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β-Robust scheduling for single-machine systems withuncertain processing timesRICHARD L. DANIELS a & JANICE E. CARRILLO ba School of Management, Georgia Institute of Technology , Atlanta, GA, 30332-0520, USAb Olin School of Business, Washington University , St. Louis, MO, 63130-4899, USAPublished online: 30 May 2007.

To cite this article: RICHARD L. DANIELS & JANICE E. CARRILLO (1997) β-Robust scheduling for single-machine systems withuncertain processing times, IIE Transactions, 29:11, 977-985, DOI: 10.1080/07408179708966416

To link to this article: http://dx.doi.org/10.1080/07408179708966416

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lIE Transactions (1997) 29, 977-985

f3-Robust scheduling for single-machine systems withuncertain processing times

RICHARD L. DANIELS) and JANICE E. CARRILL02

1School of Management, Georgia Institute of Technology, Atlanta, GA 30332-0520, USA-ou« School of Business, Washington University. St. Louis, MO 63130-4899, USA

In scheduling environments with processing time uncertainty, system performance is determined by both the sequence in whichjobs are ordered and the actual processing times of jobs. For these situations, the risk of achieving substandard system perfor­mance can be an important measure of scheduling effectiveness. To hedge this risk requires an explicit consideration of both themean and the variance of system performance associated with alternative schedules, and motivates a p-robustness objective tocapture the likelihood that a schedule yields actual performance no worse than a given target level. In this paper we focus onp-robust scheduling issues in single-stage production environments with uncertain processing times. We define a general p-robustscheduling objective, formulate the ,B-robust scheduling problem that results when job processing times are independent randomvariables and the performance measure of interest is the total flow time across all jobs, establish problem complexity, and developexact and heuristic solution approaches. We then extend the p-robust scheduling model to consider situations where the uncer­tainty associated with individual job processing times can be selectively controlled through resource allocation. Computationalresults are reported to demonstrate the efficiency and effectiveness of the solution procedures.

1. Introduction

In this paper we consider a single-machine scheduJingenvironment where the processing times of individualjobs are uncertain. An immediate consequence of pro­cessing time uncertainty in this environment is that sys­tem performance, as measured by the scheduling criteriaof interest) can vary with both the sequence in which jobsare ordered and the actual processing times ofjobs. Givenprobabilistic processing time information, the distribu­tion of achievable system performance associated withany sequence can be determined, thus providing a meansfor evaluating alternative schedules. Stochastic schedul­ing models (see, for example, [1-4]) typically assessschedule quality by focusing on the mean of the outcomedistribution, and seek to identify the schedule whose ex­pected system performance is optimal. However, formu­lations that anchor on expected system performance donot explicitly address the concerns of decision makerswho may reasonably be more interested in minimizing therisk of achieving unacceptably poor system performancethan in optimizing average performance. These concernsare particularly relevant when there is a clear distinctionbetween acceptable and substandard performance (e.g.,when a promised delivery date must be met, or whenthere is a lead time threshold beyond which customers arelost), and the scheduling objective is to ensure as far aspossible that actual system performance is acceptable. In

0740-817X © 1997 "liE"

these situations, a decision maker may strongly prefer aschedule with suboptimal average performance, but lowperformance variability) over an alternative schedule withoptimal average performance and significantly highervariance. More generally, the risk of achieving unac­ceptably poor system performance reflects a schedule'seffectiveness in hedging against the underlying processingtime uncertainty, and motivates a scheduling approachthat considers both average system performance andperformance variability in determining the optimalschedule. Such an approach is consistent with the robustdecision-making formulations presented by Gupta andRosenhead (5], Rosenhead et al. [6], Rosenblatt and Lee[7], Kouvelis et al. [8,9], and Daniels and Kouvelis [10].

In this paper we develop scheduling approaches forsingle-stage production environments with uncertainprocessing times. In Section 2 we define a generalfJ-robustness measure that is based on the likelihood ofachieving system performance no worse than a giventarget level, formulate the ,B-robust scheduling problemthat results when the processing times of individual jobsare independent random variables and the performancemeasure of interest is the total flow time across all jobs,and establish problem complexity. A branch-and-boundalgorithm and a heuristic approach for solving thefJ-robust scheduling problem are presented in Section 3.In Section 4 we extend the ,B-robust scheduling model toconsider situations where the uncertainty associated with

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978 Daniels and Carrillo

individual job processing times can be selectively con­trolled through resource allocation. Section 5 reports theresults of a set of experiments designed to test the com­putational performance of the solution approaches. Sec­tion 6 concludes with a summary and suggestions forfurther research.

