a mixed (continuous + discrete) time-cost trade-off model considering four different relationships...

KSCE Journal of Civil Engineering (2013) 17(2):281-291DOI 10.1007/s12205-013-1506-3

− 281 −

www.springer.com/12205

Construction Management

A Mixed (Continuous + Discrete) Time-Cost Trade-Off ModelConsidering Four Different Relationships with Lag Time

Jaeho Son*, TaeHoon Hong**, and Sangyoub Lee***

Received March 8, 2011/Revised June 12, 2012/Accepted July 6, 2012

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Abstract

Numerous Time-Cost Trade-Off (TCTO) models have been developed to identify the best combination of time and cost in a criticalpath network. Although there are four relationships in the critical path network, most of the models developed so far have consideredonly the Finish-Start (F-S) relationship. Thus, an advanced TCTO model that considers all four relationships between activities wasdeveloped in this study to accurately present a project, and is presented in this paper. The model also takes into account the lag timebetween activities. Previous TCTO models minimize the total project cost based on the given crash scenario. Moreover, to enhancethe practicality, the model was developed to work with various TCTO scenarios such as continuous, discrete, and even mixed(continuous + discrete). The combined scenario reflects the most realistic situation. Two independent scenarios cannot be combinedwithout mathematical modifications since a rudimentary mixing of two scenarios may provide an incorrect solution. A newformulation technique is introduced to merge the two independent scenarios mathematically and it guarantees the optimal solution.Keywords: Time-Cost Trade-Off, critical path method, optimization models, linear programming, integer programming, costs

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1. Introduction

Crash is defined as shortening the duration of an activity toshorten a project’s duration. It is accomplished using extraresources and overtime (Stevens, 1990). In general, if a crash isused, the indirect cost is reduced since the crash shortens theproject duration. On the other hand, the direct cost increases dueto the extra resources and overtime (Moder et al., 1995). Thistrade-off between time and cost has an important role in findingan optimal schedule for producing the minimum total cost of theproject. This problem is known as the Time-Cost Trade-off (TCTO)problem. Numerous TCTO models have been developed toidentify the best combination of time and cost in a critical pathnetwork. These models minimize the total project cost based onthe given crash scenario.

Although there are four relationships in a critical path network,most of the models developed so far consider only the Finish-Start relationship. Some TCTO models (Adeli and Karim, 1997;Ezeldin and Soliman, 2009) consider other relationships in theirmodel, however, consider only the continuous TCTO scenario.Thus, a simple and flexible TCTO model that considers all fourrelationships and lag times between activities was developed inthis study to accurately present various and practical TCTO projectscenarios. This model is presented in this paper. There are twotypical TCTO model scenarios: the continuous and the discrete

TCTO model scenarios. These models will be explained in detailin the following section. Also, a mixed (hybrid) model (discrete+ continuous) could exist to practically represent the project. Forexample, each alternative in the discrete TCTO model could havea continuous TCTO model. Some previous TCTO models coveredthis mixed TCTO scenario (Liu et al., 1995; Moussourakis andHaksever, 2004; Yang, 2007). None of them, however, considersall four relationships between activities, which are important torepresent the practical network scheduling.

To enhance the practicality, the model was developed to workwith various TCTO scenarios such as continuous, discrete, andeven mixed TCTO scenarios. The combined scenario reflects themost realistic situation. By simply combining two independentscenarios, it may provide an incorrect solution. A new formulationtechnique is required to merge them mathematically in order toguarantee the optimal solution. This feature is presented in thispaper.

2. Literature Review of Existing TCTO Models

2.1 Existing TCTO Model based on the Continuous Sce-narios

The continuous TCTO model assumes that each activity has anormal and a crash duration. The normal duration is the durationof completion of an activity in normal conditions, and the crash

*Member, Associate Professor, School of Architectural Engineering, Hongik University, Chungnam 339-701, Korea (E-mail: [email protected])**Associate Professor, Dept. of Architectural Engineering, Yonsei University, Seoul 120-749, Korea (Corresponding Author, E-mail: [email protected])

***Member, Associate Professor, Dept. of Real Estate Studies, Konkuk University, Seoul 143-701, Korea (E-mail: [email protected])

Jaeho Son, TaeHoon Hong, and Sangyoub Lee

− 282 − KSCE Journal of Civil Engineering

duration is the duration of the shortening of the activity to itsminimum practicable duration. The normal and crash durationsalso have corresponding costs, which are the Normal Cost and theCrash Cost. These assume that time and cost have a continuousrelationship (Kerzner, 2009). The objective of the continuousmodel is to identify the optimal duration between the crash andnormal durations to minimize the total project cost. Fig. 1 showsa typical example of a linear continuous relationship betweentime and cost. This assumption is reasonable because moreresources and crews reduce the normal duration but increase thecost. Many previous models were developed based on continuousTCTO scenarios (Baker, 1997; Li and Love, 1997; Leu andYang, 1999; Li et al., 1999; Que, 2002).

2.2 Existing TCTO Model based on the Discrete ScenarioThe discrete TCTO model assumes that each activity in the

project has several alternative construction methods, and thateach method has its own duration and cost (Feng et al., 1997;Demeulemeester, 1998; Hegazy and Ayed, 1999; Leu et al.,1999; Zheng et al., 2004; Chen and Weng, 2009; Eshtehardian etal., 2009). This assumption is justified because an activity canhave alternatives to be finished with different methods, crews,equipment, etc. Of course, each alternative has its correspondingcost and duration. The discrete TCTO model identifies the optimalcombination of the construction methods for each activity inorder to minimize the total construction cost. Fig. 2 shows atypical example of a discrete relationship between time and cost.

