Scheduling independent tasks with multiple modes - ScienceDirect

DISCRETE APPLIED MATHEMATICS Discrete

Scheduling

Applied

Mathematics

independent

62 (1995) 35-50

tasks with multiple modes

L. Biancoavbv*, P. Dell’Olmob, M. Grazia Speranzac ‘IASI bDepartment ‘Department

CNR, Vi&

Manzoni 30, I-00185 Rome, Italy University of Rome “Tor Vergata”, Viale Manzoni, 30-I-00185 Rome. Italy Universify of Brescia, Corso Mameli 27, I-25122 Brescia. Italy

of Electronic Engineering,

of Quantitative Methods, Received

21 March

1992; revised 5 January

1994

Abstract In this paper a nonpreemptive scheduling problem is studied in which a set of independent tasks must be processed on a set of discrete and renewable resources. Each resource can be used at any time by a single task at most. Each task can be carried out in several given alternative modes, that is, by using different resource sets and with different processing times. The objective of the problem is to determine a mode and a starting time for each task in such a way that the makespan is minimized. The problem instances are represented by means of a graph model. The complexity of the problem is studied and several particular cases are identified which remain NP-hard. Upper and lower bounds on the optimal value of the objective function are identified which correspond to some dimensional parameters of the graph. A heuristic iterative solution procedure is sketched and a numerical example is presented.

1. Introduction

In this paper we consider a scheduling problem in which a set of discrete and renewable resources is always available over time and a set of tasks has to be carried out using these resources. At any time each resource can be used by a single task at most. Each task can be carried out in one among a set of predefined alternative modes, where each mode corresponds to a set of resources and a processing time. No preemption is allowed. We assume that tasks are independent, that is, no precedence constraints are allowed among them. The problem consists in identifying a mode for each task and a starting time for its processing in such a way that the makespan is minimized. Real examples of this problem arise whenever a group of tasks or activities have to be scheduled over time with the objective of completing them as soon as possible. The

*Corresponding

author.

0166-218X/95/$09.50 0 1995-Elsevier SSDI 0166-218X(95)00003-8

Science B.V. All rights reserved

36

L. Bianco et al. / Discrete Applied Mathematics

makespan

may be often interpreted

the activities.

Among the potential

of project scheduling fact that an activity

as a measure application

62 (1995) 35-50

of the cost of the resources

areas, of particular

and the area of flexible manufacturing can be carried

out in several alternative

interest

used by

are the area

systems, In both cases, the modes is often a crucial

characteristic of the problem. For instance, in the case of project scheduling, happen that an excavation can be carried out either by using 1 big excavator,

it may 1 truck

and 3 people in 16 hours or, alternatively, using 2 small excavators, 2 trucks and 5 people in 10 hours. In most cases, the processing time of an activity is shorter if it is carried out by using the most powerful resources or a large number of equivalent resources. Obviously, different activities which require the same resource must be sequenced and cause an increase of the completion time. In the case of flexible manufacturing systems, jobs can be routed on the machines in several alternative ways, as each machine is equipped with a set of tools which make it flexible. This means that each task of a job can be carried out in several modes. In general, the problem is to decide which is the mode and starting time for each task which allow the tasks to be carried out in the shortest possible time, or, in other words, which allow the resources be released as soon as possible. Two main characteristics make this problem different from classical machine scheduling problems. The first is that each task simultaneously requires a set of dedicated resources and the second is that each task can be processed in one among a set of alternative processing modes. The interest in problems in which each task requires a set of dedicated resources, with few exceptions, is more recent. In the pioneering paper by Bozoki and Richard [7] a solution algorithm is proposed for the problem in which a single mode is defined for each task and each task requires a set of dedicated resources. The same problem has been analyzed in more detail and different solution algorithms are proposed in the later papers by Cangalovic and Schreuder [S] and Bianco et al. [S]. The particular case in which only three resources are available has been studied in [6]. Scheduling problems in which each activity requires a set of resources and activities can be carried out in multiple modes are dealt within the project scheduling area (see, for instance, [14-161). However, in those papers the interest

