Annals of Operations Research 57(1995)191-201 191

Single machine group scheduling with ordered criteria*

A d a m Janiak

Institute of Engineering Cybernetics, Wroclaw Technical University, ul. Janiszewskiego 11/17, 50-372 Wroclaw, Poland

and

Mikhai l Y. K o v a l y o v

Institute of Engineering Cybernetics, Belarus Academy of Sciences, Surganova 6, 220012 Minsk, Belarus

The single machine group scheduling problem is considered. Jobs are classified into several groups on the basis of group technology, i.e. jobs of the same group have to be processed jointly. A machine set-up time independent of the group sequence is needed between each two consecutive groups. A schedule specifies the sequence of groups and the sequence of jobs in each group. The quality of a schedule is measured by the criteria F1 . . . . . F m ordered by their relative importance. The objective is to minimize the least important criterion Fm subject to the schedule being optimal with respect to the more important criterion Fro_ l which is minimized on the set of schedules minimizing criterion Fro-2 and so on. The most important criterion is FI, which is minimized on the set of all feasible schedules. An approach to solve this multicriterion problem in polynomial time is presented if functions Fl ..... Fm have special properties. The total weighted completion time and the total weighted exponential time are the examples of functions Fi ..... Fro-1 and the maximum cost is an example of function Fm for which our approach can be applied.

Keywords: Single machine scheduling, multicriterion scheduling, group technology, maximum cost, total weighted completion time.

1. Introduction

The pr inciples o f group technology [8] are wide ly used in m a n y produc t ion

systems. In mechan ica l parts manufactur ing, for example , it is c o m m o n to adopt

these pr inciples , whereby the factory layout is such that the mach ines are g rouped

into cells. Each cell then produces several groups of jobs with s imilar product ion

*The research of the authors was partially supported by a KBN Grant No. 3 P 406 003 05, the Fundamental Research Fund of Belarus, Project N ~60-242, and the Deutsche Forschungsgemeinschaft, Project Schema, respectively. The paper was completed while the first author was visiting the University of Melbourne.

© J.C. Baltzer AG, Science Publishers

192 A. Janiak, M.Y. Kovalyo~; Single machine group scheduling

requirements. No machine set-ups are needed between two consecutively scheduled jobs from the same group, although a set-up is required between jobs of different groups. In group technology, all jobs from the same group are scheduled contiguously.

Group scheduling problems are studied by several authors (see, for example, [5, 14,9, 13, 10,2, 11,6]). All these papers deal with problems in which the quality of a solution is measured in terms of a single criterion. In practice, however, quality is a multidimensional notion. We study a single machine group scheduling problem in which the quality of a solution is measured by several criteria ordered by their relative importance.

The single machine group scheduling problem with ordered criteria is stated as follows. A set { 1, 2 ..... j ..... n} of jobs has to be scheduled on a single machine which can handle at most one job at a time. All jobs are available for processing at time zero. On the basis of group technology, jobs requiring similar operations are classified into a specific group and have to be processed contiguously. The set of groups is { 1, 2 ..... 1 ..... G }, where G < n. All the groups are nonintersecting and each job has a group it belongs to. Each job j requires for its processing an uninterrupted processing time pj > 0. For each group I, a set-up time tl > 0 independent of the group sequence is needed immediately before the jobs of this group are processed. It is assumed that there is no machine idle time between the jobs, so a schedule is completely characterized by the sequence of groups and the sequence of jobs for each group. Given a schedule, a completion time Cj is easily determined for each job j.

The quality of a schedule is measured by the cost functions F I ..... F,,, which are minimized on the sets Z o .. . . . Zm_ l, respectively. Here, Z0 is the set of all feasible schedules and Z1 ..... Z,~_l are recursively determined as follows: Zj is the set of schedules minimizing Fj on the set Zj_l. It is evident that Zm_l C_ Zm-2 C ... C Zo.

The objective is to minimize F m on the set Z,,_ I.