2.3 Existing and Improved TCTO Model based on theMixed Scenario

Liu et al. (1995) used the mixed scenario in their TCTOmodel. In their paper, they used the following table to show thecombination of the continuous and discrete scenarios. Fig. 3 andTable 1 show the time-cost relationship for activity 2. This meansthat activity 2 can be completed in (1) anywhere between 15 and18 days; (2) 20 days exactly; or (3) between 23 and 25 days.

Also, Moussourakis and Haksever (2004) used various mixedscenarios in their TCTO model. Fig. 4 shows one of the mixedcrash scenarios that they used in their paper.

Yang (2007) used similar mixed scenarios that are basicallyvariations of previous mixed scenarios.

Although they used various mixed crash scenarios, the rangesof the duration and cost of one alternative did not overlap withthose of the other alternatives. In other words, each alternativehad its own specific range of duration and cost and did not share

Fig. 1. Continuous Case Fig. 2. Discrete Case

Fig. 3. Mixed Scenario I in the Existing TCTO Models

Table 1. Mixed Scenario I used in the Existing TCTO ModelsActivity

description(1)

ActivityNo.(2)

Option(3)

Duration(days)

(4)

Cost(dollars)

(5)

Sloperelationship

(6)Forms and rebar 2 Method 1 15 3,000 CForms and rebar 2 Method 2 18 2,400 -Forms and rebar 2 Method 3 20 1,800 DForms and rebar 2 Method 4 23 1,500 CForms and rebar 2 Method 5 25 1,000 -

A Mixed (Continuous + Discrete) Time-Cost Trade-Off Model Considering Four Different Relationships with Lag Time

Vol. 17, No. 2 / March 2013 − 283 −

it with other alternatives. In reality, however, different alternativescould share ranges of duration and cost. In other words, morethan one alternative could have the same cost in the specificrange of the duration, or vice versa. This new mixed crash scenariois shown in Fig. 5 with the overlapping areas. Thus, this new mixedscenario is more versatile than the previous mixed scenarios. Themodel presented in this paper uses the new mixed TCTOscenario. The advantage of using this new mixed scenario is thatit can also represent all the previous mixed scenarios.

3. Model Formulation for Various TCTO Scenar-ios

Complete project scheduling network information could berepresented with the sum of a unit that consists of a precedingactivity, a succeeding activity, and the relationship between theactivities. The unit is called a network-unit in this paper, and isshown in Fig. 6. Fig. 6 illustrates a network-unit based on all the

possible relationships with the lag time.Activity A precedes activity B, and Activity B succeeds activity

A. Except for the first and last activities, all the succeedingactivities precede some activities in the scheduling network.Therefore, if the formulation of the network-unit is completed,the entire network can be fully formulated. In this section, theinformation on the preceding activity, the succeeding activity,and the relationship between the activities is used to formulatethe network-unit using mixed integer programming. The objectivefunction of the TCTO model is to minimize the total project cost.The total project cost consists of direct costs and indirect costs.The direct costs can be calculated based on the crash scenariosand the indirect costs are generally proportional to the period ofthe project. Constraints are used to maintain the network logicand requirements. The following section will explain how toformulate the TCTO model when various crash scenarios are used.

3.1 TCTO Model for Continuous ScenarioThe objective function for the continuous TCTO model is as

follows:

(1)

Where NC_Acti is the normal cost of activity i; CS_ Acti is thecost slope of activity i; CR_ Acti is the number of days of thecrash of activity i; PD is the project duration; and IDC is theindirect cost rate ($/day) that is generally given.

The modification of the objective function is not required whendifferent relationships between activities are used. However, theconstraints have to be modified for each different relationship.The modifications of the constraints for four different relationshipsare as follows.

Min NC_Actii 1=

n

∑ CS_Actii 1=

n

∑+ CR_Acti× IDC+ PD×

Fig. 4. Mixed Scenario II used in the Existing TCTO Models

Fig. 5. Mixed Scenario in the Proposed TCTO Model

Fig. 6. Four Relationships with Lag Time



3.1.1 Finish-Start (F-S)Compression (crashing) of an activity itself does not determine

the starting time but the finishing time of the activity. Thus, in theF-S relationship, the compression of the preceding activity’sduration affects the starting time of the succeeding activity. Thefollowing constraints have to be added:

(2)(3)

Where S_A and S_B are the starting times of activities A andB, respectively; D_A is the duration of activity A; Lag_A_B isthe lag (lead) time between activities A and B; ND_A is thenormal duration of an activity; CD_A is the crash duration ofactivity A; and Lag time delays the start time of the succeedingactivity.

3.1.2 Start-Start (S-S)The compression of the preceding activity’s duration does not

affect the starting time of the succeeding activity. The followingconstraint has to be added:

(4)

As seen from the preceding constraints, the crash of activity Adoes not affect the scheduling of activity B in the case of the S-Srelationship. The lag time delays the starting time of the succeed-ing activity.

3.1.3 Finish-Finish (F-F)The compression of the duration of the preceding and succeed-

ing activities affects the finishing time of the succeeding activity.The following constraints have to be added:

(5)(6)(7)

As seen from the aforementioned constraints, the crash ofactivities A and B does not affect the scheduling of activity B inthe case of the F-F relationship. The lag time delays the finishingtime of the succeeding activity.

3.1.4 Start-Finish (S-F)The compression of the duration of the preceding activity does

not affect the finishing time of the succeeding activity. The com-pression of the succeeding activity itself affects its finishing time.The following constraints have to be added:

(8)(9)

The lag time delays the finishing time of the succeeding activity.

