A REVIEW OF TIME COST AND QUALITY TRADEOFF ANALYSIS MODELS IN CONSTRUCTION
Abstract
For any construction project, time, cost and quality of a construction are three most significant factors that affect the construction lifecycle. An important part of project management is balancing these three conflicting factors depending on the availability of resources and project constraints. The ultimate goal is to minimize the time and maximize the quality as per client satisfaction, and at the same time minimize the cost to increase profits. Over the past two decades, with the development of computing technologies, the methods for optimizing timecostquality tradeoff have become increasingly sophisticated. The objective of this paper is to review three models, viz. Linear Programming Model, MultiObjective Genetic Algorithm Optimization Model, Fuzzy MultiObjective Particle Swarm Optimization Model. Applications, advantages and disadvantages of these models in construction projects shall be studied and key factors involved in timecostquality tradeoff shall be highlighted. This study shall not include the environmental and energy effects of timecostquality tradeoff.
Background
Time, cost and quality of construction are three interdependent and conflicting factors that determine the success of a construction project. Crashing an activity to achieve a timeline increases the cost and may result in poor quality of output. Excessive compromise on the cost of construction leads to elongated timeline and inferior quality of construction. Finally, too much focus on the quality may cause cost and time overrun, along with imposition of penalties. Constraints on time, cost and quality are imposed by a variety of instruments like type of contract, availability of resources, project timeline, competitive bidding, contractor’s profit. Thus, an important part of Project Management involves balancing these factors as per the availability of resources and project constraints.
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A lot of research has been carried out on the timecost tradeoff in construction, as time and cost can be assumed to be deterministic and interdependent on each other. Mathematical and heuristic models have been developed to deal with this 2dimensional problem. These models too however have their own advantages and drawbacks. Although mathematical models provide a rigorous mathematical solution to the timecost trade of problem, they require formulation of objective functions and constraints which is often a difficult and timeconsuming process. The heuristic models on the other hand provide easy and practical solutions but lack the scientific rigor and range of possible solutions. The quality aspect of the tradeoff, however, and its relationship with cost and time of the project has been studied only recently. The main reason for this is the fact that it is difficult to quantify quality of construction. The quality of the product at the end of a construction activity is largely subjective and depends on the inspector’s approach, experience and psychology. In recent years however, a number of standards have been developed to determine the quality of construction.
The objective of this paper is to review three timecostquality tradeoff analysis models, their application and their merits and demerits.
TimeCostQuality Tradeoff Analysis Models
The three timecostquality tradeoff analysis models reviewed in this paper are:
 Linear Programming Model
 MultiObjective Genetic Algorithm Optimization Model
 Fuzzy MultiObjective Particle Swarm Optimization model
Each of these methods can be executed in two phases:
 Model Development
 Model Optimization
The model development phase consists of mathematical formulation of the time cost quality problem, quantification of construction quality and formulation objective function. In the optimization phase, as the name suggests, objective function is optimized to get the best timecostquality balance.
Linear Programming Model
This analysis model was one of the earliest attempts at studying the timecostquality trade off. All the models preceding this considered only the timecost relationship in activity crashing. In this approach, three separate models are developed to optimize the three factors, one at a time, by assigning constraints on the other two factors. Each activity is assigned a value on a continuous scale between zero and one for quality. The overall project quality is then a function of qualities of each activity. This approach assumes that this function is arithmetic mean, geometric mean or minimum value. ^{[2]}
Model development:
For a project with M number of events, N number of activities and D number of dummy activities, each event is given an index K=1 to M, each activity is given an index L=1 to N and each dummy activity is given an index LCD =N to N+D. A normal time, cost and quality given by NTIME(L), NCOST(L) and NQUAL(L) is associated with each activity. Also, each activity has a crash time, cost and quality given by CRTIME(L), CRCOST(L) and CRQUAL(L) depending on the maximum allowable crash. The normal and crash time, cost and quality of the individual activities are based on the past experience and act as an input data for the model.
