the complexity of two group scheduling problems

JOURNAL OF SCHEDULINGJ. Sched. 2002; 5:477–485 (DOI: 10.1002/jos.118)

The complexity of two group scheduling problems

Jacek Blazewicz 1 and Mikhail Y. Kovalyov 2;∗;†

1Instytut Informatyki; Politechnika Poznanska; and Instytut Biochemii;Polska Akademia Nauk; Poznan; Poland

2Institute of Engineering Cybernetics; National Academy of Sciences of Belarus; Minsk; Belarus

SUMMARY

The problems of scheduling groups of jobs under the group technology assumption are studied. The tworemaining open questions posed in the literature a decade ago about the computational complexity ofthese problems (J. Oper. Res. Soc., 1992; 43:395–406), are answered. The parallel machine problem tominimize the total job completion time is proved to be NP-hard in the strong sense, even if group set-uptimes are equal to zero. A dynamic programming algorithm is presented to solve this problem, which ispolynomial if the number of machines is �xed. The two-machine open-shop problem to minimize themakespan is proved to be NP-hard in the ordinary sense, even if there are zero group set-up times andequal job processing times on both machines. It is shown that the latter problem under the conditionthat any group can be split into batches does not reduce to the problem where the splittings are notallowed, i.e. the group technology assumption is satis�ed. Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: scheduling; group technology; parallel machines; open shop

1. INTRODUCTION

Group technology (GT) is an approach to manufacturing and engineering management thatseeks to achieve the e�ciency of high-volume production by exploiting similarities of di�er-ent products and activities in their production=execution. Principles of GT are used by manymanufacturing companies to improve productivity and competitiveness, see, for example, Ref-erences [1–6].The �rst publications on scheduling in group technology environments are due to Petrov

[7] and Yoshida et al. [8]. In recent publications, a scheduling model of this type is oftenreferred to as a family scheduling model, see, for example, References [9–11].

∗Correspondence to: Mikhail Y. Kovalyov, Institute of Engineering Cybernetics, National Academy of Sciences ofBelarus, Surganova 6, 220012 Minsk, Belarus.

†E-mail: [email protected]

Contract=grant sponsor: Polish Government; contract=grant number: KBN 8T11A01618Contract=grant sponsor: International Association for Promotion of Cooperation with Scientists from Former SovietUnion; contract=grant number: INTAS 00-217

Copyright ? 2002 John Wiley & Sons, Ltd.

478 J. BLAZEWICZ AND M. Y. KOVALYOV

The problem of scheduling groups of jobs under GT assumption can be formulated asfollows.There are n independent jobs available at time zero, to be scheduled for processing by

a system consisting of m; m¿2; machines. The jobs are a priori partitioned into F; F¿2;groups such that jobs of the same group have similar requirements for setting up the machines.On a machine, all jobs of the group must be processed contiguously. No group can be splitinto subgroups which are processed separately on the same machine or on di�erent machines.Since the nature of the jobs, groups and machines is immaterial in our studies, we use naturalnumbers for their notation. In this paper, we assume that processing of each group f ispreceded by a sequence independent and machine independent set-up time sf. No job can beprocessed by a machine while a set-up on this machine is being performed.Processing systems of two types are considered. In the parallel-machine system, each job

can be completely processed by any of the machines. We consider identical parallel machines.The processing time of job j on any of the machines is the same and equal to pj. In thetwo-machine open-shop system, each job has to be processed by two machines A and Bpassing them in arbitrary order. The processing times of job j on the machines are equalto pj;A and pj;B, respectively. For the open-shop system, anticipatory and non-anticipatoryset-ups are distinguished. The presence of a job of group f on a machine is needed to startthe corresponding non-anticipatory set-up. Anticipatory set-up may start irrespective of the jobpresence. In both systems, a job cannot be processed on two or more machines at a time, amachine cannot process more than one job at a time and preemption of job processing on amachine is not allowed.A schedule speci�es sequences of jobs on the machines, or more speci�cally, an assignment

of groups to the machines, processing order of groups on each machine and processing orderof jobs within each group.Given a schedule, let Cj denote the completion time of job j. It is assumed that a job

completes immediately after its processing is �nished. For the parallel-machine problem, theobjective is to �nd a schedule that minimizes the total job completion time,