To indicate the importance of group scheduling with set-up times, we give three examples. The first example, which is present in [1 I], involves the production of coloured plastics on the same machine. Customer orders for many different colour shades may await production. These orders can be divided into major colour groups, such as reds, blues, etc., and within a colour group, say red, they may range from very light to dark red. Set-up times between colours from the same group are small, since it would be usual for production to graduate from lighter to darker shades. However, a large set-up is required when production switches from reds to blues, for example, since a thorough cleaning of the machine is necessary between the current colour (dark red) and the next colour (light blue). Because of these time-consuming and costly set-ups between different colour groups, machine efficiency is maximized by choosing a long run-length for each colour group, i.e. by adopting the principles of group technology.

A second example is the scheduling of computer systems. In this problem, a collection of tasks is to be processed on a computer, where each task requires specific utilities such as compilers, drivers, input-output programs, etc., to be resident in the

A. Janiak, M.Y Kovalyov, Single machine group scheduling 193

computer 's memory. If all necessary utilities are resident, then the task may start processing immediately; otherwise, a set-up time is incurred to bring the relevant utilities into memory. One of the strategies to solve this problem is to partition the tasks into families so as all tasks from the same family require the same set of utilities and then schedule all tasks of a family contiguously.

A final example described a situation which often occurs in mechanical parts of manufacturing. Consider the production of many different items on a multipurpose machining center. These items can be divided into several types on the basis of the machine tools to be required. A set-up time is needed to re-tool the machine center whenever there is a switch from the processing of an item of one type to an item of another type. In this situation, it is common practice to schedule all items of the same type jointly.

Let us assume that the quality of a solution in the above examples is measured by work-in-process inventories, noted by Conway et al. [3] as one of the most important scheduling criteria. Also, suppose that many jobs have similar or equal processing requirements, which is natural for mass production. In this situation, one can expect many different optimal solutions and there is an opportunity to choose among them a solution performing well on another criterion, for example, observance of due dates. In our paper, we present some techniques to realize this opportunity.

Adopting the notation of Graham et al. [4], we denote our problem by 1/GTI(FI ..... Fro), where acronym GT indicates that group technology constraints have been specified.

The remainder of the paper is organized as follows. In the next section, we establish properties of functions F l ..... Fm- 1 which allow us to reformulate in O(mn 2) time our multicriterion problem as a single criterion group scheduling problem IlGT, prec/F m of minimizing Fm subject to specifically determined precedence constraints. In the following section, we show that the total weighted completion time Y.w~l)ci and the total weighted exponential time ~w~l)exp(aCi) are examples of functions FI, l = 1 ..... m - 1, which have the above properties. We generalize Lawler's [7] O(n z) algorithm presented for the problem ltprec/fmax to solve the problem I/GT, prec/fmax in O( Gn 2) time. Thus, the problem 1/GT/(F l . . . . . Fro) where F t E { ~,w~l) ci, Y.w!/)exp(ctC/) }, l = 1 ..... m - 1 and Fm =fmax is solved in O((m + G)n 2) time. Some concluding remarks are presented in the last section.

. Formulating the multicriterion problem as a single criterion problem with precedence constraints

In this section, we define properties of functions F 1 ..... Fm-l which allow us to formulate our multicriterion problem 1/GTI(FI ..... Fro) as a single criterion problem I/GT, preclFm) of minimizing Fm subject to specifically determined precedence constraints.

194 A. Janiak, M.Y Kovalyov, Single machine group scheduling

We f'n-st determine the job interchange property for function Ft, l = 1 ..... m - 1.

DEFINITION I

The job interchange property is satisfied for function Ft if there are values at(i), i = 1 . . . . . n such that for any pair of jobs i, k from the same group,

at(i) < at(k) implies Ft(S) < Ft(S') (1)

for any pair of schedules S, S" from Z t_l, where S is a schedule in which job i immediately precedes job k and S ' is obtained from S by swapping i and k.