was to capture

most characteristics

of the real problem

and, as a result, the

models are very complex and no structural property of the models is identified. The aim of this paper is to investigate the basic characteristics and properties of scheduling problems in which the existence of alternative execution modes is a relevant feature. In particular, the complexity of particular cases of the problem is studied and new and known complexity results are summarized. The problem of obtaining bounds on the objective function is faced and some lower and upper bounds are obtained. Finally, a heuristic solution procedure which makes use of the bounds is sketched. In Section 2 the problem is defined and a representation of the problem instances by means of a graph model is given. The problem is NP-hard and several particular cases of it remain NP-hard, as it is shown in Section 3. In Section 4 lower and upper bounds on the optimal value of the objective function are given. The complexity of the

L. Bianco et al. / Discrete Applied Mathematics

problem

and the presence

task sequencing, together

of different decisions,

calls for a heuristic

with a numerical

iterative

62 (1995) 35-50

namely

approach

37

the mode assignment which is sketched

and the

in Section

5

example.

2. Problem definition and graph representation AsetB={R,,...,R,} of m discrete and renewable resources is available. People, trucks and machines are some examples of the types of resources we consider. Each resource R,, E B is always available over time. A set of n tasks F = { Tr , . . . , T,,} has to be carried out. Each task T E 9 can be undertaken in several modes, each mode requiring a set of resources for a given processing time. We consider here each mode different from any other mode, even if two modes of different tasks are associated with the same resource set and require the same processing time. Let J&‘(T) be the set of modes for task z and mi = 1A(T)I the number of modes of task T. Moreover, let JZ = UTIEF JzZ(~) be the set of all modes, with 1~2’ = Cimi. The set of resources required by mode Mj E J%‘(T) will be denoted by I c 92 and the

possible

processing time by pj = p(Mj). A single mode has to be assigned to each task and a starting time has to be defined for each task in such a way that each resource is used by a single task at a time. The processing of a task cannot be interrupted and restarted later. The objective of the problem, which will be referred to as the multiple modes scheduling problem (MMSP), is the minimization of the time within which all the tasks are completed. More formally, let us define by

an “assignment” function to d(Tj) and by

which assigns to each task T a mode Mj = d( Ti) belonging

Y:9--+% a “sequencing”

function

which assigns to each task T a starting

denote

by ci = Y(T,) + p(Oe(T,))

Se(T).

Then the MMSP z* = miy”, c(d,

9)

d(T)

E A(T)

Y(X)

+ p(d(Ti))

the completion

time ti = 9’(T).

time of task T, carried

We

out in mode

can be stated as = mt;, VT

my

Y(Z)

(mode constraints),

< ,4p(Ti,) or Y(T,)

if B(&(Ti))n9(&(T,))

+ p(d(T,)),

#

+ p(d(Tr))

d Y(T)

0 (resource constraints),

Y(T)

2 0

VTi.

In the sequel we refer to the objective function z as makespan and we will denote functions for which this value is minimum as x2* and Y*. A suitable representation

the of

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L. Bianco et al. / Discrete Applied Mathematics

62 (1995) 35-50

MMSP instances can be obtained as follows. Let us consider a mode graph $9 = (V, E), where the vertex set T/ is partitioned as V = Vr, . . . , V,,. The set of vertices Vi is identified with the set of modes &‘(Ti) of task Ti. An edge (j, k) E E, with j E l$ and k E Vi,, exists if W(Mj)nB(Mk) # 8, that is, if modes of different tasks share some resource. No edge exists between vertices which belong to the same class l$, that is, between vertices associated to the same task. The weight Pj is associated to each vertex j E l$. In other words the processing time of mode Mj of task Ti is associated to the corresponding vertex. In the following we will denote by r(j) the set of vertices which are adjacent to vertex j. In Fig. 1 a MMSP instance is given together with the corresponding mode graph. We now give on the graph representation the definition of the graph structures corresponding to the functions d and 9. 1. A set I/’ E V is a partition covering if it is a set of vertices such that for each Vi there exists one and only one j E V’.

