Open access peer-reviewed chapter

A New Model to Improve Project Time-Cost Trade-Off in Uncertain Environments

Written By

Mohammad Ammar Al-Zarrad and Daniel Fonseca

Submitted: 12 October 2017 Reviewed: 15 January 2018 Published: 22 February 2018

DOI: 10.5772/intechopen.74022

From the Edited Volume

Contemporary Issues and Research in Operations Management

Edited by Gary P. Moynihan

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Abstract

The time–cost trade-off problem (TCTP) is fundamental to project scheduling. Risks in estimation of project cost and duration are significant due to uncertainty. This uncertainty cannot be eliminated by any scheduling or estimation techniques. Therefore, a model that can represent uncertainty in the real world to solve time–cost trade-off problems is needed. In this chapter, fuzzy logic is utilized to consider affecting uncertainties in project duration and cost. An optimization algorithm based on time-driven activity-based costing (TDABC) is applied to provide a trade-off between project time and cost. The presented model could solve the time–cost trade-off problem while accounting for uncertainty in project cost and duration. This could help generate a more reliable schedule and mitigate the risk of projects running overbudget or behind schedule.

Keywords

  • scheduling
  • fuzzy logic
  • time–cost trade-off
  • cost estimating
  • risk management

1. Introduction

Operation management (OM) is vital to achieve success in many disciplines, particularly in a field which requires dealing with large amounts of information such as the construction industry. Most construction projects are a collection of different activities, processes and requirements, involving different factors and aspects to consider. In this way, making decisions in such environments can be a hard task. For these reasons, the need for OM to assist the characterization of such complex scenarios arises. OM could help project managers to improve their decision regarding project time–cost trade-offs (TCTP) [1]. To expedite the execution of a project, project managers need to reduce the scheduled execution time by hiring extra labor or using productive equipment. But this idea will incur additional cost; hence, shortening the completion time of jobs on critical path network is needed. According to several researchers, time–cost trade-off problem (TCTP) is considered as one of the vital decisions in project accomplishment [2]. Usually, there is a trade-off between the duration and the direct cost to do an activity; the cheaper the resources, the larger the time needed to complete an activity. Reducing the time on an activity will usually increase its direct cost. Direct costs for the project contain materials cost, labor cost and equipment cost. Conversely, indirect costs are the necessary costs of doing work which cannot be related to a specific activity and in some cases, cannot be related to a specific project. The total project construction cost can be found by adding direct cost to indirect cost. When the trade-off of all the activities is considered in the project then the relationship between project duration and the total cost is developed as shown in Figure 1. Figure 1 shows that when the duration for the project is reduced, the total cost becomes quite high and as the duration increases, the total cost increases [3]. The literature review of current practices reveals a shortage of existing tools and techniques specifically tailored to solve the time–cost trade-off problem while accounting for uncertainty in project time and cost. The objective of this research is to develop a model to find time–cost trade-off alternatives using TDABC and fuzzy logic. The next sections discuss these analytical methods.

Figure 1.

Project cost and time relationship.

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2. Time-driven activity-based costing

The activity-based costing (ABC) concept was first defined in the late 1980s by Robert Kaplan and William Burns [4]. At first, ABC was utilized by the manufacturing industry where technological expansions and productivity developments had reduced the proportion of direct costs but increased the proportion of indirect costs [5].

ABC was developed as a method to address problems associated with traditional cost management systems, which tend to be usable to accurately determine actual production and service costs or provide useful information for operating decisions. ABC is defined as “a method for tracing costs within a process back to individual activities” [6].

ABC has been used in the construction industry for cost estimating [7]. Further, ABC has been used to forecast the optimum duration of a project as well as the optimum resources required to complete a defined quantity of work in a timely and cost-effective manner [8]. Although traditional ABC systems provide construction managers with valuable information, many have been abandoned or never were implemented fully [3]. The traditional ABC system is costly to build, requires time to process, is difficult to maintain and is inflexible when needing modification [3]. These problems are particularly acute for small companies that are not likely to have a sophisticated information processing system. Further, ABC is very expensive for medium-sized-to-large companies.

