E L S E V I E R Fuzzy Sets and Systems 83 (1996) 291 299

FUZZY sets and systems

Fuzzy priority heuristics for project scheduling

M a c i e j H a p k e , R o m a n S l o w i n s k i *

Institute q f Computing Science, Poznan UniversiO' c~f Technolog3', ul. Piotrowo 3a, 60-965 Poznan, Poland

Received March 1995; revised September 1995

Abstract

This paper presents a generalization of the known priority heuristic method for solving resource-constrained project scheduling problems (RCPS) with uncertain time parameters. The generalization consists of handling fuzzy time parameters instead of crisp ones. In order to create priority lists, a fuzzy ordering procedure has been proposed. The serial and parallel scheduling procedures which usually operate on these lists have also been extended to handle fuzzy time parameters. The performance of the method is presented on an example problem.

Keywords: Project scheduling; Heuristics; Priority rules; Fuzzy scheduling procedure

1. Introduction

A continuing preoccupation of operational researchers is realistic modelling of decision prob- lems with their inherent uncertainty. There are two major approaches to the handling of uncertainty: stochastic and fuzzy. The latter one is especially well suited to the uncertainty having epistemic character.

One of decision problems that in practice often involves uncertain information is resource-con- strained project scheduling (RCPS). It has been proved that RCPS problems belong to the class of NP-hard problems [9]. Therefore, the use of heu- ristic methods for solving them is well justified. The problem with crisp time parameters has been inves- tigated since late 1950's; many papers proposing

* Corresponding author.

many different solutions were written. Some com- parisons and surveys of heuristic methods can be found in [1,3 6, 12, 14].

The RCPS problem consists in allocation of re- sources to activities that optimizes some project performance measure. When scheduling activities, it appears that some of them are in resource con- flict, In order to solve this conflict, a priority is calculated for the activities, taking into account some time, resource and/or succession character- istics. Then, a parallel or a serial scheduling proced- ure can be used to set up feasible start and finish times of activities. There exist many methods based on this idea, called priority heuristics. They ap- peared to be very efficient and have been regularly applied in practice.

The CPM and PERT models have been ex- tended as first ones to fuzzy time parameters [8]. They neglect, however, the project resource con- straints. A method for solving the RCPS problems

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292 M. Hapke, R. Slowinski / Fuzzy Sets and Systems 83 (1996) 291 299

with fuzzy time parameters have been then pro- posed by Hapke and Slowinski [10] and Hapke et al. [11]. It is based, however, on an early trans- formation of the non-deterministic problem to a set of its deterministic associates which are then solved using known deterministic procedures.

The aim of this paper is to propose a generalized fuzzy priority heuristic with fuzzy serial and parallel scheduling procedures which, instead of transform- ing the fuzzy problem to a set of its deterministic associates, handles the fuzzy data directly, in par- ticular steps of the procedure.

The considered model of RCPS with fuzzy time parameters is described in Section 2. Section 3 pre- sents some introductory material: uncertainty mod- elling using fuzzy sets, some operations on fuzzy numbers, comparison of fuzzy numbers based on the compensation of the areas determined by their membership functions and a version of the C P M for fuzzy time parameters. In Section 4 a set of fuzzy priority heuristics is presented and Section 5 de- scribes the generalized fuzzy scheduling proced- ures. Section 6 gives an account of an application of the method to an example problem; in particular, graphical presentation of fuzzy Gant t charts and resource usage profiles is explained. The final sec- tion groups conclusions.

2. The model

The project is characterized by the following components S = {R, Z, t , C} [17, 18], where

R - the set of renewable resources, Z the set of project activities,

- the precedence constraints in set Z, C - the set of project performance measures

(criteria, objectives). Set R is composed of p types of renewable

r e s o u r c e s R 1 , . . . , R p with the usage limited to N~, . . . ,N v units in every period t'. Set Z is com- posed of n activities which have discrete resource requirements. Performance of activity Zj is defined by vector r = [ q , . . . , rp] , whose elements deter- mine the usage of renewable resources R~,. . . , Rp.