Activity cost curve and activity quality curves are developed for each activity with slopes and intercepts as follows:
$\mathrm{S}\left(\mathrm{L}\right)=\mathrm{}\frac{\mathrm{NCOST}\left(\mathrm{L}\right)\u2013\mathrm{CRCOST}\left(\mathrm{L}\right)}{\mathrm{NTIME}\left(\mathrm{L}\right)\u2013\mathrm{CRTIME}\left(\mathrm{L}\right)}$
(1)
$\mathrm{INTERCPT}\left(\mathrm{L}\right)=\mathrm{CRCOST}\left(\mathrm{L}\right)\u2013\mathrm{S}\left(\mathrm{L}\right)\times \mathrm{CRTIME}\left(\mathrm{L}\right)$
(2)
$\mathrm{SQ}\left(\mathrm{L}\right)=\mathrm{}\frac{\mathrm{NQUAL}\left(\mathrm{L}\right)\u2013\mathrm{CRQUAL}\left(\mathrm{L}\right)}{\mathrm{NTIME}\left(\mathrm{L}\right)\u2013\mathrm{CRTIME}\left(\mathrm{L}\right)}$
(3)
$\mathrm{INTERCPTQ}\left(\mathrm{L}\right)=\mathrm{CRQUAL}\left(\mathrm{L}\right)\u2013\mathrm{SQ}\left(\mathrm{L}\right)\times \mathrm{CRTIME}\left(\mathrm{L}\right)$
(4)
Project bounds functions UBTIME, UBCST, LBQAV are set depending on the maximum allowable time and cost and minimum quality requirements for the project.
The activity completion times X(L) are the decision variables and Earliest time for event Y(K) are the auxiliary variables.
The functions are defined as:
$\mathrm{OBJFNT}:\mathrm{Y}\left(\u2018{\mathrm{N}}^{\u2018}\right)=\mathit{OBJT}$
(5)
$\mathrm{OBJFNC}:\mathrm{SUM}\left(\mathrm{L},\mathrm{INTERCPT}\left(\mathrm{L}\right)+\mathrm{S}\left(\mathrm{L}\right)\times \mathrm{X}\left(\mathrm{L}\right)\right)=\mathit{OBJC}$
(6)
$\mathrm{OBJAVQ}:\frac{\mathit{SUM}(L,\mathit{INTERCPTQ}\left(L\right)+\mathit{SQ}(L)\times X\left(L\right))}{N}=\mathit{QAVQ}$
(7)
$\mathrm{ETIME}:\mathrm{Y}\left(\u2018{\mathrm{I}}^{\u2018}\right)+X\left(L\right)\u2013Y(\u2018{J}^{\u2018})\le 0$
(8)
$\mathrm{LBTACT}:\mathrm{X}\left(\u2018{\mathrm{L}}^{\u2018}\right)\ge \mathit{CRTIME}\left(L\right)$
(9)
$\mathrm{UBTACT}:\mathrm{X}(\u2018{\mathrm{L}}^{\u2018})\le \mathrm{NTIME}\left(\mathrm{L}\right)$
(10)
$\mathrm{UBTIME}:\mathrm{OBJT}\le \mathrm{UBTIME}$
(11)
$\mathrm{UBCOST}:\mathrm{OBJC}\le \mathrm{UBCST}\left(\mathrm{L}\right)$
(12)
$\mathrm{LBQUAV}:\mathrm{QAVQ}\le \mathrm{LBQAV}$
(13)
Model Optimization:
The three linear programming models developed to study the timecostquality tradeoff based on the above formulation are:
 Minimizing the project duration: Minimize crash time objective OBJT subject to project time objective function OBJFNT, earliest time constraint ETIME, activity bounds LBTACT and UBTACT, project bounds UBCST and LBQAV.
 Minimizing the project cost: Minimize crash cost objective OBJC subject to project cost objective function OBJFNC, earliest time constraint ETIME, activity bounds LBTACT and UBTACT, project bounds UBTIME and LBQAV.
 Maximizing the project quality: Maximize crash (average) quality objective OAVQ subject to project quality objective function OBJAVQ, earliest time constraint ETIME, activity bounds LBTACT and UBTACT, project bounds UBTIME and UBCST.
This method assumes a linear relationship between timecostquality of construction project which may not be appropriate. However, the linearity of relationships makes computation easy. Also, this model can be used to accommodate nonlinear relationships which makes it a foundation for the models developed in the future.
