Scheduling with bicriteria: total flowtime and number of tardy jobs

international journal of

product!on economics ELSEVIER

Int. J. Production Economics53 (1997)91 99

Scheduling with bicriteria: total flowtime and number of tardy jobs Suna K6ksalan Kondakci*, Tuncay Bekiroglu IE Department, Middle East Technical University, Ankara, 06531, Turkey

Received 5 June 1996; accepted26 June 1997

Abstract

in this paper the problem of minimizing total flowtime and number of tardy jobs on a single machine is considered. Some properties of the nondominated solutions are discussed. Computational results on the usefulness of developed properties for problems having up to 30 jobs are reported. Keywords: Scheduling; Bicriteria; Single machine

1. Introduction

There has been a number of publications in multiple criteria scheduling during the last two decades. Most of these studies consider two criteria (see Heck and Roberts, 1972; Van Wassenhove and Gelders, 1980; Lin, 1983; Sen and Gupta, 1983; Azizoglu et al., 1991; Kondakci et al., 1996). In this paper we study the properties of efficient schedules with respect to flowtime and the number of tardy jobs. Flowtime behaves as a measure representing manufacturer's concerns. Minimizing flowtime implies minimizing work-in-process inventory. Minimizing the number of tardy jobs, on the other hand, can be considered as a measure of customer satisfaction, i.e., it represents customer's concerns. Hence we included measures representing both the customer's and the manufacturer's concerns.

*Corresponding author. Tel.: (90)(312)2102289: Fax.: (90)(312)20 1268.

The complexity of the hierarchical scheduling problem where total flowtime is minimized while keeping the number of tardy jobs at its minimum value is not known. The simultaneous minimization of the two criteria is a more general problem and its is also open (Lee and Vairaktarakis, 1993; Chen and Bulfin, 1993). The available approaches for these problems try to reduce the problem size and find properties to eliminate inferior schedules in order to solve moderate size problems in a reasonable amount of time. Emmons (1975) develop such a BAB approach for the hierarchical problem having the number of tardy jobs as the primary criterion. Kyparisis and Douligeris (1993) extend Emmons' approach where the problem is to select and sequence only a given number M of the available N jobs. Emmons (1975) develops a polynomial algorithm for the special case where the jobs that will be tardy are given. Nelson et al. (1986) address the simultaneous minimization problem and develop a BAB procedure to generate all efficient solutions. Kiran and

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S.K. KondakcL T. Bekiroglu/lnt. J. Production Economics 53 (1997) 91-99

Unal (1991) present some properties of the efficient solutions. In this paper, we develop new theorems, that can be used to reduce the problem size and eliminate inferior schedules. Five of the theorems construct pairwise precedence relations between jobs and one identifies the job which can be sequenced last in an efficient schedule. Some other properties of efficient solutions are also discussed. We define the problem in the next section. We then present precedence theorems and some other properties of efficient solutions. We next report the computational results. Finally, we give our concluding remarks.

2. Problem definition Consider n jobs ready to be processed on a single machine at time zero. Let Pi and di denote the processing time and the due date of job i, respectively for i = 1, ..., n. Let Ci(S) denote the completion time of job i in schedule S. Define F(S) as the total flowtime of schedule S:

F(S) = ~ Ci(S); i=1

nt(S) is the number of tardy jobs in schedule S: nv(S) = ~ Ui(S) i=l

where U~(S) = {~

ifdi
Both total flowtime and the number of tardy jobs problems can be solved optimally in polynomial time. The Shortest Processing Time (SPT) rule (i.e., ordering the jobs in non-decreasing order of their processing times) minimizes total flowtime and an algorithm due Moore (1968) minimizes the number of tardy jobs. A schedule S is efficient with respect to flowtime and the number of tardy jobs if there does not exist a schedule S' such that

F(S') <~F(S), nT(S') ~< nx(S)

with at least inequality. If we say that exist such an schedule.

one of the above holding as a strict there exists such a schedule, S', then S' dominates S. If there does not S', then we say that S is an efficient