To overcome the difficulties inherent in traditional ABC, Kaplan and Steven presented a new method called “time-driven activity-based costing (TDABC).” The new TDABC has overcome traditional ABC difficulties, offering a clear, accessible methodology that is easy to implement and update [4]. TDABC relies only on simple time estimates that, for example, can be established based on direct observation of processes [9].

TDABC utilizes time equations that directly allocate resource costs to the activities performed and transactions processed. Only two values need to be estimated: the capacity cost rate for the project (Eq. (1)) and the capacity usage by each activity in the project (Eq. (2)). Both values can be estimated easily and accurately [4]. Kaplan and Steven (2007) further define the capacity cost rate and the capacity usage as follows:

Capacity cost rate = Total estimated cost ÷ Working hours × Efficiency rate E1
Capacity usage rate = Capacity cost rate × Activity duration × Quantity E2

Although TDABC has many advantages over ABC, TDABC is not flawless. There are many difficulties associated with this deterministic TDABC approach. TDABC is unable of accounting for any variation or uncertainty in the project cost and duration (Hoozée and Hansen, 2015). Research carried out in TDABC, so far, has applied deterministic approaches. But, because of uncertainty present in the estimation of project cost and duration, a fuzzy TDABC would lead to more accurate results [10].

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3. Fuzzy logic

Fuzzy logic is a technique that provides a definite conclusion from vague and inaccurate information. Fuzzy set theory was first introduced by Zadeh in 1965. He was motivated after witnessing that human reasoning can utilize concepts and knowledge that do not have well-defined boundaries [11].

A useful method for investigating many everyday problems is fuzzy approximate reasoning or fuzzy logic. This technique is founded on the fuzzy set theory that allows the elements of a set to have variable degrees of membership, from a non-membership grade of 0 to a full membership of 1.0 [12]. This smooth gradation of values is what makes fuzzy logic tie well with the ambiguity and uncertainty of many everyday problems.

Fuzzy logic has become an important tool for many different applications ranging from the control of engineering systems to artificial intelligence. Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may range between completely true and false [13]. Fuzzy logic and fuzzy hybrid techniques have been used to capture and model uncertainty in construction, thereby improving workforce and project management. Fuzzy logic can effectively capture expert knowledge and engineering judgment and combine these subjective elements with project data to improve construction decision-making, performance and productivity [14].

Among the various shapes of fuzzy numbers, the triangular fuzzy numbers (TFNs) are the most popular [15]. A triangular fuzzy number μA(x) can be defined as a triplet (a1, aM, a2). Its membership function is defined as follows [16]:

μA x = x a 1 a M a 1 for a 1 x a M x a 2 a M a 2 for a M x a 2 0 otherwise E3

where [a1, a2] is the interval of possible fuzzy numbers and the point (aM, 1) is the peak. This parameter (a1, aM, a2) signifies the smallest possible value, the most promising value and the largest possible value, respectively [17]. Figure 2 illustrates a TFN.

Figure 2.

Triangular fuzzy number.

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4. Fuzzy time-driven activity-based costing model

This model utilizes TDABC as a tool for tracing costs and time within a project back to individual activities. TFNs are proposed as a logical approach to manage uncertainty in the deterministic TDABC system. TFNs were used to signify vagueness of TDABC because of their simplification to formulate in a fuzzy environment. Further, they are potentially more intuitive than other complicated types of fuzzy numbers such as trapezoidal or bell-shaped fuzzy numbers [16]. This model has the ability to fuzzify the project cost and duration by transferring these values from crisp numbers to fuzzy sets. A crisp number has a specific value while a fuzzy set has a possible range of values [15]. Then after applying a fuzzy rule, the model will defuzzify the cost and duration of the project to transfer these values back to crisp numbers. Figure 3 shows the fuzzy logic process that has been used in this model, as suggested by [14]. The fuzzy TDABC model consists of three stages as follows:

Figure 3.

Fuzzy logic controller.