Definition

Thus, a partition covering which induces a subgraph 3’ of 9 is equivalent to a function JZ! in the MMSP definition. In Fig. 2 a partition covering is shown. Any partition covering and any acyclic orientation of the resulting $9’ identify a solution of the MMSP. In particular, the function 9’ associated to an acyclic orientation of 3’ can be found assigning starting time ti = Y(F) = 0 to each task T such that j E 6 is a source node in $9’and deriving recursively the starting time of all other tasks, by means of the algorithm used for PERT networks. Vice versa, any solution of the MMSP identifies an assignment of a single mode to each task which is

(2)

T3

Tz

Tl

a Ml

(2)

M2

(3)

M4

(2) (2)

MS

M3

Fig. 1. A problem

Tl

T2

Fig. 2. A partition

instance.

5

covering.

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62 (1995) 35-50

39

a partition covering and a partial order of the tasks which corresponds to an acyclic orientation of Y’. The length of the maximum path is the makespan of the solution of the MMSP. The MMSP can now be defined as the problem of finding a partition covering and an acyclic orientation of the induced subgraph 3’ such that its maximum oriented path has minimum length. Given an assignment function d, the problem of finding the optimal sequencing function Y can also be formulated as the interval coloring problem, that is, the problem of finding the interval chromatic number on the vertex weighted graph [lo].

3. Computational

complexity

In the simplest case, that is, if a single mode is given for each task, the MMSP problem is reduced to the problem of finding the optimal function Y and thus the complexity results for the interval coloring problem hold. The interval coloring problem is known to be NP-hard even if the weights are restricted to 1 and 2 and the graph is an interval graph, and solvable in polynomial time on superperfect graphs and in particular on comparability graphs [lo]. Moreover, the same problem remains NP-hard if only three resources are given, while it becomes solvable in linear time in the case of two resources [6]. The complexity of some other particular cases of the MMSP is now investigated. Theorem 1. Zfl~(Mj)l

= 1, VMj E A’(z),

VT, the MMSP

is NP-hard.

Proof. The MMSP has, in this case, the R /IC,,, scheduling problem, that is, the problem of scheduling tasks on unrelated machines with the objective of minimizing the makespan. This corresponds to the case in which for each task Ti and for each resource Rh there exists a mode Mj E A(q) such that BT(Mj) = {R,,}. As the R 11C,,, is NP-hard (see [9]), the proof is completed. 17 Theorem 2. Zf IW(Mj)I = 1, VMj E A’(T), MMSP is NP-hard.

VT,

and pj = pi, VMj E A(T),

VT,

the

Proof. In this case the P (1C,,,

scheduling problem, that is, the problem of scheduling tasks on identical machines with the objective of minimizing the makespan, is a particular case. The proof is analogous to the proof of Theorem 1. The P 11C,,, is known to be NP-hard (see [9]). 0 We already said that in the case I A( T)l = 1 Vi, and the mode graph is a comparability graph, the MMSP problem can be solved in polynomial time. Now, we investigate the complexity of the problem in a more general case, maintaining the assumption that the mode graph is a comparability graph.

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40

Theorem 3.

62 (1995) 35-50

Zfthe mode graph is a comparability graph, the MMSP is NP-hard.

Proof. Consider the graph representation of the P I(C,,, problem which is, as above mentioned, an NP-hard particular case of the MMSP ( see Fig. 3). Each mode of a task shares exactly one resource (machine) with one mode of all other tasks. The mode graph results to be the “union” of disjoint complete graphs and thus is a comparability graph. 0

Before giving a result on a polynomial case, we recall the definition of autonomous set. 2. A set fi is an autonomous set if the following condition holds: if there exists (j, k), j E Vi, k E Vi, then there exists (j’, k) Vj’ E Vi. Definition

Theorem 4. If6 is an autonomous set, Vi, and the mode graph ‘29is a comparability graph, then the MMSP can be solved in polynomial time.