3.2 TCTO Model for Discrete ScenarioThe objective function for the discrete TCTO model is as

follows:

(10)

Where n is the number of activities in the network; mi is thenumber of alternatives of activity i; and Xik is a binary variable towhich 0 or 1 is assigned. The meaning of Xik = 1 is that construc-tion method (alternative) k was chosen for activity i; and C_Actik

is the budget cost when the alternative k is chosen for activity i.Again, the modification of the objective function is not required

when different relationships are used with the discrete TCTOscenario. However, the constraints have to be modified for eachdifferent relationship. The modifications of the constraints forfour different relationships are as follows.

3.2.1 Finish-StartThe following constraints have to be added:

(11)

(12)

Where mA is the number of alternatives of activity A and XAk isa binary variable. The meaning of XAk = 1 is that the kth

construction method (alternative) is chosen for activity A. Themeaning of is that only one construction alternative isallowed for activity A. D_Ak is the duration of activity A whenthe kth alternative is chosen.

3.2.2 Start-StartThe following constraint has to be added:

(13)

3.2.3 Finish-FinishThe following constraints have to be added:

(14)

(15)

(16)

3.2.4 Start-FinishThe following constraints have to be added:

(17)

(18)

3.3 TCTO Model for Mixed ScenarioAs mentioned in the introduction, it is difficult to represent an

S_B S_A≥ D_A CR_A– Lag_A_B+ +CR_A ND_A CD_A+≤

S_B S_A Lag_A_B+≥

S_B D_B CR_B+ + S_A≥ D_A CR_A– Lag_A_B+ +CR_A ND_A CD_A–≤CR_B ND_B CD_B–≤

S_B D_B CR_B–+ S_A≥ Lag_A_B+CR_B ND_B CD_B–≤

Min C_Actikk 1=

mi

∑i 1=

n

∑ Xik× IDC+ PD×

S_B S_A D_Ak XAk×( )k 1=

mA

∑ Lag_A_B+ +≥

XAkk 1=

mA

∑ 1=

XAkk 1=

mA

∑ 1=

S_B S_A Lag_A_B+≥

S_B D_Bk XBk×( )k 1=

mB

∑+ S_A D_Ak XAk×( )k 1=

mA

∑ Lag_A_B+ +≥

XAkk 1=

mA

∑ 1=

XBkk 1=

mB

∑ 1=

S_B D_Bk XBk×( )k 1=

mB

∑+ S_A Lag_A_B+≥

XBkk 1=

mB

∑ 1=


Vol. 17, No. 2 / March 2013 − 285 −

activity in the network using only discrete or continuous TCTOscenarios. Also, the previous mixed scenario did not cover somepractical TCTO scenarios. Thus, a new representation of theactivity in the TCTO model is introduced in this section. As seenin Fig. 5, activity A combines the discrete case with the continuouscase. Activity A has three alternative construction methods, eachof which has a normal duration, a normal cost, a crash duration, acrash cost, and a crash cost rate. In other words, each alternativehas a continuous crash scenario. The following objective functionand constraints were formulated to combine the discrete case andthe continuous case. The objective function for the mixed TCTOmodel is as follows:

(19)

Where NC_Actik is the kth alternative’s normal cost of activity i;CS_Actik is the cost slope of the kth alternative of activity i; andCR_ Actik is the number of days of the crash of activity i when thekth alternative is selected.

Again, the modification of the objective function is not requiredwhen different relationships are used with the mixed TCTOscenario. Unlike the continuous and discrete scenarios, however,the mixed scenario has not only a relationship constraint but alsosome special constraints. The special constraint is not changedaccording to the relationships between the activities. The constraintensures that when one alternative is selected, the crashing scenariosof the other alternatives are automatically aborted. The constraintis expressed as follows:

(20)

, (21)

(22)

where k = 1, 2, …, m; m is the number of alternatives of activityA; M is a big number where M = 1,000; where XAk ={0, 1}; and YAk = {0, 1}.

Activity A in Fig. 5 was utilized to help explain equations (20)to (22). The equations when activity A in Fig. 5 is used are asfollows:

CR_A2 + CR_A3 − 1000YA1 ≤ 0 (23)CR_A1 + CR_A3 − 1000YA2 ≤ 0 (24)CR_A1 + CR_A2 − 1000YA3 ≤ 0 (25)XA1 + 1000YA1 ≤ 1000 (26)XA2 + 1000YA2 ≤ 1000 (27)XA3 + 1000YA3 ≤ 1000 (28)

Assume that the first alternative was chosen in the process ofinteger programming. If the first alternative is selected, the valueof XA1 is one. If XA1 = 1, the value of YA1 never becomes one tosatisfy the constraint (26). In other words, the value of YA1 mustbe zero. Consequently, CR_A2 and CR_A3 must both be zero to

satisfy the constraint (23). The constraints ensure that when onealternative is chosen, the crashing scenarios of the other alternativesare automatically neglected. When the second or third alternativeis selected, the same logic is applied.

However, the constraints for the relationship have to bemodified for each different relationship. The modifications of theconstraints for four different relationships are as follows.

3.3.1 Finish-StartThe following constraints have to be added:

(29)

(30)

Where CR_ Ak is the number of days of the crash of activity Awhen the kth construction method (alternative) is chosen; ND_Ak

is the normal duration of activity A when the kth constructionmethod (alternative) is selected; and mA is the number ofalternatives of activity A.