It follows from definition 1 that if function F t has the job interchange property, then for any pair of jobs i, k from the same group, at(i ) = al(k ) if and only if FI(S ) = Ft(S') for any pair of schedules S, S ' mentioned in this definition.

Assume that functions F 1 ..... Fro_ l possess the job interchange property. We now determine a precedence relation ~ on the set of jobs for each group.

DEFINITION 2

For any pair of jobs i, k from the same group, i ---> k if and only if al(i ) < al(k), where l E {1 ..... m - 1 } is the smallest index for which at(i ) <a l (k ) or at( i )> at(k) is satisfied.

We note that if the above index l exists, then ar(i) = ar(k) for r = 1,. . . , l - 1. Otherwise, ar(i) = ar(k) for r = 1 ..... m - 1.

It is easy to see that precedence relation ----> can be determined in O(mn 2) time for all groups if values at(i), i = 1 ..... n, l = 1 ..... m - 1 are calculated in advance.

We say that a schedule is feasible with respect to the relation ---> if, for any pair of jobs i, k form i ---> k, it follows that i precedes k in this schedule.

The following theorem holds.

THEOREM 1

If the sequence of groups is fixed, then the set Z m_ I coincides with the set of schedules feasible with respect to the relation --->.

Proof

We first show that every schedule from Z m_ 1 is feasible with respect to --->. Assume that there exists a schedule S EZra_ l in which there are two jobs i and k of the same group such that i immediately precedes k in this schedule but k ---> i, i.e. at(i ) > at(k), where l ~ { 1 ..... m - 1 } is the smallest index for which at(i) < al(k) or at(i ) > at(k) is satisfied. It follows from the definition of the relation k ---> i and from

A. Janiak, M.Y. Kovalyov, Single machine group scheduling 195

the job interchange properties of functions Fl ..... F t that swapping i and k decreases the value of Ft and does not change values of FI ..... Ft-l. Hence, schedule S is not optimal for the problem of minimizing Ft on Zt_ l, i.e. S ~ Zt. Since Zm_ 1 C Zt, we have S ~ Zm-1. This contradiction shows that our assumption k---> i is not correct. Repetition of this interchange argument shows that every schedule from Zm_ 1 is feasible with respect to the relation --->.

Conversely, if a schedule is feasible with respect to --->, then the same argumentation shows that there is no interchange of jobs which decreases the value of function Ft for a certain l E { 1 ..... m - 1 } and does not increase the value of function Fr for a certain r < 1.

Thus, every schedule which is feasible with respect to the relation ----> is from the set Zm-l and vice versa, as required. []

Theorem 1 shows that if functions FI ..... Fm-l possess the job interchange property, then we can limit ourselves to schedules which are feasible with respect to

the relation --->. We now define the group interchange property for function Fl, l = 1 ..... m - 1.

Assume that all schedules under consideration are feasible with respect to the

relation --->.

DEFINITION 3

The group interchange property is satisfied for function F l if there are values At(l), I = 1 .... , G such that for any pair I, K of groups,

At(l) < At(K) implies Ft(S) < FI(S') (2)

for any pair of schedules S, S ' from Zt-1, where S is a schedule in which group I immediately precedes group K and S' is obtained from S by swapping I and K.

Clearly, if function Ft has the group interchange property, then for any pair of groups I, K, At(l) = Al(K) if and only if Ft(S) = F I ( S ' ) for any pair of schedules S, S" mentioned in definition 3.

Assume that functions Fl ..... Fm-l possess the group interchange property. A precedence relation =o on the set of groups is determined as follows.

DEFINITION 4

For any pair of groups I, K, I ~ K if and only if At(I)< At(K), where l E {1 ..... m - 1 } is the smallest index for which At(l)<At(K) or At(l)>AI(K) is

satisfied.

If index l exists, then At(l) = Ar(K) for r = 1 ..... l - 1. Otherwise, Ar(l) = Ar(K) for r = 1 ..... m - 1. Precedence relation ~ can be determined in O(mG 2) time if values At(l), I = 1 ..... G, l = 1 ..... m - 1 are calculated in advance.