Tl R(MJ

= machine 1

R(M~ = machine 2

6s e

R(kf 3) = machine 3

R(kf4) = machine 4 Fig. 3. The mode graph

Table 1 Complexity

classification

= = = =

1, pj 1, 1, m 1, pj A(T)I = 1, Y a( =1 a( = 1, P comparability 9 comparability pj

pj

= 1 or 2, 9 interval graph 1, IW(Mj)l = 2 = 3 = 1, m = 5 comparability graph =

=

pi, VMj E %4!(7J graph graph,

to the P 11C,,,

problem.

of MMSP

Problem A’(K)1 A(TJl A(F)I A(

associated

K autonomous

set

Complexity

Reference

NP-hard NP-hard NP-hard P P NP-hard NP-hard NP-hard P

Interval coloring [lo] Edge coloring [ 111 3-partition [6]

ISI [lOI R 11C,,

(this paper)

P IICm,, (this paper) p IIC,., (this paper) (This paper)

L. Bianco et al. / Discrete AppliedMathematics62 (1995) 35-50

41

Proof. As vi is an autonomous

set, then for each 6, r(j) = r(j’) with j, j’ E vi. Having each mode of a task the same set of conflicting modes, the optimal assignment function d* is given by d* = Mj with pj = minklM,,,~/i(T,~pk. For the hereditary property, the graph induced by d* is a comparability graph and thus an optimal Y* can be found in polynomial time. 0 It is worth observing that, for general graphs, the MMSP remains NP-hard even if all K are autonomous sets. Indeed, all problem instances in which I&‘(T)1 = 1, VT, are such that all Vi are autonomous sets. In Table 1 the known complexity results on the MMSP are summarized. For the NP-hard cases the problem used for the proof is cited.

4. Lower and upper bounds The problem formulation given in Section 2 emphasizes the interaction of two problems in the MMSP, namely an assignment problem and a sequencing problem. As we already mentioned, any function LZZcan be thoroughly evaluated only by finding the best sequencing function Y. This last problem was shown to be NP-hard, so we investigate the possibility of evaluating & by means of some approximation of the solution, deriving lower and upper bounds. 4.1. Lower bounds Given an assignment function d, we denote by R,,,(d) the maximum resource usage determined by @‘,i.e., the maximum total time during which a resource is used by the modes d(Ti), and by IV,,,,,(&) the weight of the maximum weighted clique of the graph induced by ZZZ on the mode graph 9. Moreover, let z*(d) = min(,,,, c(d, 9). As tasks whose modes define a clique, i.e., are mutually in conflict, must be linearly sequenced, it follows that

Moreover, as the tasks whose modes share the same resource define a clique, it also follows that

The calculation of R,,,(d) of calculating W,,,(Y)

can be performed in O(nm). On the contrary, the problem is NP-hard [9]. Polynomial algorithms exist only for

Fig. 4. The graph

P3.

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62 (1995) 35-50

comparability graphs and (graphs admitting an orientation with no induced subgraphs isomorphic to the graph P3 shown in Fig. 4) other classes of perfect graphs (see [l, 3]), while more recent exact and heuristic algorithms have been presented in [2]. Taking the minimum on the assignment functions d of the three terms of the above inequality, we can derive a similar inequality for the optimal solution of the MMSP, observing that z* = min, min, c(&, Y), thus m: &,,(a

) < min W,,,(R~) < z*. .v!

Unfortunately, calculating any of these lower bounds is an NP-hard problem. Calculating min,R,,,(d) is an NP-hard problem, as it contains the multiprocessor scheduling problem, that is, the problem of scheduling tasks on parallel machines with the objective of minimizing the makespan, as a particular case. The multiprocessor problem, which is NP-hard (see [12]), is obtained in the case each activity can be processed by any single resource (machine) and with the same processing time. Observe that calculating W,,,(d) on a heuristically determined assignment function XZ’does not in general guarantee a lower bound on z*. In some particular cases the above lower bounds provide the value of the optimal solution of the MMSP. Proposition

1. IflW(Mj)I

= 1, VMj E A(T,),

VT, th

Proof. If each mode requires one resource only, tasks using the same resource are linearly sequenced and tasks using different resources can be processed in parallel. Hence, given the assignment function, the makespan coincides with the maximum resource usage. 0 Proposition