3.3.2 Start-StartThe following constraint has to be added:

(31)

3.3.3 Finish-FinishThe following constraints have to be added:

(32)

(33)

(34)

3.3.4 Start-FinishThe following constraints have to be added:

(35)

(36)

4. Example: Comprehensive TCTO Model For-mulation

This section shows how a given scenario is easily formulatedand how the model produced a viable solution from the generalnetwork description. The sample precedence diagram in Fig. 7will be used as an example. The activity time, cost data, andcrash scenario are provided in Table 2. The crash scenario iseasily interpreted by the project manager, who must have a basicunderstanding of the precedence diagram and Critical Path

Min NC_Actikk 1=

mi

∑i 1=

n

∑ Xik× CS_Actikk 1=

mi

∑i 1=

n

∑ CR_Actik× I+ DC+ PD×

CR_Aii 1=

k 1–

∑ CR_Aii k 1+=

m

∑ M–+ YAk× 0≤

CR_Aii 1=

0

∑ 0= CR_Aii m 1+=

m

∑ 0=

XAk M+ YAk× M≤

XAkk 1=

m

∑ 1=

S_B S_A≥ ND_Ak XBk× CR_Akk 1=

mA

∑ Lag_A_B+–k 1=

mA

∑+

XAkk 1=

mA

∑ 1=

S_B S_A Lag_A_B+≥

S_B ND_Bk XBk×( )k 1=

mB

∑ CR_Bkk 1=

mB

∑–+ ≥

S_A ND_Ak XAk×( )k 1=

mA

∑ CR_Akk 1=

mA

∑– Lag_A_B+ +

XAkk 1=

mA

∑ 1=

XBkk 1=

mB

∑ 1=

S_B ND_Bk XBk×( )k 1=

mB

∑ CR_Bkk 1=

mB

∑–+ S_A Lag_A_B+≥

XBkk 1=

mB

∑ 1=



Method (CPM). It has nine activities and uses all four relation-ships and lag times. The relationships and the lag time betweenthe activities are shown in Table 3. Also, the crash scenarios foreach activity are shown in Table 3. The continuous, discrete, andmixed TCTO scenarios are included in this example to show all themodules of the formulation. The proposed model is programmedand tested on the sample project. Here, the optimization model isdemonstrated with the example to show the details of theproposed optimization model. The complete objective function

and constraints for Example 1 are listed in Appendix I. The totalcost of the sample project has two components: direct and indirectcosts. The direct cost of each activity can be calculated based onthe crash scenario shown in Table 2. The indirect cost is generallyrepresented as a single cost per time period (e.g., day or week).In this example, the indirect cost is assumed to be $2,500/day.This assumption is made since the indirect cost of the projectvaries depending on the project’s size, type, region, etc. There isno fixed cost for the indirect cost. The indirect cost of $2,500/dayis chosen to experiment and prove the model with examples inthis paper. Nevertheless, in the proposed model, the indirect costcan be easily modified according to the requirements of the project.In this paper, due to financial reasons, the computer program thatproduced the optimal solution for the TCTO model could not bedeveloped. In this research, however, if the information on thecrash scenarios and the relationships between the activities weregiven, all the procedures for formulating the objective functionand the constraints of the TCTO model were developed andsuggested in detail. If the project manager has a basic understand-ing of mixed integer programming, he/she can set up the specificTCTO model based on the proposed model and solve the problemwith a commercial linear programming package that includes theinteger-programming module. LINDOTM was used for programcoding and to solve the sample TCTO problem. Thus, the develop-ment of the procedure for providing an efficient and practicalmeans of solving construction TCTO problems was regarded asmeaningful. If this idea will be used in model development, a prac-tical TCTO add-on to commercial project management software,such as P3 or MS Project, can be programmed in the future.

5. Analysis of Results: Optimal Solution for theExample

The optimal solution is summarized as follows.The minimized total cost is $141,500 and the corresponding

project duration is 28 days. The specific decisions that must bemade for each activity to produce the optimal solution aresummarized in Table 4.

Fig. 7. Network Diagram of the Example 1

Table 2. Project Information on the Example

ActivityNormalduration(days)

Normal cost($)

Crash duration(days)

Crashcost($)

Crash cost rate

($/day)A↑ 6 7,000 3 9,400 800

B↑↑↑

B1 8 5,000 5 8,000 1,000B2 10 8,000 8 9,200 $600B3 15 5,000 12 7,100 $700B4 18 4,000 14 6,000 $500

C↑ 10 8,000 6 14,000 $1,500

D↑↑↑

D1 6 5,000 5 6,000 $1,000D2 8 3,000 5 5,700 $900D3 10 1,000 4 5,800 800

E↑↑↑

E1 16 10,000 11 17,500 $1,500 E2 14 8,000 10 15,000 $1,750 E3 15 8,000 12 12,500 $1,500 E4 10 16,000 8 20,000 $2,000

F↑↑

F1 3 10,000 NA NA NAF2 6 8,000 NA NA NAF3 8 6,000 NA NA NAF4 10 5,000 NA NA NA

G↑↑↑

G1 10 6,000 6 12,000 $1,500G2 12 7,000 9 10,000 $1,000G3 8 8,000 4 13,000 $1,250

H↑ 5 10,000 2 16,000 2,000

I↑↑I1 9 9,000 NA NA NAI2 7 13,000 NA NA NA

Note: ↑Activities A, C, and H have a continuous TCTO scenario. ↑↑Activi-ties F and I have a discrete TCTO scenario. ↑↑↑Activities B, D, E, and Ghave a mixed (discrete + continuous) TCTO scenario.