A schedule S is weakly efficient if there does not exist another schedule S' such that

F(S') <~F(S), nT(S') < n~(S). Note that the set of efficient schedules is a subset of the set of weakly efficient schedules. Let u be a nondecreasing objective function of the two criteria and Q be the set of all possible schedules. Then the problem may be defined as (P)

Min s.t.

u(F(S),nT(S)) S6Q

3. Some properties of efficient solutions It is well known that in any multicriteria problem, for any objective function, u, which is a nondecreasing function of the multiple criteria, the optimal solution is an efficient solution. In this section we will discuss some properties of efficient solutions. Define Bj as the set of indices of all jobs that are shown to precede j o b j at any point in time, Aj is the set of indices of all jobs that are shown to follow job j at any point in time and A) the complement of Aj. n

Theorem 1. If dr ;i Y~i=lPl where p, = maxi{pl}, then there exists an efficient schedule where job r is sequenced last. Proof. Consider a schedule in which job r is not scheduled last. Moving job r to the last position will not increase the number of tardy jobs since job r is early at all positions and completion times of all other jobs either decrease or stay same. Total flowtime value of the resulting schedule is also at least as good since Pr = maxi{pi} and we know that the SPT rule minimizes total ftowtime. []

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S.K. Kondakci, T. Bekiroglu/Int. J. Production Economics 53 (1997) 91 99

Jobs satisfying the above stated conditions should initially be removed from the problem. In doing so Epi is reduced and other jobs may successively become eligible for removal. A common aspect of Theorems 2-5 is that they require the interchange of jobs j and k where PJ ~ Pk. We consider schedule S in which job k is sequenced before job j and schedule S' which is obtained by interchanging these two jobs (Fig. 1). In the proofs of the theorems we will show that schedule S' is at least as good as schedule S in terms of total flowtime and the number of tardy jobs. The part of the proof related with total ftowtime is identical in Theorems 2-5. Let W and C be the start time of job k and the completion time of job j in schedule S, respectively. Jobs that are scheduled before W and after C are not affected by the interchange. The jobs between jobs j and k have their completion times decreased by Pk -- Pr >~ O. Hence their number tardy as well as their flowtime values do not increase. So we need to evaluate the two schedules only in terms of number tardy and flowtime values of jobs j and k. If we consider the total flowtime o f j o b s j and k, we see that the flowtime of job k increases by C - ( W + Pk) while the flowtime of job j decreases by C - (W + pj). The resulting decrease is C-

( W + Pr) - C + ( W + Pk) = Pk-- Pr >~ O.

Hence schedule S' is at least as good as schedule S in terms of total flowtime. Note that, for all the theorems stated Ak ~ A r (considering only the precedence relations constructed up to that point) is

required for job k to be interchanged with job j. We next give the proofs of Theorems 2-5 in terms of the number of tardy jobs criterion. Theorem 2. I f p r ~ Pk, dr ~

dk and d r - Pr >>"dk - - Pk then there exists an efficient schedule where job j precedes job k.

Proof. The theorem is the same as Nelson et al.'s (1986) theorem developed for minimizing number of tardy jobs and maximum tardiness. They use the theorem in their BAB procedure while allocating jobs starting from the last position. The proof is also similar. Let T r and T~ be the tardiness of job j in schedule S and S', respectively. Then Tj = max{0, W + pj - dr} << T k max{0, W + Pk --dk}. Thus, if job j is tardy in schedule S', then job k is tardy in schedule S. Similarly, Ti, = max{0, C - dk} ~< r r - max{0, C - dr}. Thus, if job k is tardy in schedule S' then job j is tardy in schedule S. Then schedule S' is at least as good as schedule S. [] =

Corollary 2.1. I f Pr <~ Pk, dr ~ dk for all k ~ j and d r >~ p j, then there exists an efficient schedule where job j is the first.job in that sequence.

Proof. The proof is similar to the proof of Theorem 2. Let job k be assigned to the first position in schedule S. Note that, W = 0 for the first position. Then we have T'j = 0 <~ Tk = max {0, Pk -- dk}

T~ = max{0, C - dk} <~ Tj

W

C

max{0, C - dj}.

