efficient priority rules for the resource-constrained project scheduling problem

Efficient priority rules for the resource-constrained project scheduling problem

Rainer Kolisch *

-.-- -.~-- -- -. Abstract

Resource-cor.s:i.‘ined prqject scheduling is a rorc probicm for many in;f-ctrial appiicaiions. e.g. finite scheduling systems and the planning oi businc;; re-eneenecring IUXYS. In order to sol\c these Froblems commercial <>stems employ classical priority rulei. We review one of them, the well-known RSM priority rule. A deTailed analysis of this rule rcveats that it does no: ~rflc~:. ?he finite cap,:~;ry of resources properly. Conscouen!ly. this leads to poor solutions. !-irnce. in improved RSM priority rirle is derived and two new I 1 --r3v rules are developed which extend the well-known precedence , 5a,cd Minimum SLacK priority rule !&<SLK) to a precedence and resou~cc hasec! sl:rck priority NIZ, respectivc:y. On the basis of a comprehansive c’xi &mental in\e&gation we provide new i,lsight irlto the performance of cl~rss1~11 ,md new rules.

Furthermore we show that on.: of the new rufcs is signiii~army hettcr than tiie classrcal ruler and hence should be cmpio\~+.d ir. pr~;r*~t bchcduling system<.

----. -

I. introduction

Project plannir,g and scheduling hat. become an important management tool for today’s complex business and manuiacturing systems. M*>dels and methods f;om project planning play a Vito.! role in such different tasks as, e.g., the re-design of pusiness and work processes and finite scheduling of manufacturing systems (cf. (Adelsberger and Kanet, 1991; Tobias, 1991; Stadtler and Wilhelm, 1993)). The co;e of project planning is the Resource-Constrainsd Project Scheduling Problem (RCPSP). It addressc:.

--- * Fax: +49-431-8110- 1531: e-mad: [email protected].

the question of how activities which arc interrelated by technological and multiple capacity constraints have to be time-phased in order to accomplish a pre-specified management goal.

On behalf of the great managerial relevance of the RCPSP. numerous efforts have been undertaken in order to rtcvise efficient 4ution procedures. Broadly, the methods car, be classified into (simple) priority rules, advan ,cd heuristics, and exact procedures. Al- though proprietary. commercial project managenlent software, e.g. Primavera Project Planner, and tinite scheduling systems. e.g. the AHP Lcithtand, usually employ simple priority rules for solving the problem. Priority rules have the advantage of being robust, intuitive, and fast in terms of the computational effort. Furthermore, commercial systems enable the

O272-6963/96/$15.OO 0 1996 Eluvier Science B V. All nghts rce,encd SSDI 0272-6963(95)000!2- I

u,cr to define their own priority rules in older IO SCICC~ iActivities for schcduiing. Hence. managers should know. under what conditions >+hich priorit) rules u-ill provide good results.

In this context the paper focuses on the prcscnla- tion of three new priority rules and XI cxpcrimental investigation which will provide insight into the performance of clasGcal and new priority rules.

The remainder is organised as foilows. Section 2 introduces the RCPSP and sivcs an overview c!t available solution proccdurcs. Section 3 presents the parall 4eduling scheme as a framl~work for priority ruies IO ..olvc the RCPSP. Based on a cla.qsifica- tion scheme. classical priority rules are introduced afterwards. It will be bcc’n that none of them consid- crs in a proper manner tht: fact th,t the capacil) ot rckourccs is finite. Hcncc. hascd on the classical RSM priority rule u’c will present three new rules u hich appropriately take into account resource con- jlr;:hk:r 5ecGc.n -l is devoted to an extensive compu- ca:j!r? ,I X‘;‘ .smcnt of the new rules. Furthcmlore. based on 3 plscisc classillcation :f prohicms via parameter; WC frill provide insight into the perfor- Irldl:~~.. ’ f “OTL Llnssic.7: an2 the new --iority nllcs. -J-t-‘, .‘l,,jlfJ gi:,; q?;in.qcrc ;: ; -::cr u5tz :rstan&ig i i.

rl\e proper selection of priorit:: rule> cmploycd within commercial scheduling systems. The paper ends with overall conclusions given in Section 5.

2. Problem formulation and solution procedures

The classical Resource-Constrained Project Scheduling Pro&m (RCPSP) can be stated as follows. We consider a single project which consists of i = I,...,J activities with a non-preemptable duration of <I, periods. respectively. The activities arc intcrrc- lated hy two kinds of constraints: precedence constraints - as known from traditional CPM-analysis - force an activity i not to be started before all its predecessors f, are finished. Additionally, resource constraints arise as follows. In order to he processed, activity i requires k,, units of resource rype r E R

during every per& of its duration. Since resource r

is only available with the constant period availability of K, units for each period, activities might not he scheduled at their earliest (precedence feasible) start time but later. The objective of the RCPSP is to

assign cnch nctisity j ;I Finr~h T’I!I~ !-Yj :I~!I that precedence and rcsourcc i:itnstraint,s arc ohcycd and the makespan of lhc project is minim&d.

%kespa;; mi;limi:iation i< a proper objecti\‘e for several reasons. ii) The majority of income payments occurs a: the cnJ of a project. Hence. linishing the pro-jcct early reduces the amount of tied-up capital. (II) I% quality ~11. forecasts tends to detcrioratc with the distance into the future of the period for which they art‘ m:llfc. itili~imlcatic~rr of the makcspan re- ducch the planning horiron and thus the uncertainty of d:!ta. (iii) Finishing products as car& as possible lowers in the ligh: of lean capacities the probability of violating deadlines. (iv) Generally. capacity is freed in NC future if ;-csourccs arc used irs early as possihlc. This provides cornpanics with more tleui- hility in orLr to cope with changes of the economic environment.

