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IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 43, NO . 4, NOVEMBER 1996 411

Time-Resource Tradeoff ProblemP. Simin P u l a t and Steven J. Horn

Abstruct- Given a project network with a set of tasks tobe completed according to some precedence relationship, theobjective is to determine efficient project schedules for a rangeof project realization times. This problem is referred to as thetime-resource tradeoff problem. Associated with each task are itsnormal duration, maximum allowable crash range, and resourcecost per time unit for each resource. A multiple objective linearprogramming (MOLP)model is presented. The time-cost tradeofftechnique is extended to solve the time-resource tradeoff problem.The methodology assumes that the project managers (the deci-sion maker) utility function over the resource consumption costsis linear with unknown weights for each resource. Enumerativeand interactive algorithms utilizing Geoffrions P(A) approach,are presented as solution techniques. It is demonstrated that bothversions have desirable computational times.

1. INTRODUCTIONROJECT scheduling deals with the determination of ac-P ivity (task) realization times in a way which minimizes

the project completion cost. The cost of completing an ac-tivity of a project is actually an aggregated cost of usage ofseveral resources required to perform the task. A time-costtradeoff problem generates minimum cost project schedules asa function of project realization times. It provides the decisionmaker w ith information on the minimum achievable increasein project cost when the project realization time is decreasedby one time unit . Generally, the minimization of the projectcost is not the only desired objective. There will be somecritical resources for which it is desirable to minimize theamount of consumption. A time-resource tradeoff problemprovides project managers with information on how much eac hresources consumption (or its cost) will be increased if projectduration is decreased by one time unit. Instead of generating asingle time-cost tradeoff curve, one has to generate an efficientset of time-resource tradeoff curves for the project manager.Each time-resource tradeoff curv e corresponds to the optimalcurve for the time-cost tradeoff problem with the cost definedas the weighted sum of the resource usage costs for eachactivity for each resource.A multi-objective linear programming appro ach for the

project scheduling model is also discussed [I] . The examplegiven in the paper involved three objectives: 1) minimizationof project cost, 2) minimization of project dura tion, and 3)minimization of realization time of a specific node (nodek ) . The time-resource tradeoff problem defined in our paperassumes that the set of objectives is comprised of: 1) mini-mization of resource cost for each activity and 2) minimization

Manuscript received A ugust 22, 1995; revised January 1996. Review ofP. S. Pulat is with the School of Industrial Engineering, University ofS. J. Horn is with MPSI Systems, Inc., Tulsa, OK 74133 USA.Publisher Item Identifier S 00 I8-9391(96)08823-X.

this manuscript was arranged by Editor-in-Chief D. F. Kocaoglu.Oklahoma, Norman, OK 73019 USA.

of the project completion time. In order to solve the moregeneral problem defined [ l ] by our technique, one needsto define the objective concerning the minimization of therealization time of node k in terms of activity realizationtimes. They have suggested the use of ADBASE [9], amultiple objective linear programming (MOLP) code, to solvethe problem. Other researchers [3], [5]-[7] have suggestedgoal programming formulations for the project schedulingproblem. Rahman [8] describes another methodology for thetime-resource tradeoff problem which deals with the efficientsolutions in the objective spa ce rather than the decision space.The algorithm is limited to planar project networks since ituses the dual network of a given project network. Reference [SIalso introduces the co ncept of an efficient cut. While Ra hmansalgorithm is much faster than ADBASE it is much slowerthan the algorithm presented in our paper. Section V discussesthe computational comparisons of the two techniques. Thecomputational t ime required by our method o n a 100-nodeproblem is only 5 s.