MultiObjective Genetic Algorithm Optimization
Genetic algorithm (GA) is systematic series of steps used in searching the optimal or near optimal solution out of a large space of solutions for problem using concepts of evolution like survival of the fittest and structured exchange of genetic materials among population over successive generations. This method of timecostquality tradeoff was developed to give construction planners a tool that can generated optimal resource utilization plans that optimize time, cost and quality by considering and quantifying quality in construction optimization, and that can help in visualizing the threedimensional representation of the tradeoffs so that planners can evaluate the effect of various utilization plans on the overall performance of construction. ^{[5]}
Model Development:
The development of model for this method is twofolds:
1. Determining the major decision variable affecting the resource utilization problem.
2. Formulating the optimization objectives
The characteristics of construction activities on which time, cost and quality depend are:
 construction methods (m) which consists of availability of materials and/or methods that can be used
 crew formation (f) which represents the feasible sizes and configurations of construction crews
 Crew overtime policy (p) which represents the capacity of each crew to work overtime and/or at night.
These three factors are combined to form a single variable, n, that is a representative of the resources utilized for an activity. Thus, a construction activity can have more than one resource utilization option depending on which construction methods, crew and crew overtime policy is available as shown in figure. An expected daily productivity, cost and quality is associated to each of these resource utilization options.
Fig.1 Timecostquality tradeoff optimization model. ^{[5]}
Thus, a project with N activities, each activity having l number of utilization plans, generates N^{l }approaches to complete the project.
To minimize the project duration and cost, and maximize the quality of the project, three objective functions are defined. The model uses these functions to measure the impact of the various resource utilization plans on time, cost and quality of the project. The functions to be optimized are:

$\mathit{Project\; Time}=\sum _{i=1}^{l}{T}_{i}^{n}$
(14)
Where,
${T}_{i}^{n}$
: duration of activity (i) on the critical path using resource utilization n

$\mathit{Project\; Cost}=\sum _{i=1}^{l}{(M}_{i}^{n}+{D}_{i}^{n}\times {R}_{i}^{n}+{B}_{i}^{n})$
(15)
Where,
${M}_{i}^{n}$
: material cost of activity (i) using resource utilization n
${D}_{i}^{n}$
: duration of activity (i) using resource utilization n
${R}_{i}^{n}$
: daily cost rate of activity (i) in $/day using resource utilization n
${B}_{i}^{n}$
: subcontractor lump sum cost activity (i) for resource utilization (n)

$\mathit{Project\; Quality}=\sum _{i=1}^{l}{\mathit{wt}}_{i}\sum _{k=1}^{K}{\mathit{wt}}_{i,k}\times {Q}_{i,k}^{n}$
(16)
Where,
${Q}_{i,k}^{n}$
: performance of quality indication (k) in activity (i) using resource utilization (n)
${\mathit{wt}}_{i,k}$
: weight of quality indicator compared to other indicators in activity (i)
${\mathit{wt}}_{i}$
: weight of activity (i) compared to other activities in the project
It is important to note that the performance of different quality indicators may be measured on construction sites and expressed in different units. These performances are then converted to a value ranging between 0 and 100% before inserting in the project quality equation. The weights assigned to each quality indicators may be determined based on its contribution to the quality of the activity. The weights assigned to each activity can be determined based on the contribution of the activity to the quality of the entire project.
Model optimization:
The model optimization using GA consists of three parts:
 Initialization
 Fitness Functions Evaluation
 Population Generation
Initialization:
In this stage, the project and GA parameters are defined and initialized. The project parameters consist of:
 Project size, which is determined by the number of activities in the construction project.
 Activity precedence information, which is described by the job logic.
 Available resource utilization options for each activity as described before and the associated time, cost and quality performances.
The GA parameters include:
 String size, which is GA equivalent to project size and depends on the number of activities
 Number of generations, which is the number of iterations the GA must perform
 Population size, which is the number of alternatives to complete a project in a given iteration. The population size is chosen depending on the computing power available.
 Mutation rate
 Crossover rate
The mutation and crossover rate are determined depending on population size and selection method used in the algorithm. For the first generation, i.e. iteration, random solutions are generated for the population. Each solution represents a construction option to complete the project.
Fitness Function Evaluation:
In this stage, time, cost and quality associated to each solution in the generation is calculated using the project time, cost and quality functions to determine the fitness of the solution.
