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Cornput. & Op. Res. Vol. 11, No. 3, pp. 27X84, 1984
Printed in the U.S.A.
0305-0548/84 $3.00 + .a0 0 1984 I’.qamon Rcss Ltd.
OPENSHOP AND FLOWSHOP SCHEDULING TO MINIMIZE SUM OF COMPLETION TIMES-f
I. ADIRIS and N. AMIT§ Technion, Faculty of Indust~a~ En~n~~ng and Mana~ent, Technion City, Haifa S2ot), Israel
Abstract-This paper deals with efficiently solvable special cases of openshop and permutation- Aowshop scheduling where the objective function is minimum sum of completion times.
Two 0(mn) algorithms for openshop scheduling where all operations have equal processing times, are presented. The first constructs a no-wait schedule and the second a schedule where both criteria (sum of completion times and schedule length) take on their minimal values.
For ~~utation-flowshop scheduling where processing times satisfy domin~cy and/or ordered relations, SPT rules are proved to be optimal.
The formulations of openshop, flowshop and permutation-flowshop scheduling are: n jobs
(J,, Jz, . . . , J,) have to be processed by m machines (M,, M,, . . . , AI,). Job Ji, i = 1,2,. . _ , n, consists of at most m operations CO,,, Q,, . . . , 0,). Operation 0, has to be processed uninte~upted for tti time units on Mi- til is a non-negative integer-t, > 0 if the operation exists and tii = 0 if the operation is missing. I/
Openshop. The order by which the operations of a job are processed is immaterial. Flowshop. Processing of operation 0, + , may start only after 0, has been completed. PermutationfEowshop. Flowshop where all machines process the jobs in the same order. Two operations of the same job cannot be processed simultaneously and a machine
cannot process more than one job at a time. The problem is to find the operation schedule on each machine obeying the constraints and optimizing a given objective function.
Scheduling problems are designated n/m/y, IT/S 7 where, for the cases under discussion in this paper, y = (0, F, P}. 0 stands for openshop scheduling, F-for flowshop sched- uling, P-for permutation-flowshop scheduling; I?--assumptions dropped and/or added (for example, t# = 1 means that all operations have equal processing times, or “no-wait”- once the processing of a job starts it is processed continuously with no waiting time between machines); ii-the objective function, for the cases dealt with in this paper: sum of completion times CC, where Ci is the completion time of job Ji,
The following definitions are used in succeeding sections: -Job Ji is smaller [larger] than J1: if fik I tik[tik 2 tjk] for 1 I k I m. In abbreviated
notation J, -C JIIJI > J;il. -Job J, is strictly smull [strictly large] if Ji < JiEJi > 41 for 1 I j I n. -A scheduling problem with ordered jobs is one where it is possible to arrange the jobs
in a decreasing order, J,, > J2, > . 3 . > J,,,. -Machine 1M, is smaller [larger] than M, if tik I ti,[tik 2 ti,] for 1 I i 2 n. In abbreviated
notation Mk < M,[M, > M,]. -Machine iM, is strietIy small [strictly large] if fMk < &fAMk > Mr] for 1 < r 5 m. -A scheduling problem with ordered machines is one where it is possible to arrange
the machines in a decreasing order M,, > M2’ > .* . > M,,,,. -An ordered scheduling problem is one with both ordered jobs and ordered machines.
Machine kfk dominates M, if min tik 2 max t!,. In abbreviated notation Mk+ kf,_ f ,
7Operation Research Statistics and Economics, Mimeograph Series No. 286. JIgal Adiri is Associate Professor of Operations Research at the Faculty of Industrial Engineering and
Management, Technion, Haifa, Israel. Research interests include deterministic and stochastic scheduling and queueing theory. His papers have appeared in Operations Research, Management Science, Mathematics of Operations Research, Journal of the Association for Computing Machinery, Naval Research Logistic Quarterly.
$Neta Amit is a Ph.D. student in Computer Sciences at Yale University, New Haven. Received BSc. and MSc. in Computer Sciences from the Technion-Israel Institute of Technology, Haifa, Israel.
j/For further elaboration of the meaning and influence of zero processing times, see Hefetz and Adiri[4]. IFor notation and classification of scheduling problems we follow Lenstra[fl and Rinnooy Kan[8].