For each activity Zj the following time parameters are known: duration (processing time) /S j; ready time cij; due date dj. It is assumed that all time

parameters of activities are, in general, uncertain and modelled by fuzzy numbers (denoted by ~).

In this paper the project network is represented by an Activity-On-Node (A-O-N) model.

The scheduling of the project consists in such an allocation (considered in time) of resources from set R to activities from set Z that all activities are completed, the constraints are satisfied and the best compromise between criteria from set C is reached.

Before the solution of the problem is presented, some concepts useful in the following sections should be introduced.

3. Some useful concepts

3.1. Uncertainty modelling

Let ~ be a fuzzy number, i.e. a normalized con- vex fuzzy subset of real line ~: M = {(x,/~M(x))[ x e {R}, where J~v(X) is the membership function, taking values from [0, 1], specifying to what degree x belongs to M.

The x-level set or a-level cut of M is the set I(A4, ~) = {x ~ R I ~M(X) >>- ~}, where ~ ~ (0, 1].

The lower and upper bounds of any a-level set I(M, c0 are equal to infx~ I(M, ~) and supx~ I(.M, x), respectively, and we assume here that both are finite.

Dubois and Prade [7] have shown that a conve- nient representation of fuzzy numbers is an L R type flat fuzzy number denoted as follows:

= (_m, ~, ~, /3k~

where [t_n, n~] core of M, _m, tfi - lower and upper modal values of M, ~,fi left-hand and right-hand spreads.

The membership function of ~r is expressed by means of symmetric, bell-shaped reference func- tions L and R, such that L ( O ) = R ( O ) = 1 and L(1) = R(1) = O:

f l u ( x ) =

L[(_m - x)/c~] if x < ~V,

1 if m e [_m,n~],

R [ ( x - m ) / f i ] if x>,_n, ~,13eR +.

M. Hapke, R. Slowinski / Fuzz), Sets and Systems 83 (1996) 291 299 293

In this paper, according to the practical way of getting suitable membership functions of fuzzy data [15], it is proposed to model flat fuzzy numbers with up to five linear pieces, as shown in Fig. 1. It should be noticed that such a representation allows for modelling the convex and concave reference functions of fuzzy numbers. A symbolic definition of this representation is as follows:

1~ = (mL m ~, m, r~, ~fia, tfi ~ )

3.2. Some operations on fuzzy numbers

If J` and 9 are defined as five-linear-pieces fuzzy numbers, then the following operations can be defined: • Addition

=(q~ + b_~,q ;~ + b;,a_ _ + b,a_ + D,~ ~ + b~,a ~ + [~).

• Subtraction

J ` - 9

= (q~ - _b':, q ~ - _b~, _a - _ b , ~ - K ~ ~ - ~ , a ~ - ~ ) .

• Maximum

max(j`, 9) = (max(q ~, _b~), max(q ~, b_ ;'), max(a, _/2),

max(a, {)), max(a a, b;'), max(a ~, f)~)).

• Minimum

min(j`, 9) = (min(a ~, _b~), min(a x, ha), min(a, _/2),

min(& b), min(d x, ha), min(a% b~)).

It should be noted that fuzzy CPM (presented in Section 3.4) makes sense only if subtraction is the inversion of the addition. In contrast to extended subtraction (cf. [8]) thus defined subtraction satis- fies this requirement.

3.3. Comparison offi4zzy numbers

While solving problems with fuzzy data there often appear situations in which two or more fuzzy parameters are to be compared. In the problem considered in this paper, the comparison of fuzzy

m ~ ( x )

H

m ~ m ~ m N ~ ° ~

Fig. 1. Five-linear-pieces fuzzy number.

parameters is done in three circumstances. While creating a priority list of activities (the list is ordered taking into account various fuzzy para- meters), while solving problems of precedence of fuzzy events and while comparing fuzzy results of optimization.