Table 3. Relationships and Lag Times

Activity Successors Relationship Lag time(days)

Crashscenario

AB F-S 2

ContinuousC S-S 0D F-S 0

B F F-F 4 MixedC E F-S 0 ContinuousD G F-F 3 Mixed

EG S-S 5

MixedH F-F 3

F I F-S 0 DiscreteG I F-S 0 MixedH I F-S 0 ContinuousI None N/A N/A Discrete


Vol. 17, No. 2 / March 2013 − 287 −

The interpretation of Table 4 is as follows. Activity A’s normalduration (6 days) is used. For activity B, alternative 4 is chosen,and its normal duration (18 days) is used. For activity C, 4 daysare crashed from the normal duration (10 days). Thus, the newduration (6 days) is used. For activity D, alternative 3 is selected,and its normal duration (10 days) is used. For activity E, alterna-tive 2 is chosen, and 2 days are crashed from the normal duration(14 days). Thus, the new duration (12 days) is used. For activityF, alternative 4 is chosen, and its duration (10 days) is used. Foractivity G, alternative 1 is chosen, and its normal duration (10days) is used. For Activity H, its normal duration (5 days) isused. For activity I, alternative 2 is chosen, and its duration (7days) is used.

Parameters of the proposed model are as follows:1. Indirect cost2. Number of alternatives of each activity3. Cost slope that is determined by normal cost, normal duration,

crash cost, and crash durationAll the parameters are dependent on the specific network

diagram, the company’s competency, and the resource available.In particular, b and c are the parameters related to each activitynot directly to the project duration. Network scenarios expressedin Table 2 show the range of those parameters indirectly. Thus,this example has different ways of sensitivity analysis with thoseparameters already because the decision-making process isimplemented from the range of those parameters. Indirect cost isthe parameter related to project duration, not to each activity.Sensitivity analysis with indirect cost is carried out. The schedulethat produces the optimal solution can be changed when theindirect cost is changed. Thus, a sensitivity analysis, whichchanges the indirect cost rate, must also be implemented to seethe range of the project duration with the example and the givenscenario. The longest and shortest project durations were foundfrom the sensitivity analysis, and their values are 36 days and 24days, respectively. The longest project duration is 36 days, andthe shortest project duration is 24 days. If the indirect cost rate ismore than or equal to $4,550/day, the shortest project duration isthe optimal schedule. If the indirect cost rate is less than or equalto $1,450/day, the longest project duration is the optimal schedule.The specific decisions that must be made for each activity toproduce those optimal schedules are summarized in Table 5.

6. Validation of Proposed Model

The project duration ranges from a minimum of 24 days to amaximum of 36 days. Firstly, authors identify all possible projectschedules that generate the maximum duration (i.e. 36 days), andmanually calculate the total cost of each schedule. The total costsare compared, and the least-cost schedule is marked. Authorsiterate the procedure until the specific project duration reachesthe minimum of 24 days. Table 6 summarizes the marked (least-cost) schedules for each project duration day. In detail, the 2nd

column in Table 6 shows each selected activity and its number ofdays that crashed. For example, E4(2) represents that Activity E4was selected and crashed for 2 days. As seen in Table 6, theoptimal total cost is $141,500, when the project length is 28 days.The corresponding schedule scenario is {A(0), B4(0), C(4), D3(0),E2(2), F4(0), G1(0), H(0), I2(0)}, which is identical to the optimalplan from the proposed model. Since the minimum cost schedulefrom Table 7 is the same as the optimal one from the proposedmodel, the proposed model is further validated.

The purpose in using a simplified sample network is to clearlyshow all the formulations for possible crash scenarios in regardsto the Time-Cost tradeoff problem. Since all the potential modularformulations have been established, the model can easily beapplied to the expanded project example. For instance, a medium-sized network project (See Fig. 8, Table 7, and Table 8) showsthe expandability of the model. The proposed model is pro-grammed and then simulated to find the optimal schedule for theproject network. The same indirect cost rate ($2,500/day) is usedas in the previous example. The summary of the optimal solutionis as follows:

The minimum total cost is $400,900 and the correspondingproject duration is 54 days. The specific details in order for eachactivity to produce the optimal result are {A2(6), B2(0), C(5),D2(1), E1(0), F(0), G3(0), H(3), I1(4), J2(0), K3(0), L(6), M2(0),N3(0), O(0), P1(0)}.

7. Conclusions

This paper presented a mixed integer programming TCTO

Table 4. Optimal Schedule of the Example

Activity Decision based onthe discrete scenario

Decision based onthe continuous scenario

Crashscenario

A N/A CR_A=0 ContinuousB XB4=1 CR_B4=0 MixedC N/A CR_C=4 ContinuousD XD3=1 CR_D3=0 MixedE XE2=1 CR_E2=2 MixedF XF4=1 N/A DiscreteG XG1=1 CR_G1=0 MixedH N/A CR_H=0 ContinuousI XI2=1 N/A Discrete

Table 5. Shortest and Longest schedules of the example

Activity

Shortest project duration Longest project durationDecision based on the discrete

scenario

Decision based on the continu-

ous scenario

Decision based on the discrete

scenario

Decision based on the continu-

ous scenarioA N/A CR_A=2 N/A CR_A=0B XB4=1 CR_B4=0 XB4=1 CR_B4=0C CR_C=4 CR_C=0D XD3=1 CR_D3=0 XD3=1 CR_D3=0E XE4=1 CR_E4=2 XE2=1 CR_E2=0F XF4=1 N/A XF4=1 N/AG XG3=1 CR_G3=2 XG1=1 CR_G1=0H N/A CR_H=0 N/A CR_H=0I XI2=1 N/A XI1=1 N/A



model that considers all the possible relationships betweenactivities and is based on the hybrid (continuous + discrete) TCTOscenario. Those features enable the proposed model to representthe project more realistically.