215
276 I. hIR1 and N. AMIT
--Machine Mk strictly dominates if Mk.> M, for 1 I r 5 m, r #k. -A pair of tightly joint series of increasing and decreasing dominating machines is one
where M, <*M2 <.- . . -c~M,~> . . -*> AI,,,, -A pair of joint series of increasing and decreasing dominating machines is the same as
one of tightly joint series, except that M,, is larger than or smaller than Mh + r(Mh > M,, + ,
or MMK+d. We denote by SPT(M,) an arrangement according to the shortest-processing-time (SPT)
order on M,, and by tl*j” the processing time of the i-th job on Mj for SPT(M,). Unary NP-Completeness of n/2/0/Xi and n/2/F/CCit was proved by Achugbue and
Chin[l] and Garey et al. [3], respectively. Thus, it is most unlikely that there exists an efficient algorithm (polynomial-bounded) solving openshop or flowshop or permutation- flowshop problems for two or more machines to minimize the sum of completion times. This opens up the need for developing efficient algorithms for special cases of the above problems. At the same time, very few articles can be found in literature on this subject. Adiri and Pohoryles[2] proved that SPT(M,) is optimal for n/m/P, no-wait/Xi with a decreasing series of dominating machines, and that SPT(M,) is optimal for n/m/P, no-wait/CC, with an increasing series of dominating machines, except the first job which satisfies mini {Vi = n IS:km:; t$” + (i - l)tEm - If,;: f tGm}. Sarin and Eybl[9] proved that SPT(M,) is optimal for n/2/F/X,, M, > M2, and proved that n121FICCi, M, c M2 is unary NP-Complete. Panwalker and Kahn[7] proved that SPT is optimal for ordered- n/m/P/CC,.
This paper studies efficiently solvable special cases of openshop and permutation- flowshop scheduling problems where the criterion is minimum sum of completion times.
In Section 1 we develop two O(mn) algorithms minimizing the sum of completion times for openshop scheduling where all operations have equal processing times. The first minimizes the sum of completion times and disregards the schedule length but the jobs are scheduled continuously and thus satisfy a no-wait constraint as well; the second constructs an optimal schedule where both criteria-sum of completion times and schedule length- take on their minimal values. In Section 2 it is proved that the following arrangements are optimal: (i) SPT(M,) for n/m/P/Xi, M, -+ * * <*M,,-> * * -> M,, except for the first job which satisfies mini Vi as above with h replacing m; (ii) SPT(M,,) for n/m/P/X,, M, <-, ‘. <.M/, > Mh+,.> . * ..> M,,, and a strictly small job on the first h + 1 machines; (iii) SPT(MJ for n/m/P/CC,, M, c.. * * <.Mh < M,,+ ,Q . . -> M,,,, SPT(M,,) = SPT(M,, + i) and a strictly small job on the first h + 1 machines; (iv) SPT(M,,) for n/m/P/CC, with ordered jobs, Mh strictly large and a subproblem ordered-n/h/P/X,, h I m. The results obtained ((i)-(iv)) hold also for permutation-flowshop, no-wait, scheduling. Section 3 is devoted to a discussion and concluding remarks.
1. OPENSHOP/EQlJAL PROCESSING TIMES/MINIMUM SUM OF COMPLETION TIMES
In this section two O(mn) algorithms minimizing the sum of completion times for openshop scheduling where all operations have equal processing times, are presented.
1.1 n/m/O, no-wait, tli= l/Xi and n/m/O, tii= l/Xi We present Algorithm 1, which constructs in O(mn) steps an optimal schedule for both
n/m/O, no-wait, tv= l/ZCi and n/m/O, tii= l/ZCi. Let us first examine the relations among the following problems: (1) n/m/O/Xi;
t,-processing’ time of Ji on Mj, namely of operation O,, (2) n/m/O, prmt/CCi; the same as problem 1 where preemptions are allowed (prmt); (3) n/@/prmt/Xi; @-m identical
parallel machines, where the processing time of job Ji is ti = i p, tu, and tii is that of problem
1; (4) n/@//Xi; the same as problem 3 where preemptions are prohibited. Let 7ck denote the value of min Xi in problem k, k = 1,2,3,4.
tn/2/F/ZC, and n/2/P/X, are equivalent.