Population Generation:
In this stage, three types of populations are generated, viz. Parent, Child and Combined population. Population generation stage is executed in the following series of steps:
 For each solution in the parent population P_{g} of size S, a Pareto optimal rank is calculated based on the time, cost and quality of project. A solution with the best time, cost and quality is considered dominant with respect to other solutions. Crowding distance of each solution is calculated, which expresses the closeness of neighboring solutions to the solutions under consideration.
 A new Child population is generated by selection, crossover and mutation operations of parent population. The solutions with high optimal rank and large crowding distance are selected. The selected solutions are then paired at randomly selected points and the variables stored in the strings are swapped resulting in two new individuals by the crossover operation. The mutation operation, then changes one of the variable values. The mutation operation thus prevents convergence of solution to local optimum. The time, cost and quality functions are calculated for the child population to determine the fitness of these newly generated solutions.
 The parent and child populations together form a new combined population of size 2S. This preserves the solutions generated in the parent population and expands the solution space by adding new child population.
 Similar to step 1, a Pareto optimal rank is assigned to each solution of the combined population of size 2S and their crowding distance is calculated.
 The combined population is then sorted based on their Pareto optimal rank. As a tiebreaker, the solution with higher crowding distance is given preference.
 The top S number of solutions from the combined population are kept as parent population for the next generation.
Steps 3 to 6 are repeated until specified maximum number of generations, i.e. iterations are achieved.
Fig. 2 Flowchart of Genetic Algorithm for TimeCostQuality Tradeoff Problem ^{[5]}
FuzzyMultiObjective Particle Swarm Optimization
This method incorporates the uncertainty in timecostquality relationship associated with various construction methods for each activity by considering the traditional timecostquality problem as a fuzzy timecostquality problem and solving it using a fuzzymultiobjective particle swarm optimization algorithm. The time, cost and quality are considered as fuzzy numbers. This approach of fuzzy timecostquality analysis also takes into account the goodnessoffit between a construction method and the associated performance which is highly dependent on the other factors such as uniqueness of construction, site conditions, weather conditions, material supply etc.^{ [3]}
Model Development:
Attribute utility is a measure of the desirability of outcomes associated with alternatives. ^{[1]} In this method, each construction approach is considered as an alternative to be evaluated based on composite attribute utility function that is a combination of single attribute functions representing all the criteria of performance i.e. time, cost and quality. A weight is assigned with each criterion depending on the significance of the criterion in that approach such that sum of weights is 1. The composite attribute utility function for a construction approach is thus given by
$U={w}_{T}{u}_{T}+{w}_{C}{u}_{C}+{w}_{Q}{u}_{Q}$
(17)
${u}_{T}=\frac{{T}^{+}\u2013T}{{T}^{+}\u2013{T}^{\u2013}}$
(18)
${u}_{C}=\frac{{C}^{+}\u2013C}{{C}^{+}\u2013{C}^{\u2013}}$
(19)
${u}_{Q}=\frac{{Q\u2013Q}^{\u2013}}{{Q}^{+}\u2013{Q}^{\u2013}}$
(20)
Where
${w}_{T}{,w}_{C},{w}_{Q}$
: Weights of criteria
${T}^{+}{,C}^{+},{Q}^{+}$
: Maximum time cost and quality of each activity
${T}^{\u2013}{,C}^{\u2013},{Q}^{\u2013}$
: Minimum time cost and quality of each activity
$T,C,Q$
: Time cost and quality of each activity
The uncertainty associated with the construction methods and the corresponding performance outcome is incorporated into the analysis by using convex and normalized fuzzy sets called fuzzy numbers. ^{[4]} The fuzzy set theory was developed to develop mathematical models of real phenomenon involving vagueness of colloquial language, in case of timecostquality tradeoff problem, the vagueness involved in time cost and quality quantification and relationships. For example, the quality measured on site is a highly subjective performance criterion which depends on the engineer’s judgement among many other factors. Thus, an activity may be quantified as of highest quality by one engineer and very high by another. A fuzzy set, a subset of universal set, is defined by a membership function ${\mathbf{\mu}}_{\mathit{A}}$