∑Cj; and, for the

two-machine open-shop problem, a schedule that minimizes the makespan, Cmax = max{Cj},where summation and maximum are taken over all jobs. All data, i.e. set-up times andjob-processing times are assumed to be positive integers.Recall that the parallel-machine problem to minimize

∑Cj without GT assumption, i.e.

when there is a single group, is solved in O(n log n) time by renumbering the jobs in theshortest processing time (SPT) order such that p16 · · ·6pn and assigning job j; j=1; : : : ; n,to the �rst available machine (see Reference [12]). The two-machine open-shop problem tominimize Cmax without GT assumption is solved in O(n) time by the algorithm of Gon-zales and Sahni [13], which constructs a schedule achieving an analytical lower bound forCmax.A decade ago Potts and Van Wassenhove [9] noted that the computational complexities

of the parallel-machine problem and the two-machine open-shop problem for the above-mentioned criteria and GT assumption are not known. To the best of our knowledge, theseare the remaining open questions posed in the literature about the complexity of schedulingproblems under GT assumption. The complexity of the single-machine problem to minimizethe number of late jobs under GT assumption was also indicated as unknown by Potts andVan Wassenhove. However, this problem was recently proved to be strongly NP-hard by Liuand Yu [14].

Copyright ? 2002 John Wiley & Sons, Ltd. J. Sched. 2002; 5:477–485

TWO GROUP SCHEDULING PROBLEMS 479

There are complexity results available for the problems, in which groups are allowed to besplit into batches, each batch is preceded by a non-zero sequence-independent group set-uptime. Cheng and Chen [15] proved that the parallel-machine problem to minimize

∑Cj is

NP-hard in the ordinary sense for m=2 and pj=p for all jobs j. Webster [16] proved thatthis problem is NP-hard in the strong sense for m variable and the condition that jobs of thesame group have identical processing times, which may, however, di�er among the groups.Kleinau [17] proved that the two-machine open-shop problem to minimize Cmax is NP-hardin the ordinary sense for anticipatory set-ups.In the remainder of this paper, we assume that GT assumption is satis�ed. In Section 2,

the parallel-machine problem to minimize∑Cj is proved to be NP-hard in the strong sense

even for the case of zero set-up times and unit job processing times. This result impliesthat the general case of the problem is strongly NP-hard as well. The problem remains NP-hard in the ordinary sense if the number of machines m is �xed, set-up times are equalto zero and pj=1 for all j. A dynamic programming algorithm is presented to solve thegeneral case of the problem in O(n log n+ mF min{n; 2F}m−1) time. Note that the two latterresults demonstrate that a polynomially solvable problem may have an NP-hard sub-problem.Liaee and Emmons [18] proved that, under GT assumption, the parallel-machine problem

to minimize∑Cj is polynomially solvable when m is �xed and each group comprises the

same number of jobs. They also proved that this problem is NP-hard in the ordinary sensewhen the group cardinalities are di�erent, m=2 and pj=p for all jobs j. Our results are moregeneral. An important issue is that their NP-hardness proof does not imply NP-hardness of theoriginal problem with arbitrary job processing times. Moreover, if in their NP-hard problem weassume that pj are arbitrary, then according to our algorithm, the problem becomes solvablein O(n log n+Fn) time! Based on the results of Liaee and Emmons, one can think that equalor unequal numbers of jobs in each group change the complexity status of the problem. Butthey do not a�ect it.In Section 3, the two-machine open-shop problem is proved to be NP-hard in the ordinary

sense for the case of zero set-up times and pj;A=pj;B for all j. It is also shown that thetwo-machine open-shop problems with and without GT assumption are not equivalent. Thisquestion was indicated by Potts and Van Wassenhove as a key issue for scheduling groupsof jobs in a two-machine open shop.