196 A. Janiak, M.Y Kovalyov, Single machine group scheduling

We say that a schedule is feasible with respect to the relation ~ if, for any pair of groups I, K, from I ~ K it follows that I precedes K in this schedule.

We now establish the main result of this section.

THEOREM 2

The set Zm_l coincides with the set of schedules feasible with respect to the relations ---> and =~.

Proo f

Assume that the sequence of jobs in each group is feasible with respect to the relation ---> and fixed. To prove the theorem, it is sufficient to show that, in this case, the set Zm-1 coincides with the set of schedules feasible with respect to the relation ~ . To prove the latter statement, we can completely repeat the proof of theorem 1, inter- changing notions of jobs and groups and replacing relation ~ by the relation ~ . []

Let 1/GT, ---~, ~ / F m denote the single machine group scheduling problem of minimizing F,,, subject to precedence constraints imposed by the relations ---> and =~. Theorem 2 shows that the problems 1/GTI(FI . . . . . F m) and 1/GT, --->, ~ ~Fro are equivalent if every function FI . . . . . Fm_ 1 possesses the job and group interchange properties. The procedure of formulating the problem 1/GT,--->, ~ / F m requires O(mn 2) time.

3. Polynomially solvable cases

In this section, we study special cases of the problem 1/GT/(FI ..... Fro) which are solved in polynomial time.

The criteria considered in this section are the following:

• the total weighted completion time F l = ~w~i)cj;

• the total weighted exponential completion time FI = ~,w}i)exp(aCj);

• the maximum cost Fm~x = max{~(Cj)}.

Here, w} i) > 0 is a weight denoting the relative importance of job j in criterion l, a ~ 0 is a certain coefficient, Jj(t) is a nondecreasing cost function. Each summation and maximum is over all jobs j.

We first show that the total weighted completion time and the total weighted exponential completion time criteria possess the job and group interchange properties.

THEOREM 3

The job and group interchange properties are satisfied for the total weighted completion time criterion Ft = ]~w}i)cj if values at(i), i = 1 . . . . . n, and At( l) , I = 1 . . . . . G, are defined as follows:

A. Janiak, M.Y Kovalyov, Single machine group scheduling 197

al( i) = pi/w~ i),

At(I) = PtlWCt O,

i = 1 ..... n,

I = 1 . . . . . G,

where P! = t! + ]~j~: pj and W~t)= Y.j~twJ i).

Proof The job interchange property is easily proved using the argumentation of

Smith [12] for the problem of minimizing the total weighted completion time. The group interchange property is similarly proved as follows.

Consider an arbitrary pair of g r o u p s / , K and an arbitrary pair of schedules S, S ' such that I immediately precedes K in S and S ' is obtained from S by swapping I and K. Assume At(l) < At(K ), i.e. Pt/WCt t) < Pr/WC~ ). The latter inequality is equivalent to PIW~ ) < PrW} t). It is easy to see that Fl(S) - Ft(S') = PtW~ ~- PKW~ ). We conclude that At(l) < At(K) is equivalent to Ft(S) < FI(S'). Thus, the group interchange property is also satisfied for the total weighted completion time criterion. []

We now prove a similar theorem for the total weighted exponential completion time criterion.

T H E O R E M 4

The job and group interchange properties are satisfied for the total weighted exponential completion time criterion F t = ~,w~t)exp(aCj) if values al(i ), i = 1 ... . . n, and At(I ), I = 1 ..... G, are defined as follows:

at(i) = w~t)exp(trpi)/(1 - exp(api)) , i = 1 ..... n,

At(1 ) = ~t0/(1 - exp(aPt)) , I = 1 ..... G,

where Vt(t)=Y~=lw~J)exp(a(tt+ ~.J=lPio)) and (il ..... it) is the sequence of jobs of group I ordered so that at(ij) < al(ij+l), j = 1 ..... r - 1 .