total flow time over all jobs. Define Xik to indicate theposition of job i in sequence rt, i.e., Xik = 1 if 1t(k) = i andXik = 0 otherwise, with X = {Xik : i = 1,2, ... .n andk = 1,2, ... , n }. Then the total flow time of sequence ngiven .processing time scenario A can be expressed as

n 11

cp{X,pi.) = L L(n - k + l)p;·xik. (2)i=1 k=1

X;k E {O, I}, i = 1,2, ... ,n, k = 1,2, ... ,n. (8)

Problem (,B-RSP) thus recognizes the effect of sequenc­ing on both expected flow time performance and flow time

n 11

q;(X) = L L(n - k + 1)J1;Xik, (3)i;: I k= 1

(5)n 11

L L(n - k + 1)2a;xiki= I k=1

n n

T - L L(n - k + l)J1iX ik

j;: I k= Imax

n

LX;k=l, k=I,2, ... ,n; (7);= 1

n

s.t.LxiA:=I, i=1,2, ... ,nj (6)k;:1

({J-RSP)

n 11

(12[q>(X)] = L L{n - k + 1)2c1xik ' (4);=1 k= I

The associated standard-normal statistic, z(X, T)= [T - q;(X)]/J{a2[cp(X )]}, then yields the likelihood ofachieving actual flow time performance no worse than thespecified target level.

The quality of the normal approximation of cp(X,pJ·)depends both on the number of jobs and on the charac­teristics of the individual job processing time distribu­tions. Hines and Montgomery [11] indicate that closeapproximations should be obtained for even small valuesof n (n ~ 4) when processing time distributions are uni­modal and nearly symmetric, whereas a moderate num­ber of jobs (n:::::: 12) is required to validate theapproximation for processing time distributions with noprominent mode and nearly uniform density.

The P-Robust Scheduling Problem (,B-RSP), i.e., id­entifying the schedule with the maximum likelihood ofachieving flow time performance no greater than targetlevel T, can be formulated as:

Because the total flow time of sequence tt is calculatedas the weighted sum of a set of independent randomvariables (with the weights determined by associated as­signment vector X), by the Central Limit Theoremq>(X, pi.) has an approximate normal distribution asn ~ 00. The mean and variance of this distribution canbe expressed as

rfJ(n, T) = Prob[q>(n, pi.) s T] = Ly(p)·j). (1)

j=1

Consider a set {I, 2, ... ,n} of n independent jobs to beprocessed on a single machine. We assume that the pro­cessing times of individual jobs can be specified onlyimprecisely, due to, e.g., incomplete information on theprocessing requirements of the jobs. This processing timeuncertainty is captured within the set of processing timescenarios, A. Each scenario A E A represents a unique setof job processing times, which can be realized with somepositive probability. Let p{ denote the processing time ofjob i in scenario A E A, pi. = {p{ : i = 1,2, ... , n} thevector of job processing times associated with scenario A,and y(p).) the probability of realizing the processing timescorresponding to scenario A.

Let Q represent the set of permutations sequences thatcan be constructed from the n jobs, and let1t = {1[(1) 1 n(2), ... , n(n)} denote a given sequence in Q,with n(k) specifying the job that occupies position k insequence n. Suppose that system performance is evalu­ated by using measure q>(n, p).) (note that due to pro­cessing time uncertainty, q>( 1t, p;.) varies with both the jobsequence and the actual processing times of the n jobs).Suppose also that alternative schedules are evaluatedaccording to the corresponding likelihood that actualsystem performance will be no worse than a given targetlevel T, which might represent, e.g., a promised deliverydate or a limit on tolerable waiting time. Without loss ofgenerality, assume that the elements of A can be num­bered, A = {AI, A2, ... ,AlAI}' for e~ch sequence tt suchthat Aj<Aj' if and only if cp{1t,p;·j)$q>(1t,p)i). ThenP{1t, T), the probability that the actual system perfor­mance associated with sequence tt will be no greater thanthe specified target level, can be calculated by identifyingscenario j. such that j+ = max{j : cp(1t, p).j) ~ T}:

2. Problem setting

The scheduling objective is to determine the sequence thatmaximizes the likelihood of achieving system perfor­mance no worse than target level T, with this (J-robustschedule, np, defined such that f1(1tp, T) = maXnEfJp(n, T).

To illustrate the development of a detailed solutionapproach for determining the p-robust schedule, assumethat the actual processing time of each job i, pi, is anindependent random variable with mean /li and variance(1;, and that the performance criterion of interest is the

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f3-Robust scheduling for single-machine systems

variability across processing time scenarios, and deter­mines the best schedule by considering performance acrossboth dimensions. Note that the p-robust schedule alsominimizes total flow time subject to Prob[<;o(n,pi_)]= f3( np, T), i.e., when the sequencing decision is based onthe performance of alternative schedules at the f3(np, T)­confidence level of the associated flow time distribution.Thus, the IJ-robust scheduling approach can be tailored toreflect the level of risk that an individual decision maker iswilling to bear in hedging against processing time uncer­tainty. This differs significantly from an approach thatexclusively seeks to minimize expected total flow time bysequencing jobs in nondecreasing order of the Ili to formthe shortest expected processing time (SEPT) schedule(see, for example, [12]). Because the SEPT schedule mayalso exhibit high flow time variance, an alternativeschedule with higher expected flow time but lower variancemay be preferred by a risk-averse decision maker.