2. PARALLEL-MACHINE SCHEDULING PROBLEM

In this section, the problem of scheduling groups of jobs on parallel machines to minimize∑Cj under GT assumption is studied. Firstly, strong NP-hardness of the problem with zero

set-up times and unit job processing times is established. When the number of machinesm¿2 is �xed, this problem is shown to remain ordinary NP-hard. A dynamic programmingalgorithm is derived for the general case of the problem.

Theorem 1The parallel-machine scheduling problem to minimize

∑Cj under GT assumption is NP-hard

in the strong sense, even if set-up times are zero and pj=1 for all j.



ProofA transformation from the 3-PARTITION problem, which is NP-complete in the strong sense, tothe decision version of the scheduling problem is used.3-PARTITION: Given positive integers e1; e2; : : : ; e3k and E satisfying E=4¡ej¡E=2; j=

1; : : : ; 3k; and∑3k

j=1 ej= kE; does there exist a partition of the set {1; : : : ; 3k} into k subsetsX1; : : : ; Xk such that

∑j∈Xl ej=E for l=1; : : : ; k?

Let an instance of 3-PARTITION be given. Construct an instance of the parallel-machinescheduling problem, in which there are m= k machines and F =3k groups. Group j includesej unit-time jobs. All group set-up times are equal to zero.We prove that 3-PARTITION has a solution if and only if there exists a schedule for the

constructed instance of the scheduling problem with objective value∑Cj6(kE + kE2)=2.

It is easy to verify that our transformation is both polynomial and pseudopolynomial withrespect to the problem with unit or identical job processing times within a group and it ispseudopolynomial only with respect to the problem with arbitrary job processing times.‘If ’. Suppose �rst that a schedule with objective value

∑Cj6(kE+kE2)=2 exists. Let (il1; : : : ;

ilrl) be the sequence of groups on machine l and let Yl be the set of all jobs assigned to machinel. Let us state the contribution of the set Yl to the total completion time. We have

∑j∈YlCj =

[1 + 2 + · · ·+ eil1

]+[(eil1 + 1

)+(eil1 + 2

)+ · · ·+

(eil1 + eil2

)]

+ · · ·+[(eil1 + eil2 + · · ·+ eilrl−1

+ 1)+(eil1 + eil2 + · · ·+ eilrl−1

+ 2)

+ · · ·+(eil1 + eil2 + · · ·+ eilrl−1

+ eilrl

)]

=1+ eil12

eil1 +1 + 2eil1 + eil2

2eil2 + · · ·+

1+ 2eil1 + · · ·+ 2eilrl−1+ eilrl

2eilrl

Denote by Xl the set of groups scheduled on machine l; l=1; : : : ; k. From the aboveequation, we obtain

∑j∈YlCj=

∑j∈Xl

ej=2 +

( ∑j∈Xl

ej

)2/2

Therefore, for the total completion time of all jobs, we must have

∑Cj=

k∑l=1

∑j∈Xl

Cj= kE=2 +k∑l=1

( ∑j∈Xl

ej

)2/26(kE + kE2)=2

It follows that

k∑l=1

( ∑j∈Xl

ej

)26kE2



Assume that the solution to the above inequality is∑

j∈Xl ej=E + �l; where �l is arbi-

trary (negative or non-negative) integer, l=1; : : : ; k. From∑k

l=1

∑j∈Xl ej= kE; we deduce

that∑k

l=1 �l=0. Then we must have

k∑l=1

( ∑j∈Xl

ej

)2=

k∑l=1(E2 + 2E�l + �2l )= kE

2 + 2Ek∑l=1�l +

k∑l=1�2l = kE

2 +k∑l=1�2l6kE

2

It follows that �l=0; l=1; : : : ; k. Therefore,∑

j∈Xl ej=E; l=1; : : : ; k.‘Only if ’. Suppose now that 3-PARTITION has a solution and X1; : : : ; Xk are the required

sets. Construct a schedule, in which groups of the set Xl are assigned to machine l and aresequenced there in arbitrary order for l=1; : : : ; k. Similar to the proof of part ‘if’, we have

∑Cj= kE=2 +

k∑l=1

( ∑j∈Xl

ej

)2/2= (kE + kE2)=2

A similar transformation from the NP-hard in the ordinary sense problem PARTITION can beused to prove the following theorem.