Openshop and flowshop scheduling to minimize sum of completion times 271
LEMMA 1
Proof. The set of all feasible solutions for problem 1 is included in that of 2, which is in turn included in that of 3, thus rrl 2 7c2 2 7r3. z3 = n, by McNaughton’s theorem[6]. 0
ALGORITHM 1 [1] Schedule job Ji, i = 1,2, . . . , n, continuously, starting at the earliest possible time
on Mirno+) until M,, then move to the earliest possible time on M, and schedule continuously the remaining operations of Ji.
The complexity of Algorithm 1 is O(mn).
THEOREM 1 Algorithm 1 constructs an optimal no-wait schedule for n/m/O, tv = l/Xi. Proof. Let n = yk + 1, k = 0, 1,2,. . . , . Algorithm 1 constructs k compact blocks
(without idle intervals). In block r (r = 1,2, . . . , k) m jobs start at time (r - 1)m and finish at rm. The last I jobs start at time km and finish at (k + 1)m. We have
mm +2mm +3mm -j-s.. + kmm + (k + 1)ml = m(1 + k)(mk + 21)/2. (1)
The minimum sum of completion times in appropriate n/@//Xi problem with ti = Zj’!! , tg = m, Vi, is as per (1). The optimality of Algorithm 1 is a direct consequence of Lemma 1. cl
Example 1. A no-wait schedule for 13/8/O, tl = l/Xi. Implementing Algorithm 1 we have Fig. 1.
M, 1 87654329 13 12 1.1 10 M2 2 1 8 7 6 5 4 3 10 9 13 12 11 M3 3 2 1 8 7 6 5 4 11 10 9 13 12 M4 4 3 2 1 8 7 6 5 12 11 10 9 13 MS 5 4 3 2 1 8 7 6 13 12 11 10 9 % 6 5 4 3 2 1 8 7 13 12 11 10 9 M7 7 6 5 4 3 2 1 8 13 12 11 10 9 M8 8 7 6 5 4 3 2 1 13 12 11 10 9
Fig. I. Optimal no-wait schedule for 13/8/O, tii = l/Xi; Xi = 144, C,, = maxi C, = 16.
1.2 n/m/O, tij = l/(~CinC~~). Algorithm 1, presented previously, minimizes the sum of completion times but
disregards the schedule length, C,, = maxi Cj. Algorithm 2, which constructs in O(mn) steps an optimal schedule where both criteria-Xi and C,,--take on their minimal values, is presented below.
We distinguish here two cases: (i) number of jobs greater than or equal to that of machines, n > m; (ii) number of jobs smaller than that of machines, n < m.
ALGORITHM 2
(i) m<n=km+I, k=1,2 ,...,.
Phase I [l] Schedule the first (k - 1)m jobs according to Algorithm 1. For simplicity we renumber the remaining jobs from 1 onward to n’ = m + I, and
consider the starting time of the kth block as zero.
278
Phase II
I. ADIRI and N. h4lT
[l] Schedule job .Ji, i = 1,2, . . . , n’, at time i (place i) on M,. [2] Schedule job Ji, i = 1,2, . . . , n’ -m, continuously to time m. [3] Schedule job J,, i = n’ - m + 1, . . . , n’, continuously to time n’. [4] Schedule job Ji, i = 2,3, . . . , n ’ - m, continuously from time 1 on M, + 2 _ i to time
i - 1 on M,. [5] Schedule job J,, i = n’, n’- 1, . . . , m + 2, continuously from time m + 1 on
M Zm+2_i to time i- 1 on M,,,. [6] Schedule continuously the unscheduled operations of job Ji, i = n’,
n’-l,... , n’ - m + 2, starting at the latest possible time on the lowest indexed available machine.
(ii) m > n = I, k = 0. [l] Schedule the 1 jobs according to Algorithm 1. The complexity of Algorithm 2 is O(mn).
THEOREM 2
Algorithm 2 constructs an optimal schedule for n/m/O, tii = l/(ZCi fl C,,). Proof. (i)n=km+I,k=1,2 ,...,.
Feasibility Phase I implements Algorithm 1 and constructs a feasible schedule for the first (k - I)m
jobs. We have to prove that phase II constructs a feasible schedule for the remaining m + I jobs (renumbered from 1 to m + 1). Let us schedule job 4, i = 1,2, . . . , m + I, con- tinuously, starting at time i on M, until completion at time i + m - lon M,,,, see Fig. 2. Algorithm.2 shifts sectors D to A (step [4]), G to D (step [S]), F to B (step [6]) and E to C (step [6]). Since all shifts are horizontal, there are no “horizontal conflicts” (every job is processed once by every machine). The fact that there are no “vertical conflicts” either (no job is processed at the same time by two machines) is demonstrated by Fig. 3.