Whereas the SEPT schedule to minimize expected flowtime can be constructed in O(nlogn) time, the f3-robustscheduling problem is NP-hard, as established by thefollowing result.

Theorem 1. Problem (f3-RSP) is NP-hard.

Proof: Consider optimal solution x P to problem ({J-RSP),and let { = I if job i is assigned to position k in thissolution (with x~ = 0 otherwise) for i = 1,2, ... , nandk = 1,2, ... .n. Suppose that the variance in total flowtime for the schedule associated with solution x P is givenby a2 [<;o (XP)] = 2::7=1 2::~= I (n - k + 1)2a;xfk = V. Then,solution XfJ could be determined by solving the followingequivalent formulation of problem (fJ-RSP).

n n

min L L(n - k + l)l1ixiki== 1 k= I

n n

s.t. L L(n - k + 1)2a;xik ~ V,i=1 k=1

(6), (7), and (8).

Setting Cik = (n - k + I )JL;, rik = (n - k + 1)2o}, andb = V, this problem is immediately recognized as an as­signment problem with a single side constraint, which hasbeen shown to be NP-hard (see, for example, [13,14]) .•

3. Solution approaches for the p-robust schedulingproblem

Given processing time information (Ji.i and aT) for eachjob i and a specified target level of flow time performanceT, problem (fJ-RSP) can be solved by using a standardbranch-and-bound approach that systematically assignsjobs to alternative sequence positions. Each nodegenerated at the Ith level of the tree corresponds to thejob assigned to position I in the associated partial

979

sequence. One branch emanating from this node is gen­erated for each of the n - 1jobs not occupying the first 1positions in the associated partial schedule. Consideringin this manner all n jobs for assignment to each sequenceposition implies a total of n! possible schedules. However,the number of schedules requiring explicit evaluation canbe reduced considerably by using the following result.

Theorem 2. Let ip+ represent the expected total flow timeassociated with the SEPT schedule.

(a) If Ji.i ~ Ili' and a; ~ uf" then there exists a {j-robustschedule in which job i precedes job i' for anyT 2: cp*.

(b) If J.1.i ~ 11;/ and aT 2: a~, then there exists a fJ-robustschedule in which job i precedes job i' for anyT < ip*.

Theorem 2 indicates that both mean and variance infor­mation must be considered to determine job precedenceunambiguously. When T 2: ip*, the fJ-robust scheduleshould reflect risk aversion, with shorter and more certainjobs favored for positions early in the schedule. At theextreme, the SEPT schedule yields an optimal hedgeagainst processing time uncertainty for any T ~ cp* whenjob processing time variance is a nondecreasing functionof the mean processing time, i.e., when longer jobs arealso uniformly more uncertain. In contrast, when T < cp*,the p-robust schedule should reflect risk seeking, and jobswith low mean processing times but higher variancesshould be scheduled earlier. The proof of Theorem 2 iseasily obtained using a straightforward interchange ar­gument (see [15]).

The search for the {j-robust schedule can be furtherenhanced by computing an upper bound on the objectivevalue for a given partial schedule. To illustrate, consideran assignment of jobs to the first 1 positions in sequence,represented by partial schedule np ' A lower bound on theexpected flow time of any schedule that starts with partialsequence 1rp can be determined by constructing scheduleJip , consisting of partial sequence 1rp followed by the re­maining n - I jobs sequenced in nondecreasing order of Ji.;:

n

LBiP('rrp ) = L(n - k + 1)JLnlp(k)' (9)k=l

Similarly, if T ~ cp*, a lower bound .on the flow timevariance associated with any schedule that starts withpartial schedule 1!p can be determined by constructingschedule 1£;, in which np is followed by the n - I un­scheduled Jobs sequenced in nondecreasing order of ar:

n

LBa2[lp](1rp ) = L(n - k + 1)2~;(k}' (10)k=1

.If T < ip" then the n - I unscheduled jobs in n~ should besequenced in nonincreasing order of aT, and expression(10) represents an upper bound on the flow time variance

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of any schedule starting with partial sequence np ' In ei­ther case, an upper bound on the value of objectivefunction (5) associated with partial sequence np is ob­tained by combining LBq,(np ) and LBu2[l,O](1tp) :

UB(rep ) = IT - LBq;(rep ) ] / JLBu'I",] (rep). (11)

Since tt'. and nil are feasible schedules, they also providep p . I 1 .lower bounds on the optima so ution.