In this paper, we present the time-resource tradeoff problemwith two resources or two groups of resources. The MOLPmodel of the problem has three objective functions, 21, 2 2 ,z3 which are the minimization of the total cost of resourceconsumption for each resource and the minimization of projectduration, t,,, respectively. The project completion time t ,varies between t,,(normal) and t,(min), which are the normaland the minimum project completion times, respectively. Letz = (zl,2 , z3) represent the objective function vector.The utility function U ( z l , 2 , ) : R2 + R1 is assumed tobe linear and hence defined by U ( z 1 , z 2 ) = Xz l + (I -X)zz with unknown parameters X an d (1 - A) values, 0 5X 5 1 for each z:j = t,. A methodology that utilizesGeoffrions P(X)approach [4] to determine the set of efficientsolutions for t,(min) 5 ,rL t,(normal) for 0 5 5 1 ispresented. An interactive version of the methodology is alsodiscu ssed. Computational results demonstrate the efficiency ofboth enumerative and interactive methods.

Section I1 introduces the time-resource tradeoff model andthe methodology used to determine all efficient solutions tothe problem. An illustrative example is given in Section 111.Section IV describes an interactive version of the methodol-ogy followed by computational results and comparisons inSection V. Conclusions are given in Section VI.

TI . TIME-RESOURCERADEOFF ODEIA project network of a given project consists of a set ofnodes indicating realizations of certain milestoncs 1, 2 ;and a set of arcs , A, referring to the tasks to be performedin the project. Each arc ( i ,j) E A refers to a unique task(activity). The project network is drawn so that all precedence

relationships among tasks are observed.0018-Y391/96$0S.00 0 996 IEEE

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412 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 41, NO . 4, NOVE MBE R 1996

1Activity(ij) a;i &

Fig. I . An example CP S

The resource cost and the project duration are assumed tohave a linear relationship. Each task ( i , j ) E A has associatedwith it a normal duration Mi3 and an allowable crash rangeSi,. The objective is to determine the set of efficient projectschedules for t, , where t , ,(min) 5 t,r,5 t,(normal). A projectschedule is efficient if there does not exist any other projectschedule Tor the same t,, with a better resource cost for allresources for a given range of values of A .

T he MOLP model for the time-resource tradeoff problemfo r k resources can be written as follows:

M i n z l = ~ : ~ y i ,M in zg = ~ ? ~ y ; j

( 2 , ~E

(",7)A

Min Z I ~ a{:yi,( i , ) -4

M in z K + l = t ,subject to

t j - i+ g ig 2Mfj for all (i, ) E Ayl , 5 S for all (2 , j ) E AY i j 20 for all ( i , ) t At i 2 0 for all i = l . . . - , n

wheret ,M734;P i3a!.

realization time of milestone (node) i ;normal duration of activity (i , ) ;allowable crash on activity ( 2, j ) ;amount that activity (i; J') is crashed;cost of additional unit of resource k required tocrash activity (i , ) by one unit,set of activities in the network;number of nodes in the network.

73

k = I ? , . " , K . ;A71

Objective functions 21 , . . , z~ minimize the cost of con-suming resources 1 through K while Z K + ~ minimizes the

Fig. 2 . The resource cost plots as a function of A.that is, it varies the value of t , from t,(normal) to t,(min),and generates the set of efficient solutions corresponding toeach t,. The variation of Z K + I from t,,(normal) to t,(min)follows the same process as in time-cost tradeoff approach.That is, cheape st set of activities are determined and crashed totheir limit, which in turn reduces the project completion time.The constraints preserve the precedence relationships betweenactivities and assure that activities will not be crashed beyondtheir l imits. The minimum project completion time is reachedwhen all activities on a path from node one to node n. havereached their allowable crash times.

Let {uij; (,L; j ) E A } be a solution to the above MOLPproblem with fixed t rL . he n z k = C(i, j) tAnfl ,y~j or all k .If the project manager's utility function is defined by

k = lwhere

KCXk=lk = l

then given {Ak ; k = 1, . . . . K } ne can determine the optimalduration of the project. The procedure enumerates on z ~ + 1 , schedule for the project manager. H owever, Xk's are generally

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PULAT AND HORN. TIMk. RESOURCh T R A D E O R PROBLEM 413

(4-A 2)(4

Fig. 3. Labeling and residual capacity illustrations.