Theorem 2The parallel-machine scheduling problem to minimize

∑Cj under GT assumption is NP-hard

in the ordinary sense, even if the number m of machines is equal to two, set-up times areequal to zero and pj=1 for all j.

In contrast to the above statement, we now show that the more general problem with a�xed number of machines m¿2 and arbitrary set-up and job processing times can be solvedin polynomial time.Consider the parallel-machine scheduling problem to minimize

∑Cj with arbitrary job

processing times. For the single-machine case of this problem, Potts and Van Wassenhove [9]observe that there exists an optimal schedule in which:

(i) jobs within a group are sequenced in SPT order,(ii) groups are sequenced in non-decreasing order of Pf=qf, where Pf= sf+

∑j∈f pj is the

total processing time of group f plus its set-up time, qf= |f| is the number of jobsof group f.

For the parallel-machine problem, property (i) evidently holds and property (ii) is satis�edfor each machine. Assume that the jobs of each group are sequenced in SPT order and thegroups are numbered so that P1=q16 · · ·6PF=qF . Thus, the problem reduces to �nding anassignment of the groups to the machines. Our dynamic programming algorithm DP, statedbelow, �nds an optimal assignment.In algorithm DP, group sequences on the machines are constructed backwards. Suppose

that the current group sequence on machine l comprises kl jobs (not groups) and groupf= {if1 ; : : : ; ifqf} is added to the beginning of the sequence. Then the contribution of thisgroup to the total completion time is∑

j∈fCj= qf(sf + pif1 ) + (qf − 1)pif2 + · · ·+ pifqf + Pfkl (1)



This contribution depends only on the group and on the number of jobs on machine l.Moreover, the �rst qf summands in (1) do not depend on the schedule and can be includedin∑Cj prior to the execution of the algorithm. In DP, we recursively compute the value

of Tf(k1; : : : ; km); which is the minimal total completion time, provided that groups F; F −1; : : : ; f are scheduled and the total number of jobs on machine l is kl; l=1; : : : ; m. A formaldescription of the algorithm is given below.

Algorithm DPStep 1 (Initialization). Number jobs of each group f in the SPT order such that pif16 · · ·

6pifqf; f=1; : : : ; F; and number the groups so that P1=q16 · · ·6PF=qF . Calculate TF+1(0; : : : ; 0)

=∑Ff=1 [qf(sf+pif1 )+ (qf− 1)pif2 + · · ·+pifqf ] and Tf(k1; : : : ; km)=∞ for (f; k1; : : : ; km) �=(F +

1; 0; : : : ; 0).Step 2 (Recursion). For f=F; F−1; : : : ; 1 and each tuple (k1; : : : ; km) such that kl ∈{0; 1; : : : ;∑Fg=f qg}; l=1; : : : ; m; and

∑ml=1 kl=

∑Fg=f qg; compute the following:

Tf(k1; : : : ; km)= minl=1; :::; m

{Tf+1(k1; : : : ; kl−1; kl − qf; kl+1; : : : ; km) + Pf(kl − qf)}:

If the above minimum is reached for l= l∗; then in the corresponding partial schedule groupf is assigned to the beginning of the group sequence on machine l∗.Step 3 (Solution). Calculate optimal solution value

T ∗= min{T1(k1; : : : ; km) | l=1; : : : ; m; kl ∈{0; 1; : : : ; n};

m∑l=1kl= n

}

and backtrack to �nd the corresponding optimal schedule.