The dominance property and bounding approach areeasily incorporated into the branch-and-bound algorithmfor' the ,B-robust scheduling problem. Partial sequencesare fathomed when Theorem 2 is violated, or when theassociated upper bound is exceeded by the best lowerbound encountered thus far. Complete sequences areevaluated by computing objective function (5) for thatschedule, with the best sequence and objective valueconsistently retained. At termination, the algorithm yieldsthe ,B-robust schedule, 1r.p, and the associated maximumprobability of achieving flow time performance no worsethan target level T, fJ( 1tp, T).

The rationale behind the heuristic for the ,B-robustscheduling problem is to generate discrete processing timescenarios that are likely to yield flow time performancenear the ,B-confidence level for any sequence. This isachieved by defining zp = [T -lP·]/J{(J2[cp(SEPT)]}(where a-2[q>(SEPT)] represents the flow time variance ofthe SEPT schedule), and setting Pi = J1, + (zp/cx)J(a;),i = 1, 2) ... , n, for several values of control parameter a.Sequencing the jobs in shortest processing time (SPT) orderyields the corresponding optimal schedule. This approachensures that Theorem 2 is not violated, e.g., pj ~ J1,"' ando} ~ u~ implies that Pi + (zp/ex)J((J~) ~ pjl + (zp/a)J(ut,) for zp ~ 0 (T ~ cp.), and hence job i will precedejob i' in the associated heuristic schedule since Pi ~ Pi'.The SPT schedule is then evaluated by using objectivefunction (5), and the best schedule encountered in thesearch over alternative values of ex is consistently retained.

Procedure jJ-HeuristicInput. Mean processing time, J1i' and processing time

variance, a;, for each job i = I) 2) ... .n, targetlevel of flow time performance, T, mean flowtime, lP·, and flow time variance, a 2[cp (SEPT )),associated with the SEPT schedule, and selectedparameter values, cx}, j = I) 2, ... 1 f.

Output. Heuristic sequence, 7th, that provides a lowerbound, LB, on the probability of achieving flowtime performance no worse than target level T.

Step 1. Set j = 0, LB = 0, and zp = [T - ~.]/

J{ (J2rq>(SEPT)]}.Step 2. Set j = j + I. If j > 1, stop. Otherwise, set

Pi = J1, + (zp/C1.j)J(a;) for i -= 1,2, ... In.Step 3. Construct schedule Ttj by sequencing the jobs in

nondecreasing order of the pi, and set:

Daniels and Carrillo

n

L(n - k + 1)2(J;j{k)·k=1

If LBj > LB, set LB = LBj and 7th = 7tj_ Return toStep 2.

Procedure ,B-Heuristic requires a total of f iterations, withSteps 2 and 3 of each iteration performed in O(n) andO(nlog n) time, respectively. The complexity of the heu­ristic is therefore O[1(nlogn)). Clearly, the number ofvalues of parameter a considered by the procedure alsoaffects solution quality. The computational results re­ported in Section 5 indicate that close approximations ofthe fJ-robust schedule are obtained in very few iterations.

4. Variance reduction and p-robust scheduling

The formulation of problem ({J-RSP) can be extended toinclude resource-allocation decisions that reduce pro­cessing time uncertainty, e.g., managerial time and effortspent interacting with customers to better understand thedetailed requirements of a particular job. We model theselective reduction of processing time uncertainty by as­suming that there exists a single resource, available inlimited supply (R), that can be applied to individual jobsto linearly decrease the associated processing time vari­ance. The processing time of job i is again represented asan independent random variable with mean pj and vari­ance a; = if; - QiYi, where a; denotes the normal variancein the processing time ofjob i, a, denotes the rate at whichthe application of resource reduces the processing timevariance ofjob i, and Yi represents the amount of resourceallocated to job i. We also assume a minimum processingtime variance for job i, Q~, so that sr.,2 ~ (1~ ~ at-

Assigning resource among the n jobs thus reduces flowtime variance without affecting the expected performanceof any given schedule, thereby allowing investigation ofthe impact of variance reduction on the schedule-selectionprocess. Note that opportunities to decrease processingtime uncertainty would not be considered by decisionmakers focused solely on optimizing expected systemperformance, who simply use the SEPT priority rule todirectly determine the optimal schedule. However, vari­ance reduction is an important concern for decisionmakers interested in hedging the risk of unacceptablypoor system performance, because these resource alloca­tion decisions ultimately affect any schedule's perfor­mance variability. If T again represents a given targetlevel of flow time performance, then the ,B-RobustScheduling Problem with Variance Reduction ({J-RSPVR)can be formulated to simultaneously determine the

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[j-Robust scheduling for single-machine systems 981

optimal job sequence and allocation of resource amongthe n jobs:

11

s.t. LYi ~ R, (13)i=l

a? - a?o~ Yj ~ I -I, i = 1~ 2, ... , n, (14)aj

s.t. (13) and (14).