4.1. Model stage one

The first step in stage one is to transfer the three-point estimate of project duration from crisp values to the fuzzy set. This can be done by calculating the estimated project duration using one of the traditional scheduling techniques (i.e., CPM) [18]. This value will be called the moderate duration and will use the notation DM. Then the pessimistic duration (the maximum project duration) should be calculated using expert opinion. The pessimistic duration notation is DP. Finally, the optimistic duration (the minimum project duration) should be calculated also using expert opinion. The optimistic duration notation is DO.

The second step is to transfer the three-points estimate of project cost from crisp values to the fuzzy set. This can be done by calculating the estimated project cost using one of the traditional cost estimation techniques (i.e., unit area cost estimate, unit volume cost estimate or parameter cost estimate) [18]. This value will be called the moderate cost and will use the notation CM. Then, the pessimistic cost (the maximum project cost) should be calculated using expert opinion. The pessimistic cost notation is CP. Finally, the optimistic cost (the minimum project cost) should be calculated also using expert opinion. The optimistic cost notation is CO.

During this step, each activity’s moderate duration, optimistic duration and pessimistic duration should be determined. The notations for an activity moderate duration, optimistic duration and pessimistic duration are dm, do and dp, respectively. The third step is to calculate the fuzzy capacity cost rate (CCR) using Eq. (4):

CCR = C P D O C M D M C O D P E4

Then, the fuzzy capacity usage rate (CUR) should be calculated as a triangular membership function (TMF) using the following equations:

CUR = C P D O C M D M C O D P d o d m d p Q Q Q E5
CUR = C P D O d o Q C M D M d m Q C O D P d p Q E6

where Q = Number of Each Activity (quantity).

The fourth step is to defuzzify the triangular membership function (TMF) to get crisp CUR values. Available defuzzification techniques include a max-membership principle, a centroid method, a weighted average method, a mean-max membership method, a center of sums, a center of largest area, the first of maxima or last of maxima [19]. Among these, a centroid method (also called Center of Gravity [COG]) is the most prevalent and physically appealing method [20]. The α-cut method is a standard method for performing arithmetic operations on a Triangular Membership Function [21]. The α-cut signifies the degree of risk that the decision-makers are prepared to take (i.e., no risk to full risk). Since the value of α could severely influence the solution, its choice should be carefully considered by decision-makers. Figure 4 shows a TFN with α-cut. The higher the value of α, the greater the confidence (α = 1 means no risk) [21].

Figure 4.

Triangular fuzzy number with α-cut.

By using the center of gravity (COG) defuzzification technique and = 0.1 , crisp CUR values (cost values) can be calculated for each activity using the following formula:

CUR COST = CUR O CUR M μA x xdx + CUR M CUR P μA x xdx CUR O CUR M μA x dx + CUR M CUR P μA x dx E7

where:

CUR COST = Improved cost estimate of an activity at = 0.1
CUR O = C P D o d O Q = Optimistic cost at = 0.1
CUR P = C O D P d P Q = Pessimistic cost at = 0.1
CUR M = C M D M d m Q = Moderate cost

The crisp CUR COST value that is calculated in this step is the improved cost estimate for an activity at = 0.1 and its notation is iac 0.1 .

The fifth step is to repeat the same process to get the improved cost estimate for all project activities. Finally, add the improved cost estimate for all the activities to get an improved cost estimate for the project at = 0.1 . The project improved cost estimate will be abbreviated as I PC 0.1

I PC 0.1 = i Project improved activities cost at = 0.1 E8

4.2. Model stage two

The first step in stage two is to calculate the fuzzy capacity cost rate (CCR) using the new I PC 0.1 cost and the following equation:

CCR = D O IPC 0.1 D M IPC 0.1 D P IPC 0.1 E9

The second step is to calculate the fuzzy capacity usage rate (CUR) as a triangular fuzzy function using the following equation:

CUR = D O IPC 0.1 D M IPC 0.1 D P IPC 0.1 iac 0.1 i ac 0.1 iac 0.1 Q Q Q E10
CUR = D O IPC 0.1 iac 0.1 Q D M IPC 0.1 iac 0.1 Q D P IPC 0.1 iac 0.1 Q E11

where iac 0.1 = The improved activity cost at = 0.1 (it is already calculated in stage one).