Optimality Schedule length. A lower bound on schedule length is C,,, 2 n. In phase 1, (k - 1)m
jobs are scheduled in (k - 1) compact blocks such that in each block m new jobs are processed from start to completion. In phase 2, the remaining m + 1 are processed from start to completion in a compact block of length m + 1. Thus, C,,, = km + I= n meets the lower bound and is optimal.
Sum of completion times. For the (k - 1)m jobs scheduled in phase I,
(k - I)m
mm + 2mm + . . . + (k - 1)mm = m2k(k - 1)/2. (2)
In phase II, 1 jobs are completed at km and the remaining m at (km + I). Thus, for the
I.1 i-1 i E 1 “‘1
; D il.1 I I G m’ I
Fig. 2. Schematic description of algorithm 2.
Openshop and flowshop scheduling to minimize sum of completion times
1 1
2 ... ,_, , . . . 1-l 3.’ ml-1-l
279
I Oh1 I
1 : ml-l ,
I . I ml :,
I-
m-1.1
1
/
l-l
,
1 2 3 *I. 1 lel *.* WI;
_ l-l _ _m-1
F-B and E-C (8tep tti)
Fig. 3. Demonstration of shifts performed by algorithm 2.
(m + I) jobs scheduled in phase II we have
km+/
2 Ci = km1 + (km + 1)m. (3) i=(k- l)m+ I
Summing (2) and (3), we obtain for the sum of completion times the same value as per (I), thus it is optimal.
(ii) n = I < m. Implementing Algorithm 1 yields a feasible schedule where Ci = mVi, which is a lower
bound on the completion time of job Ji. Thus, it is optimal for both criteria-X, and c
-~~~rn~~~ cl
2. 13/8/O, tij = l/(ZCi tl C,,,). Implementing Algorithm 2, we have Fig. 4. (Compare the optimal schedule constructed
by Algorithm 1 for 13/8/O, t = l/X, (Fig. 1) with that of Algorithm 2 for 13/8/O, tli= l/(XC,fl C,,,) (Fig. 4d).
2. SPECIAL CASES OF ~i~lPl~~i
We study nlrn~P~~Ci with dominancy and/or ordered relations.
2.1 n/m/P/Xi with series of dominating machines In this section we analyze n/m/P/Xi with a pair of tightly joint and a pair of joint
series of increasing and decreasing machines.
THEOREM 3 SPT(M,,) is optimal for n/m/P/Xi, M, <- - . -c.M,,*> . * *a> hi,,,, 1 I h 2 m, except the
first job which satisfies min, (Vi = n Zjg,’ t$ + it;h” - XC:,, t$}.
Ml
1 23
4561
8 9
10
11
12
13
M,
1 2
3 4
56
7 8
9 10
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13
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2 3
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8 9
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1 2
3 45
6
I 8
9 10
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, 1
2 3
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6 7
8 9
10
11
Mj
1 2
34
5 6
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11
M
4 1
2 3
4 5
6 7
8 9
10
M4
1 2
3 4
5 6
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9 10
M
S 1
2 3
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9 M
, 5
1 2
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8 9
%
1 2
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M6
4 5
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M7
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* (a
) 6
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2 3
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6
M,
1 .2
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10
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12
13
M,
1 2
3 4
M2
1 2
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5 6
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9 10
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M
2 13
1
2 3
M3
1 2
3 4
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, 12
13
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2 w
1
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M4
II
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M,
5 1
2 3
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9 A
4,
5 10
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M
6 4
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8 M
6
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M
7 3
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M,
3 4
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M*
2 3
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1 10
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,I
2 13
6
M,
2 3
4 5
(c)
5 6
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34
5
6 7
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8 91
0 1
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1
2 3
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13
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8 91
0 1
2 11
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6
7 7
8 9
1 10
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12
13
6
(d)
Fig.
4.
Pa
rtia
l sc
hedu
le
at
the
end
of:
(a)
step
13
1, (
b)
step
[4
], (c
) st
ep
[5]
and
(d)
step
[6
]- -
optim
al
sche
dule
, X
C,
= 14
4,
C,,,
, =
13.