The RCPSP is a generalisation of the static job shop prohlcm and hence belongs to the class of NP-hard problems (cf. (Blazewicz et al.. 1983)). On account of the inherent intractability of the RCPSP, a multitude of cx’~.ct and hcdristic approaches were proposed

Optimal procedu. es arc rfwrun~ic pro~~rcrnmirrg

(cf. (Carruthers and Battershy. 1966)). ;e~)-~ne [pro- grtrnmiq (cf. (Bowman. 1959: Pritsker et al.. 1969; Patterson and Huher. 1973: Patterson and Roth. 1976)). as well as implicit enumeration with hrmch

crrtn hound (cf. (Balas. 1971; Davis and Heidorn. 1971; Hastings, 1972: Stinson et al., 197X; TJbot and Patterson, 197X; Radcrmachcr. 1985/ 1986; Christofides et al., 1987: Bell and Park, 1990: Car- lier and Latapic, 1991; Mingozzi et al., 1994; De- meulemcester and Herroelen, 1992. 1995)). Cur- rently. the branch and bound approach of Demeule- mecster and Herroelcn (1992, 1995) seems to be the most powerful optimal procedure available.

With the pioneering works of Kelley (1963) and Brooks (cf. (Bedworth and Bailey, 1982)) priority rules marked the beginning of heuristic methods able to solve the RCPSP. In the following years a multitude of new priority rules were proposed and tested experimentally (cf. (Pascoe. 1966; Miiller- Merhach, 1967; Gonguet, 1969; Cooper, 1976, 1977; Davies, 1973; Patterson. 1973; Davis and Patterson, 1975: Tbesen, 1976; Patterson, 1976; Whitehouse and Brown, 1979; Elsayed, 1982; Lawrence, 1985;

L~USOY and t-jrdatl~ar. l%Y: Alvarez-Valdcs and Tamarit, 1989n.b: Boctor. 1990: Valls et al.. 1992’1). Recently. the r-search interests shifted to more clah- o<ate heuristic?: like truncated branch and hound (cf. (Alvarez-Valdes and Tamarit, 19893)). integer programming based heuristics (cf. (Oguz and Bala. lYY4)), disjunctive arc concepts (cf. (Shaffer Ed al.. iY65: Alvarez-Valdes and Tamarit, IYXYa: Bell and Hr:n, 199111, and local search techniques (cf. (Samp- son 2nd Weiss. 1993; Leon and Balakrishnan. 1995)). With an - in terms of the employed CPU-time - greater r”ffort these heuristics derive superior solutions than simple priority rule procedures.

Ncverth4ess. for sc\eral reasons priority rules are of utmost importance in order to solve the RCPSP. (i) They are ihe core of new highly efficisnt local search based hcuri4cs. E.g., based on the idea of Storer et al. (19 !Z). Leon and Balakrishnan (199% devised a local SC trch procedure which repeatedly emplovs a priority r4e in order to obtain so!lltions with perturbed input dita. Within sampling h. istic< (cf. (Levy et al.. 1962. Wlest. 1967; Cooper. lY76: Alvarez-Valdes and Tamarit. lY89h: KtJisch, 1995a)) the priority values of the xhedulahle activities ars biased. That wzj different rntutions can be generated and the solution with the LXSI objcct~vc function value is sclectzd. It is intuibivc’ and was clperimen- tally &kcd up by Kolisch (19‘?Sb) that the prrfor- mancc . . sampling heuristics dcF :nds on MC good- ness of the priority rule cmp!oycd. (ii) As outlined in Section I, commercial project plannmg and sehcdul- ing software make use of priority rules in order to rapidly derive feasible schedules in the presence of resource constraints. (iii) Finally. priority rules have to be cmploycd when very large problems with more than 1000 activities have to IX solved 17 a fast manner (cf. (Valls et al., 1993)). Hence, it can be summarised that there is a great need for good priority rules.

3. Classical and new priority rules for the RCPSI:

Generally, a priority rule based scheduling heuristic is made up ol two components, a schedule generation scheme and a priority rule. Two different schemes can be distinguished. The so-called serial and the parallel scheduling scheme (where we use

Kolisch (1995b3 shcjucd that - \vhen con\iderinz a regular measure of performance - frrr :ilIl,‘:? PC!-\. proccdums (as studicd in this pape, i the i:. *.i:IcI method is superior to the serial method. Hence, we restrict our focus to the parallel method.

Today. two algorithm\ XC‘ as>ociateU aith the so-called parallel method. The algorithm of Kelley I lY63) and the one of Brooks fci. (Bedworth and Bailey. 1982)) which is also termed Brooks AIGo- rithm CBXGf. Ltkc in the malority of publrcations. the scheduling scheme as proposed b) Brooks is employed her&.

The parallel scheduling scheme generates a feasi- blc schedule by extending a partial schrdulc (i.e.. a scheJulc lvhcre only a subxt of the activltics has been assigncci a finish time) in a stage-wise fashion. There arc at most J stages in each of which a set of activities (which might he empty) is cchedulcd. As- hociated with each stage 11 is a schedule time I, and three disjoint activity-sets. Activities which are com- pleted $1~ to the schedule time are it-, the kwpkre set C,. Acti:ities which arc already scheduled. fxt during the schedule time stdl active arc’ in the Ac!r! I set A,. Finally. activities which are available for scheduling w.r.t. preccdencc and resource constraints hut yet unscheduled arc in the Drcision SC’: 0,. The partial schedule of each stage is made up by the activities in the complete set and the active set. The schedule time of a stage equals tt,e earliest compie- lion time of activities in the actike set of the ances- tral stage.