The developed model provides a framework for formulationwith which a project manager with basic knowledge of integerprogramming can solve various TCTO problems that consist ofpractical TCTO scenarios. The model presented in this paper canbe a viable option for project managers over conventional modelsthat use sophisticated algorithms, because the proposed modeluses a relatively simple formulation. Project managers who havebasic knowledge of integer programming and related softwarecan set up the model for their project without any difficulty. Asprevious researchers mentioned in their papers, however, mixedinteger programming models have significant computing demand(Feng et al., 2000; Leu et al., 2001; Moussourakis and Haksever,2004). Therefore, small- to medium-sized projects are moresuitable for this model. Fortunately, since computing time andmemory capacity have been significantly enhanced and techniquesfor decomposing the project network have been studied (Taha,2006), the proposed model would be able to handle biggerprojects in the future.

Using an example, the model formulation and the analysis ofthe resulting solution were presented. The output can be easilyinterpreted in the form of the precedence diagram scheduling andcan help a project manager make a viable decision for the successof a project. Sensitivity analysis was implemented to find thepossible range of the project duration, which includes the longestand shortest durations of the project. This information is alsoimportant for project managers to be able to effectively manage aproject. Iteration of finding possible schedules and calculatingthe corresponding total costs was implemented to validate theproposed model. The minimum cost schedule from the iterationis the same as the optimal one from the proposed model.Additionally, the expandability of the model was tested with themedium sized project.

The automated input system was investigated and proven to

Table 6. Total Cost & Minimum Cost Schedules for Possible Project DurationsProject Duration Minimum Cost Scenario Total Cost

36 ↑A(0), B4(0), C(0), D3(0), E2(0), F4(0), G1(0), H(0), I1(0) 148,00035 A(0), B4(0), C(1), D3(0), E2(0), F4(0), G1(0), H(0), I1(0) 147,00034 A(0), B4(0), C(2), D3(0), E2(0), F4(0), G1(0), H(0), I1(0) 146,00033 A(0), B4(0), C(3), D3(0), E2(0), F4(0), G1(0), H(0), I1(0) 145,00032 A(0), B4(0), C(4), D3(0), E2(0), F4(0), G1(0), H(0), I1(0) 144,00031 A(0), B4(0), C(4), D3(0), E2(1), F4(0), G1(0), H(0), I1(0) 143,25030 A(0), B4(0), C(4), D3(0), E2(2), F4(0), G1(0), H(0), I1(0) 142,50029 A(0), B4(0), C(4), D3(0), E2(1), F4(0), G1(0), H(0), I2(0) 142,25028 A(0), B4(0), C(4), D3(0), E2(2), F4(0), G1(0), H(0), I2(0) 141,50027 A(0), B4(0), C(4), D3(0), E2(3), F4(0), G1(1), H(0), I2(0) 142,25026 A(0), B4(0), C(4), D3(0), E2(4), F4(0), G3(0), H(0), I2(0) 142,00025 A(1), B4(0), C(4), D3(0), E4(1), F4(0), G3(1), H(0), I2(0) 144,55024 A(2), B4(0), C(4), D3(0), E4(2), F4(0), G3(2), H(0), I2(0) 146,100

Note: ↑Selected activity (number of days crashed)

Table 7. Project Information of Example 2

ActivityNormalDuration

(days)

NormalCost($)

CrashCost($)

CrashDuration

(days)

Crash CostRate

($/day)

AA1 10 20000 24000 8 $2,000A2 12 18000 24000 6 $1,000

BB1 9 10000 11500 6 $500B2 10 8000 10000 8 $1000B3 15 5000 8600 12 $1200

C 10 8000 12000 5 $800

DD1 6 5000 6000 5 $1000D2 8 3000 5700 5 $900

EE1 12 15000 N/A N/A N/AE2 15 12000 N/A N/A N/A

F 12 10000 12000 8 $500

GG1 10 12000 N/A N/A N/AG2 12 10000 N/A N/A N/AG3 15 8000 N/A N/A N/A

HH1 12 20000 N/A N/A N/AH2 16 15000 N/A N/A N/AH3 18 12000 N/A N/A N/A

II1 18 20000 26000 14 1500I2 24 16000 22400 16 800

JJ1 12 28000 34000 8 1500J2 16 23000 30500 10 1250J3 20 20000 26400 12 800

KK1 14 35000 N/A N/A N/AK2 18 32000 N/A N/A N/AK3 20 30000 N/A N/A N/A

L 16 20000 32000 10 2000

MM1 8 24000 27000 6 1500M2 12 20000 35000 6 2500

NN1 16 25000 30000 14 2500N2 20 22000 35500 14 2250N3 25 20000 40000 15 2000

O 12 15000 21000 8 1500

PP1 12 10000 NA NA NAP2 16 8000 NA NA NA


Vol. 17, No. 2 / March 2013 − 289 −

enable a user to generate a lengthy formula with simple input re-quirements and without any error. A Windows-based Java pro-gram is under consideration to enhance the user interface of themodel. The proposed model did not consider nonlinear relation-ships and multiple criteria at the time due to various circumstances.Future research could involve the non-linear relationships betweenthe cost and the time in the proposed TCTO model. 4D dynamicmanagement model including resource utilization and multi criteriahas been developed in other civil engineering areas (Wang et al.,2004; Zhao et al., 2006). Resource utilization can be added to theproposed model for multi-objective function for future research.This could make the model more versatile. The decompositiontechnique of the project network could also be investigated toreduce the complexity of the computation. Genetic algorithmand fuzzy technique (Cheng et al., 2002) can also improve theproposed model to enhance the efficiency of the computation.

Acknowledgements

This work was supported by 2011 Hongik University ResearchFund.

References

Adeli, H. and Karim, A. (1997). “Scheduling/cost optimization andneural dynamics model for construction.” J. Constr. Engrg. andMgmt., Vol. 123, No. 4, pp. 450-458.

Baker, B. M. (1997). “Cost/time trade-off analysis for the critical pathmethod: A derivation of the network flow approach.” J. OperationalRes., Vol. 48, No. 12, pp. 1241-1244.