~

Thus, j o b j is not tardy in schedule S' but job k may be tardy in schedule S and if job k is tardy in schedule S' then j o b j is tardy in schedule S. Hence, schedule S' is at least as good as schedule S. [] Theorem 3. For any two jobs j and k having pj ~ Pk, there exists an efficient schedule where job j precedes job k if dk < Y,i~BkPi + Pk.

S':

W

C

Proof. T k = W d- Pk -- dk ~ ~i~B~ Pi + Pk dk > 0 and T~ = C - dk > W + Pk dk > O. Thus job k is -

Fig. 1. Interchangeof jobs j and k.

-

-

-

94

S.K. Kondakci, T. Bekiroglu/Int. J. Production Economics 53 (1997) 91 99

tardy in both schedules. On the other hand, Tj = max(0, W + p j - d~) ~< T~ = m a x ( 0 , C - dj). Hence schedule S' is at least as good as schedule S. [] Corollary 3.1. I f Pk ~ Pj, for all j ~ k and d k < Pk, then there exists an efficient schedule where job k is sequenced last. Proof. dk < ~ieB, Pi + Pk is always satisfied since dk < Pk. Job k succeeds all other jobs due to Theorem 3 since Pk/> P3 for allj. That is, job k is the last job in the schedule. [] Theorem 4. For any two jobs j and k having pj <~Pk, there exists an efficient schedule where job j precedes job k if dk >~Z~a~ pi. Proof. Job k is not tardy before and after the interchange since dk >~Y~i~A~,Pi >~ C. The completion time ofjobj decreases after the interchange. Hence, if job j is tardy in schedule S, it may become not tardy in schedule S'. So the schedule obtained by interchanging these jobs, schedule S', is at least as good as the original schedule, schedule S. [] Similarly, jobs k and j are not tardy at both positions if dk >~dj >1Y~i~Aj,pl >~ C, giving rise to the following theorem. Theorem 5. For any two jobs j and k having p~ <<.Pk, there exists an efficient schedule where job j precedes job k if dk ~ dj ~ ~ieAj Pi. Theorem 6. I f di >~~,~RjP, + PJ where R j is the set of indices of the jobs having processing times less than or equal to pj excluding job j, then there exists an efficient schedule where job j precedes job k if dj < ~i~e~pi + pj + Pk. Proof. Nelson et al. (1986) extend a theorem developed by Emmons and state that if a job is early in the Shortest Processing Time (SPT) sequence, then it is early in the optimal sequence that minimizes F. Hence such a job precedes any job which makes it become tardy. []

Remark 1. The schedule obtained by Moore's algorithm may be inefficient. Consider the case where di >1~ pj for all i, i.e., all possible sequences yield zero tardy jobs. Then the schedule obtained by Moore's algorithm is the same as the Earliest Due Date (EDD - sequencing the jobs in non-decreasing order of their due dates) schedule. Assume that the first job in the EDD schedule, i.e., the job with the earliest due date, has the longest processing time which will yield a schedule having a smaller total flowtime (and no tardy jobs) when interchanged with the last job. Hence Moore's algorithm may yield a weakly efficient but not an efficient schedule. Remark 2. The schedule obtained by the SPT rule may be inefficient. Consider a case where the processing times of all jobs are equal. Then the SPT schedule will be any arbitrary schedule. Different schedules may yield different number of tardy jobs as due dates are different. Hence, the SPT schedule may yield a weakly efficient but inefficient schedule. Remark 3. The maximum number of efficient schedules is Min{F(MOORE) - F(SPT) + 1, nT(SPT) -

nT(MOORE) + 1}

assuming that the processing times and due dates are all integer values. The worst possible value for flowtime in any efficient schedule is F(MOORE) since in this case we obtain the best possible case for the number of tardy jobs, nT(MOORE). Therefore, F > F(MOORE) correspond to inefficient schedules. Clearly, the best possible value of flowtime is F(SPT). Then the permissible values of flowtimes for efficient schedules are F(SPT), F(SPT)+ 1. . . . . F(MOORE) which correspond to F ( M O O R E ) - F(SPT)+ 1 distinct values. Similarly, permissible values of number of tardy jobs is nT(MOORE), nT(MOORE) + 1. . . . . nT(SPT) which correspond