/[4,71 \r L07=JJ -

Fig. 4. An example project network.

not known in a dvance. Then, one either generates a set of al lefficient solutions for 0 5 XI 5 1 for all k , or develops aninteractive procedure where the utility function of the projectmanager is implicitly refined as he/she is iteratively askedto indicate preference between schedule pairs. The solutionapproach uses Geoffrions P(X) method [4].The time-cost tradeoff algorithm is discussed in detail by[2]. One can also refer to Fulkersons work 11 I ] for the devel-opment of the technique. We will assume familiarity with thetime-cost tradeoff problem and extend it to the time-resourcetradeoff problem with two resources. Further extension to Kresources ca n be carried out by using interactive techniques torefine the A-space. The utility function fo r the decision ma kerfor the two resource time-resource tradeoff problem is given by

U ( X l , 2 ) = X X l + (I - X)zzwhere 0 5 X 5 1. The third objective function is X J = t,.Initially, all activity durations are set at normal levels andt,(normal) is computed. This schedule corresponding to t,, =t , (normal) is optimal for 0 5 X 5 1 with z1 = 0 an d252 = 0. The activity crash cost for activity ( i , j ) is givenby the function:

a&) = Xa$ + (1-

Once the critical path subnetwork (CPS) is determined, onecan use the maximum flow algorithm with uij(A) as the arccapacity for ( i , ) n order to find the cheapest set of arcsto crash which will reduce t , by one unit. The cheapest setof arcs to crash will be different for different X ranges. Forexample, suppose Fig. 1 represents the CPS for a given projectnetwork. The plots of ai3(A)are given in Fig. 2.Using theseplots, one can use the labeling procedure and determine theflow augmentings paths. After checking the arc capacities as afunction of A on Fig. 2, it is concluded that the two A-rangeswhich need to be considered are 0 5 X 5 1 and 2 5 X 5 1.For 0 5 X 5 i, he node labels and the updated arc capacitiesare shown in Fig. 3(a) and (b), respectively. The node labelfor node j s given as ( y 3 ; c) where q.j is the maximum flowthat can be delivered to node j from node one and is theimmediate predecessor node from which node j received itslabel. ( 3 A) units of flow are augmented along thc path 1-2-3-4 which saturates arc (3, 4). The labeling procedure on thenetwork of Fig. 3(b) leads to a flow of (3 - 4X) units alongthe path 1-2-4. Furthe r labeling results in a nonbreakthrough,indicating that arcs (2, 4) and (3, 4) are the cheapest set ofcritical a rcs to crash in orde r to expedite the project completiontime. Similarly, the procedure can be carried o ut for 5 X 5 1

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414 IEEE TRANSACTIONS ON ENGINEERING MANAGEMENT, VOL. 43, NO. 4, NOVE MBE R 1996

( 9 2 , j3,8)Fig. 5. Efficient solutions for the time-resource tradeoff problem

which will lead to arc (1, 2) being the cheapest arc to crash.The calculations which determine the amount by which tocrash the selected arcs are the same as the calculations for thetime-cost tradeoff technique described in [2].

Th e algorithm to determ ine the efficient t ime-resource trade-off curves as a function of X for t ,(min) < t,, < t,(normal)is explained next.