Chen, P. H. and Weng, H. (2009). “A two-phase GA model for resourceconstrained project scheduling.” Automation in Construction, Vol.

Fig. 8. Network Diagram of the Example 2

Table 8. Relationships and Lag Times (Example 2)

Activity Successors Relationship Lag Time(days)

CrashScenario

AB S-S 3

MixedC F-S 0D F-S 0

B E F-S 0 Mixed

CE F-S 0

ContinuousF F-F 4

DG F-S 0

MixedJ F-S 5

EH S-S 0

DiscreteK F-F 5

FH F-S 0

ContinuousI F-S 0J S-S 0

GL S-S 4

DiscreteM F-S 0

HL S-S 0

DiscreteN F-F 0

I L F-S 0 Mixed

JL F-S 0

MixedM F-F 0

K N F-S 0 Discrete

LN F-S 0

ContinuousP F-S 3

MO S-S

MixedP F-S

N P F-S 0 MixedO P N/A N/A ContinuousP - - - Discrete



18, No. 4, pp. 485-498.Cheng, C. T., Ou, C. P., and Chau. K. W. (2002). “Combining a fuzzy

optimal model with a genetic algorithm to solve multi-objectiverainfall-runoff model calibration.” Journal of Hydrology, Vol. 268,Nos. 1-3, pp. 72-86.

Demeulemeester, E., De Reyck, B., Foubert, B., Herroelen, W., andVanhoucke, M. (1998). “New computational results on the discretetime/cost trade-off problem in project networks.” J. OperationalRes.,Vol. 49, No. 11, pp. 1153-1163.

Eshtehardian, E., Afshar, A., and Abbasnia, R. (2009). “Fuzzy-basedMOGA approach to stochastic time–cost trade-off problem.” Auto-mation in Construction, Vol. 18, No. 5, pp. 189-198.

Ezeldin, S. A. and Soliman, A. (2009). “Hybrid time-cost optimizationof nonserial repetitive construction projects.” J. Constr. Engrg. AndMgmt., Vol. 135, No. 1, pp. 42-55.

Feng, C. W., Liu, L., and Burns, S. A. (1997). “Using genetic algorithmsto solve construction time-cost trade-off problems.” J. Comp. in Civ.Engrg., Vol. 11, No. 3, pp. 184-189.

Feng, C. W., Liu, L., and Burns, S. A. (2000). “Stochastic constructiontime-cost trade-off analysis.” J. Comp. in Civ. Engrg., Vol. 14, No. 2,pp. 117-126.

Hegazy, T. (1999). “Optimization of construction time-cost trade-offanalysis using genetic algorithms.” Can. J. Civ. Engrg., Vol. 26, No.6, pp. 685-697.

Hegazy, T. and Ayed, A. (1999). “Simplified spreadsheet solutions:Models for critical path method and Time-Cost-Tradeoff analysis.”Cost Engrg., Vol. 41, No. 7, pp. 26-33.

Kerzner, H. (2009). Project management: A systems approach toplanning, scheduling, and controlling, Wiley, John & Sons, Inc.,New York.

Leu, S. S., Chen, A. T., and Yang, C. H. (2001). “GA-based fuzzyoptimal model for construction time-cost trade-off.” Int. J. Proj.Mgmt., Vol. 19, No. 1, pp. 47-58.

Leu, S. S. and Yang, C. H. (1999). “GA-based multicriteria optimalmodel for construction scheduling.” J. Constr. Engrg. And Mgmt.,125, No. 5, pp. 420-427.

Li, H., Cao, J. N., and Love, P. (1999). “Using machine learning and GA

to solve time-cost trade-off problems.” J. Constr. Engrg. And Mgmt.,Vol. 125, No. 5, pp. 347-353

Li, H. and Love, P. (1997). “Using improved genetic algorithms tofacilitate Time-Cost optimization.” J. Constr. Engrg. And Mgmt.,Vol. 123, No. 3, pp. 233-237

Liu, L., Burns, S. A., and Feng, C. W. (1995). “Construction Tim-CostTrade-Off analysis using LP/IP hybrid method.” J. Constr. Engrg.And Mgmt.,Vol. 121, No. 4, pp. 446-454.

Moder, J. J., Phillips, C. R., and Davis, E. W. (1995). Project managementwith CPM, PERT, and precedence diagramming, 3rd Ed., Blitz,Middleton, Wis.

Moussourakis, J. and Haksever, C. (2004). “Flexible model for time/costtradeoff problem.” J. Constr. Engrg. And Mgmt., Vol. 130, No. 3, pp.307-314.

Que, B. C. (2002). “Incorporating practicability into genetic algorithm-based time-cost optimization.” J. Constr. Engrg. And Mgmt., Vol.128, No. 2, pp. 139-143.

Stevens, J. D. (1990). Techniques for construction network scheduling,McGraw-Hill, New York.

Taha, H. A. (2006). Operations research: An introduction 4th edition,Prentice Hall, New Jersey.

Wang, H. J., Zhang, J. P., Chau, K. W., and Anson, M. (2004). “4Ddynamic management for construction planning and resourceutilization.” Automation in Construction, Vol. 13, No. 5, pp. 575-589.

Yang, I. T. (2007). “Using elitist particle swarm optimization to facilitatebicriterion time-cost trade-off analysis.” J. Constr. Engrg. andMgmt., Vol. 133, No. 7, pp. 498-505.

Zhao, M. Y., Cheng, C. T., Chau, K. W., and Li, G. (2006). “Multiplecriteria data envelopment analysis for full ranking units associated toenvironment impact assessment” International Journal of Environ-ment and Pollution, Vol. 28, Nos. 3-4, pp. 228-464

Zheng, D. X. M., Ng, S. T., and Kumaraswamy, M. M. (2004). “Applyinga genetic algorithm-based multiobjective approach for time-costoptimization” J. Constr. Engrg. And Mgmt., Vol. 130, No. 2, pp.168-176.