S.K. Kondakci, T. Bekiroglu/Int. J. Production Economics' 53 (1997) 91 99

to n T ( S P T ) - nv(MOORE) + 1 distinct values. Therefore, the upper bound on the total number of efficient solutions is the minimum of the above distinct values. The following example illustrates the use of some of the theorems.

• Job 1 precedes job 3 since

Example. Consider the following six jobs having distinct processing times and due dates.

• Job 1 precedes job 6 since

Job i

1

Pi

10 30

dl

2

3

4

5

6

F(SPT) = 750

45 50

60 90 nT(SPT) =4 37 36 245 55 10080

F(MOORE) = 820 nT(MOORE) =3

There are 15 possible pairwise precedence relations and 6 ! - - 7 2 0 possible sequences. Following relations are constructed through our theorems.

Pl ~ P4, dl ~< d4 and ~>d4-P4.

(Theorem 2)

• Job 2 precedes job 4 since P2 ~< P4, d2 ~< d4 and d2 - P2 ~> d4 - P4.

(Theorem 2)

• Job 3 precedes job 4 since P3 ~< P4, B4 = { 1,2},

d4
(Theorem 3)

• Job 6 is the last job since P6 >~ Pi, i = 1, ..., 5 and d6 < P6. (Corollary 3.1) • Job 2 precedes job 3 since P2 ~< P3, A; = {1, 2, 3, 5} and d3 > P l q- P2 q- P3 -t- P s .

(Theorem 4)

• Job 1 precedes job 2 since dl >~ Pl and dl < P2 + Pl.

(Theorem 6)

• Job 1 precedes job 5 since d~ >1 Pl and dl < P5 + P~.

dl ~> Pl and dl < p6 + Pl.

(Theorem 6)

(Theorem 6)

Above relationships identify job 6 as the last job and job 1 as the first job (since it precedes all other jobs). Using all precedence relations, we need to evaluate four sequences; 1-5-2-3-4-6, 1-2-5-3-4-6, 1-2-3-5-4-6, and 1-2-3-4-5-6 to find Min{820 - 720 + 1,4 - 3 + 1} = 2 efficient schedules or the optimal solution.

4. Computational results

• Job 1 precedes job 4 since

dl-pl

dl >~ pl and d 1 < P3 q- Pl.

95

(Theorem 6)

2400 test problems are run for problem sizes n = 10, 15,20 and 25. Processing times are generated from a discrete uniform distribution within the ranges [1,25], [1,50] and [1,100]. Due dates are generated from a discrete uniform distribution over the range [P(1 - r - R/2), P(1 - ~ + R/2)] where P = ~Pl, R is the range of due date and r is the average tardiness factor (Fisher 1976). r is a coarse measure of proportion of potentially tardy jobs and R is the maximum possible due date range expressed as a fraction of the expected value of the length of the schedule. An increase in R means an increase in the range of values for due dates which implies high variations in their values while a decrease in R results in a narrower range of values which implies that the due date values are closer to the mean and to each other. On the other hand, an increase in r, reduces the due date values which implies an increase in the number of tardy jobs. By varying R and r, we try to incorporate different problem settings to reduce bias in case the theorems favor certain environments. Four different problem sets are chosen as in Sen and Gupta (1983) with different R and z values. We present the problem sets we use in Table 1. As can be seen from the table, problem sets I and IV have

96

S.K. Kondakci, Z Bekiroglu/Int. ,£ Production Economics 53 (1997) 9 1 - 9 9

wider ranges leading to more variety in due dates. On the other hand, problem sets III and IV have bigger average due date values. Fifty runs for each problem set and a given range of processing time values resulting in 600 runs for each problem size and 2400 runs in total are performed. All the theorems except Theorem 1 construct pairwise precedence relations. Theorem 1, on the other hand, identifies the location of the last job, hence reduces the problem size. Notice that for