2. If ‘3 is a comparability graph then

rn> IV,,,(&) = z*. Proof. If an assignment

function induces on $ a comparability graph, it holds W,,,(d) = z*. Then, as for the hereditary property any induced subgraph of 9 is a comparability graph, taking the minimum on the assignment functions of the above equality, the result follows. 0 We now establish a different lower bound on the optimal value of the objective function of the problem. We say that two sets vi and vi. are satured if (j, k) E E, Vj E vi and Vk E I$. Let 97 be a maximal clique of 9, and w(w) the weight of %?. - Theorem 5. Consider the graph $7 = (V, E) with i E V identified with the task T and weight of vertex i, pi = minjEvip(Mj), and (i, i’) E E if and only if v and 6, are satured

L. Bianco et al. 1 Discrete Applied Mathematics 62 (1995) 35-50

43

in 9, Then

Proof. Obviously, a lower bound on z* is obtained relaxing the problem, that is, for instance eliminating edges in Y. If all edges are eliminated between any pair of vertices Vi and Vj which are not satured, then all nodes of a set Vihave the same conflicts with the nodes of any set I’j. In this condition it is possible to select the optimal mode of each task in the relaxed problem as the mode with minimum weight. The so obtained graph is B and the maximum weighted clique of it is a lower bound on the optimal solution of the relaxed problem and thus on z*. 0

Although, in general, the calculation of W,,,,,(S) is an NP-hard problem, a polynomial lower bound can be obtained by heuristically finding a clique of large weight. 4.2. Upper bounds For a vertex x of a graph 3’ = (I”, E’) induced by an assignment function d we denote by S(x) the subgraph induced by x and its neighborhood T(x) and with P(S(x)) = PX+ Cypr(& the weight of S(x). Furthermore, given an acyclic orientation of 3’ and a vertex x of %“,we say that a path p of 3’ is contained in S(x) if every vertex of p is also a vertex of S(x). We denote by S,,,(&‘) = maxXGy,P(S(x)). Theorem 6. Given an assignment function

d,

if pj = 1 Lfj, then

Proof. In the case of unit processing times, it is easy to see that any optimal solution to the vertex coloring problem of the graph induced by & corresponds to an optimal schedule of the tasks. In other words,

z*w1 = Y(d), where y(d) is the chromatic number of the graph induced by d. As any graph of maximum degree h can be colored with h + 1 colors (see [4]) and in the case of unit processing times it is: S,,,(d)

= h + 1,

it follows z*(d)

G Srn,,(~).

cl

The upper bound of Theorem 6 holds for general values of pj if a condition is assumed on the structure of the graph. We need to introduce the following definition.

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62 (1995) 35-50

3. A pseudo-transitive orientation of a graph $9’is an acyclic orientation which contains no induced subgraph isomorphic to the graph P4 and to the graph P6. shown in Fig. 5. Definition

Lemma 1. Given any acyclic orientation of a graph Y, if there exists a path p which is not contained in any S(x), x E cY, then the oriented graph contains a P4 or a Pi (see Fig. 5). Proof. Let us denote by ‘C!%’ = (I/‘, E’) the oriented graph. Let al, . . . , a, be the vertices

of the path p. Let us suppose C!?does not contain any P4 or a PL. Let us suppose (aI, a3) $ E. In this case (az, ak) E E’, for k = 4, . . . ,s. If not so, al, a2, a3, ak would be a P4 or a Pi. If (a2, ak) E F’, for k = 4, . . ..s. S(az) would contain the whole path p. Thus, (al, a3) E E’. With analogous reasoning, we can show (al, ak) E E’, for k = 2, . . . , s. If so, S(a,) would contain the whole path p. Thus, g-’ must contain a P4 or a Pk. q Theorem 7. Given an assignment function -c4, let 9’ be the induced subgraph. If 3’ admits a pseudo-transitive orientation, then

Proof. Let us suppose that there exists an acyclic orientation of 3’ which neither contains a P4 nor a PA. Let x be a vertex for which S(x) has maximum weight in 9’. Let us suppose z* > S,,,(&). In this case there must exist a path p, whose weight is greater than S,,,(Y). Hence, p cannot be contained in S(x) and, in general, in any S(y), with y E 9’. Hence, due to Lemma 1, the path p must contain a P4 or a PL. q

It can be observed that if, for instance, the graph 9’ is a complete graph, the upper bound is tight. If in ‘3’ a node with degree n - 1 exists then the above upper bound obviously holds.