Proof Consider an arbitrary pair of jobs i, k from the same group and an arbitrary

pair of schedules S, S ' such that i immediately precedes k in S and S" is obtained from S by swapping i and k. Assume that at( i )< at(k), i.e.

w(t) exp(ap i )/(1 - exp(ap i ) ) < w(k l) exp(apk) / (1 - exp(apk )). i

The latter inequality is equivalent to

w~ t) exp(api ) . (1 - exp(apk )) < w~ l) exp(Ctpk)" (1 -- exp(ap i )).

198 A. Janiak, M.Y. Kovalyov, Single machine group scheduling

Since

F/ (S ) - F / ( S ' ) = e x p ( a P ) (w~ l) exp( txpi ) (1 - exp(tXpk ))

- w(k 0 exp(ap~ ) (1 - exp(ap / ) ) ) ,

where P is the sum of processing times for all the jobs preceding i and k, then Ft(S) < Ft(S" ) follows at(i) < at(k). Thus, the job interchange property is satisfied.

Assume that relation ---) is imposed by the values at(i), i = 1 ..... n. It is easy to see that value At(l) is the same for any sequence of jobs of group I feasible with respect to the relation --->, i.e. it depends only on the index of the group.

Consider an arbitrary pair of g r o u p s / , K and an arbitrary pair of schedules S, S" feasible with respect to the relation ---> such that I immediately precedes K in S and S" is ob ta ined f rom S by swapping I and K. We have F t ( S ) - Ft(S') = exp(otP)(~0(1 - exp(tzPK) - V~0(1 - exp(aPl)). Assume that At(l) < At(K). As above, the latter inequality is equivalent to Ft(S) < Ft(S'), i.e. the group interchange property is also satisfied. []

We now describe polynomially solvable cases of the problem 1/GTI(FI ..... Fro). The simplest polynomially solvable case arises when all m functions F 1 .. . . . Fm

have the job and group interchange propert ies , for example , Ft E {~w)l)c j , ~,w(l)exp(aCj)}, l = 1,.. . ,m. In this case, relations --) and ~ can be determined in O(mn 2) time so that any schedule feasible with respect to these relations is optimal for the problem 1/GT/(F1 ..... Fro).

We now assume that functions F1 .... , Fro- 1 possess the job and group interchange properties and Fm =fmax. In this case, the problem 1/GT/(FI ..... Fro-l, fmax) is equivalent to the problem 1/GT, ---), ~/fmax, where relations ---> and ~ are defined as shown in the previous section. We present an algorithm which solves 1/GT, ---), ~/ fm~x in O(Gn 2) time. Our algorithm is a straightforward modification of Lawler's [7] algorithm for the problem of minimizing maximum cost subject to arbitrary precedence constraints. This Modified Lawler Algorithm is based on the following observation. Let M denote the set of groups that have no successors with respect to the relation ~ and let U denote the sum of processing times of all jobs plus the sum of set-up times of all groups. Consider the problem of scheduling jobs of some group I to minimize fmax subject to the relation ---> and the condition that all jobs are available for processing at t ime U - (tl + ~,j~ tPj). Let z~ l denote an optimal sequence of jobs for this single group problem and let Ft(U) denote the corresponding optimal objective value. If Fr(U) = min{F~(U)lI ~ M } , then there exists an optimal schedule for 1/GT, ---), ~ / fmax in which jobs of group K are sequenced last in the order imposed by zr r . The formal description of the algorithm is given below.

MODIFIED LAWLER ALGORITHM

Step I. Compute U = •tG=1(tt+ ~j~IPj)" Set R = {I .... . G}.

A. Janiak, M.Y. Kovalyov, Single machine group scheduling 199

Step 2.

Step 3.

Step 4.

Determine the set M containing the groups that have no successors in R with respect to =*.

Solve the single group problem and find ~r t and Ft(U) for all I ~ M. Choose K E M that has minimal Fr(U) value, settling ties arbitrarily; jobs of group K are processed from time U - ( t r + ~,j~rPj) to time U in ~r r order.