The result of comparison is trivial if two different fuzzy numbers do not overlap each other.

Generally, two different fuzzy numbers can over- lap each other in two different ways. The examples are presented in Fig. 2. Looking at Fig. 2(a), one can easily notice that the respective values of both lower and upper bounds of any a-level set of 9 are greater than those of J`. It means that max(J,,/~) = B. In such a case we will say that 9 is strongly greater than or equal to J`. The relation of strong inequality will be denoted by 9 >> = J`.

Fig. 2(b) presents another case of overlapping of two fuzzy numbers J` and/~ where the relation of strong inequality does not occur. In such a case it is proposed to apply the comparison based on the compensation of areas determined by the member- ship functions (cf. [2, 13, 16]).

Let J`,/~ be fuzzy numbers and SI.(J` >~ 9), SR(J` >~ B) the areas determined by their member- ship functions according to the formulae:

SL(J` >J /~) = fv, ~,.h) I i n f l ( j ` ' ~ ) - i n f l ( 9 " ~ ) ] d : ¢ x ~ R x~

where U~(J`,/~) is the subset of [e, 1] defined by {~ ] infx~ I(j`, ,~) ~> inf ,~ 1(9, c~)}, and

SR(J`>JB)=fv Vsup l ( j ` ,~ ) - sup l (9 ,~ ) ]dc¢ l(a.b) [_ x ~ x ~

294 M. Hupke, R. Slowinski /' Fuzzy Sets and Systems 83 (1996) 291 299

b)

a)

X

I #(x) A B

U

Fig. 2. Examples of overlapping of two fuzzy numbers.

where VI(A,B) is the subset of [c, 1] defined by {~1 supx~u I(A, ~) ~> supper I(/~, ~)}.

Sr(ft <~ B) and SR(~] ~< /~) can be defined analo- gously. According to [-16], the degree to which

~>/~ is defined as follows:

C(A/>/~) = max {SL(/~ >~ /~) + SR(/~ >~ /~)

- > - SR(/T >

We will consider that A ~>/~ as soon as C(/~,/~) > 0 and will say that A is weakly greater than or equal to B (the relation of weak inequality).

It is easy to note that the relation of weak inequality is included in the relation of strong inequality.

where Pj the set of predecessors of Z i. The order- ing of nodes of the A-O-N project network guaran- tees that VZi ~ Pa ¢: O, i < .].

The earlyflnishin9 time (EFT) of activity Z j:

~f ~S tj = ti +/~J.

The earliest project completion time (critical path length):

T,,, = max{{5 I j E Z}

where Z is the set of nodes. The latest finishing time (LFT) of activity Z a is obtained analogously:

s~~ S~. if S j¢O, ~fj =_ m i n ~ T i - - P i l Z i ~ j~,

L , if s j = 0,

where S i is the set of successors of Zi. The latest starting time (LST) of activity Zi:

The slack time (total.float) of activity Z j:

~s = - t j .

If J~ = 0 then Zj is on the critical path. The operat- ions of addition, subtraction, maximization and minimization applied in above calculations are extended to fuzzy arguments (cf. Section 3.2). The fuzzy CPM time parameters are then taken into account while solving resource conflicts in fuzzy RCPS. Fuzzy priority heuristics and fuzzy schedul- ing procedures are described in the next sections.

3.4. Fuzz3, CPM 4. Fuzzy priority heuristics

Dubois and Prade [-8] have shown that classical CPM can be generalized to handle uncertain activ- ity durations. |n this point we will add to generaliz- ation another fuzzy time parameter ready time. The way of computing all the fuzzy CPM para- meters is presented below.