Table 1 The problem sets used in the experiments Set

R

r

[P(1 - r - R/2), P(1 - z + R/2)]

I

0.4 0.2 0.2 1

0.8 0.8 0.65 0.5

[0,0.4P]

II III IV

[0.1 P, 0.3P] [0.25P,0.45P] [0.3P, 1.3P]

Table 2 Nodes examined by BAB procedure utilizing developed theorems n

Prob. set

Pi ~ D U l l , 2 5 ]

Aver. I II 10 III IV I II 15 III IV I II 20 III IV I II 25 III IV

Min.

34.1 (77.40) a 14.9 (73.67) 24.7 (70.49) 133.4 (81.76)

0

516 (92.08) 118.4 (88.91 ) 789.5 (88.21) 7531.6 (90.89)

0

9115.7 (96.28) 1993.1 (97.12) 5321.5 (97.31) 269981.7 ( > 97.80) 125728.7 ( > 99.00) 37600.9 ( > 99.70) 44116.4 ( > 99.60) .

Pi ~ D U [ I , 100]

Pi ~ D U l l , 5 0 ] Max.

Aver.

Min.

104 (85.33) 86 (82.05) 69 (86.44) 438 (91,47)

35.4 (69.87) 15.2 (68.66) 26.9 (66.58) 134.4 (86.28)

0

5855 (91.02) 500 (94.90) 23995 (83.80) 256087 (71.97)

317.7 (91.32) 106 (80.31 ) 523.2 (90.69) 2361.7 (96.42)

0

2080

243909 (89.43) 25161 (95.30) 88849 (97.96) 5251019

5134.4 (97.23) 874 (97.98) 2491.7 (98.85) 148390.3 ( > 98.80)

736

2396572

0

1515248

0

1040886

64825.4 ( > 99.50) 11170.1 ( > 99.90) 53705.8 ( > 99.50) .

0 0 0

0 0 146 0 0 0

.

.

.

Max.

Min.

131 (71.15) 85 (79.06) 167 (79.78) 577 (94,48)

36.4 (74.53) 13.8 (73.61) 24.7 (80.75) 146.5 (73.16)

0

1498 (96.68) 386 (90.76) 4871 (92.14) 17253 (97.60)

296.7 (95.45) 106.6 (90. l 7) 210.3 (94.08) 6758.9 (89.32)

0

1862

76040 (95.71 ) 5895 (99.45) 28005 (99.02) 2621784

4042.6 (98.00) 734 (98.50) 1539.2 (99.66) 100268 ( > 99.80)

2166

818856

0

253328

0

671421

12156 ( > 99.90) 3679 ( > 99.97) 9654.8 ( > 99.92)

0 0 0

0 0 145 308 0 0

.

Aver.

.

0 0 0

0 0 0

Max. 142 (87.11) 68 (69.78) 92 (93.11) 770 (81.63) 1617 (98.22) 386 (96.94) 1096 (97.98) 209260 (57.44)

0

65013 (68.64) 5485 (96.44) 6266 (99.05) 1544010

0

4066819

2080

36843

736

100558

0 146 0

.

aNumbers in paranthesis represent the percentage reduction (BAB with theorems versus with BAB without theorems) in the number of nodes due to theorems.

S.K. Kondakci, T. Bekiroglu/lnt. J. Production Economics 53 (1997) 91-99

Theorem 1 to hold, a job is required to have a due date greater than or equal to the total processing time (P) of all the unscheduled jobs. This requirement is only attainable for problem set IV where the due dates are generated over the range

97

complete schedule (the incumbent), whereas the second one uses theorems in addition to the incumbent solution. 2400 test problems are run using both procedures. Results of the runs are presented in Table 2. The missing entries in the table correspond to problems that could not be completed in an hour whereas the other problems were solved within several minutes. The table lists the average

l-0.3P, 1.3P]. In order to test the efficiencies of Theorems 2 - 6 we use two branch and bound algorithms. The first one eliminates nodes only using the best known