Each stage is made up of tv o steps. (Ii The new schedule time is determined and activities with a finish time equal to the (new) schedule time are removed from th- active set and put into the complete set. This. in turn, may place a number of activities into t’le decision set. (2) One activity from the decision set is selected with a priority rule (in case of tics the activity with the smallest label is chosen) and scheduled, respectively started, at the current schedule time. Afterwards, this activity is removed from the decision set xd put into the active set. Step (2) is repeated until the decision set is emnty. i.e.. activities have been scheduled or ‘are no

longer available for scheduling w.r.1. resource constraints. The prirallcl method terminates when all activities are in the complete or active set. Appendix A provides a formal description of the parallel scheduling scheme.

3.2. Clussiwl priority rides

Within the parallel scheduling scheme (cf. Ap- pendix A) the first line of step (2) states that for each activity in the decision set a priority value c(j) is computed and the activity with the cxtremun; (i.e. the maximum or minimum) priority value is selected. In case of ties, the activity with the smallest activity label is chosen. The number of priority rules proposed to calculate L’ is abundant. For the RCPSP, a review can be found in (Lawrence, 1985; Alvarez- Valdes and Tamarit, 1989a).

where the priority value i?; a lorcer btrzrrtd or makes use of a lower bound on the one side and rules where no lower bound is part of the priority value (11, the other side. For classification purposes we use the following notation ~/p/x/ii with u = (N,T.R). /3 = {S.D). ,v = (LG) and 6 = (B. - ). The capital letter uniquely identifies the characttzl>ltc via the bold letter of the adjective.

Pr’.)rity rules can be classified according to different criteria. It must be said that the classes provided by thcc~, ,riteria are neither exhaustive nor exclusive and p*e just one way of characterising the abundance of ruled presented. (i) A straightforward discrimination w.r.t. the informatior pr ccsced, 1s tfb classify priority rules into network, time and resource based priority rules (cf. (Lawrence, 1985; Alvarez-Valdes and Tamarit, 1989a)). (ii) Another distinction is w.r.t. the dynamic nature of rules: Priority rules which return the same priority value for a certain activity - regardless of the stage they are performed in - are called static, whereas priority rules which may produce different values are called dymnic. (iii) A further discrimination of priority rules is due to the amount of information processed. Rules which employ a small amount of input, usually w.r.t. the activity under consideration only, are regarded to be local or myopic, while rules which make use of a large amount of information are called global, (iv) Finally, priority rules can be classified into rules

In order to select the best classical priority rules the computational studies of Davis and Patterson (I 975). Alvarez-Valdes and Tamarit (1989a), Valls ct al. (1092) and 3octor (1990) hrtve been analysed. Table 1 presents the priority .lcs which ranked among the three best rules in at least one of the studies. The fi~>i coluni~~ contains the name and the abbreviation of the rule, the second column its a,‘P/X/&classification. the third column indicates if the activity with the maximum or minimum priority value is selected. and the last column gives the formal description of the rule. Additional to the notation already introduced, !?, CS,> denotes the set of all (immediate) successors of activity j and LFI’, &ST,) denctes the Latest Finish Time (Latest Start Time) of acti\ Ity j &S derived by traditional backward rcculsion (cf. (Elmaghraby, 1977)). Note that for the parallel scheduling scheme the Minimum SLacK priority rule (MSLK) equals the Latest Start Time priority rule (LST) which has been proven by Davis and Patterson (1975). Table 2 summarises the ranking of the three best priority rules in all four publications, respectively, where Q > b defines that priority rule a has a better performance than rule 6.

3.3. A detuiled analysis of the RSM priority rttle

The only rule listed in Table 2 which is not considered in Table 1 is the so-called RSM priority rule. The fundamental idea of this classical priority rule stems from the disjunctive arc based Resource

Table I The best clwicd pnority rules

Ptiorrty Rule

Most Total Successors @ITS)

Greatest Rank Positional Weight (GRPW)

Latest Finish Time (LFT) Minimum SLacK (MSLK)

Classification extr.

N/S/G/- m3x

N.T/S/G/- milx

N.T/S/G/B min N,T/D/G/B min

c(j)

iq

4 l c 4 I E s,

LFI; LST, - I,,

Table 2 Ranking of priwity ruks in the ltterature

Liternture Ranking of pnonty rules

(Davis and Patterson, iY75) MSLK + LFT > RSM (Boctor, 1YYO) MSLK > LFT > RSM (Alvarez-Valdes and Tamarit. 198%) GRPW t LFT > MTS (Valls et al., IYQ MSLK z- GRPW t MTS

