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A Single-Machine Scheduling Problem with Random Processing Times S. Chakravarthy Department of Science and Mathematics, GMI Engineering and Management Institute, Flint, Michigan 48502-2276 We consider the problem of scheduling n jobs with random processing times on a single machine in order to minimize the expected variance of the completion times. We prove a number of results, including one to the effect that the optimal schedule must be V shaped when the jobs have identical means or variances or have exponential processing times. 1. INTRODUCTION We consider the problem of scheduling n independent jobs on a single machine. The processing times of the jobs are assumed to be random. We are interested in the schedule that minimizes the expected variation of the completion times of the n jobs. This type of problem has applications in many manufacturing settings, notably in computer systems, where it is often desirable to provide uniform response time to users (jobs). Previous work on this problem concentrated on the deterministic processing times, and here we draw particular attention to the papers of Merten and Muller [3], Schrage [4], Eilon and Chowdhury [l], and Kanet [2]. In the deterministic case, although no explicit characterization of an optimal schedule is available, except when there are up to five jobs, a number of necessary conditions on the nature of the optimal schedule have been established [ 1,4]. For example, Eilon and Chowdhury [ 11 have shown that the optimal schedules must be V shaped. A schedule is V shaped if the jobs are arranged in descending order of processing times if they are placed before the shortest job, but in ascending order if placed after it. Heuristic algorithms to find extremely good solutions have been proposed in [ 11 and [ 2 ] . In this paper, we show that some of the results in the deterministic case can be extended to the stochastic case with suitable assumptions on the underlying random variables. Examples are provided where the results need not be true, otherwise. The paper is organized as follows: In Section 2, we describe the model and set up the notations. Some preliminary results, including a type of duality theorem are established in Section 3, and two results on the form of the optimal schedule are proved in Section 4. Concluding remarks are given in the last section. 2. THE MODEL AND THE NOTATIONS We consider a single machine, working without failure, on which n jobs are to be processed. All the jobs are available at the beginning. The processing times of these n jobs are independent random variables, all having (possibly) different distributions with finite means and variances. The completion time of job i is defined as the time at which the processing of that job finishes. Since the processing times are assumed Naval Research Logistics Quarterly, Vol. 33, pp. 391-397 (1986) Copyright 0 1986 by John Wiley & Sons, Inc. CCC 0028- 1441 /86/030391-07$04.O0 392 Naval Research Logistics Quarterly, Vol. 33 (1986) to be random, the completion times are also random. The jobs are processed according to a given schedule and once the processing begins no jobs are pre-empted and the schedule remains unchanged until all jobs are finished. We are interested in the schedule that minimizes the expected variance of the completion times. Let X i , for 1 I i 5 n , denote the processing time of job i . Denote by pi and af, respectively, the mean and variance of X i , 1 I i I n. We assume that 0 < pi < w and 0 I a' < w, for 1 I i 5 n. A schedule S = { i , , i2 , . . . , in} is some permutation { i l , i2 , . . . , in} of { 1 , 2 , . . . ,n} and gives the order in which the n jobs are to be processed. For a given schedule S = { i l , . . . ,in}, let c k ( S ) denote the completion time of job i,. That is, C,(S) is the completion time of the kth job in the schedule S. For notational convenience we shall simply use ck for C,(S). In terms of X , , ck is given by k ck = c xi; ( 1 ) r = l If c = ( l / n ) ck denotes the mean of the completion times, then we have . n 1 - C = - c. ( n - k + l ) X t t . k = l The variance V ( S ) of the completion times for a given schedule S is defined to be 1 v = V(S) = - c (C, - C)2. k = l ( 3 ) Note that V is a random variable. In the following we shall let E ( . ) stand for expectation of a random variable. Thus E(XJ = pi. Our problem of interest is to study the schedule S*, where S* is such that E(V(S*)) = min[E(V(S))], (4) and the minimum is taken over all (namely n!) possible schedules. is, For later use we shall let yi denote the second moment of X i , for 1 5 i 5 n. That ( 5 ) yi = p' + a:. 