The third step is to defuzzify the triangular membership function (TMF) using the center of gravity (COG) defuzzification technique. Using COG and = 0.1 , a crisp CUR value (time value) can be calculated for each activity using the following formula:

CUR TIME = CUR O CUR M μA x xdx + CUR M CUR P μA x xdx CUR O CUR M μA x dx + CUR M CUR P μA x dx E12

where:

CUR TIME = Improved time estimate of an activity at = 0.1
CUR O = D O IPC 0.1 iac 0.1 Q = Optimistic time at = 0.1
CUR P = D P IPC 0.1 iac 0.1 Q = Pessimistic time at = 0.1
CUR M = D M IPC 0.1 iac 0.1 Q = Moderate time

The crisp CUR TIME value that is calculated in this step is the improves duration for an activity at = 0.1 and its notation is iad 0.1 .

The fourth step is to repeat the same process to get the improved duration for all project activities. Finally, add the improved duration for all the activities to get an improved duration for the project. The project improved duration will be abbreviated as IPD 0.1

IPD 0.1 = i Project improved activities duration at = 0.1 E13

4.3. Model stage three

In stage three, a sensitivity analysis should be performed to investigate the variability of the results obtained with respect to the choice of the α-cut value. Sensitivity analysis is “the study of how the uncertainty in the output of a model can be apportioned to different sources of uncertainty in the model input” [22]. One of the simplest and most common approaches to sensitivity analysis is changing the α-cut value, to see what effect this produces on the project cost and duration. To achieve that, stage one and two should be repeated using α-cut values equal to 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0. The results obtained from the different α-cut values will be saved as shown in Table 1. The sensitivity analysis will help investigate various levels of confidence associated with each time–cost alternative.

α-cut Improved project cost Improved project duration
0.1 IPC 0.1 IPD 0.1
0.2 I PC 0.2 IPD 0.2
0.3 I PC 0.3 I PD 0.3
0.4 IPC 0.4 IPD 0.4
0.5 IPC 0.5 IPD 0.5
0.6 IPC 0.6 IPD 0.6
0.7 IPC 0.7 IPD 0.7
0.8 IPC 0.8 I PD 0.8
0.9 IPC 0.9 IPD 0.9
1.0 IPC 1.0 I PD 1.0

Table 1.

Project time and cost at each α-cut.

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5. Fuzzy time-driven model verification and validation

To illustrate an application of the fuzzy TDABC model, a case study of seven activities proposed initially by Zheng et al. (2004) was used [23]. The case study illustrates a construction project that has seven activities as shown in Table 2. The letters O, M and P in Table 2 signify optimistic, moderate and pessimistic time and direct cost. The assumed value for indirect cost per day is $1000, $1150 and $2000 for optimistic, moderate and pessimistic values, respectively. The calculated project duration is (60, 81 and 92) days for optimistic, moderate and pessimistic, respectively.

Activity Predecessor Time (Days) Direct cost ($)
O M P O M P
A 14 20 24 23,000 18,000 12,000
B A 15 18 20 3000 2400 1800
C A 15 22 33 4500 4000 3200
D A 12 16 20 45,000 35,000 30,000
E B, C 22 24 28 20,000 17,500 15,000
F D 14 18 24 40,000 32,000 18,000
G E, F 9 15 18 30,000 24,000 22,000

Table 2.

Activities duration and cost.

The first step is to calculate the total cost of the project by adding the indirect cost to the direct cost. Table 3 shows the optimistic, moderate and pessimistic total cost.

Total cost ($)
P M O
296,772 238,169 205,192

Table 3.

Project total cost.

Applying stage one of the fuzzy TDABC model begins by using Eq. (4) to calculate the fuzzy CCR as shown in Table 4.