The starting time of an activity Zj is obtained from the following formula:

~s fsmax{clj, t~q-pi[ZiEPj} , if P j @ O ,

t J = ( t o , if P i = 0 ,

Priority heuristics using crisp time parameters were found efficient by many researchers (cf. [3 6, 12, 14]). Since their computational complexity is low it is worth, therefore, to apply a set of priority heuristics instead of one. Such an approach was applied in the FPS system [,10], where a beam of heuristics was used. For a large class of problems this approach has appeared to be very efficient.

A review of priority rules used in these heuristics can be found in [1]. Those that appeared to be good in minimizing makespan are presented in

M. Hapke, R. Slowinski / Fuzzy Sets and Systems 83 (1996) 291-299

Table 1 Priority rules giving good results in makespan minimization

No. Rule Name Formula

1 EST Early start time min [~ 2 EFT Early finish time min [~ 3 LST Late start time min T) ~ 4 LFT Late finish time min T~ 5 MINSLK Minimum slack minJ~ 6 SPT Shortest processing time min/~j 7 LPT Longest processing time max/~t 8 LRPW Least rank positional weight rain fit + ~z~sj/~J 9 GRPW Greatest rank positional weight max/~ + Xz~st/5i

10 LIS Least immediate successors mini S t [ 11 MIS Most immediate successors max [ Sj I 12 GRD Greatest resource demand max/~ Z~= 1 r~k

Note: i~ is the earliest start time of activity Zj /5 t is the processing time of Z t [5 is the earliest finish time of Z t S t is the set of successors of Zj T~ is the latest start time of Z t ] d means cardinality of a set 7~ is the latest finish time of activity Z t J~k is the requirement for resource Rk

is the slack time of activity Zj

295

Table 1. After a p r io r i ty list is set up, a s ingle-pass serial or paral le l schedul ing p rocedure gives a single solut ion. Using the 12 pr io r i ty heurist ics one ob ta ins a set of different fuzzy solut ions. The m e t h o d of c o m p a r i s o n of fuzzy numbers , presented in Sect ion 3.3, is used to choose the best one.

5. Fuzzy scheduling procedures

The heuris t ic me thods for solving R C P S p rob - lems can be genera l ly d iv ided into serial and pa ra l - lel. The serial a p p r o a c h derives its name from the fact tha t act ivi t ies are cons idered sequential ly. The para l le l a p p r o a c h cons iders in one m o m e n t all the activit ies which could be scheduled in that moment . In bo th app roaches the activi t ies are ran- ked in some o rde r and then scheduled in that o rde r accord ing to the ava i lab i l i ty of resources. The main difference between them consists in the way of solving the resource conflicts. If an act ivi ty canno t be scheduled in m o m e n t t" because of resource con- straints , the serial p rocedure looks for the earl iest m o m e n t in which all the requi red resources are avai lab le and then schedules the activity. In the same s i tuat ion, the paral le l p rocedure takes the

act ivi ty with the next highest p r io r i ty from the list and tries to schedule it in m o m e n t 7.

In o rde r to general ize bo th heurist ics to fuzzy var iables one should know how to de te rmine the sequence of fuzzy t ime mome n t s in which new ac- t ivity (activities) is (are) to be considered. It should be no ted tha t in o rder to keep resource const ra ints , every current fuzzy t ime m o m e n t ~ must be greater than or equal to the previous fuzzy t ime moment , in the sense of s t rong inequali ty. The next m o m e n t is, therefore, the m a x i m u m over the current m o m e n t

and the earl iest moment , in the sense of weak inequal i ty , chosen from the set of those ones in which any resource is re leased or any act ivi ty is ready to be scheduled.

Moreover , in o rde r to satisfy the precedence con- s t ra ints in m o m e n t 7, a set of those activit ies whose immed ia t e predecessors have been comple ted will (only) be considered. It will be assumed that an act ivi ty has been comple ted by t ime ~ if ~ is s t rongly greater than or equal to the fuzzy finish t ime of the activity.