Algorithm 1-Enumeration:Step 0:

WhileStep 1:Step 2:

Step 3:

Initialization step. De term ine t,(normal), t,(min).Set all activity durations to normal durations. Sett,, = t , ( n o r in a l ) , A m i n = 0, A = 1.f ,, > t,,(min) for any X rangeDetermine the CPS.Using labeling procedure, determine the cheapestset of arcs to crash on the CPS fo r each A-range. LetAI < A 2 < X 3 < . . . < A, and define the A-rangeswith X I = A m i n an d A, = A, Let Ajj define these t of cheapest arcs to crash for Ai < X < A,.For each ( i >), se A,, to determine the maxim umcrash amount for t , and let & ( 2 , j) denote thisamount. Cra sh the activit ies in A,, by that amount.Determine the new value for t,,, tn(2, ) = t , ~S t , ( i , j), and the corresponding z l (2 ) an d z z ( 2 )values.

Step 4: For each ( 2 ; j), se t t,, = tT, ( i ; ), rn i t l = X i an dA = X j . Process steps 14 recursively for each(2 ; j ) individually (i.e., the A-ranges defined byeach ( i , ) pair may each be broken down intoseveral ranges of A, and each of those rangesbroken down further, in tum, until t , (min) isreached for each final individual A-range).

End while.Step 5 : Stop. The time-resource tradeoff curves fort , , (min) 5 t,rL t,, (normal) for all 0 5 X 5 1have been found.

111. AN EXAMPLEConsider the project network shown in Fig. 4. For each

activity ( i , ) , he normal duration Mij, he minimum durationMij - i; , and the crash costs for resource types 1 and 2 , u:;and a:,, respectively, are indicated on the figure. The normaland minimum project completion times are ts(norma1) = 17an d ts(rr i in) = 8, respectively. T he critical path is 1-2-3-5.For 0 5 A 5 0.5, ar c (3, 5 ) is the cheapest to crash. For0.5 5 X < 1, arc (2, 3) is the cheapest arc to crash. Arc (3,5 ) is crashed by one unit since path 1-4-5 then also becomescrit ical. The project completion time is t,, = 16. Similarly,arc (2, 3) is crashed by one unit before path 1-2-4 become s

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PULAT AND HORN: TIME-RESOURCE TRADEOFF PROBLEM

Number of Arcs I Sample S i x Enumc tive Method8 10 Data Set 1 Mean Std. Deviation0.007

seconds

Interactive MethodMean I Std. Deviation0.114 I 0.006

Algorithm 1(Enumerative Method)

123456789101112

415

0.0620.0680.0720.0630.0770.0900.0840.2340.4531.8194.6874.969

-I-RahmansMethod

SetFig. 6. Computational time comparisons.

Data Set12345678910I I12

TABLE IDESCRIPTIONF TESTPROBLEMSNumber of Nodes

5677811111520304050

TABLE I1CP U TIME IN SECONDS)

101112151523253565100100

1010

critical. One continues in this fashion until t , = 8. Fig. 5summarizes the iterations in a tree diagram. Once x1, 22 , x3values are determined f or all efficient solutions at xg = t , , onecan check to see if the sam e solution was generated m ore thanonce. If that is the case, the two solutions can be merged bytaking the union of ihe individual A-ranges as show n in Fig. 5.IV. AN INTERACTIVEIME-RESOURCERADE-OFF ETHOD

The interactive version also allows the project manager toinput a de5ired completion time since he/she may not wantto incur the cost of reduction to the minimum. This versionof Algorithm 1 presents the project manager two efficientsolutions for the same t , and asks him he r for the preference.Assuming that the project manager is a rational person andmakes consistent decisions, one can refine the A-range basedon the given answer. The modification over the previousalgorithm is that the While loop is only fo r t,, > t , (rnin)

0.0090.0100.0120.0160.0060.0170.0860.2230.65 11.4971.623

0.1150.1 160.1180.1200.1210.1250.1260.1260.1410.1560.184

0.0070.0060.0090.0070.0050.0080.0070.0070.0100.0090.045

(o r t , > t,, (desired) if the project managers desired time isgreater than the minimum achievable). Step 3 is still insertedafter Step 3 and Step 4 still replaces Step 4.Step 3: Ask the project manager to select one of the twosolutions defined by [z1(1),zz(l), 231 for A1