mapping Universal set $\textcolor[rgb]{}{\mathit{U}}$
into a closed unit interval [0,1], where for $\textcolor[rgb]{}{\mathit{x}}\textcolor[rgb]{}{\mathit{\in}}\textcolor[rgb]{}{\mathit{U}}$
,
${\mathrm{\mu}}_{A}\left(x\right)=0,\mathit{if}\mathrm{x}\notin A$
(21)
${\mathrm{\mu}}_{A}\left(x\right)=1,\mathit{if}\mathrm{x}\in A$
(22)
${\mathrm{\mu}}_{A}\left(x\right)\in \left(0,1\right),\mathrm{}\mathrm{if}\mathrm{x}\mathrm{possibly\; belongs\; to}\mathrm{A}\mathrm{but\; it\; is\; not\; sure}$
(23)
For the last case – the nearer to 1 the value ${\mathrm{\mu}}_{A}\left(x\right)$
is, the higher is the possibility that $\mathrm{x}\in A$
. Triangular fuzzy numbers, denoted as A (a_{1}, a_{2}, a_{3}), are used to express the time, cost and quality of an activity with membership functions given by:
(24)
Where,
a_{1} is the minimum possible value, a_{2} is the most likely value and a_{3} is maximum possible value of time and cost. In case of quality associated an activity, however, since the description on site is usually in linguistic terms “The highest (TH)”, “high(H)”, “very high (VH)”, “medium (M)”, “low(L)”, “very low(VL)” or the “lowest (TL)”, fuzzy partitions are used to associate a value to each of these terms. Each linguistic term is assigned a minimum (a_{1}), maximum(a_{3}) and most likely value (a_{2}) to be used in the membership functions.
Fig. 3 Relationship between linguistic quality and a series of fuzzy numbers. ^{[3]}
The sum of fuzzy activity duration and cost gives the fuzzy project duration and cost respectively. The sum of product of fuzzy quality of activities and the corresponding weights associated to the quality activities gives the fuzzy project quality.
Substituting the fuzzy values of time cost and quality of a construction approach, and applying constrained fuzzy arithmetic operations in the composite attribute utility, we get the fuzzy composite attribute utility for the approach, U= (U1, U2, U3) comprised of fuzzy single attribute utility functions given by,
(25)
It is important to note that the above constrained fuzzy arithmetic operations are unlike the regular arithmetic operations and can be expressed as follows:
${A\u2013B}_{\mathit{CO}1}=\left({a}_{1},{a}_{2},{a}_{3}\right)\u2013\left({b}_{1},{b}_{2},{b}_{3}\right)=({a}_{1}\u2013{b}_{3},{a}_{2}\u2013{b}_{2},{a}_{3}\u2013{b}_{1})$
(26)
${C\xf7D}_{\mathit{CO}2}=\left({c}_{1},{c}_{2},{c}_{3}\right)\xf7\left({d}_{1},{d}_{2},{d}_{3}\right)=({c}_{1}\xf7{d}_{3},{c}_{2}\xf7{d}_{2},{c}_{3}\xf7{d}_{1})$
(27)
Where,
CO1 is the constraint on fuzzy subtraction arithmetic operation and is defined as
( ${a}_{1}\ge {b}_{1},\mathit{}{a}_{2}\ge {b}_{2},\mathit{}{a}_{3}\ge {b}_{3})$
and ( ${a}_{2}\u2013{a}_{1}\ge {b}_{2}\u2013{b}_{1})$
CO2 is the constraint on fuzzy division arithmetic operation and is defined as
( ${d}_{1}\ge {c}_{1},\mathit{}{d}_{2}\ge {c}_{2},\mathit{}{d}_{3}\ge {c}_{3})$
and ( ${c}_{3}\u2013{c}_{2}\ge {d}_{3}\u2013{d}_{2})$
Without using the above constraints to perform fuzzy arithmetic, it is possible that the values of utility functions may come out to be out of bounds i.e. less than 0 and greater than 1. ^{[4]}
The composite attribute utility functions for a variety of construction approaches are evaluated and ranked using Graded Mean Integration Representation (GMIR) denoted as $\mathrm{P}\left({A}_{i}\right)\mathrm{}$
for a fuzzy number ${A}_{i}({a}_{1},{a}_{2},{a}_{3})$
. GMIR is a simple and accurate way of ranking fuzzy numbers and can be calculated using the formula:
(28)
A larger GMIR means a larger fuzzy number. The optimal construction approach is then selected by the conditional check,
$\mathit{Optimal}=I\mathit{P}\left(I\right)\ge P\left({A}_{i}\right),\mathit{i}=1\mathit{to\; N},\mathit{I}\in [1,N]$
(29)
Model Optimization:
The optimal construction approach is selected using particle swarm optimization (PSO). Each combination of construction approaches for all activities is considered as particle with characteristics, viz. position and velocity, in an N dimensional space. The number of dimensions is determined by the number of activities in the project. The position of a particle is a candidate solution to the fuzzy timecostquality tradeoff problem, and velocity of the particle is used to update the position of the particle until it achieves a local best position. The local best positions of all the particles are compared at each iteration to determine the global best position, i.e. the optimal solution to the timecostquality tradeoff problem.