M a n y researchers work ing on R C P S prob lems have conc luded that the para l le l p rocedures give genera l ly bet ter results than the serial ones. So, we will concen t ra te on the former. The general scheme

296 M. Hapke, R. Slowinski / Fuzzy Sets" and &,stems 83 (1996) 291 299

of the fuzzy parallel procedure is presented below in a Pascal-like form:

The fuzzy parallel procedure: /':-- F,, repeat

Compose a set Q(~) of those activities which have not been scheduled yet and whose im- mediate predecessors have been completed by time ~. for each activity Z~ from Q({), in the order of the priority list do begin

if Z[s resource requirements ~< resource availabilities then

if & ~ = t" then begin

Zi's start time: tT:= t" Z[s finish time: -r. -s t i .= t i Jr- Pi

remove Zi from Q({) allocate required resources to Zi insert {[ into set T

end else

insert a~ into set T end t := max0", l) if l = t'~ then

update resource availabilities remove I from set T

until all activities from the priority list are completed.

where: l'- the least (in the sense of weak inequality) value from the set T,

f ~f" [~li: VZi ~ A ( t ' ) } , set T = t t i . VZiE S( t ' ) } w SO') - the set of activities that have been sched- uled before ~, ~ ~ = ~, A(/') - the subset of Q(/) of those activities that are not ready in moment/~, c~ ~> ~.

Generalization of the serial procedure would be anaologous.

6. Numerical illustration

We will use an example problem for a numerical illustration. The precedence graph of the problem is

2 5

1 3

Fig. 3. The precedence graph of an example problem.

Table 2 Activity parameters

Activity No. Duration Resource requirement

1 42,45, 50, 54, 54, 61 8 2 36,44,46,52,61,66 17 3 49, 51,54, 62, 71,79 12 4 34, 40, 45, 46, 53, 59 3 5 16,22,30,32,33,42 13 6 43,49,51,57,57,57 7 7 52,52,58,62,65,69 16

presented in Fig. 3. The activity parameters are summarized in Table 2. The objective is to mini- mize the project makespan. For simplicity, the pro- ject consists of seven activities with zero ready times. Each activity requires one renewable re- source whose availability is constrained to thirty units in every moment.

The calculation process begins with setting up the priority lists using the 12 fuzzy priority rules. Next, the fuzzy parallel heuristic schedules activ- ities and computes the project makespan. The pri- ority lists, activity sequences in the schedules and project makespans of corresponding priority rules are given in Table 3.

Applications of the 12 priority heuristics gave three different schedules with three different values of the makespan. It should be noted that the opti- mal procedure had found the same solutions. The best (optimal) result has been obtained by rules LFT, G RP W and GDR (cf. Table 3).

In Table 4 activity start and finish times obtained by the LFT rule are given. Fig. 4 presents the

M. Hapke, R. Slowinski / Fuzzy Sets and Systems 83 (1996) 291 299 297

Table 3 Priority lists and project completion times for 12 priority rules

No. Rule Priority list Sequence in scheduling Fuzzy project makespan

1 EST 1,2,3,4,5,6,7 1,2,3,4,5,6,7 2 EFT 1,4,2,3,5,6,7 1,4,2,3,5,6,7 3 LST 1,2,3,4,6,5,7 1,2,3,4,6,5,7 4 LFT 1,3,4,2,6,5,7 1,3,4,2,6,5,7 5 MINSLK 6,2,3,1,5,7,4 1,2,3,5,4,6,7 6 SPT 5,1,2,6,7,4,3 1,2,5,4,3,6,7 7 LPT 2,3,4,5,1,6,7 1,2,3,4,5,6,7 8 LRPW 1,5,3,2,6,7,4 1,3,2,5,4,6,7 9 GRPW 3,4,7,1,5,2,6 1,3,4,6,2,5,7