Let S1t(i) denote the position of job i in sequence n ( i.e.,S1t(i) = k if and only if 1t(k) = i), and suppose without lossof generality that the jobs in sequence 1t are renumberedsuch that if (n - S1t(i) + 1)2a; ~ (n - sn(i') + 1)2ai" theni < i' (with ties broken arbitrarily). Let critical (renum-bered) job i' be identified such that",j'-I[(-2 2 ] r -2 2L,..i= 1 (1; - Q; )/aj 5:. Rand L:i= d(a j - Qj )/a;] > R.Then the optimal allocation of resource across the re­numbered jobs in sequence 1r is given by:

{

(iif - fl})/ai, for i < r,.. )

Y; = R - t[(iT~ - Q})/a;j, for i = i'; (16)j=l

0, for i > r .

11

LBu2f<pl(np ) = :L(n - k + 1)2((1~(k) - a~(k)Y~(k)). (17)k=l

If T < ip*, then the n - I unscheduled jobs should simplybe sequenced in nonincreasing order of ift, and expression(17) (with Y~I(k) = 0 for I ~ k ~ n) represents an upper

bound on th~ flow time variance of "ltp . In either case, anupper bound on the value of objective function (l2) isobtained by combining LBij)(1tp) and LB0'21<p) (1tp) , i.e.,UB(1tp) = [T - LB~(1tp)l/ J[LBall<pJ(1tpH.

The evaluation process, dominance property, andbounding approach are easily incorporated into thebranch-and-bound algorithm for problem (P-RSPVR).At termination, the algorithm yields the [J-robust sched­ule, np, the associated assignment of resource across jobs,Y1tp(k) = {Y~6(k) : k = 1,2, ... ,n}, and the resulting maxi­mum probability of achieving flow time performance noworse than target level T, f1(1rp, T).

Theorem 3. Let i/>* represent the expected total flow timeassociated with the SEPT schedule.

(a) If ,u; ~ /1;1, af $lf7;, Z ~ Q];, and a, ~ ail, then thereexists an optimal solution to problem (P-RSPVR) inwhich job i precedes job i' for any T ~ ip*.

(b) If J1j ~ jljl and Cir 2:: a;', then there exists an optimalsolution to problem (P-RSPVR) in which job i pre­cedes job i' for any T < ip*.

The proof of Theorem 3 again utilizes a straightforwardinterchange argument (see [15]).

We employ a bounding process similar to that developedin Section 3. To illustrate, consider partial schedule 1tp inwhich jobs have been assigned to the first I positions insequence, and note that a lower bound on the expectedflow time of any schedule starting with 1tp , LBij>(1tp), isagain given by expression (9). If T ~ ip*, a lower boundon the flow time variance of 1rp can be determined byconstructing schedule 1tp" , in which partial sequence 1tp isfollowed by n - I artificial jobs. The makeup and se­quence of the artificial unscheduled jobs is such that

if~~(k) s a~~(JrI)' a,r~(k) ~ an'~(k'), and ~~(k) s ~~(k') 'V I + 1~ k < k' ::; n. Expression (16) can then be used to deter­mine the corresponding optimal allocation of resourceacross jobs in artificial schedule 1t;, from which a lowerbound on flow time variance can be calculated as

So long as a consistent mapping is maintained between therenumbered jobs in expression (16) and the original jobsin sequence 1t, the complexity of the process for evaluatingcomplete sequence "It is O(nlog n). Note also that a similarmarginal analysis yields the optimal allocation of resourceacross jobs for a given sequence whenever at is a convex,nonincreasing function of Yi for i = 1,2, ... )n, An exten­sion of the dominance result in Section 3 can be used toreduce the actual number of sequences requiring explicitevaluation by using expression (16).

(15)

n

T - L(n - k + l),u1t(k)k=l

n 11

T - L L(n - k + l)ll;x;k;=Ik=l

n

L(n - k + 1)2(O-;(k) - an(k)Y1t(k))k=l

(1t-EVAL) max

(6), (7), and (8).

Problem ({J-RSPVR) thus maximizes the probability ofachieving a given target level of system performance whilelimiting the amount of resource allocated among the njobs to the available supply, and ensuring that i~ a; ~ if; for each job i. Since problem ([J-RSPVR)generalizes problem ({J-RSP) (with a, = 0 fori = 1,2, ... ,n), problem ([J-RSPVR) is clearly NP-hard.

