Single Machine Scheduling with Controllable Processing Times and Number of Jobs Tardy

Single Machine Scheduling with Controllable Processing Times and Number of Jobs TardyAuthor(s): Richard L. Daniels and Rakesh K. SarinSource: Operations Research, Vol. 37, No. 6 (Nov. - Dec., 1989), pp. 981-984Published by: INFORMSStable URL: http://www.jstor.org/stable/171479 .

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SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES AND NUMBER OF JOBS TARDY

RICHARD L. DANIELS and RAKESH K. SARIN Duke University, Durham, North Carolina

(Received February 1987; revisions received March, July 1988; accepted October 1988)

Most scheduling research has treated individual job processing times as fixed parameters. In many practical situations, however, a manager may exert processing time control through the allocation of a limited resource. We consider the problem of joint sequencing and resource allocation when the scheduling criterion of interest is the number of tardy jobs. Theoretical results are derived that aid in developing the tradeoff curve between the number of tardy jobs and the total amount of allocated resource.

Jn this note, we consider a problem in which both the sequencing of jobs and their processing times

are under managerial control. The processing times of individual jobs can be reduced by employing a limited resource, such as overtime or the application of additional manpower. For the single machine case, we provide theoretical results that are useful in developing a procedure for constructing the tradeoff curve between the number of tardy jobs and the total amount of allocated resource. For related work, see Vickson (1980a, b) and Van Wassenhove and Baker (1982) who provide results for the scheduling criteria mean flow time and maximum tardiness.

1. PROBLEM FORMULATION

Let J = {1, 2, .. ., n} denote the set of jobs to be scheduled on a single machine. For each i E J, pi denotes the normal and pi the minimum processing time ofjob i, a, denotes the reduction in the processing time of job i associated with the allocation of one unit of resource to job i, and xi denotes the number of units of resource allocated to job i. The final processing time of job i is then given by pi =

maxfPi - aix1, pi). The due date of job i is denoted by di and its completion time by C1. A job is tardy if Ci - di > 0; nT measures the number of tardy jobs associated with a given schedule.

Our objective is to construct the tradeoff curve between nT and the total amount of allocated resource

(X = Z;. xi). This tradeoff curve may be generated by solving a series of integer programming problems, one for each value of nT. Computationally, however, such an approach proves impractical for problems of even modest size. The next section presents results that simplify the joint sequencing/resource allocation problem and suggests an alternative approach for constructing the relevant tradeoff curve.

2. THEORETICAL DEVELOPMENT

Let L C J represent the set of jobs permitted to be tardy in a particular schedule. The complement of L is denoted E and represents the set of jobs that are constrained to be early (Ci-di - 0). In evaluating a given late set L, we are determining the minimum amount of resource required to achieve on-time completion of each job in the associated early set E. Throughout, we assume that jobs are numbered in earliest due date (EDD) order.

To characterize the optimal sequence associated with late set L, note that the jobs in L should be sequenced after the jobs in E, thus avoiding any unnecessary delays in the completion time of jobs required to be nontardy in the corresponding schedule. In addition, the jobs in E should be sequenced in EDD order because this sequence is guaranteed to minimize X subject to C, 6 di, i E E (see Daniels 1986). Jobs in L then can be sequenced in any order.

The next task is to identify the minimum resource

Subject classification: Production/scheduling: deterministic sequencing, single machine.

Operations Research 0030-364X/89/3706-0981 $01.25 Vol. 37, No. 6, November-December 1989 981 ? 1989 Operations Research Society of America



982 / DANIELS AND SARIN

allocation policy such that all jobs in E obtain zero tardiness. To solve this problem, we focus solely on the jobs in E as follows.

Initially, the first job (job i) that incurs the maximum nonzero tardiness of the jobs sequenced in EDD order is identified. Job k is identified next such that

k = max{j: j < i, pj> pj

and aj - a,, for all m < i, Pm > P I.

Job k will thus reduce the tardiness of job i most efficiently. If no such job exists, then it is not possible to reduce the tardiness of all jobs in E to zero. Job 1, / < k is identified next as the first job that incurs the maximum tardiness among the first (k - 1) jobs in the EDD sequence. Units of resource Xk are allocated to job k where

(P (p- P k) (T - TI) xl=minV al 'k ak

The processing time of job k is revised, P1k = Pk - akxk,,

and the process is repeated until no job, i E F, incurs stiictly positive tardiness. The total resource expended by this process is the minimum amount required to ensure that all jobs in E are nontardy.