3. PRELIMINARY RESULTS In this section we shall derive expressions for the expected variance of a given schedule and for the amount of change in the expected variance when two jobs are exchanged in a given schedule. Also, a type of duality theorem is presented. THEOREM 3.1: For a given schedule S = { i l r i 2 , . . . ,in}, the expected value of the variance of the completion times is given by " ( n - j + l)(j - 1) Yi, n2 E(V) = c. j = 2 (j - l ) ( n - r + 1) n - l n Pi,Pi,. (6) + 2 c j = 2 r = j + l c n2 where yi is as defined in (5 ) . Chakravarthy: Scheduling Problem 393 PROOF: On using Eqs. (1) and (2), V as defined in (3) can be written as which, after expanding, simplifies to ( j - l)(n - r + 1) n - l n nz x ; + 2 c 2 nz XI,XI,. (7) (n - j + l ) ( j - 1) v=c j = 2 r = j t l 1 = 2 By taking expectation on both sides of (7), we get (6). REMARKS: (1) Using Eq. (6) it can easily be seen that in the two-job case, the optimal schedule is to schedule the job with the larger second moment first. (2) For more than two jobs, the fact that scheduling the job with the largest mean or the largest variance or even the largest second moment need not be optimal can be seen from the following examples. EXAMPLE 1: Here n = 3. The means and variances are given by Job no. 1 2 3 P 6 0.01 0.005 U2 5 50 20 For this example an optimal schedule is {2,1,3}. EXAMPLE 2: Here we have n = 3 and Job no. 1 2 3 P 10 6 4 0 2 4 90 120 An optimal schedule is {2,1,3}. These examples demonstrate the difficulty of finding a characterization, if any, of an optimal schedule. However, in the next section we shall give two results on the nature of the optimal schedule with suitable assumptions on the underlying random variables. The remainder of this section is devoted to establishing two results that are very useful in searching for an optimal schedule. The first is a type of duality result given in Theorem 3.2, which is a direct generalization of the one in [3]. The second result gives an expression for the amount of change in the expected variances when two jobs are exchanged in a given schedule. THEOREM 3.2: The expected variance of the completion times for the schedule S = {ill&, . . . , in- l , in} is the same as that for the schedule = { i l , inr . . . ,i3,i2). 394 Naval Research Logistics Quarterly, Vol. 33 (1986) PROOF: Let V and v, respectively, denote the variances of the completion times for the schedules S and 3. We first note that Eq. 6 is independent of i l . To prove that E(V) = E(V) , it is sufficient to show that the coefficients of yi,, 2 5 j 5 n, in E ( V ) and E(V) are equal and that the coefficients of pi,pi,+,, 1 5 k 5 n - j , 2 5 j 5 n - 1, in E ( V ) and E ( V ) are equal. This follows immediately on noting that the schedule s is such that the jth job of s is (n - j + 2)th job of S and the (j + k)thjobinSis(n - j - k + 2 ) t h j o b o f S , f o r l < k < n - j , 2 5 j s n - 1. REMARK: For every schedule S there is a dual 3 that yields the same expected variance. Hence, there are at least two optimal schedules. Let S = { i l , i2 , . . . ,in} and S be a schedule obtained from S by exchanging jobs ij and ik, for k > j 2 1 . Denote by V and V, respectively, the variances of the completion times for the schedules S and S. Then we have the following. THEOREM 3.3: Let D ( j , k ) = V - V and d = k - j . Then we have j - 1 n 2 ~ ( ~ ( j , k ) ) = (pi, - ~ 5 ) r # j L PROOF Let Cr and C:, respectively, denote the completion times of the rth job in the schedules S and S. Then we have r = 1,2, . . . j - 1, r = k,k + 1, . . . ,n. + Xi, - Xi,, r = j j + 1,. . . ,k - 1, Clearly, the mean c = ( l /n) 2;=, C; is given by - d n C = c + - (x i , - Xi,). By definition, n n D ( j , k ) = {(c: - C) - (Cr - C)} [c: - cr + c - c][c: + c, - c - C ] , r = 1 n - = r = 1 which, on using (9) and the fact that k- 1 j - 1 k - 1 C C r = d C xi, + r = j r = 1 r = j (k - r ) xi,, Chakravarthy: Scheduling Problem 395 can be simplified to k - 1 n 2 D ( j , k ) = 2(Xlk - XI, ) Xl , + ( k - r ) X, , - d ? (11) (n - k + J)d 2 + Since X i , 1 5 i 5 n, are independent by assumption, (XI, - X,,) and X I , , r # j , k are also independent. Thus, on taking expectation on both sides of (1 1) and after some elementary manipulations, we get the stated result. 