As in Section 3, a branch-and-bound algorithm can bedeveloped for solving problem (P-RSPVR) given pro­cessing time (Ili and at) and variance-reduction (aj andfl}) information for each job i, and given a target level offlow time performance (T). Nodes in the search tree againrepresent the assignment of jobs to sequence positions,and branches in the tree correspond to feasible partialsequences. Complete schedules are evaluated by deter­mining the allocation of resource that maximizes the as­sociated likelihood of achieving actual performance noworse than the target level. Sequence tt is thus evaluatedby solving:

({J-RSPVR) max

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The heuristic developed for problem (!3-RSPVR) is adecomposition approach that separately determines anappropriate job sequence given an estimate of each job'soptimal processing time variance, and the optimal allo­cation of resource across jobs given a sequence. An initialjob sequence, 1t1, is constructed using procedure fJ-Heu-.. . h 2 -2 f . 1 2 S .nstic wit a; = a,. or I = , , ... , n. equence 1t1 IS

evaluated by using expression (16) to determine theassociated optimal allocation of resource, Yn l = {Y;\(k) :

k = 1,2, ... , n}; the resulting probability of achieving flow

time performance no worse than target level T is then

computed as [3(1t I ' T) = [T - I:~= 1(n - k + l)J.lnl(k)] -7­

J[L~c:J(n - k +1)2(a;l(k)-an,(k)Y;lk))]' Because this op­timal allocation of resource implies a set of processingtime variances that may be different from those used toconstruct sequence nl, we update the estimate of theoptimal variance for each job, a;1 (k) = 8;\ (1') - anI (k)Y;1 (k)

for k = 1,2, ... .n, and generate a new candidate jobsequence, 1t2, using procedure p-Heuristic. This newsequence is evaluated as described above, with the bestencountered solution consistently retained. The processcontinues until a previously encountered sequence isgenerated, as further search would simply result in cycling.

Decomposition HeuristicInput. Processing time, 11; and a}, and variance-reduc­

tion, a, and g}, information for each jobi = l , 2, ... , n, R units of avaiJable resource, andtarget level of flow time performance, T.

Output. Heuristic sequence, 7th, and resource-allocation,Y: (k)' k = 1,2, ... , n, that provide a lower bound,LB, on the probability of achieving flow timeperformance no worse than target level T.

Step 1. Set a7 = (j~ for i = I, 2, ... .n, S = 0, j = 0, andL8=0.

Step 2. Set j = j + 1. Construct schedule 1tj by usingProcedure p-Heuristic applied to u, and (1~ fori = I, 2, ... .n. If 1tj E S, stop. Otherwise, setS = S + {nj} and go to Step 3.

Step 3. Evaluate schedule 1tj by determining the associ­ated optimal resource-allocation vector,{Y;[(k) : k = 1,2, ... , n}, by using expression (16),and set:

n

L(n - k + 1)2(U;Ak) - anj{k)Y;Ak))'k=1 •

If LB) > LB, set LB = LB), ith = it), and.. - * • k - ] 2 S t 2 _-2Yn/t(k) - Ynj{k) lor - , , 1 n. e a7tj (k) - anAk)

- Qnj(k)Y;j(k) for k = 1, 2, 1 n and return toStep 2.

Daniels and Carrillo

The effectiveness of the decomposition heuristic is clearly afunction of the number of job sequences generated by theprocess. The computational results reported in the nextsection indicate that the heuristic yields close approxi­mations to the optimal solution while considering very fewdistinct sequences and resource-allocation policies.

5. Computational results

To explore the computational performance of the solu­tion approaches, procedures were coded in FORTRANand implemented on a 486 personal computer. The firstexperiment involved test problems consisting of n = 10,15, and 20 jobs. The mean processing time for each job iwas first randomly drawn from a uniform distribution ofintegers on the interval J.lj E [10, 50bd, where parameterl5} controls the variability in the average processing timesacross jobs in a given test problem. The processing timevariance of each job i was then randomly drawn from auniform distribution of integers on the intervalat E [0,bJ.ltb2], where parameter c52 controls the variancein the ratio )1,./ Jar across jobs in a given test problem,and l52 ~ 1 ensures that virtually all of the mass of theprocessing time distribution for each job i lies to the rightof Pi = O. Three values of c5 1 (0.4, 0.7, and '1.0), threevalues of c52 (0.4, 0.7, and 1.0), and three values of zp(1.04, 1.65, and 2.33, corresponding to target levels wherethe SEPT schedule yields acceptable performance withprobability approx. 0.85, 0.95, and 0.99) were included inthe experimental design to simulate a wide range ofscheduling environments. Ten replications were generatedfor each combination of n, l5), b2 , and zp, resulting in atotal of 810 test problems.