Appendix

Min NC_A + NC_C + NC_H + 800CR_A + 1500CR_C + 2000CR_H + 10000XF1 + 8000XF2 + 6000XF3 + 5000XF4 + 9000XI1+ 13000XI2 + 5000XB1 + 8000XB2 + 5000XB3 +4000XB4 + 1000CR_B1 + 600CR_B2 + 700CR_B3 + 500CR_B4 + 5000XD1+ 3000XD2 + 1000XD3 + 1000CR_D1 + 900CR_D2 + 800CR_D3 + 10000XE1 + 8000XE2 + 8000XE3 +16000XE4+ 1500CR_E1 + 1750CR_E2 + 1500CR_E3 + 2000CR_E4 + 6000XG1 + 7000XG2 + 8000XG3 + 1500CR_G1 + 1000CR_G2+ 1250CR_G3 + 2500PD (37)

ST! Relationships between activities:S_A > 0 S_B > S_A + ND_A - CR_A + LAG_A_B S_C > S_A S_D > S_A + ND_A - CR_A S_E > S_C + ND_C - CR_C S_F + 3XF1 + 6XF2 + 8XF3 + 10XF4 > S_B + LAG_B_F S_G + 10XG1 - CR_G1 + 12XG2 - CR_G2 + 8XG3 - CR_G3 > S_D + 6XD1 - CR_D1 + 8XD2 - CR_D2 + 10XD3 - CR_D3 + LAG_D_GS_G > S_E + LAG_E_GS_H + ND_H – CR_H > S_E + 16XE1 - CR_E1 + 14XE2 - CR_E2 + 15XE3 - CR_E3 + 10XE4 - CR_E4 + LAG_E_H


Vol. 17, No. 2 / March 2013 − 291 −

S_I > S_F + 3XF1 + 6XF2 + 8XF3 + 10XF4PD > S_I + 9XI1 + 7XI2 (38)

! Force the unselected activities’ crash duration to 0 in mixed scenario.! All X and Y variables are binary variables.

CR_B2 + CR_B3 + CR_B4 – 1000YB1 < 0 XB1 + 1000YB1 < 1000CR_B1 + CR_B3 + CR_B4 – 1000YB2 < 0 XB2 + 1000YB2 < 1000CR_B1 + CR_B2 + CR_B4 – 1000YB3 < 0 XB3 + 1000YB3 < 1000CR_B1 + CR_B2 + CR_B3 – 1000YB4 < 0 XB4 + 1000YB4 < 1000

CR_D2 + CR_D3 – 1000YD1 < 0 XD1 + 1000YD1 < 1000CR_D1 + CR_D3 – 1000YD2 < 0 XD2 + 1000YD2 < 1000CR_D1 + CR_D2 – 1000YD3 < 0 XD3 + 1000YD3 < 1000

CR_E2 + CR_E3 + CR_E4 – 1000YE1 < 0 XE1 + 1000YE1 < 1000CR_E1 + CR_E3 + CR_E4 – 1000YE2 < 0 XE2 + 1000YE2 < 1000CR_E1 + CR_E2 + CR_E4 – 1000YE3 < 0 XE3 + 1000YE3 < 1000CR_E1 + CR_E2 + CR_E3 – 1000YE4 < 0 XE4 + 1000YE4 < 1000

CR_G2 + CR_G3 – 1000YG1 < 0 XG1 + 1000YG1 < 1000CR_G1 + CR_G3 – 1000YG2 < 0 XG2 + 1000YG2 < 1000CR_G1 + CR_G2 – 1000YG3 < 0 XG3 + 1000YG3 < 1000 (39)

! Alternative selection: if one alternative is selected, the others are discarded.! All X variables are binary variables:

XB1 + XB2 + XB3 + XB4 = 1 XD1 + XD2 + XD3 =1 XE1 + XE2 + XE3 + XE4 = 1XF1 + XF2 + XF3 + XF4 = 1 XG1 + XG2 + XG3 =1 XI1 + XI2 =1 (40)

! Normal Duration of activities in which a continuous scenario is applied.

ND_A = 6 ND_C = 10 ND_H = 5

! Normal duration of activities in which a discrete and mixed scenario is applied.! This duration is input directly in the formulation.! Here, the duration is presented to help readers understand.

!ND_D1 = 6 !ND_D2 = 8 !ND_D3 = 10 !ND_E1 = 16 !ND_E2 = 14!ND_E3 = 15 !ND_E4 = 10 !D_F1 = 3 !D_F2 = 6 !D_F3 = 8!D_F4 = 10 !ND_G1 = 10 !ND_G2 = 12 !ND_G3 = 8 !D_I1 = 9!D_I2 =7

! Normal Cost of activity A, C, H:NC_A = 7000 NC_C = 8000 NC_H = 10000

! Crash Time Range:CR_A < 3 CR_C < 4 CR_B1 < 3 CR_B2 < 2 CR_B3 < 3CR_B4 < 4 CR_D1 < 1 CR_D2 < 3 CR_D3 < 6 CR_E1 < 5CR_E2 < 4 CR_E3 < 3 CR_E4 < 2 CR_G1 < 4 CR_G2 < 3CR_G3 < 4

! LAG Time input:LAG_A_B=2 LAG_B_F=4 LAG_D_G=3 LAG_E_G=5 LAG_E_H=3

END

!Here, statement starting with !(exclamation mark) is a remark line.

a mixed (continuous + discrete) time-cost trade-off model considering four different relationships...

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