CCR ($): Phase I
O M P
2938 1791 1229

Table 4.

Fuzzy capacity cost rate (CCR).

Then, the fuzzy capacity usage rate (CUR) is calculated as a cost function using Eq. (5). Table 5 shows the CUR values.

Activity CUR ($): Phase I
O M P
A 41,137 35,815 29,489
B 44,075 32,233 24,574
C 44,075 39,396 40,547
D 35,260 28,652 24,574
E 64,643 42,978 34,403
F 41,137 32,233 29,489
G 26,445 26,861 22,117

Table 5.

Fuzzy capacity usage rate (CUR).

Next, α-cut values of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 are applied to the CUR values in Table 5. This will generate new CUR values associated with each α-cut. Table 6 shows the CUR values that are associated with each α-cut for each activity in the project.

α Fuzzy
CUR ($)
Activities
A B C D E F G
0.1 CURo 40,604 42,891 43,607 34,599 62,477 40,246 26,487
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 30,121 25,340 40,432 24,982 35,261 29,763 22,591
0.2 CURo 40,072 41,707 43,139 33,938 60,310 39,356 26,528
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 30,754 26,106 40,317 25,390 36,118 30,038 23,065
0.3 CURo 39,540 40,523 42,671 33,278 58,144 38,466 26,570
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 31,387 26,872 40,202 25,797 36,976 30,312 23,540
0.4 CURo 39,008 39,338 42,204 32,617 55,977 37,575 26,611
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 32,019 27,638 40,087 26,205 37,833 30,587 24,014
0.5 CURo 38,476 38,154 41,736 31,956 53,811 36,685 26,653
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 32,652 28,404 39,972 26,613 38,691 30,861 24,489
0.6 CURo 37,944 36,970 41,268 31,295 51,644 35,795 26,695
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 33,284 29,170 39,857 27,021 39,548 31,135 24,963
0.7 CURo 37,411 35,786 40,800 30,634 49,477 34,904 26,736
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 33,917 29,936 39,741 27,428 40,405 31,410 25,438
0.8 CURo 36,879 34,602 40,332 29,973 47,311 34,014 26,778
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 34,550 30,701 39,626 27,836 41,263 31,684 25,912
0.9 CURo 36,347 33,418 39,864 29,313 45,144 33,124 26,820
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 35,182 31,467 39,511 28,244 42,120 31,959 26,387
1.0 CURo 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURM 35,815 32,233 39,396 28,652 42,978 32,233 26,861
CURP 35,815 32,233 39,396 28,652 42,978 32,233 26,861

Table 6.

CUR value at each α-cut ($): Phase I.

Using Eq. (7), crisp CUR values associated with each α-cut are determined for each activity. These CUR values are the improved cost estimate for each activity at the associated α-cut. By adding the improved activities’ costs, the project improved cost estimates are determined as shown in Table 7.

Crisp CUR Values ($) - Phase I
α-cut Activities Improved project cost ($)
A B C D E F G
0.1 35,622 34,868 42,040 30,049 50,132 35,266 24,590 252,567
0.2 35,617 34,504 41,744 29,869 49,226 34,905 24,837 250,703
0.3 35,620 34,158 41,449 29,695 48,345 34,550 25,085 248,902
0.4 35,628 33,829 41,154 29,527 47,490 34,200 25,335 247,164
0.5 35,643 33,517 40,860 29,366 46,663 33,857 25,586 245,491
0.6 35,665 33,223 40,566 29,210 45,863 33,519 25,839 243,885
0.7 35,693 32,947 40,273 29,061 45,094 33,188 26,092 242,348
0.8 35,727 32,690 39,980 28,918 44,356 32,863 26,347 240,882
0.9 35,768 32,452 39,688 28,782 43,650 32,545 26,604 239,488
1.0 35,815 32,233 39,396 28,652 42,978 32,233 26,861 238,169

Table 7.

Project improved cost estimates.