Each problem was solved using the branch-and-boundalgorithm and the p-Heuristic procedure (with parame­ter (X set to 1.0, 2.0, 4.0, and 00) discussed in Section 3.Note that (J. = 00 corresponds to Pi = 11; fori = 1, 2, ... , n, so that the Implications of ignoring pro­cessing time variance information and scheduling ac­cording to the SEPT priority rule can be investigated.The performance of the procedures is reported in Table1, which provides information on the average CPU time(in seconds) for the branch-and-bound algorithm, theexpected flow time of the p-robust schedule, ijJ(np),compared with the optimal expected flow time, ip*, andthe approximation errors incurred by the SEPT and [3­Heuristic solution approaches. Note that approximationerrors are calculated to represent the additional risk ofachieving substandard system performance if a heuristicschedule is selected instead of p-robust schedule 1tp. Theresults in Table I are aggregated over the three values ofl5 1 and the three values of D2 included in the experi­mental design.

The results in Table I show that over the 810 testproblems, the expected flow time of the p-robust schedule

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fJ-Robust scheduling for single-machine systems 983

Table 1. Computational performance of p-robust solution procedures

Branch-and-bound Percentage above optimal solution

[ep( 7l:p) - cp.]/ tp. SEPT p-Heuristic

n zfJ CPU Mean Max. Mean Max. Mean Max.

10 1.04 0.2 0.1 0.8 4.0 24.0 0.6 6.01.65 0.2 0.3 1.9 11.1 54.8 0.9 13.22.33 0.3 0.6 2.5 32.0 135.4 1.2 14.5

Avg. [Max.] 0.2 0.3 [2.5] 15.7 [135.4] 0.9 [14.5]

15 1.04 1.0 0.1 0.5 3.4 14.8 1.2 9.31.65 1.7 0.2 1.0 10.9 33.7 1.4 11.92.33 2.1 0.4 1.9 26.2 83.1 2.4 16.0

Avg. [Max.] 1.6 0.2 [1.9] 13.5 [83.1] 1.6 [16.0]

20 1.04 ]2.6 0.1 0.3 3.6 11.7 1.5 6.71.65 33.6 0.1 0.7 . 10.2 38.2 2.3 9.92.33 38.2 0.2 1.1 25.1 90.2 3.1 22.6

Avg. [Max.] 28.1 0.1 [1.1] 13.0 [90.2] 2.3 [22.6]

averaged 0.2°10 above the optimal expected flow time, witha maximum deviation of 2.5°/0. This indicates that the {J­robust schedule maximizes the probability of achievingflow time performance no worse than the specified targetlevel while maintaining excellent performance with respectto expected flow time. The results in Table 1 also showthat close approximations to the fJ-robust schedule wereobtained using the p-Heuristic procedure. Over all 810 testproblems, the risk level associated with the best heuristicsolution averaged 1.6°/0 above the optimal solution, with amaximum deviation of 22.60/0. Because only four itera­tions of the heuristic were required to identify consistentlygood schedules, we note that further improvement in so­lution quality could be realized with a moderate increasein computational effort by judiciously including addi­tional values of cx. In contrast, the poor performance(l4.1 % average error; 135.4°/0 maximum error) of theSEPT schedule illustrates the substantial additional riskassociated with a scheduling approach that ignores pro­cessing time variance information.

A second experiment was designed to investigate theperformance of the solution approaches developed forproblem ({J-RSPVR). Problem sizes (n = 10, 15, and 20jobs) and mean processing times (Pi E [10,50tSd) weredetermined as in the first experiment. Processing timevariance data were next randomly generated from uni­form distributions of integers such that i1~ E [0,~ p;] anda, E [0,a;], with the minimum variance of each job set toO. Resource levels were then randomly generated from auniform distribution of integers on the intervalR E [0, tJ2 2:7=1 a} / ai], with parameter tS 2 included tocontrol resource availability. Three values of b1 (0.4, 0.7,and 1.0), three values of b2 (0.4, 0.7, and 1.0), and threevalues of zp (1.04, 1.65, and 2.33) were again included inthe experimental design, and ten replications for each

combination of parameters resulted in a total of 810 testproblems.

Each problem instance was solved using the branch­and-bound algorithm and the decomposition heuristic(with parameter CI. set to 1.0, 2.0, 4.0, and 00 in thefJ-Heuristic procedure used to generate alternative se­quences) discussed in Section 4. The optimal allocation ofresource across jobs in the SEPT schedule was also de­termined to highlight the added risk that results from asequencing approach that ignores processing time vari­ance information, even when available opportunities toreduce processing time uncertainty are exploited througha separate resource allocation decision. The performanceof the solution procedures is presented in Table 2, whosestructure is identical to that of Table 1 with the exceptionthat information on the average number of iterationsrequired before termination of the decomposition heu­ristic is also reported.