(a)

(b) Fig. 5. (a) The graph P4 and (b) the graph Pi.

L. Bianco et al. / Discrete Applied Mathematics

Corollary 1. If, f or any assignment function &, a pseudo-transitive orientation, then

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45

the induced subgraph Q’ admits

where Smin = min, S,,,(JZI). Proof. As the result of Theorem 7 can be applied to any assignment function d,

min z*(d) d

= rn> S,,,(d).

q

At the best of the authors’ knowledge, graphs which admit a pseudo-transitive orientation have not been explicitly studied. Only for some particular class of graphs it can be seen that the hypothesis of Corollary 1 is satisfied. Corollary 2.

If $9is a

comparability graph, then

Proof. For the hereditary property of comparability

graphs, any induced subgraph is a comparability graph. Moreover, as comparability graphs admit a transitive orientation, they obviously admit a pseudo-transitive orientation. 0 Unfortunately, not all graphs admit a pseudo-transitive orientation. The finite triangle free graphs with large chromatic number studied in [ 131 represent an example of graphs for which a pseudo-transitive orientation does not exist. While, given an assignment function d, the problem of finding S,,,(&‘) is easy, the problem of finding Smincan be shown to be NP-hard. Proposition

3. Given a mode graph ‘3, the problem ofjnding

Smin is NP-hard, even on

comparability graphs.

of the P 11 C,,, problem (see Fig. 3). As the mode graph results to be the “union” of disjoint complete graphs, the problem of finding Sminis equivalent to the problem of finding Wminwhich, in turn, is equivalent to solving the P 11C,,,. q Proof. Consider the graph representation

Observe that a heuristic procedure for calculating an approximate value of Sminin general does not guarantee an upper bound on z*. Obviously, any heuristic procedure for the solution of the MMSP produces an upper bound on z* and in particular the following result holds. Theorem 8. Given a collection of independent sets I 1, . _., I, of $3 such that U I,, contains a partition covering, then z* < ~~rnax~~~,,p~.

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62 (1995) 35-50

Proof. A schedule is found whose makespan is 1, maxj.l,, pj by processing in parallel

all tasks which belong to the independent stable set and sequencing the tasks which belong to different independent sets. 0

5. An iterative heuristic

In this section we present an heuristic solution procedure for the MMSP. Due to the nature of the functions d and 9, the objective function of the MMSP can be rewritten as min min c(&, 9). .& 9 The solution procedure is based upon the heuristic investigation of the space of the functions & and upon solving the sequencing problem for the identified assignment functions. A similar approach has been presented in [ 141 for a more general problem. This is obtained by means of an iterative procedure, whose scheme is shown in Fig. 6. After an assignment function has been identified on the basis of the procedure assignment applied on graph 9, a sequencing problem is solved by the procedure sequencing on the subgraph 3’ induced by it. Then, on the basis of the so-obtained solution, a variation of 9 is produced by vary and a new assignment function is possibly identified by applying the procedure assignment on the variated 3. The procedure iterates this scheme until a stopping condition is verified. The above procedure is a general framework for algorithms whose actual implementation depends upon the way in which the above procedures are implemented. Giving a detailed description of such procedures is out of the scope of this paper. In the sequel, we give the basic ideas on the structure of the procedures above mentioned.

procedure multi-mode(V1, . . . , V,, E,pl, . . . , ~1~1) begin Q” := G

con&ne:=true while continue begin assignment(input:&?;output:@) sequencing(input$‘;output:9+) vary(input:@‘$+;output:G”) if(stopping rules) then continue:=false end end Fig. 6. The scheme of the decomposition

procedure.