In principle, we can construct the tradeoff curve between the total amount of resource required, X, and the number of tardy jobs, nT, by evaluating all possible late sets using the above process. However, the number of late sets may be as large as 2 n. The following results are useful in reducing the number of late sets that require explicit evaluation.

Suppose that for a specified late set L and an associated early set E, the optimal sequence is obtained and the process described above has provided the corresponding processing time reductions for each job so as to minimize X subject to C1 < di, i E F. Suppose that job j is the first job in this schedule to which some resource is allocated, that is, x > 0 and Xk = 0

for all k < j. The following theorem defines a set of conditions under which the augmentation of late set L by some job I can be excluded in favor of an alternative augmentation by some job m.

Theorem 1. For a specified late set L, suppose that job j is the firs, 'ob in the corresponding schedule to which some resource is allocated to minimizeXsubject to Ci < di, i E E. If jobs 1 and m are alternatives to augment late set L and: (i) 1 j, (ii) m < j, and (iii) P > p,, then late set L U {m} will provide a

solution to the resource minimization problem no worse than that obtainedfrom late set L u {1}.

Proof. For a specified late set L, job j is the first job to which resource is allocated. Hence, C, < di, i < j, i E E. If job m is selected for inclusion in the expanding late set rather than job 1, where 1, m E E, / < j, m < j, and Pin ?,: P,, then the completion times of all jobs k - j, k E E, are reduced by PAn rather than p,. The total amount of resource, X, required to ensure C, < di, i i L U I } must be no greater than the amount needed to guarantee C1 < di, i (4 L U {1l. Therefore, late set L U {1l need not be considered for evaluation. Furthermore, since additional common jobs included in L U {l} and L U {m} have identical impact on the completion times of jobs in the associated early sets, late set L U {l} and all its descendants can be eliminated from consideration.

The effect of Theorem 1 is to significantly reduce the number of late sets that require explicit evaluation in order to construct the tradeoff curve between nT and X. This property may be utilized as part of a tree procedure (see, e.g., Nelson, Sarin and Daniels 1986) where specified late sets are systematically augmented one job at a time. In this context, Theorem 1 represents a vertical elimination scheme, where information obtained at one level of the tree is utilized to eliminate late sets to be created at later levels of the tree, for example, evaluation of late set L suggests that L U {m} be evaluated in favor of L U {1l. We next examine a horizontal elimination scheme, determining conditions where information obtained during the evaluation of late set L U {m establishes that late set L U {1l need not be evaluated.

Theorem 2. Suppose that for a specified late set L U {ml, the resource minimization problem is solved and no resource is allocated to job 1, 1 4 L U { m, P1 > _,, in that solution. If: (i) d,7, < dl, (ii) Pin > Pl) and (iii) a,, - al, then late set L U {m} will yield a solution to the resource minimization problem no worse than that associated with late set L U {1l.

Proof Consider late set 1 with jobs in complement E sequenced in EDD order, represented by UmVIW, where U, V, and W are subsets of the jobs in E and I and m are jobs currently being considered to augment L. The sequence associated with late set L U {m is represented as Sin U VlW, while that associated with late set L U {l} is given as SI UmV W. The task is to demonstrate that the total amount of resource required to eliminate tardiness from jobs in S,7,



Technical Note / 983

is no greater than that associated with S,. First, note that jobs in U are unaffected by the choice of jobs I and m. In addition, the completion times of jobs in W are no greater in Sm than in SI because P5m a Pil. Since no resource is allocated to job I to ensure that jobs in W are completed on time and a, > am, the total amount of resource required to ensure that all jobs in W are completed on time will be no greater in S,, than in SI. The completion times of jobs in V in sequence Sm are less than in sequence S, by an amount equal to Pm. Thus, the completion time of every job in V is reduced by fPm (the maximum possible reduction that job m can provide in S,) at no cost. Hence, the maximum tardiness associated with any job k in U, V, and W is no greater in sequence Sm than in sequence S and requires no greater amount of resource to eliminate. If job I incurs the maximum tardiness in Sm, then the job occupying the corresponding position in S, must incur even higher tardiness. Exclusive of jobs I and m, identical jobs are available for reduction to eliminate tardiness in se- quences S, and Sm; since no resource is allocated to job 1, jobs in U and V must exist that are more efficient than job 1. Since a, > am, these same jobs must be more efficient than job m. Since the maximum tardiness associated with sequence Sm is no greater than that of sequence S,, and the relative efficiency of jobs in Sm to eliminate tardiness is at least as great as those in SI, it follows that the total amount of resource associated with Sm is no greater than that associated with S,, and, thus, late set L U {I} need not be evaluated.

