Ready-Time Scheduling with Stochastic Service Times

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<ul><li><p>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: .Accessed: 09/05/2014 12:50</p><p>Your use of the JSTOR archive indicates your acceptance of the Terms &amp; Conditions of Use, available at .</p><p> .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</p><p> .</p><p>INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.</p><p> </p><p>This content downloaded from on Fri, 9 May 2014 12:50:22 PMAll use subject to JSTOR Terms and Conditions</p><p></p></li><li><p>READY-TIME SCHEDULING WITH STOCHASTIC SERVICE TIMES </p><p>THOM J. HODGSON, RUSSELL E. KING, AND PAUL M. STANFIELD North Carolina State University, Raleigh, North Carolina </p><p>(Received October 1992; revisions received October 1993, May 1994; accepted May 1995) </p><p>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. </p><p>M any marketing/production processes are subject to the following scenario. As part of the sales process, </p><p>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 stochas- tic, the question is: What should the "ready" time be if the due date must be met with a given probability? The prob- lem 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 produc- tion sequence and latest job ready times given the on-time completion confidence level constraints. </p><p>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 air- craft 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. </p><p>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 </p><p>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 process- ing by an Erlangian server. Total customer waiting and server idle costs are minimized, but job due dates are not considered. </p><p>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.) </p><p>1. MODEL DESCRIPTION </p><p>Consider n jobs to be serviced by a production system. Each job i has a deterministic due date (di) and a stochas- tic nonnegative service time, (Xi). The distribution of the service time (Fxr) is known and is independent of the ser- vice 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): </p><p>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. </p><p>Area of review: MANUFACTURING, OPERATIONS AND SCHEDULING. </p><p>Operations Research 0030-364X/97/4605-0779 $05.00 Vol. 46, No. 5, September-October 1997 779 ?) 1997 INFORMS </p><p>This content downloaded from on Fri, 9 May 2014 12:50:22 PMAll use subject to JSTOR Terms and Conditions</p><p></p></li><li><p>780 / HODGSON, KING, AND STANFIELD </p><p>Yi random variable corresponding to the completion time of job i, e.g., in the single facility case, Yi is 14=1 Xj; </p><p>Yi(k) Yi variate given sequence k; ri latest ready time to complete jobs 1,..., i with </p><p>confidence level ai by due date di; ri(k) ready time as determined by constraint equation </p><p>i in sequence k; a-job sequence 1, 2, .. ., i, j, . . ., n; and </p><p>a' job sequence 1, 2, . . , j, i, . .., n. </p><p>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: </p><p>Maximize r </p><p>Subject to Fy, (di - ri) a oi V1i = 1, . ,n, (1) </p><p>r r i = 1, ...,n. (2) </p><p>The maximum ri for each of the type (1) constraints occurs when the distribution function is equal to the re- spective ai. Thus, one only need consider where the distri- butions are equal to the respective air's; i.e., Fy,(di - ri) = ai Vi = 1, . . ., n. The common ready time r is the mini- mum of {r1, r2,..., rn}. The equation that renders this minimum ri for a particular sequence is termed the con- straining 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. </p><p>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 se- quence can be determined a priori, then at most n proba- bility 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. </p><p>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). </p><p>2. SINGLE FACILITY </p><p>The purpose of single-facility analysis is to determine when and to what degree the number of potentially optimal se- quences and potentially binding equations may be re- duced, 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: </p><p>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. </p><p>Fact 1. If FA(X) = FB(y) and if A is stochastically greater than B (since their distributions are monotonically increas- ing), x : y. </p><p>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 vari- ables X1, ... , Xn. </p><p>2.1. Differing Due Dates </p><p>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. </p><p>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. </p><p>Proof. Consider an arbitrary ready time r &lt; 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 tar- diness 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 </p><p>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 inver- sion of the corresponding distribution function. The distri- bution to be inverted is the convolution of the service times of all jobs due on or before the considered di. </p><p>2.2. Differing Confidence Levels </p><p>Now consider scheduling n jobs with differing confidence levels ai and common due date d. The constraint equa- tions may be generally stated in terms of the Yi random variable, i.e., Fy(d - ri) = ai Vi = 1, ... , n. </p><p>Consider the sequences o- (i -&gt; j) and a-' (j -- i), which differ only in the order of adjacent jobs i and j with confi- dence 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 be- cause 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, </p><p>This content downloaded from on Fri, 9 May 2014 12:50:22 PMAll use subject to JSTOR Terms and Conditions</p><p></p></li><li><p>HODGSON. KING. AND STANFIELD / 781 </p><p>and as a result, decrease the ready time. The optimal job sequence can be determined as follows. </p><p>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. </p><p>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-)) = </p><p>Fy,(.) (d - ri(o')) = ai. Since Yi(o') :ST Yi(ur): </p><p>d - ri(a-) S d - ri(o-'); and </p><p>ri (a) 3ri ('). </p><p>Thus, a- is better if job i is constraining for the sequence. Next, consider the situation where job j is constraining </p><p>for a. Since ai 3- a1, Fyj(d - rj()) Fy&lt; (d ) and since Fy() and FY(, are equivalent: </p><p>d - r1(or) s d - ri(o-'); and </p><p>rj (a) 3ri ('). </p><p>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 re- sulting optimal sequence would be in DCL order. Eli </p><p>The number of inversions required to solve for r is the number of unique confidence levels. </p><p>2.3. Differing Due Dates and Confidence Levels </p><p>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 re- sulting sequence yields the maximum ready time. The order- ing 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. </p><p>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 pre- cedes job j. </p><p>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: </p><p>"normal" and "hot." The two-confidence level system is discussed later. </p><p>3. COMMENTS ON SINGLE FACILITY MODEL </p><p>The model exhibits intuitive behavior: jobs should be se- quenced 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: </p><p>* For the sequence with the maximum ready time, con- sider job i. The constraint that corresponds to this job is Fyi(di - ri) = ai. Recall that the value of ri represents the latest ready ti...</p></li></ul>


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