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62 (1995) 35-50

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5. I. Assignment

An algorithm should be identified which selects a promising assignment function. Obviously, it is not possible to define the concept of makespan in this phase in which the tasks are not being sequenced. Thus, the algorithm might reasonably tend to the minimization of a bound on the objective function. Recalling that Wmin has been shown to be a “good” lower bound on z*, a reasonable approach to the problem can be to identify an assignment function which induces a graph Y’ whose maximum weighted clique has minimum weight. Obviously, this selection criterion does not guarantee an optimal assignment function. For this reason, the algorithm should be fast. Thus, as finding Wmin is in general a NP-hard problem, the use of heuristic procedures should be investigated. 5.2. Sequencing After an assignment function has been identified, the sequencing problem has to be solved which consists in sequencing the tasks in such a way that the makespan is minimized. For the sake of clarity, we briefly summarize the most relevant features of such an algorithm. As already mentioned, a polynomial algorithm exists for the solution of the sequencing problem, in the case the graph ‘9’ is a comparability graph. In the case 9” is not a comparability graph, an algorithm for the optimal or approximate solution of the sequencing problem can be applied. Optimal solution algorithms are presented in [S, 7,8]. The schedule obtained by any solution algorithm for the sequencing problem determines an acyclic orientation of the graph 99’.We construct the transitive closure of the so-obtained oriented graph and denote it by ‘9’. 5.3. Graph variation The above algorithm gives an orientation of 9’. Note that if ‘9”is not a comparability graph, then the transitive closure 9’ of the oriented graph has some additional arcs with respect to 9’ which represent precedence relations between tasks which were not in conflict in the mode graph. This fact can be interpreted as follows: The assignment function selected in the previous step implicitly induces additional conflicts between tasks. This remark suggests a way for modifying the graph 9” in the procedure vary. Such a graph can be obtained by adding in 9” an arc between each pair of modes linked in the graph 9’. 5.4. Stopping rules If any of the following stopping conditions is verified the procedure of Fig. 6 ends: (1) the induced subgraph 9’ is a comparability graph;

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L. Bianco et al. / Discrete Applied Mathematics

(2) the assignment produces

the same solution

(3) the sequencing produces

a solution

the sequencing

with respect to 3’. Therefore,

problem

in two subsequent

worse than the previous

In the first case, as any acyclic orientation orientation,

62 (199.5) 35-50

produces

of a comparability

iterations; solution.

graph is a transitive

a graph %+ in which no arcs are added

the graph Y’ is not modified

and the procedure

must be

stopped.

5.5. An example In this section we show a simple example of how the iterative procedure works. Let us consider the problem instance with 5 tasks and 6 resources shown in Fig. 7 by means of its mode graph. The resources required by the different modes are a(M,)

= {R,, &1, W(M,) = {&), g(M3) = {&> R3, R& g(W) = (R3, h), = {R4, R5}, B(M6) = {R,, R5}. Only task T1 has more than one mode, namely modes M, and M2. Modes M3, M4, MS and M6 are the modes of tasks T2, T3, T, and T,, respectively. At the first iteration of the procedure multi-mode, the procedure assignment assigns mode Ml to task T,, as this choice gives rise to a maximum weighted clique with weight equal to 30, while the choice of mode M2 would provide a weight equal to 31. Then the procedure sequencing is solved and the optimal orientation of 3’ is shown in .G@(M,)

Fig. 8(a), while the transitive closure 3 + is shown in Fig. 8(b). The corresponding schedule has a makespan equal to 40. The graph 3” produced by the procedure vary is shown in Fig. 9. Then the execution of the procedures is iterated. The procedure assignment assigns mode M2 to task Tl and the procedure sequencing finds the optimal orientation, with makespan equal to 31. As the graph Y’ is in this case a comparability graph, the procedure ends.

Fig. 7. A problem

instance.

L. Bianco et al. / Discrete Applied Mathematics 62 (1995) 35-50 T

5

4

T3

49

:i;oj

(10)

(10) (a)

(b) Fig. 8. (a) A valid orientation and (b) the transitive closure.

M2 / (21)

Fig. 9. The new graph 9”.

6. Conclusions In this paper we deal with a scheduling problem in which each task can be processed in one among a set of predefined modes, where each mode requires the availability of a set of discrete and renewable resources and a given processing time. The problem instances are represented by means of a so-called mode graph which is

50

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used as a basis for the investigation the analysis

of the complexity

and upper bounds. problem

is sketched

be devoted

A heuristic together

to the evaluation

of properties

of some particular solution

approach

with a numerical and implementation

62 (1995) 35-50

of the problem

and, in particular,

cases and for the derivation based upon a decomposition example.

Further

of this solution

research

for

of lower of the

efforts will

approach.

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