Theorem 2 is also valid when late sets L U {m} and L U {I} are augmented by identical sets of jobs; in such instances, late set L U {m} is universally favored over late set L U {1}.

The next result demonstrates that when late set L is augmented by job m and all jobs 1 M L, I > m obtain zero tardiness, late sets L U / 1, 1 > m need not be evaluated.

Theorem 3. Consider late set L U {ml, m 4 L. Sup- pose that prior to resource allocation, C, - di for all 1( L, 1 > m. Then late setsL U {l}, 1> m, willprovide solutions to the resource minimization problem no better than that provided by late set L u {m}.

Proof. The result is obtained by noting that by placing job m in the expanding late set, the completion times of jobs j > m are reduced by P,n, and these jobs all attain zero tardiness. Therefore, any resource allocated

in the resulting schedule is applied to reduce the tardiness of jobs k, k 4 L U {m}, k < m, to zero. Selecting job I > m for inclusion in the late set cannot reduce the tardiness of jobs j > m further and, in fact, may increase the tardiness of these jobs. The tardiness of jobs k, k 4 L U {m}, k < m must still be reduced to zero, as above. Therefore, L U III cannot provide a schedule with a lower total amount of allocated resource than L U {ml.

The next result utilizes a lower bound on the amount of resource required to eliminate tardiness among a specified subset of jobs to exclude late sets from consideration.

Theorem 4. Consider late set L with complement E. For m E E, define z = min X subject to C, < di, i < m, i E E. If late set L U {k}, k < m, k E E, yields a solution to the resource minimization problem smaller than z, then late sets L U {1l, 1 , m, 1 e E, need not be consideredfor evaluation.

Proof. Suppose that X represents the total amount of resource associated with late set L U {k}. Furthermore, suppose that z is the amount of resource required to ensure that, for a specified job m > k, all jobs i < m, i, m E E, are completed on time. Thus, z represents a lower bound on the total amount of resource required to eliminate tardiness among jobs i $ L U {II, 1 , m. If z a X, then L U {I} need not be considered for evaluation for all 1 > m.

Finally, the minimum (T) and maximum (4T)

cardinality of late sets that must be considered are given by the Moore (1968) schedule using processing times pi and Pi, respectively.

3. CONCLUSIONS

This note has presented a joint sequencing/resource allocation model for single processor scheduling, where job processing times are treated as decision variables that may be controlled through the assign- ment of an additional resource. Our theoretical results suggest a constructive procedure for developing the tradeoff curve between the number of tardy jobs and the total amount of allocated resource. The managerial usefulness of this research lies in facilitating the scheduling task by identifying profitable resource allocation strategies to improve customer service as measured by the number of tardy jobs.



984 / DANIELS AND SARIN

ACKNOWLEDGMENT

The authors are grateful to the two anonymous re- viewers for their helpful-comments on an earlier draft of this paper.

REFERENCES

DANIELS, R. L. 1986. Joint Sequencing/Resource- Allocation Scheduling With Multiple Performance Measures. Ph.D. Thesis, University of California, Los Angeles, Calif.

MOORE, J. M. 1968. An n Job, One-Machine Sequencing

Algorithm for Minimizing the Number of Late Jobs. Mgmt. Sci. 15, 334-342.

NELSON, R. T., R. K. SARIN AND R. L. DANIELS. 1986. Scheduling With Multiple Performance Measures: The One-Machine Case. Mgmt Sci. 32, 464-479.

VAN WASSENHOVE, L. N., AND K. R. BAKER. 1982. A Bicriterion Approach to Time/Cost Trade-offs in Sequencing. Eur. J. Opnl. Res. 11, 48-54.

VICKSON, R. G. 1 980a. Two Single-Machine Sequencing Problems Involving Controllable Job Processing Times. AIIE Trans. 12, 258-262.

VICKSON, R. G. 1980b. Choosing the Job Sequence and Processing Times to Minimize Total Processing Plus Flow Cost on a Single Machine. Opns. Res. 28, 1155-1167.



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Single Machine Scheduling with Controllable Processing Times and Number of Jobs Tardy