Scheduling Method (RSM) proposed in (Shaffer et al., 1965). The RSM rule was considered in s!udies undertaken by Davis and Patterson (1975). Alvarez- Valdes and Tamarit (1989a), Boctor (1990), Valls et al. (1992) and Ulusoy and &damar (1989). The idea of the RShl rule is as follows. For each pair of activities i and j in the decision set. the number of periods activity i is delayed beyond its (precedence based) latest stat: time if it is scheduled a%er activity j is calculated. The activity j which mtiuces the smaliest delay of every other acti-:itv in the decision set is scheduled. In order to write the RSM pile formally we denote w;~II AP,, the set of all ActieJtty Pairs in the decision set at stage n, i.e. AP,,: = {(i,j)li$j E D,,, i Z j). Now, the RSM priority value of activity j, j E 0,. is

t’(j) = max(O,r, + d, - LST,‘( i,j) E AP,}, (1;

and the activity with the minimum value is selected. Stated this way, the RSM priority rule implicitly makes use of the following assumption. For each pair of activities i and j, one activity has always to be delayed until the end of the other activity. Let us look at two examples in order to show that this assumption does not hold in general and hence might produce poor results. Consider the projec. displayed in Fig. I where we have IRI = I resource type with a period availability of K, = 4 units and 6 activities. The associated latest start times derived by backward recursion from the earliest finish time of activity 6, I.e. Ln;i = Em6 = 5. are given in Table 3.

At stage n = 2 the following situation arises: t, = 0, ITK, = 4, C, = (I), A,, = 0, Dn = {2,3,4,5), and

Table 3 Latest start times for the example project

i I 2 3 4 5 6

LST, 0 3 1 2 0 5

IRi=l K, z 4

Fig. I. Example project.

AP, = (j1.3). (2.4). (2 51, (3,4), (3,5), (4,511. The R,SM priority value fcr activity 2 is calculated to c(2) = max(O,tl + dL - LST,, tz + d2 - LST,,r? + d2 - LST,) = max(O.O t- 2 - I,0 + 2 - 2,0 + 2 - 0) = maxf0,1.0.2) = 2. Presuming that the above stated assump:icn holds, scheduling activity 2 would in- duce a delay of two periods for the latest stcrt time of activity 5. Obviously. this is not the cast ;ince each of the activity pairs containing activity 2, i.\.. (2,3), (2.4! and (2,5), can be started simultaneously at the schedule time.

Let us now consider that activities 2 and 3 were scheduled and consequently at stage n = 2 the partial schedule is as shown by Fig. 2, i.e. t, = 0, TK, = 2, C, =(I), A, = (2,3), D, = (4.5), and AP, = ((4,5)). Actually, at this stage the activity pair (4,5) in the decision set cannot be jointly started. The RSM priority values turn out to be v(4) = max{O,r, + dJ - LST,) = m;x(O,O + 3 - 01~ 3 and t:(5) = max{O,r, + d, - LST,] = max(O,tJ + 5 - 2) = 3. A look at Fig. 3 reveals that scheduling activity 4 indeed forces

2 4

Fig. 2. Partial schedule of the example project.

Fig. 3. Fcasihle solution of the example pro$ct

activity 5 to be delayed until the end of activity 4, i.e.. three periods beyond its latest start time as rndi, ated by u(4).

But Fig. 4 reveals that scheduling activity 5 post- pone activity 4 only two periods beyond its latest Starr time. Hence, the assumption implicitly stated by the RSM priotity rule is assoc:iated with severe drawbacks. Conxquentl~. ihe ,jril rity rule ~‘am)c)t suf~st properly which activity has to be scheduled first.

3.4. Three new priority rules for the RCPSP

Let us now refine the RSM priority rule as follows. The set of activity pairs AP, is divided into three disjoint subsets, The set of Generally Forbid- den Pairs (GFP”) contains activity pairs which - due to resource constraints - can never be scheduled simultaneously, the set of Temporarily Forbidden

Kl4 4 ’ .~.~.-.--. .--__.-

5

r-- :--- -- -I- 1 - ---- .-~- _---

/ . ..‘. ._. .l..-.-.--.. 4

i . :--.---- ..__ i -.IT-- ___ -.

-1 I 1 --+ I

2 4 5 7

Fig. 4. Optimal solution of the example fkojecl.

Pairs !TFP,,) contains pairs of activities which ir, general can be scheduled simultaneously but due to limited left over capacity cannot be scheduled simultaneously at the schedule time. Finally the set of Currently Schedulable Pairs (,_‘SP,) contains the activity pairs which can be scheduled simultaneously at the current schedule time. LMore formally. we denote

GFP,,: = {(i,j)l(i,j) E AP,,,

3rEH:.bi,F+kjr> K,}, (2)

TFp,:={(i,j)j{i,j)~AP,.(i ‘\4GFF’,,

3rER:k,,+kj,>7:K,},

CSP,: = ((i,j)l( i,j) E APO,

(3)

Vr E R: k;, + kj, s nK,). (4)

Since we have three disjoint subsets AP, = GFP” UTFP, UCSP, holds. Let us look at our example. AL the beginning of stage n = 2 we have GFP” = VP” = W and CSP, = AP” = {(2.3), (2.4). (2,5), (3.41, (3.9, (4,5)). After activities 2 and 3 were scheduled we have GFP,, = 0, TFP” = AP, = ((4.5)), and CSP, = 0.

Obviously, the earliest time two activities of an activity pair can be started simultaneously is the schedule time t, if (i,j) E CSP, and = if (i,j) E GFP”. The question is at what period the two activities of a pair can be scheduied simultaneously if they belong to the set TFP”. Let us consider the following. At stage n, Il;i,j,, the earliest time the temporarily forbidden activity pair (i,j> can be processed simultaneously w.:.t. resource type r only, is given by

r ki, + kj, ,; = r “,..., T),

(i,j) ETFI?,. (5)

Then, lJojr, the earliest time to process two activities (i,j) of a temporarily forbidden activity pair simultaneously w.r.t. to all resource types, arises to

n,i*j): = max(ll[i,jJrE: RI. (i,j) ET’FJ’“. (6)

Now. we can define ~ci.,~, the earliest time for any activity pair (i.j) to be scheduled simultaneously, as

i

Xv if( i,j) E GFP. nci.jj,: = Il(i,j,, if (ivj) E ‘UT. (i.j) E AP,.

t tl’ if ( i,j) E CSP.