The results in Table 2 emphasize the excellent expectedflow time performance of the p-robust schedule withvariance reduction, which averaged 0.3°/0 above the op­timal expected flow time (with a maximum deviation of3.1 °/0) over the 810 test problems. Table 2 also indicatesthat good approximations to the p-robust schedule withvariance reduction were obtained in very few iterationsusing the decomposition heuristic. Over all 810 testproblems, the risk level associated with heuristic solutionsaveraged 4.1 % above the optimal solution, with a maxi­mum deviation of 31.9°/0. In contrast, allocating resourceoptimally to jobs in the SEPT schedule yielded solutionswhose risk level averaged 33.0°/0 above the optimal so­lution, with a maximum deviation of 220.3°/0, illustratingagain that considerable additional risk is incurred by ascheduling approach that ignores processing time vari­ance information.

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Table 2. Computational performance of solution procedures for problem (,B-RSPVR)

Daniels and Carrillo

Percentage above optimal solution

Decomposition Heuristic SEPT with resource allocation

Number ofIterations Mean Max. Mean Max.

3.5 2.8 19.8 22.2 64.63.2 3.7 22.5 36.1 129.64.1 5.2 26.1 52.9 220.33.6 3.9 [26.]] 37.1 [220.3]

4.6 3.1 22.8 19.9 78.64.8 4.2 29.4 31.4 105.04.7 5.4 31.9 46.8 169.44.7 4.2 [31.9] 32.7 [J69.4]

5.5 3-0 9.5 19.6 94.06.8 4.4 22.7 24.6 115.36.3 5.5 28.2 43.2 151.06.2 4.3 [28.2] 29.1 [151.0]

Branch-and-bound

[q>(np) - ~*]jijJ.

n zp CPU Mean Max.

10 1.04 5.5 0.1 0.61.65 6.8 0.4 1.32.33 7.2 0.5 3.1

Avg. [Max.] 6.5 0.3 [3.1]

15 1.04 35.4 0.1 0.71.65 54.5 0.3 1.32.33 70.0 0.4 1.7

Avg. [Max.] 53.3 0.3 [1.7]

20 1.04 724.5 0.1 0_81.65 1344 0.3 1.62.33 1685 0.5 2.9

Avg. [Max.] 1251 0.3 [2.9]

6. Conclusions

This paper has focused on scheduling environments withprocessing time uncertainty, and explored the schedulingimplications of hedging the risk of substandard systemperformance by considering both the mean and varianceof system performance in evaluating alternative sched­ules. We defined a measure of schedule l3-robustness thatrepresents the likelihood of achieving system perfor­mance no worse than a given target level. For single-stageproduction environments where job processing times areindependent random variables and the performancemeasure of interest is the total flow time over all jobs, weformulated the ,B-robust scheduling problem, establishedits complexity, and developed exact and heuristic solu­tion approaches. We then considered ,B-robust schedulingissues that arise when the uncertainty associated withindividual job processing times can be selectively con­trolled through resource aJlocation. Computational re­sults indicated that ,B-robust schedules provide effectivehedges against processing time uncertainty while main­taining near-optimal performance with respect to ex­pected total flow time. Close approximations to theoptimal solution were consistently obtained using theheuristic procedures.

fJ-Robust scheduling can be an important approach forcontrolling the impact of manufacturing uncertainty onsystem performance. Further research should focus onproviding additional linkages between sources of manu­facturing variability, operational decisions that affect orare affected by these uncertainties, and the manner inwhich managers should incorporate risk into the deci­sion-making process. Examples in the scheduling domaininclude development of jJ-robust models for other

scheduling environments where the variabiJity in systemperformance associated with a schedule can be deter­mined, with consideration of alternative performancemeasures that are sensitive to manufacturing variabilityand thus exposed to the risk of substandard performance,and detailed investigation of other sources of and mod­elling frameworks for manufacturing uncertainty.

Acknowledgements

We thank the anonymous reviewers for their helpfulcomments and suggestions.

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Biographies

Richard L. Daniels is an Associate Professor of Operations Manage­ment in the DuPree School of Management at the Georgia Institute ofTechnology. He received his Ph.D. in Operations Management fromthe Anderson Graduate School of Management at UCLA in 1986. Hisresearch interests include manufacturing planning and control systems,resource flexibility and its impact on operational efficiency, and deci­sion making under uncertainty. His articles have appeared in a numberof journals, including Management Science, Operations Research. Na­val Research Logistics. and European Journal of Operational Research.

Janice E. Carrillo is an Assistant Professor of Operations Managementin the Olin School of Business, Washington University. She receivedher Ph.D. in Operations Management from the DuPree School ofManagement at Georgia Tech in 1997. Her research focuses on thedevelopment of process improvement strategies that recognize theimportance of investments in training and preparation for successfulimplementation.

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