Table 3 Nodes examined by Nelson et al.'s procedure utilizing developed theorems

n

Prob. set

Pi ~ DU[1,25]

Aver. I II 15

III IV I II 20 III IV I II 25 III IV I II 30 III

IV

49.92 (21.85)" 16.34 (14.54) 13.80 (17.46) 43.18 (18.34)

pi ~ DU[1, 50]

Min. 0 0 0 3

223.74 (23.40) 78.02 (20.63) 33.00 (11.15) 114.52 (17.95)

10

1122.36 (21.86) 283.08 (20.25) 70.00 (14.00) 435.82 (14.37)

12

3999,40 (21.87) 458.24 (18.47) 175.78 (15.69) 1495.08 (14.47)

0 0 6

0 0 21 116 15 0 62

p~ ~ DU[1,100]

Max.

Aver.

Min.

248 (45.73%) 56.00 (9.68) 85.00 (16.67) 215 1471.43)

61.50 (20.11%) 20.38 ( 14.01) 14.64 (9.29) 43.02 (11.63)

1001 (33.22) 492 (26.46) 189 (5.50) 570 (3.06)

275.74 (17.07) 69.04 ( 17.06) 32.56 (12.14) 112.24 (11.17)

11

6708 (25.85) 1412 (9.55) 395 (25.75) 4368 (12.59)

1070.56 (12.50) 248.34 (18.75) 98.84 (12.13) 391.32 (17.01)

129

32204 (13.93) 1373 (20.54) 1147 (13.11) 10118 (23.30)

3712.30 (24.38) 636.72 (18.88) 222.04 (13,03) 1140.86 " (15.89)

256

7 0 0 3

0 0 14

0 0 16

0 0 11

Max.

Aver.

Min.

369 (2.89%) 146 (22.34) 109 (2.68) 221 (0.00)

57.30 (19.99%) 28.40 (18.39) 13.02 (6.73) 37.62 (11.77)

1145 (1.89) 247 (16.84) 206 (10.82) 585 (5.03)

143.92 (9.26) 45.78 ( 17.04) 26.22 (8.77) 100.72 (15.88)

0

6307 (1.11 ) 1434 (25.74) 313 (12.32) 2043 t19.19)

919.00 (20.47) 175.98 (17.21) 108.50 (10.36) 307.16 (11.53)

59

15787 (32.07) 3749 (28.17) 1850 (6.99) 4276 (10.99)

5295.16 (12.23) 453.18 (18.42) 186.58 (11.06) 1162.82 (18.24)

15

0 0 0 4

0 0 5

0 0 14

0 0 30

Max. 323 (5.83%) 153 (11.05) 93 (12.26) 183 (19.38) 632 (26.94) 282 (12.15) 144 (15.29) 428 (19.40) 4741 (24.011 936 (6.31) 1433 (0.00) 2176 (10.86) 60872 (12.31) 1868 (19.66) 1289 (5.29) 9835 (26.65)

"Numbers in paranthesis represent the percentage reduction (Nelson et al.'s BAB with theorems versus Nelson et al.'s BAB without theorems) in the number of nodes due to theorems.