Let us estimate the running time of algorithm DP. Step 1 requires O(n log n) time. Initeration f of Step 2, there are two ways to estimate the number of di�erent values ofstate variable kl that have to be considered. On the one hand, this number does not exceed∑F

g=f qg+16n+1. On the other hand, kl may only be equal to zero or a summation of a subsetof numbers from {qf; : : : ; qF}; i.e. we need to consider at most 2F−f+162F di�erent values ofkl. Of course, in the latter case, the sets of possible values for kl used by the algorithm mustbe corrected accordingly. Thus, the number of di�erent tuples (k1; : : : ; km) needed to considerin iteration f of Step 2 is at most O(min{n; 2F}m−1). The recursive equation can be solved inO(m) time for each state if values Pf and qf are calculated in advance. Then the running timeof algorithm DP can be given as O(n log n+mF min{n; 2F}m−1). From this we conclude thatthe parallel-machine problem with criterion

∑Cj; arbitrary job processing times and set-up

times is polynomially solvable if the number of machines is �xed.Note that motivation for the problem can be found in scheduling production lines

or multi-purpose machining centres, where the number of those lines or centres is rathersmall because they are expensive. If m=2 or 3, then the algorithm runs in O(n log n+ Fn)or O(Fn2) time, respectively, and provides a solution to the group scheduling problemquickly.



3. TWO-MACHINE OPEN SHOP SCHEDULING PROBLEM

In this section, the two-machine open-shop scheduling problem with Cmax criterion, zero set-up times and equal job processing times on both machines is proved to be ordinary NP-hardunder GT assumption. The problems with and without GT assumptions are shown not to beequivalent.

Theorem 3The two-machine open-shop scheduling problem to minimize Cmax under GT assumption isNP-hard in the ordinary sense, even if set-up times are zero and pj;A=pj;B for all j.

ProofA polynomial transformation from the NP-complete problem PARTITION is used.PARTITION: Given positive integers e1; e2; : : : ; ek and E satisfying

∑kj=1 ej=2E, does there

exist a subset X ⊂{1; : : : ; k} such that ∑j∈X ej=E?Given an instance of PARTITION, construct the following instance of the decision version of

the two-machine open-shop problem. There are F = k+2 groups. Group j; j=1; : : : ; k; consistsof a single job with processing times pj;A=pj;B= ej. Group k +1 consists of two jobs a andb with processing times pa;A=pa;B=2E and pb;A=pb;B=3E. Group k + 2 consists of twojobs c and d with processing times pc;A=pc;B=8E and pd;A=pd;B=10E. All group set-uptimes are equal to zero.We show that PARTITION has a solution if and only if there exists a schedule for the con-

structed instance of the two-machine open-shop problem with objective value Cmax625E.‘If ’. Suppose that a schedule with value Cmax625E exists. Job processing times are chosen

so that their summation on either machine A or B is equal to 25E. Therefore, neither machine Anor B stands idle before time 25E. Recall that group splittings are not allowed. Then it is easilychecked that there are four possibilities for scheduling groups k + 1 and k + 2. Two of themare given in Figure 1. The other two can be obtained by interchanging the machine notations.From Figure 1, one can see that either on machine A or on machine B there are two non-

adjacent intervals of length E, where groups 1; : : : ; k should be scheduled. Denote by X theset of groups from {1; : : : ; k} scheduled in one of these intervals. We must have ∑j∈X ej=E.‘Only if ’. Suppose that set X is a solution of PARTITION. Construct one of the schedules,

with the structure for groups k + 1 and k + 2 shown in Figure 1. Schedule groups 1; : : : ; k inthe empty intervals. For such a schedule, Cmax =25E.

Figure 1. Two possibilities for scheduling groups k + 1 and k + 2.



Theorem 4The two-machine open-shop scheduling problems to minimize Cmax with and without GTassumption are not equivalent.