4. THE FORM OF THE OPTIMAL SCHEDULE In this section we prove two results on the form of the optimal schedule with suitable assumptions on the underlying random variables. First, we need the following defi- nition. DEFINITION: A schedule S = {i1, i2, . . . ,in} is said to be V shaped with respect to the second moment of the processing times if and only if there exists a k , 1 < k < n, such that yi, 2 yi, 2 *.* 2 yi, and THEOREM 4.1: The optimal schedule is V shaped with respect to the second moment of the processing times if any one of the following conditions is satisfied: (i) 0: = u2, (ii) FI = p, (iii) x, - exp(1 Ip , ) . PROOF: Suppose that 0 = u2 and that the optimal schedule is not V shaped. Then there are three consecutive jobs, say, i k - 1 , i t , and it+,, for 2 5 k < n, such that pi,-, < pi, and pi,+, < pi,. The proof is over on showing that an exchange of jobs ik and ik+ , or ik - and ik reduces the expected variance. This is equivalent to showing that (a) if E(D(k - 1,k)) > 0, then E(D(k,k + 1)) < 0, and (b) if E(D(k,k + 1)) > 0, then E(D(k - 1,k)) < 0, where D(j ,k ) is as defined in Section 3, denoting the difference in the variances for the schedule { i1 , i2 , . . . ,in} and the one obtained from it by exchanging jobs i, and i k . Using Eq. ( 8 ) and the assumption that u: = a, we see that 396 Naval Research Logistics Quarterly, Vol. 33 (1986) and The proof follows immediately from Eqs. (12) and (13). The proofs for the cases (ii) and (iii) are similar and hence are omitted. REMARKS: (1) Theorem 4.1 is a direct generalization of the result established in [ 11 for the deterministic case and is very useful in searching for an optimal schedule. (2) Note that V shapedness is not a sufficient condition for an optimal schedule. Also, that the optimal schedule need not be V shaped without assuming any of the above-mentioned conditions can be seen in the following example. EXAMPLE 3: Here there are four jobs. The means and the variances are given by Job No. 1 2 3 4 P 10 6 4 1 U2 2 8 24 100 An optimal schedule is (1, 2, 3, 4). We conclude this section with the following theorem, whose proof is left to the initiative of the reader. THEOREM 4.2: Let ki, = max,,k,n pik. Suppose that at = max,,,,, ui. Then there exists an optimal schedule where job il is processed first. That is, the job with the largest mean and variance is processed first. 5. CONCLUDING REMARKS In this paper we considered a single-machine scheduling problem with random processing times and generalized some of the results established for the deterministic processing times. The heuristic algorithms that are available for the deterministic processing times can be used in these cases. In the general case, since the optimal schedule need not be V shaped, we are currently developing efficient algorithms and these will be presented elsewhere. ACKNOWLEDGMENT My thanks are due to a referee for a careful reading of an earlier version of this article and for his suggestions that improved the presentation. Chakravarthy: Scheduling Problem 397 REFERENCES [ 11 Eilon, S . and Chowdhury, I. G., Minimizing W a h g Time Variance in the Single Machine [2] Kanet, J . J . , Minimizing Variation of Flow Time in Single Machine Systems, Management [ 3 ] Merten, A. G . and Muller, M. E., Variance Minimization in Single Machine Sequencing [4] Schrage, L., Minimizing the Time-in-System Variance for a Finite Job Set, Management Problem, Management Science, 23, 567-575 (1977). Science, 27, 1453-1459 (1981). Problems, Management Science, 18, 518-528 (1972). Science, 21, 540-543 (1975).