Suppose that a construction project has N activities, with each activity having an index j, andM_{j} number of construction methods available. Let the total number of construction approaches or combinations to complete the entire project be P.
In Particle Swarm Optimization, each construction approach (i), to complete the project is considered as a particle in space with a position and velocity. The N number of activities form the dimensions of the space that defines the position and velocity of the particle.
Thus, the position of i^{th} particle, from a population of P, in the NDimensional space for k^{th} iteration is defined by a vector,
${X}_{i}\left(k\right)=\left[{x}_{i1}\left(k\right),{x}_{i2}\left(k\right),\dots ,{x}_{\mathit{ij}}\left(k\right),\dots {x}_{\mathit{iN}}\left(k\right)\right]=[\left(1,{m}_{1}\right),\left(2,{m}_{2}\right),\dots ,\left(j,{m}_{j}\right),\dots ,\left(N,{m}_{N}\right)]$
(30)
${m}_{i}=[1,{M}_{i}]$
(31)
Where,
${m}_{i}$
: an integer representing one of the construction methods from a range of available methods to execute activity j
The velocity of the same i^{th} particle is given by a vector,
${V}_{i}\left(k\right)=\left[{v}_{i1}\left(k\right),{v}_{i2}\left(k\right),\dots ,{x}_{\mathit{ij}}\left(k\right),\dots {x}_{\mathit{iN}}\left(k\right)\right]$
(32)
Fig. 4 Particlerepresented solution for the fuzzy timecostquality problem. ^{[3]}
The updating mechanism can be formed as follows:
$\mathit{For\; i}=\mathit{i\; to\; P\; and\; k}=1\mathit{to\; K}$
${V}_{i}\left(k\right)=w\left(k\right)\times {V}_{i}\left(k\u20131\right)+{c}_{1}\times {r}_{1}\times \left({X}_{i}^{L}\u2013{X}_{i}\left(k\u20131\right)\right)+{c}_{2}\times {r}_{2}\times ({X}^{G}\u2013{X}_{i}\left(k\u20131\right))$
(33)
${X}_{i}\left(k\right)={V}_{i}\left(k\right)+{X}_{i}\left(k\u20131\right)$
(34)
${X}_{i}\left(k\right)={V}_{i}\left(k\right)+{X}_{i}\left(k\u20131\right)$
(35)
${X}_{i}^{L}=[{x}_{i1}^{L},\dots {x}_{\mathit{ij}}^{L},\dots {x}_{\mathit{iN}}^{L}]$
(36)
${X}^{G}=[{x}_{1}^{G},\dots {x}_{j}^{G},\dots {x}_{N}^{G}]$
(37)
Where,
$K$
: Maximum number of iterations to be performed i.e. iteration limit for the optimization program
${X}_{i}^{L}$
: Vector that represents the local best for a particle i
${X}^{G}$
: Vector that represents the global best for the project activities
${c}_{1}$
, ${c}_{2}$
: Positive constants called learning factors
${r}_{1}$
, ${r}_{2}$
: Random numbers between 0 and 1
$w\left(k\right)$
: Inertia weight used to control the effect of velocity in previous iteration on the current one.
${c}_{1}$
, ${c}_{2}$
, ${r}_{1}$
, ${r}_{2}$
, $w\left(k\right)$
are parameters to be selected by the construction planner, and they greatly influence the optimization of the solution. The inertia weight w(t) can be constant or variable, and Shi and Eberhart ^{[4]} justified that a constant value 0.8 is a good choice when the maximum velocity is larger than 3. ${c}_{1}$
, ${c}_{2}$
, can be assumed as 1.0 unless more precise information is available.