(7) Consequently. for the activity pair (i,j), Ecj,i), the earliest time to schedule activity i if activity j is started at the schedule time t,, is given by

EC,,;,: - min(t, + dj,il,j.j,l(i.j) E Mn). (8)

and Ec,.j,, the earliest time to schedule activity j if activity i IS started at the schedule time t,. is given by Eti,jr: = minjt, + d;, ni, ;,, jl( i.j) E A’ ‘r . (91

Now, an Impr,oved RSM priority rule (IRSM) can be stated as follotirs. Schedule me activity j which induces the smarlesi increase of the precedence :dsed lower bound for the not chosen activities in the decisron set. That is, the IRSM priority vaiue vi j) of activity j. j E 0,. is caiculated as u( j) = max(0, EC j.ii - LST,I( i,j) E AP,). (‘0) and the activity with the minimum value is selected. Note that we consider to strtit F ptivity j, j E D,,, at the current schedule time and hence calculate ,Q, for all activities i, i E D,, with i # j.

3.4.1. Observation I If at stage n all activity pairs in the decision set

are ‘Iemporarily schedulable and none of the activities in the decision set has a latest start time which is smaller than the current schedule time, then the IRSM priority value for each activity is “0” and hence the activity in the decision set with the minimum label is selected (cf. Proof of Observation 1 in Appendix B).

Utilising E,i.j, and combining it with the notion of slack another new priority rule can be formulated as follows. Schedule tie activity j which - if not chosen - has the Worst Case Slack (WCS). Mole formally, the WCS priority value of activity j, j E D,, is calculated as u(j) = LST, - max(E~,.j,l(i.j) E AP,,], (11) and the activity with the minimum value ,is selecled.

3.4.2. Observation 2 If at stage II all activity pairs in the decision set

are temporarily schedulable. i.e. resources do not impose constraints. then WCS reduces to MSLK, the best of the classical priority rules (cf. Proof of Observation 2 in Appendix B).

Employing the idea of Fadlalla et al. (1994) for the single machine problem with average tardiness performance measure we can propose the Average Case Slack priority rule (ACS). Here, the activity is selected which - if nor’ cho!en - has the largest average case slack value. Denoting with !D,I the number of activities iti the decision set, the ACS priority value of activity j, j E D,,, is calculated as

Q=LST,-& c Efi.jt* n (i.j)EAP,

(12)

and the activity with the minimum value is scheduled.

3.4.3. Observation 3 If at stage II all activity pairs in the decision s<t

are temporarily schedulable then ACS red ices to MSLK too (cf. Proof of Observation 3 in Appe.Jix B).

The three new priority rules are demonstrated by applying them to the example project of this section. Consider stage n = 2 of the partial schedule depicted in Fig. 2 with 1, = 0, nK, = 2, A, = {2,3], D,, = (4,s). The set of the temporarily forbidden pairs arises to TFP, = AP, = {(4,5)), i.e., activities 4 and 5 cannot be jointly started at the current schedule time I,. Consequently, the set of general forbidden pairs and the \et of currently schedulable pairs are empty, i.e., GFP,, = CSP,, = 0. Now, the earliest time to process activities 4 and 5 simultaneously is fI, = l7,; = 4. Then, the earliest time to start activity 5, if activity 4 is scheduled at t2, is E(,,s, = mini?, + d,.n,,] = min(O + 3,4} = 3. Correspondingly, the earliest time to start activity 4 if activity 5 is scheduied at c? is Ecs,JJ = min( t2 + d,,ii,,l = min{O + 5, 4) = 4.

Hence, the values for the IRSM rule arise to ~(4) = maxUM&,,, - LST,} = max(0.3 - 0) = 3 and u(5) = max(O,$ - 2) = 2, respectively. I.e., scheduling activity 4 (5) increases the precedence based lower makespan bound by 3 (2) periods. The IRSM

IX6 R. Koli.wh / Jg mnzul of 0perurion.s M~:n(tL~etnwr I-l (/YY6) Ii’Y- 15’2

rahle ‘I RSM-bawd pnrnity rules

Pnonry rule extr r(j)

Resouwe Schedulmg Method (RSM) mm max(O.r, +-tI!--LYiT,ki.~)s AP,,) lmpovrd RSbl (IRSM) min max(0. Et, , , - LSTJI, j) E AP,) Wont Case Slack OVCS) min LST, - rk.x{E,, j,l!i.J) E AP,,)

9verage~Case Slack (ACS) min LsJ-i - &(,,,~,,,. E(LJ) n

- Chsiknt&

NT ,‘I>/C/- N,T.R/D,‘C,‘B N.T.R/D/C/P

N.T.R/D/G/B

ule chooses activity 5 to be staved. which leads to :he optimal solution as depicted in Fig. 4.

Applying the WCS priority rule, the following Triority values are calculated: l:(4) = LST, - nax(E,,,J=2-4= -2 and &)=O-3= -3. I-hat is. if activity 4 (5) is not scheduled at tz it has a legal’vr clack of 2 (3) periods. Hence, the WCS -ru>rity rule also chooses activity 5 and thus derives ;ic optima r,>lulion.