10 LIS 6,4,7,5,2,1,3 1,4,2,3,6,5,7 11 MIS 6,4,7,2,1,5,3 1,4,2,5,3,6,7 12 GRD 7,3,5,6,4,2,1 1,3,4,2,5,6,7

207,230,250,271,290,312 220,237,258,281,300,325 207,230,250,271,290,312 186,203,229,246,266,297 207,230,250,271,290,312 220,237,258,281,300,325 207,230,250,271,290,312 207,230,250,271,290,312 186,203,229,246,266,297 220,237,258,281,300,325 220,237,258,281,300,325 186,203,229,246,266,297

Table 4 Activity start and finish times for LFT rule

Activity no. Fuzzy start time Fuzzy finish time

l 0,0,0,0,0,0 42,45,50,54,54,61 2 76, 85,95,100,107, 120 112,129, 141,152,168, 186 3 42,45, 50, 54, 54, 61 91,96, 104,116,125, 140 4 42, 45, 50, 54, 54, 61 76, 85, 95, 100, 107,120 5 112,129,141,152,168, 186 128,151,171,184,201,228 6 91,96, 104,116,125,140 134,145, 155, 173,182,197 7 134,151, 171,184, 201,228 186, 203, 229, 246, 266,297

co r r e spond ing fuzzy G a n t t chart . The G a n t t char t is usual ly the base for ob t a in ing a very i m p o r t a n t in fo rmat ion - how m a n y resource units of each type are requi red by the pro jec t in every m o m e n t of project per formance .

Cons t ruc t i on of a resource usage profi le in the case of fuzzy t ime pa r ame te r s is not s t ra ight for- ward. In o rde r to present the full i n fo rmat ion graphical ly , a 3-D char t should be used with t ime on the first axis, resource usage on the second and poss ib i l i ty of the resource usage at a given t ime on the th i rd axis. Such a p resen ta t ion would be too cumbersome , however. It is p r o p o s e d therefore to present only a par t of this in fo rmat ion - the resource usage profi le for one poss ible scenario, i.e. for all lower (opt imis t ic case) o r all upper (pessimis- tic case) bounds of ~-level act ivi ty dura t ions ,

e F0, 1]. A resource usage profile at level ~ = 0.5 in the opt imis t ic case is presented in Fig. 5.

7. Conclusions

The pape r presents a general ized heurist ic m e t h o d for solving R C P S p rob lems with respect to uncer ta in t ime pa rame te r s of activit ies mode l l ed by fuzzy numbers . The m e t h o d is based on the use of fuzzy pr io r i ty heurist ics and general ized serial or para l le l schedul ing procedure . Solu t ions of all re- la ted p rob lems (fuzzy order ing, ope ra t ions of add i - t ion and sub t rac t ion of fuzzy numbers , min and max opera tors ) have been proposed .

In au thors ' op in ion the general ized fuzzy p r io r i ty heurist ics p rov ide new tools useful for

298 ,14. Hapke, R. Slowinski ,; Fuzzy Sets and Systems 83 (1996) 291 299

activity no.

R I I r

I

7 i . . . . . . . . .

i p i

6 . . . . . . J . . . . . . . . . . . . . . . . . . . i i

b i

5 . . . . . ' . . . . . . . . . . . . . . . . . . . . . i

i i

i

4 . . . . . r - " q . . . . . . . . . . . .

i i

3 - ~ - -

i

i

t

_ _ _ A . . . . - -

I •

50 100 150 200

i

i i

250 300 time

Fig. 4. Fuzzy Gantt chart for the LFT rule.

resource usage 30

25

20

15

10

5

0

LI, i I I I

50 100 150 200 250

Fig. 5. The resource usage profile at level :~ = 0.5 in the optimistic case.

300 t ime

solving R C P S problems under uncertainty. The

research towards a cons t ruc t ion of new rules that take into account some extra informa-

tion about activities seems to be especially inter- esting.

Acknowledgements

This work was suppor ted by grant No. 8-$503

016 06 from State Commi t t ee for Scientific Re-

search (Komite t Badafi Naukowych) .

M. Hapke, R. Slowinski / Fuzzy Sets and Systems 83 (1996) 291 299 299

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