Set U = U - (t r + ~,jz K Pj), R = R - {K}. IfR # 0 , then go to step 2; otherwise, stop.

In the above algorithm, the procedure of finding n: t and FI(U) is not presented. We now describe an algorithm of finding 7r t and Ft(U) which is in fact the Lawler's algorithm [7]. This algorithm is as follows.

LAWLER'S ALGORITHM

Step 1. Step 2.

Step 3.

Step 4.

Set T = U and Y= { j l j ~ I } .

Determine set L containing the jobs that have no successors in Y with respect to --->.

Choose j ~ L that has minimal 3~(T) value, settling ties arbitrarily; job j is processed from time T - p j to time T.

Set T = T - p j , Y= Y - { j } . If Y ~ O , then go to step 2; otherwise, stop.

The following theorem holds.

THEOREM 5

The Modified Lawler Algorithm solves 1/GT,-~, =*If max in O(Gn 2) time.

Proof We adopt the proof presented by Baker et al. [1] for Lawler's algorithm [7].

Let R = { 1,...,G} be the set of all groups, and let M C_ R be the set of groups without successors. For any subset X C R, let P(X)= Y4~x(tt+ ~j~tPj) and let fm~(X) denote the cost of an optimal schedule for the problem in which X is the set of all groups. Clearly, f~ax(R) satisfies the following inequalities:

fmax(R) >- min{Ft(P(R))l I E M},

fmax(R) > f~mx(R- {I} forall I ~ R.

Now let K ~ M be such a group that

FK(P(R)) = min { Ft(P(R))II E M}.

200 A. Janiak, M.Y. Kovalyov, Single machine group scheduling

We have

* R * fmax( ) > max{FK(P(R)),fmax(R - {K})}.

The right-hand side of this inequality is precisely the cost of an optimal schedule subject to the condition that jobs of group K are processed last in lrtc order. Hence, there exists an optimal schedule in which jobs of group K are processed last in 7r K order. Since K is the group that is selected by the Modified Lawler Algorithm, repetition of the above procedure shows that the Modified Lawler Algorithm solves the problem 1/GT, --->, ~/fmax.

The same argumentation shows that Lawler's Algorithm solves the single group problem. Thus, implementation of this algorithm to find rc t and Ft(U) in step 3 of the Modified Lawler Algorithm is correct.

We now establish the time complexity of the Modified Lawler Algorithm. Steps 1, 2 and 4 require O(n + G 2) time. Step 3 requires O(Gn 2) time. Thus, the overall time complexity is O(Gn2). []

Theorems 2, 3, 4 and 5 show that our multicriterion problem 1/GT/(Fl ..... Fro) is solved in O((m + G)n 2) time if F z is the total weighted completion time or the total weighted exponential completion time criterion for l = I ..... m - 1 and Fm is the maximum cost criterion.

4. C o n c l u d i n g r e m a r k s

The single machine group scheduling problem is studied. The quality of the solution is measured by several lexicographically ordered cost functions. An approach to solve this problem is presented if the cost functions fulfill some job and group interchange properties. The main idea is to formulate and solve an equivalent single criterion problem with specific precedence constraints. The results of the paper can be used in the practical situations of production scheduling where the principles of group technology are imposed and the quality of a solution is estimated by many lexicographically structured criteria.

Further research could be undertaken to examine a more general problem where groups are allowed to be partitioned into batches of jobs and set-up times are dependent both on the current group and the group to be processed next. Consider a production line where items of different types are processed. If the group technology requirement is imposed, then all items of the same type are delivered to the customer together. However, this is not always the best strategy. In many cases, the customer's service is improved if some part of his order is satisfied immediately and the remaining part is fulfilled at some later time. Batch decisions may provide such an opportunity. As for the set-up times, it is evidently a more realistic model if they are sequence dependent.

A. Janiak, M.Y. Kovalyov, Single machine group scheduling 201

Acknowledgement

The authors would like to thank the referees for their helpful suggestions and comments on an earlier version of this paper.

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