Fin+dlv, since we have If,,1 = 2 for the above :xample the ACS priority values are calculated like tv :‘\-Cs priority values.

‘rilb!:: 4 provicies en uvcrvie\: of the classical ISM rule and the three new priority rules presented n this section.

1. Computational results

In order to evaluate the classical and the new )riority rules we employed two instance sets from .he open literature. The first one is the well-known I IO-instance set of Patterson (1984) which was orig- ,nally used to compare four exact solution proce- lures for the RCPSP. A detailed description of this Instance set will be provided in Section 4.3. Since :he Patterson-instances arc widely known and ac- :epted as benchmark problems they are employed in this study. However, there are two drawbacks associated with the instance set. Assembled from different sources the problems are not generated by using a controlled design. Furthermore, recent advances in the development of exact procedures (cf. (De- meulemeester and Herroelen, 1992)) showed that the instances can be solved in an average CPU-time of less than one second on a personal computer.

In order to have an instance set without the outlined deficiencies we also employed the 308 re-

source-constrained problems of Kolisch et al. (1995). These instances stem from the 48L -oblems which were origil .~lly generated with the parameter driven PROblem GENerator ProGen. Here, we neither considered the 120 resource-unconstrained problems nor the 52 problems for which Kolisch et al. (1995) could not verify the optimal solution. A detailed description of the instances can be found in (Kolisch et al., 1995). Briefly, each problem consists of 32 activities and 4 resource types. Each activity requires between I ?nd 4 resource types. Like the Patterson- instances. all 306 ProGen-problems were solved with the exact procedure of Ljemeulemeester and Herroe- len (1992) to optimality. Using an IBM compatible personal computer with 80386dx processor and 40 MHz clockpulse the average solution time was 45.24 CPU-seconds.

The parallel scheduling scheme and the 5 classical as well as the 3 new priority rules have been coded in PASCAL and implemented on the same computer as used for the optimal procedure.

4. I. Results for the ProGen-instances

Table 5 shows the mean (MEAN). the s!andard deviation (STDDEV) and the maximum (MAX) percentage deviation from the optimal objective function value for each of the 8 priority rules, respectively. Furthermore. the number of optimal solutions (#OPT) and the average CPU-time in seconds (CPU) is denoted. The priority rules are listed w.r.t. to ascending MEAN. It can be seen that the three new priority rules WCS, ACS and IRSM show better results than all other tested rules. Note that the performance and the ranking of the classical rules obtained by our study is comparable to the results of other studies (cf. (Davis and Patterson, 1975; Al- varez-Valdes and Tamarit, 1989a)).

Table 5 Pcrfwmnnce of priority rules for the ProGrn-insrnncrs

WCS

MEAN STDDEV XIAX #OPT CPU

4.X 4.60 26.3’) 96 0.02

A?3 4.57 4.94 IRSM 4.66 4.83 LFI- 4.83 4.17 MSLK 4.92 5.25 RSM 5.67 S.05 MTS 5.91 5.25 GRPW ‘i.29 8.17

32.81 94 0.02

27.03 9’ 0 02

‘7 37 -_. xs 0 01

32.91 93 0.01

1377 Ah_-_ 72 0.02

32.81 69 0.02

41.86 58 0.01

In order to test the significance of the results. we performed the 7-tailed Wilcoxon matched-pairs signed-ranks test for each pair of priority rules (we did not apply the t-test since for none of the priority rules the percentage deviation from the optir-:t solution was normally d stributed). Table 6 shows the confidence levels for each pair of pri rity rules. Italic entries signal that there is a significant difference between two rules at the I’%-level of significance, i.e. p < 0.01. It can be sTated that the WCS-rule significantly outperforms all classical priority rules. Also, all new RSM-based rJles are significantly better than the original RSM priority rule. Another interestmg finding is that there ar,a !wo disjoint classes: lower bound based rules, i.e., WCS, ACS, IRSM, LFI and MSLK. and priorit; rules which do not make use of a lower bound. Each rule of the first class significantly outperforms every rule in the other class.

4.2. Resdts ftw the Patterson-imtances

Tables 7 and 8 present the results for the Patter- son-instances. With one exception (LFT and MSLK)

Table 6 Z-Tiled Wilcoxon matched-pairs signed-ranks test for the Prv- Gen-instances

ACS IRSM LFr MSLK RSM MTS GRPW 6-B ;=-

WCS 0.179 0.0409 0.00% 0.0001 O.oooO O.GWO O.OOOt?

ACS, 0.5069 0.0935 Oo(hr6 O.oooO 0.0000 0.0000 IRSM 03Sl3 0.2790 (i.oooo o.cxNo o.oo00 LPI d.90’0 o.ofm o.ouoo 0.0000 MSLK 0.0005 o.oooo o.moo RSM 0.5451 o.wwcJ MTS 0. xl00

the ranking of the priority rules is identical to the one derived for the ProGcn-instances. Applying the Wilcoxorrtest to the Patterson-instances, the discrimination between lower hound based rules and other rules was not as distinctive as for the ProGen-instances. All other results were comparable to the ones already obtained with the ProGen-set.’

Table 9 gives a direct comparison of the MEAN for the two instance-sets. For each priority rule the italic entry indicates the instance set on which a better performance was obtained. From the results of optimal procedures (cf. (Demeulcmeestcr and Her- roelen. 1992)) it is expected that the performance is better for the Patterson-instances. But this holds only for the new priority rules v:hile the classical priority rules perform better on the ProGen-instances. Hence, the new priority rules are even more superior when considering the Patterson-instances.