ProofConsider the instance of the two-machine open-shop problem given in the proof of Theorem 3.Now assume that there are three groups. The �rst and second groups represent groups k + 1and k + 2, respectively. The third group consists of two jobs with processing times equal toE on any of the machines. Figure 1 shows that the minimum value Cmax =25E can only bereached if the third group is split into two batches on one of the machines.

4. CONCLUSIONS

In this paper, we have answered the two remaining open questions posed in the literatureabout the computational complexity of group scheduling problems.The parallel-machine problem to minimize the total job completion time is proved to be

NP-hard in the strong sense for the case when group set-up times are equal to zero and jobprocessing times are unit. If the number of machines is �xed, this special case of the parallel-machine problem is proved to be NP-hard in the ordinary sense. A dynamic programmingalgorithm is presented to solve the general case of this problem, which is polynomial if thenumber of machines is �xed.The two-machine open-shop problem to minimize the makespan is proved to be NP-hard in

the ordinary sense, even if group set-up times are equal to zero and job processing times areequal on both machines. Strong NP-hardness of the two-machine open-shop problem remainsan open question. It is shown that this problem under the condition that any group can besplit into batches does not reduce to the problem where the splittings are not allowed, i.e. thegroup technology assumption is satis�ed.

REFERENCES

1. Mitrofanov SP. Scienti�c Principles of Group Technology. National Lending Library, 1966.2. Ham I, Hitomi K, Yoshida T. Group Technology. Kluwer-Nijho�: Dordrecht, 1985.3. Wemmerl�ov U, Hyer NL. Cellular manufacturing in the US industry, a survey of current practices. InternationalJournal of Production Research 1989; 27:1511–1530.

4. Tatikonda MV, Wemmerl�ov U. Adoption and implementation of group technology classi�cation and codingsystems: insights from seven case studies. International Journal of Production Research 1992; 30:2087–2110.

5. Hadjinicola GC, Ravi Kumar K. Cellular manufacturing at champion irrigation products. International Journalof Operations and Production Management 1993; 13:53–61.

6. Gunasekaran A, McNeil R, McGaughey R, Ajasa T. Experiences of a small to medium size enterprise in thedesign and implementation of manufacturing cells. International Journal of Computer Integrated Manufacturing2001; 14:212–223.

7. V.A. Petrov, Flowline Group Production Planning. Business Publications: London, 1966.8. Yoshida T, Nakamura N, Hitomi K. A study of production scheduling: optimization of group scheduling on asingle production stage. Transactions of the Japanese Society of Mechanical Engineers 1973; 39:1993–2003.

9. Potts CN, Van Wassenhove LN. Integrating scheduling with batching and lot-sizing: a review of algorithms andcomplexity. Journal of the Operations Research Society 1992; 43:395–406.

10. Webster ST, Baker KR. Scheduling groups of jobs on a single machine. Operations Research 1995; 4:692–703.11. Potts CN, Kovalyov MY. Scheduling with batching: a review. European Journal of Operations Research 2000;

120:228–249.12. Conway RW, Maxwell WL, Miller LW. Theory of Scheduling. Addison-Wesley: Reading, MA, 1967.



13. Gonzales T, Sahni S. Open shop scheduling to minimize �nish time. Journal of the Association for ComputingMachinery 1976; 23:665–679.

14. Liu ZH, Yu W. Minimizing the number of late jobs under the group technology assumption. Journal ofCombinatorial Optimization 1999; 3:5–15.

15. Cheng TCE, Chen Z-L. Parallel machine scheduling with batch set-up times. Operations Research 1994;42:1171–1174.

16. Webster ST. The complexity of scheduling job families about a common due date. Operations Research Letters1997; 20:65–74.

17. Kleinau U. Two-machine shop scheduling problems with batch processing. Mathematical and ComputerModelling 1993; 17:55–66.

18. Liaee MM, Emmons H. Scheduling families of jobs with set-up times. International Journal of ProductionEconomics 1997; 51:165–176.


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