98

S.K. Kondakci, T. Bek_iroglu/lnt. J. Production Economics 53 (1997) 91-99

as well as the minimum and the maximum number of nodes generated through BAB procedure. Averages are taken over 50 replications and the values in paranthesis show percentage reduction in the number of nodes generated by the BAB procedure with the theorems. As can be observed from the table, the theorems lead to substantial reductions (between 66.58% and 99.97%) in the size of the branch and bound tree in terms of the average number of nodes examined. Same observation is valid for the maximum number of nodes generated. The relative effectivenesses of theorems increase substantially as the problem size grows. No important performance differences are observed for different problem settings. Small increases in percentage reductions are observed for wider processing time ranges for bigger problem sizes. As another test of our theorems, we incorporated them into the BAB procedure of Nelson et al. (1986). In their procedure each node identifies the set of early jobs and at each level one more job is considered to be in the early set. We again use Theorem 1 initially to reduce the problem size and use Theorems 2-5 to fathom some nodes from the branch and bound tree. The theorems construct precedence relations which are utilized in considering a job for the inclusion in the early set. Theorem 6, which is developed by Emmons, is utilized in Nelson et al.'s procedure. Hence its efficiency is not tested. We run 2400 test problems using Nelson's procedure in its original form for the similar problem settings described before. Instead of problem size 10, which is trivially solved, we included problem size 30 in our experiments. We run the same 2400 problems with the modified Nelson et al.'s procedure where we incorporate our theorems this time. The average, the minimum and the maximum number of nodes examined by Nelson et al.'s BAB procedure utilizing developed theorems and percentage reductions in the average and the maximum number of nodes generated are given in Table 3. Reductions up to 24% in the average and up to 33% in the maximum number of nodes generated are realized.

5. Conclusion

In this paper we addressed the scheduling problem on a single machine considering total flowtime and number of tardy jobs measures simultaneously. We developed several precedence theorems which identify various properties of efficient solutions. We empirically tested the effectiveness of the theorems using constructed precedence relations to fathom nodes in a branch and bound algorithm. Our extensive computational experience shows that the theorems lead to sizable reductions in the branch and bound tree. We also tested the effectiveness of our theorems by incorporating them into a BAB procedure developed in the literature. Our computational experience also showed sizable reductions in the branch and bound tree. We think that our theorems are versatile to be incorporated into different BAB procedures, or heuristics, that may be developed for this problem. We believe that the analysis of the problem stucture and construction of precedence theorems in this paper will facilitate the development of efficient heuristic procedures for the problem.

References Azizoglu, M., Kondakci, S., Kirca, 0., 1991. Bicriteria scheduling problem involving total tardiness and total earliness penalties. Int. J. Prod. Econom. 23, 17-24. Chen, C.L., Bulfin, R.L., 1993. Complexity of single machine, multicriteria scheduling problems. Eur. J. Oper. Res. 70, 115-125. Emmons, H., 1975. One machine sequencing to minimize mean flow time with minimum tardy. Naval Res. Logist. Quart. 22(3), 585-592. Fisher, M.L., 1976. A dual algorithm for the one-machine scheduling problem. Math. Programming 11,229-251. Heck, H., Roberts, S., 1972. A note on the extension of a result on scheduling with secondary criteria. Naval Res. Logist. Quart. 19(2), 403-405. Kiran, A.S., Unal, A.T., 1991. A single-machine problem with multiple criteria. Naval Res, Logist. 38, 721 727. Kondakci, S., Azizoglu, M., K6ksalan, M., 1996. Note: bicriteria scheduling minimizing flowtime and maximum tardiness. Naval Res. Logist. 43, 929-936. Kyparisis, J., Douligeris, C., 1993. Single machine scheduling and selection to minimize total ftowtime with minimum number tardy. J. Oper. Res. Soc. 44(8), 835 838. Lee, C.Y., Vairaktarakis, L., 1993. Complexity of single machine hierarchical scheduling: a survey, Complex. Numer. Optim. 19, 269-298.

S.K. Kondakci, T. Bekiroglu/lnt. J. Production Economics 53 (1997) 91-99

Lin, K.S., 1983. Hybrid algorithm for sequencing with bicriteria. J. Optim. Theory Appl. 39, 105-124. Moore, J.M., 1968. A n-job, one machine sequencing algorithm for minimizing the number of late jobs. Mgmt. Sci. 15(1), 102-109. Nelson, R.T., Sarin, R.K., Daniels, R.L., 1986. Scheduling with multiple performance measures: the one machine case. Mgmt. Sci. 32(4), 464-479.

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Sen, T., Gupta, S.K., 1983. A branch and bound procedure to solve a bicriterion scheduling problem, liE Trans. 15, 84-88. Van Wassenhove, L.N, Gelders, F., 1980. Solving a bicriterion scheduling problem. Eur. J. Oper. Res. 4. 42-48.