4.3. Effect of problem parameters on the performance of priority rules

In the last two subsections a number of interes%g observations have been made which will be ex-

Table 7 Performan..; of pricrity r&s for the Patterson-in.&mces

MEAN STDDEV MAX ,+0PT CPU

Table 8 Z-Tailed Wilcoxon matched-pairs signed-ranks &I for :he Patter- son-instaaces

ACS lRSM MSLK LFT RSM MTS GRPW

WCS 0.0575 0.0714 0.0012 0.0000 0.0000 0.0000 0.0000

ACS 0.3156 0.9280 O.OOJO O.oooO 0.0000 O.MOO IXSM 0.5789 0.00% 0.0000 0.0000 O.WOO M.%K li 154.3 o.oo90 000.36 0.m LFI 0.2514 0.0278 o.fmo RSM 0.4255 o.o#o MTS O.OtW? -.

plained in this subsection by providing insight into the problem structure. We start by providing detni!cd informations about the two sets of test instnnccs employed.

As oulined in the beginning of Section 4. the problems of the ProGen-set were generated by a controlled design of specified problem parameters. Three problem parameters were employed (for de- tads cf. (Kolisch et al., 199%). (8 The Network Co*,@exity (NC) is the ra:io of non-redundant preccdcnce relations to the number of activities. (ii) The kewurcc Factor (RF) reflects the density of the (v\;o*~i mcnsioraf- arty k,,, j - Z,...,J - 1 and r = I,,.,,IRI. That Pi, fw P” = I ~;ich non-dcmmy acti>., ity requests the full complement of the /RI resource typzs while for RF = 0 none of the activities requests any of the resource types. (iii) Finally, the Resource Strength (RS) measures the degree of rcsource-con- strainedness in the interval [O,l]. For each resource type r, r E K, the resource strength is computed as f~lhws: RS = (K, - Kyin)/( KyX - K:,“‘), where Km’” is the minimal availability of resource type r in’ order to assure feasibility of the RCPSP, i.e. KFi” = max{k,,lj = l,...,J}, and KrrnaX is the peak demand fir resource type r in a CPM-schedule. That is, for RS = 1 WC have the resource-unconstrained CPM-case, while for RS = 0 we have a highly resource constrained problem.

The ProGcn-set was generated under a full-fac- torial design employing the following problem parameter levels: NC = (1.5,1.8,2.i}, RF = (0.25, 0.59.75,1), and RS = {0.2,0.5,0.7}.

As outlined in Section 4, the Patterson-set was not generated by using a controlled test design, Table 10 gives a brief summary of the project parameter values of this instance-set in order to allow a comparison with the Pro&~-set. The majority of the 110 instances concentrates on a few levels of the problem parameters only. To wit, 90% of the instances have 20, 25 or 49 activities, 93% of the instances have 3

rcsoturce types, 92% of the instnnccs have a resource factor in the rsnge [0.97.1], 6fK of the instances have a network complexity in tit, iange [1.35,1.59], and 67% of the instances have a resource strength in the range [0.35,ft.65].

Tab& 1 I- 13 report the single-factor effects of the problem parameters on MEAN when employing the Pro&n-instances. The priority rule with the best result for a specific parameter level is indicated by an italic entry. The robustness of the new rules’ superiority under various conditions is demonstrated. With the exception of RS = 0.2, WCS performs best for uilc Ieve!< of the problem parameters. Generally, there are no major alternations within the ranking of the priority rufes. The reason for the difference in performance with rcspcct to the parameters levels of NC. RF and RS is as follows.

If thorn are only few preccdcnce constraints, i.e. NC = 1.5, then ceteris paribus there are more activities in the decision set which gives way for more potentially wrong scheduling decisions. Many prcce- dence constraints. i.e. NC = 2.1, reduce the number of scheduling alternatives but prolong the critical path. Hence, the number of possible wrong decisions is reduced but the consequences of a single wrong decision are increased. The problems where priority rules show the best performance are for a medium number of precedence constraints where the number of scheduling alternatives and the consequences of wrong decisions are balanced. respective!y.

Tnh!e I1 Effect of the Network Complexity (NC) on the performance 01 priority r&s

NC WCS ACS IKSM L5-f MSLK RSM MTS GRPW

1,s 5.Ol 5.1s 5.42 6.03 5.53 6.2 6.25 9.86 1.8 3.73 4.17 4.23 4.0’1 4.36 5.26 4.88 8.49

2.1 4.1 4.43 4.4 4.53 4.89 5.56 6.57 9.32

RF WCS ACS IRSM LFT MSLK RSM MTS GRPW

0.25 2.45 2.66 2.71 2.33 3.0-t 7 73 3.3% 5.93 0.5 6.1 6.57 6 45 6.36 6.99 7.03 7.21 10.62 0.75 ICI 4.15 4 57 5.J9 4.49 6.78 6.51 IO.12 1 -1.57 4.95 5.09 5.32 5.14 6.74 7.05 IO.98

_I-

The number of different resource types requested by each activity as measured in terms of RF influ- ences the performance of priority rules as follows. If there are only a few resource tpyas requested by each activity, i.c RF = 0.25, it is relatively easy to derive good sche&tling decisions. Incre::?.;g RF to 0.75 and further to 1 renders the problems slightly more complicated. Problems turn difficult if each activity requests half or the resource types, i.e. RF 0.5. Then selected resouze types are bottlenecks and good scheduling becomes an extremely hard task.