Constraints are established on the particle position so that the solutions represented by particle does not become infeasible. Thus, the particles that fly outside the search constraint are constrained to $[1,{M}_{i}]$
by
Similarly, constraints are established on the velocity of the particle to avoid explosion. The velocity constraint is constrained to $\left[{\u2013x}_{j}^{\mathit{min}},{\u2013x}_{j}^{\mathit{max}}\right]$
by
The algorithm for PSO updating mechanism is shown in the flowchart below. For each iteration, the local best position for each particle is determined by comparing the composite fuzzy utility function of the local best position up to the previous iteration to that of the composite fuzzy utility function corresponding to the position in current iteration. The global best is determined by comparing the composite fuzzy utility function for all particles for each iteration. The program is stopped if the either the prespecified maximum number of iterations since last global best updating or maximum total number of iterations is achieved.
Fig. 5 Flowchart of PSO for the fuzzy timecostquality tradeoff problem. ^{[3]}
Comparison of Models
Each of the above models are summarized below, along with their advantages and disadvantages:
Linear Programming Model
This model uses linear programming and simplex algorithm to solve TCQT problem. It assumes a linear relationship between time, cost and quality of activity. The objective function is optimized using algorithms like gradient descent.
Advantages:
 Since simplex algorithms provide more than one solutions, this method gives more than one solutions to the timecostquality tradeoff problem which may prove advantageous when the construction planners are looking for alternatives.
 Linear programming is a simpler mathematical concept.
 A nonlinear timecost quality relationship may be easily accommodated into the model to get more accurate results.
Disadvantages:
 It is difficult to formulate a linear program that can accurately describe the complex realworld problem like a construction project. Thus, this method can be used only for simple projects.
 Time, cost, quality relationships are complex, and cannot be represented by simple linear assumptions. A nonlinear relationship assumption on the other hand increases the computation cost.
 The gradientdescent optimization algorithm may fail to local to global optimum by terminating at a local optimum.
MultiObjective Genetic Algorithm Optimization Model:
This model uses natural evolution and reproduction process to solve timecostquality tradeoff problem. Data from past projects is used to establish relationship between time, cost and quality. The objective function is optimized using Genetic Algorithm and Pareto optimality
Advantages:
 This method is based on data collected from previous projects. Hence, an assumption regarding time, cost, quality relationship is not required.
 Genetic Algorithm is a rigorous search algorithm which is capable of locating multiple local optimums. The construction planner may then select the global optimum solution from the local optimums found.
 It can accurately model large realworld construction scenarios.
Disadvantages:
 Genetic Algorithm is a highly iterative process. It may prove to be time consuming.
 A construction planner may not be sure if the optimal solution obtained from this model is the global optimum.
FuzzyMultiObjective Particle Swarm Optimization Model:
Particle swarm optimization algorithm, which imitates the social behavior like movement of a flock of birds towards the optimum position, is used in this model. Time, cost and quality relationship for different construction methods is established using past project data. The uncertainty in time, cost and quality relationship is incorporated using Fuzzy number theory.
Advantages:
 This method is based on past project data, and hence it is not necessary to assume time, cost and quality relationships.
 The Particle Swarm Optimization algorithm with Graded Mean Integration Representation of Composite Attribute Utility functions enables this method to find the global optimal solution.
 This model can be used to find the best solution to a complex realworld timecostquality problem of construction project.
Disadvantages:
 Fuzzy arithmetic must be followed to evaluate the utility functions which may prove to be a challenge.
 Inappropriate learning factors and inertia weights, if chosen, may lead to inaccurate results.
 Swarming of particles towards an optimal solution may prove to be a time consuming and computationally expensive process.
Conclusion
The traditional timecost tradeoff analysis fails to solve the realworld construction problems where quality is an important parameter for success of a project. The threedimensional timecostquality tradeoff analysis is a more appropriate approach. Three models reviewed in this paper have merits and demerits and must be used aptly according to the complexity of the construction project. All the three models have the same structure, i.e. the model is first formulated as per the problem and then optimized using a specific algorithm.
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