Openshop and flowshop scheduling to minimize sum of completion times 281
Proof. We denote by C,, the completion time of the ith job and by tt,$ the processing time of the ith job on I%$ Since M, <.- . . -cM,*> * * Q M,,,, we have
The sum of completion times takes the form
k-l
q=n x
j=l q1li + if +i I?
i=l j=k+l
The first term on the r.h.s. of (4) is a function of the first job only; SPT(M,) minimizes the second; the third is a constant independent of the order of the jobs. Thus, SPT(Mk) is optima1 except for the first job.
Assigning the ith job in SPT(Mk) to the first position, (4) yields
i=l j=l i=I k=l ix1 j=k+l
Thus, SPT(&&) is optimal, except the first job which satisfies
h-l
Vi=n C tf+it$- j=l
E~~~~~e 3. A 4/?/P/Xi with processing times as per Table 1.
Table 1. Processing times
J, 53 J4 ~..__
M, 2 2 1 2 W 4 3 2 4 M, 5 6 7 8 M4 1 2 2 1
Since M, +I%#, <GM,-> M4, we have a pair of tightly joint series of increasing and decreasing dominating machines.
OPT is l-2-3-4. Let V, = mini Vi, thus the Ith job in SPT(M,) is assigned to the first position.
4(2 + 4) + 0 = 24; i = 1
Vj=4 i tg+it$ - j, t;; = 4(2 + 3) -f- 2.6 - (5 + 6) =21; i=2
j=1 ZE 4(1+2)-t-3.7-(5+6-t-7) =15; i=3 4(2+4)-t-4.8-(5+6+-7+8)=30; i=4
V, = min, Vi, thus the optimal sequence is 3-1-2-4 and the optimal schedule takes the form as per Fig. 5.
12 38 2.3 30
Fig. 5. Optimal schedule; i Ci = 811 ia,
282 I. ADIRI and N. AMIT
THEOREM 4
SPT(M,) is optimal for n/m/PICCi, M, < * * . * < *A4,, > IV,,+ ,’ > . . * * > M,,,, I~h~m-l,andthereisastrictlysmalljobonM;.,j=1,2,...,h+l.
Proof. Let 4 be the time elapsed between the moment the first job is ready to start on Mh and that of completion of Ji on Mh + ,.
Thus
h-l
ci = 1 q,u + 4 + f tv j=I j=h+2
~,ci=n:~ta”+~,~+~,j=~+2t, (5)
The first term on the r.h.s. of (5) is a function of the first job only and takes on its minimum value if the strictly small job is assigned to the first position. Fi is the completion time of job J, in an imaginary two-machine (namely, Mh and Mh+ ,) permutation-flowshop problem. Since &fh > Mh + , , the minimum value of X& is obtained by SPT(M,)[9], which is in agreement with assignment of the strictly small job first. The third term of (5) is a constant independent of the order of the jobs. 0
For a scheduling problem with ordered jobs there is an SPT order such that SPT = SPT(M,)Vj.
THEOREM 5
SPT(kf,) iS optimal for n/m/P/XC,, M, <.. . * <.kfh < Mh + ,.> . . ..> M,,,, 1 I h I m - 1, SPT(M,) = SPT(M,+ ,) and there is a strictly small job on M,, j = 1,2, . . . , h + 1.
Proof. Equation (5) holds here too. Panwalker and Kahn[7] proved that SPT is optimal for ordered-n/m/P/X,. Since the submatrix of the processing times of the two machines--M, and M,, + , -is ordered, SPT(Zt4,) yields the minimum value for Wi. The rest is as per Theorem 4, and we have that SPT(M,) is optimal. 0
Example 4. A 5/5/PlXi with processing time as per Table 2.
Table 2. Processing times
Ml I 2 I 2 1 J+f2 2 8 3 5 6 JQf3 3 8 4 7 7 M4 1 2 1 1 2 M, 1 1 1 1 1
Since M, <.M2 < M3.> M4.> AI5 we have two joint series of increasing and decreasing dominating machines. .J, is strictly small and SPT(M,) = SPT(M,). Thus SPT(M,), l-345-2, is optimal and the optimal schedule takes the form as per Fig. 6.
Fig. 6. Optimal schedule; XC, = 104.
Openshop and flowshop scheduling to minimize sum of completion times
2.2 n/m/P/Xi with ordered relations THEOREM 6
283
SPT is optimal for n/m/P/X, with ordered jobs, M,(h < m) strictly large, and a subproblem ordered-n/h /P/CC,.