Increasing RS enlarges the resource capacity which results in easier problems. Hence, all priority rules c’erive a better performance.

One observation given in Table 9 is the difference in the ranking between LFT and I$SLK from Pro- Gen-instances results ~omparcd to Patterson-instances results. While for the ProGcn-instances LFT outperforms MSLK, it is the other day around for the Patterson-instances. An expl anation is provided by Tables 1 l-13 where it can be seen that for the ProGen-instances with parameter levels similiar to the Patterson-instances, i.e. NC = 1.5, RF = 1 and RS = 0.5, MSLK outperforms LFI’, :oo. Hence. for problems which are characterized by these parameter levels it is obviously more beneficial to schedule activities based on MSLK than on LFT.

A second observation of Table 9 is that the new

Table 13 Effect of the Resource Strength (RSI on the pcrr~rnzu~~ of priarity rules

RS WCS ACS iRS.M Li%- MSLK RSM MTS GRPW

0.2 7.66 7.96 7.64 7.33 9.1 7.92 9.25 13.6 0.5 3.m 4.42 4.84 4.78 4.46 6.02 6.0 9.65 0.7 2.55 2.63 2.6.5 3.32 2.79 3.93 3.79 6.11

priority rules outperform the traditional priority rules and that this performance gap is even larger for the EIatterstrn-instances. An explanation is provided by the fact that only the new rules use proper information about the Fcsoui-2e const$-aints. This inf~~nnation-a~lvantage of the new rules diminishes for two extremes: (i? when problems become resource-unconstrained and (ii) when problems are highly resource constrained. For case (i) WC know from Observations l-3 that the new rules will produce the same results as the classical priority rules MSLK and smallest activity label. In cast (ii) all resouices are bottlenecks. Using local information about the resource co~~ctra~nts might then lead to inferior results than just simply considering the latest finish time of the activities.

5. Conclusions

This paper deals with the Resource-Constrained Project Scheduling Problem (RCPSP). The RCPSP is a central planning task for many probIcms arising in business and manufacturing systems. Commercial scheduling software usually employ priority rules in order to solve the RCPSP. We reviewed the best classical priority rules available in the open literature. A detailed analysis of the well-known RSM priority rule was the stepping stone for the devclop- ment of three new priority rules. By a thorough computational study with two instance-sets from the literature it was shown that on average the new rules outperform all other rules proposed so far. A detailed investigation of differer!t problem classes provided insight into the relative advantage of different priority rules. This knowledge and the new rules should further help managers to cope efficiently with the complicated planning problem addressed,

The author is gratefully indebted to Andreas Drexl, Universitat Kiel, lames pil, Patterson, Indiana Lb& versity, and to Erik Demeulcmeester, Katholieke Universiteit Leuven, for their help in this research. The comments of two anonymous referees strength- ened the focus of the paper.

Appendix A. The paraM schednhg scheme

In order to provide a formal description of the parallel scheduling scheme we have to introduce the following notation for ?TK,: the left over period capacity of the renewable rescurce type I’ at the schedule time t,, and of the decision set 0,:

7iKr:= K,- c k/,, jEA,

o,:=(jlj~(&;,CIR,},P,cC,,,

k,, 22 TK, VrE Rj.

Further, let g(j) bc a priority Value of activity j, j E 0,. Now, the parallel scheduling scheme can be descrkd as follows:

Initiabsation: n: = 1, t,: = 0, 0,: = {I), A,: = C,: = 0, ‘TTK,: = K, Vr E R, GOT0 Step (2); WI-IILE 1 A, U Czil < J DO il:tage n BEGIN

(I) t,:=min(FT,fjEA,-,};

A.:=A,-,\{~I~EA.-~,FT~=~,~;

Cn:=Cn-,U(jlj~A,-,,FTj=t,J;

COMPUTE vK, Vr E Rand D,;

(2) j’ : = ,F$ jlu(j) = ext;zmmo(i) ; II i n 1

FT,. : = t, + d,. ;

A,,:=A&{j*];

COMPtrTE x K, Vr E R and 0, ;

Since d, 2 0 holds for ail j: (8) results in

E (I.‘) = t,,, (i,j) 65 AP,. (‘4)

Furthermore, none of the activities in the decision set has a latest start time which is smaller than the current schedule time, i.e.,

LST,rt,, j~0,. (13

Substiiuting (14) into (IO) yields

I! = max(O,t, - LST,j(i,j) ( %P,,), (‘6) and on account of (15) we have l:(j) = 0, j E D,,. cl

B.2. Proof of Qbssruation 2 Since all activity pairs in the decision set are

temporarily schedulable, on account of Eq. (4) we have AP, = CSP,. Hence, Eq. (7) once more gives

fi ( I . ] ) = ‘,+ (i,j) EAP,,. (IT?

Since d, 2 0 holds for all i, (9) results in

q,.i, = cl (i,j) E AP,. (18)

Substituting (18) into (11) yields

u(j)=LST,-t,, je?D,,, 09)

which equals the calculation of the ,MSLK priority value (cf. Table I). U

B.3. Proof of Obseruation 3 Substituting (18) into (12) yields

[l(j)-LST,-t,, jEDn, (20)

which equals the calculation of the MSLK priority value (cf Table 1). 0

IF D, 0 THEN GOT0 Step (2) ELSE n : = n + 1

END; stop References

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efficient priority rules for the resource-constrained project scheduling problem

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