At this point, stage one of the model is done and stage two begins. By using the improved project costs that have been calculated in Table 7, the fuzzy capacity cost rates (CCR) are calculated using Eq. (9). Table 8 shows the CCR value associated with each α-cut.

Activity CCR - Phase II
O M P
0.1 0.000238 0.000321 0.000364
0.2 0.000239 0.000323 0.000367
0.3 0.000241 0.000325 0.000370
0.4 0.000243 0.000328 0.000372
0.5 0.000244 0.000330 0.000375
0.6 0.000246 0.000332 0.000377
0.7 0.000248 0.000334 0.000380
0.8 0.000249 0.000336 0.000382
0.9 0.000251 0.000338 0.000384
1.0 0.000252 0.000340 0.000386

Table 8.

The CCR value associated with each α-cut.

Then, the fuzzy capacity usage rate (CUR) is calculated as a time function using Eq. (10). Table 9 shows the CUR values.

α Fuzzy
CUR (Days)
Activities
A B C D E F G
0.1 CURo 9 9 10 7 12 9 6
CURM 11 11 13 10 16 11 8
CURP 13 13 15 11 18 13 9
0.2 CURo 9 9 11 8 13 9 6
CURM 12 11 13 10 16 11 8
CURP 13 12 15 11 18 13 9
0.3 CURo 9 9 11 8 13 9 6
CURM 12 11 13 10 16 11 8
CURP 13 12 15 11 18 12 9
0.4 CURo 10 9 11 8 14 10 7
CURM 12 11 13 10 16 11 8
CURP 12 12 15 10 17 12 9
0.5 CURo 10 10 12 8 14 10 7
CURM 12 11 13 10 15 11 8
CURP 12 12 14 10 17 12 8
0.6 CURo 10 10 12 9 14 10 7
CURM 12 11 13 10 15 11 9
CURP 12 12 14 10 17 12 8
0.7 CURo 11 10 12 9 15 10 7
CURM 12 11 13 10 15 11 9
CURP 12 12 14 10 17 12 8
0.8 CURo 11 11 13 9 15 11 7
CURM 12 11 13 10 15 11 9
CURP 12 11 14 10 17 12 8
0.9 CURo 11 11 13 9 16 11 8
CURM 12 11 13 10 15 11 9
CURP 12 11 14 10 16 11 8
1.0 CURo 11 11 13 10 16 11 8
CURM 12 11 13 10 15 11 9
CURP 11 11 13 10 16 11 8

Table 9.

CUR value at each α-cut (days): Phase II.

Using Eq. (12), new crisp CUR values associated with each α-cut are determined for each activity. These CUR values are the improved duration for each activity at the associated α-cut. By adding the improved activities’ durations, the project improved durations are determined as shown in Table 10.

Crisp CUR (Days): Phase II
α-cut Activities Improved project duration (Days)
A B C D E F G
0.1 10.9 10.7 12.9 9.2 15.4 10.8 7.5 77.4
0.2 11.0 10.7 12.9 9.2 15.4 10.9 7.6 77.7
0.3 11.0 10.8 13.0 9.3 15.5 10.9 7.6 78.0
0.4 11.1 10.8 13.0 9.3 15.6 10.9 7.6 78.4
0.5 11.1 10.9 13.1 9.4 15.6 11.0 7.7 78.8
0.6 11.2 10.9 13.2 9.4 15.7 11.1 7.7 79.2
0.7 11.2 11.0 13.2 9.5 15.8 11.1 7.7 79.6
0.8 10.9 10.7 12.9 9.2 15.4 10.8 7.5 77.4
0.9 11.0 10.7 12.9 9.2 15.4 10.9 7.6 77.7
1.0 11.0 10.8 13.0 9.3 15.5 10.9 7.6 78.0

Table 10.

Project improved duration.

Using the results in Tables 7 and 10, the improved project cost and the improved project duration associated with each α-cut are summarized in Table 11.

α-cut Improved project cost ($) Improved project duration (Days)
0.1 252,567 77.4
0.2 250,703 77.7
0.3 248,902 78.0
0.4 247,164 78.4
0.5 245,491 78.8
0.6 243,885 79.2
0.7 242,348 79.6
0.8 240,882 80.0
0.9 239,488 80.5
1.0 238,169 81.0

Table 11.

