and 83-88.

HUANG, C. C. AND G. WEISS. 1992. Scheduling Jobs with Sto- chastic Processing Times and Due Dates to Minimize Total Tardiness. Comm. in Stat. 8, 529-541.

JAGERMAN, D. L. 1978. An Inversion Technique for the LaPlace Transform with Application to Approximation. Bell. Syst. Tech. J. 57, 669-710.

KATOH, N. AND T. IBARAKI. 1983. A Polynomial Time Algo- rithm for a Chance-Constrained Single Machine Schedul- ing Problem. 0. R. Letts. 2, 62-65.

KiSE, H., A. SHIOMI, M. UNO, AND D. S. CHAO. 1982. An Efficient Algorithm for a Chance-Constrained Scheduling Problem, J. Opns. Res. Soc. Japan, 25, 193-203.

Ku, P. S. AND S. C. Niu. 1986. On Johnson's Two-Machine Flow Shop with Random Processing Times. Opns. Res. 34, 130-136.

LEE, C. Y. AND C. S. LIN. 1991. Stochastic Flow-Shop Sched- uling with Lateness-Related Performance Measures. Prob. in Eng. and Infi Sci. 5, 245-253.

LIAO, C. J., C. D. PEGDEN, AND M. ROSENSHINE. 1993. Plan- ning Timely Arrivals to a Stochastic Production or Ser- vice System. HIE Trans. 25, 63-73.

MAKINO, T. 1965. On a Scheduling Problem. J. Opns. Res. Soc. Japan, 8, 32-44.

PINEDO, M. L. 1982. Minimizing the Expected Makespan in Stochastic Flow Shops. Opns. Res. 30, 148-162.

PINEDO, M. L. 1983. Stochastic Scheduling with Release Dates and Due Dates. Opns. Res. 31, 559-572.

PINEDO, M. L. 1984. Optimal Policies in Stochastic Shop Scheduling. Anns. Opns. Res. 1, 305-329.

PLATZmAN, L. K., J. C. AMMONS, AND J. J. BARTHOLDI. 1988. A Simple and Efficient Algorithm to Compute Tail Proba- bilities from Transforms. Opns. Res. 36, 137-144.

STANFIELD, P. M. 1995. Production Scheduling Under Uncer- tainty. Ph.D. Dissertation, Department of Industrial En- gineering, North Carolina State University, Raleigh, NC.

TALwAR, P. P. 1967. A Note on Sequencing Problems with Uncertain Job Times. J. Opns. Res. Soc. Japan, 9, 93-97.



Documents

Ready-Time Scheduling with Stochastic Service Times