Proof. Since the jobs are ordered and M,, is strictly large, we have for SPT order that jobs are continuously scheduled on machines Mh onwards.
The completion time of the ith job is
c,, = C[tlh + 2 Z[Lli + f t[,li j=h+l j=h+l
where Crllh is the completion time of the ith job on &,, and Jlli the time elapsed between the moments of completion of job .ZrlI on M,_, and commencement of that job on Mj
The sum of completion times takes the form
i=l j=h+l i=l j=h+l
Panwalker and Kahn[7] proved that SPT is optimal for ordered-n/m/P/X,. Since the jobs are ordered and Mh strictly large, we have for SPT: Zrlli = 0, i = 1,2, . . . , n, j=h+l,..., m. Thus, SPT yields the minimum values for the first and second terms on the r.h.s. of (6) and the third term is constant, q
For h = 1, Theorem 6 yields:
COROLLARY 1 SPT = SPT(M,) is optimal for n/m/P/X, with ordered jobs and M, being strictly
large. Example 5. A 5/6/P/Xi with processing times are per Table 3.
Table 3. Processing times
Eyz-z
M, 2 4 4 2 2 M, 6 10 8 2 4 M, 3 4 4 2 3 M4 8 12 8 4 6 M5 6 6 6 2 6 % 4 4 4 4 4
Since J2 > J3 > J, > Js > J4 and M4 > M2 > M3 > M,, M4 > M,, M4 > M6, the problem is with ordered jobs, M4 strictly large, and a sub-problem ordered-5/4/P/CCi. Due to Theorem 6, SPT is optimal and the optimal schedule takes the form as per Fig. 7.
Fig. 7. Optimal schedule; 1 C, = 180. i= 1
284 I. ADmr and N. htlT
3. DISCUSSION
n/2/O/XCi and n/2/F/CCi are NP-complete, [l] and [3]. This is accepted as evidence that no efficient algorithms solving n/m/O/Xi or n/m/P/Xi, m 2 2, exist. In the present paper we develop efficient algorithms for sp+al Cases of openshop and permutation-flowshop, where the objective function is minimum sum of completion times.
(I) Openshop. It is assumed that all operations require equal processing times. For more general cases (e.g. processing times satisfying dominancy and/or order relations) development of efficient algorithms or alternatively proof of its NP-completeness, is not easy even for two machines, and further research is called for.
The optimal schedules constructed by Algorithms 1 and 2 are not unique. Moreover, any algorithm which constructs k compact blocks, where in block I (r = 1,2, . . . , k) m jobs start at time (r - 1)m and finish at rm, and the remaining I jobs start at km and finish at (k + l)m, is optimal for (mk + l)/m/O/tii = l/Xi.
(II) Permutation-flowshop. Theorems 3 + 6 yield for each of the analyzed special cases of permutation-flowshop, an algorithm that constructs an optimal schedule in O(n log n) steps (SPT rules).
The minimum sum of completion times for the cases analyzed by Theorems 3 + 6, is the same with or without a no-wait constraint. Thus, Theorems 3 + 6 are true also for permutation-flowshop, no-wait, scheduling.
Theorems 3, 4 and 6 generalize existing results. Clearly, for n/2/P/ZCi, MI > M,, Theorem 4, in agreement with Sarin and Eybl[9], yields that SPT(M,) is the optimal arrangement. For ordered-n/m/P/X,, Theorem 6, in agreement with Panwalker and Kahn[7], yield that SPT is the optimal arrangement.
Theorem 3 is also true for’n/mlP, no-wait/Xi, M, c** . - -cM,,*> * * .-> M,,,, I I h I m. For h = 1 we have a decreasing series of dominating machines and V, = mini 6 = it: - Xizl t$, = 0, thus SPT(M,) is optimal. For h = m we have an increasing series of dominating machines. These results are in agreement with those of Adiri and Pohoryles[2] for n/m/P, no-wait/CCi with a decreasing series of dominating machines and with an increasing series of dominating machines.
Note. (i) Although the optimal sequences as per Theorems 3 + 6 are the same for permutation-flowshop and permutation-flowshop, no-wait, scheduling, the optimal sched- ule might differ, as is demonstrated by Examples 3 t 5. (ii) Theorem 3 yields that SPT(M,J is optimal for n/m/P/Xi, M, c . . c-M,,+- . . Q- M,,,, 1 I h I m, with a strictly small job.
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9.
REFERENCES
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