Improved project cost and duration associated with each α-cut.

Using Table 11, a plot of the improved project costs versus the improved project durations is created as shown in Figure 5. The robustness of the new proposed TDABC model is compared with two previous models:

  1. Gen and Cheng (2000) model.

  2. Zheng et al. (2004) model.

Figure 5.

Improved project cost versus project durations.

Gen and Cheng (2004) used a genetic algorithm (GA) approach to find the best Time–Cost Trade-Offs. GA is a search method used for finding optimized solutions to problems based on the natural selection theory and biological evolution [24]. The Zheng et al. model used the modified adaptive weight approach with GA to solve the time–cost trade-off problem. The modified adaptive weight approach is a method to represent the importance of each function by assigning different weights to different functions [23].

The results of these two models are compared with the fuzzy TDABC model in Table 12.

Approaches Criteria Target
Time (days) Cost ($)
Gen and Cheng (2000) 83 243,500 Least cost
79 256,400 Least time
Zheng et al. (2004) 73 236,500 Least cost
66 251,500 Least time
Fuzzy TDABC
(This research)
81 238,169 Least cost
77 252,567 Least time

Table 12.

Fuzzy TDABC result vs. previous research results.

Figure 6 compares between the fuzzy TDABC result and the results obtained by Gen and Cheng (2004) and Zheng et al. (2004).

Figure 6.

Fuzzy TDABC result versus previous research results.

Table 12 and Figure 6 show that the fuzzy TDABC obtains better values of time and cost compared to the result obtained by Gen and Cheng (2000). However, the result obtained by Zheng (2004) is better than the fuzzy TDABC result.

To further compare the results of the fuzzy TDABC model with the past published results, a test called Wilcoxon signed-ranks test is performed. The Wilcoxon Signed-Ranks test is a non-parametric analysis that statistically compared the average of two dependent samples and assessed for significant differences. Wilcoxon signed-ranks test does not assume normality of the differences of the compered groups [25]. The Wilcoxon test has been selected because the datasets in this case do not follow normal distribution. The method to perform Wilcoxon test starts with two hypotheses. A null hypothesis (H₀) assumes that the results obtained from the three approaches are the same. An alternative hypothesis (H₁) assumes that the results obtained from the three approaches are not the same. Table 13 shows the Wilcoxon signed-ranks test result.

Source N Wilcoxon Statistic P-Value Estimated median
Time 6 21.0 0.036 77 Day
Cost 6 21.0 0.036 $246,450

Table 13.

Wilcoxon signed-ranks test result.

Table 13 shows that the p-value is 0.036. The p-value, or calculated probability, assesses if the sample data support the argument that the null hypothesis (H₀) is true. A small p-value (less or equal to 0.05) indicates solid evidence against the null hypothesis, so the null hypothesis should be rejected. A large p-value (larger than 0.05) indicates weak evidence against the null hypothesis, so the null hypothesis should not be rejected [25]. The p-value is 0.036, in this case, which is less than the significance level of 0.05. As a result, there is enough evidence to reject the null hypothesis and to conclude that the difference between the results obtained from the three approaches is significant.

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6. Conclusion

The objective of this research is to develop a model to find time–cost trade-off alternatives while accounting for uncertainty in project time and cost. The presented fuzzy TDABC model provides an attractive alternative for the traditional solutions of the time–cost trade-offs optimization problem. The presented model is simple and easy to apply compared with other approaches. Further, this model obtained a better solution when compared to the GA model that is presented by Gen and Cheng (2000). The fuzzy TDABC model could improve the reliability of the time–cost trade-off decisions. This could help construction companies mitigate the risk of projects running over budget or behind schedule.

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Written By

Mohammad Ammar Al-Zarrad and Daniel Fonseca

Submitted: 12 October 2017 Reviewed: 15 January 2018 Published: 22 February 2018