Review of Time Cost and Quality Tradeoff Analysis Models in Construction

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 life-cycle. 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 time-cost-quality tradeoff have become increasingly sophisticated. The objective of this paper is to review three models, viz. Linear Programming Model, Multi-Objective Genetic Algorithm Optimization Model, Fuzzy Multi-Objective Particle Swarm Optimization Model. Applications, advantages and disadvantages of these models in construction projects shall be studied and key factors involved in time-cost-quality tradeoff shall be highlighted. This study shall not include the environmental and energy effects of time-cost-quality 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.

A lot of research has been carried out on the time-cost 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 2-dimensional problem. These models too however have their own advantages and drawbacks. Although mathematical models provide a rigorous mathematical solution to the time-cost trade of problem, they require formulation of objective functions and constraints which is often a difficult and time-consuming 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 time-cost-quality tradeoff analysis models, their application and their merits and demerits.

Time-Cost-Quality Tradeoff Analysis Models

The three time-cost-quality tradeoff analysis models reviewed in this paper are:

• Linear Programming Model
• Multi-Objective Genetic Algorithm Optimization Model
• Fuzzy Multi-Objective 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 time-cost-quality balance.

Linear Programming Model

This analysis model was one of the earliest attempts at studying the time-cost-quality trade off. All the models preceding this considered only the time-cost 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:

$\mathrm{S}\left(\mathrm{L}\right)=\mathrm{ }\frac{\mathrm{NCOST}\left(\mathrm{L}\right)–\mathrm{CRCOST}\left(\mathrm{L}\right)}{\mathrm{NTIME}\left(\mathrm{L}\right)–\mathrm{CRTIME}\left(\mathrm{L}\right)}$

         (1) 

$\mathrm{INTERCPT}\left(\mathrm{L}\right)=\mathrm{CRCOST}\left(\mathrm{L}\right)–\mathrm{S}\left(\mathrm{L}\right)\times \mathrm{CRTIME}\left(\mathrm{L}\right)$

     (2)

$\mathrm{SQ}\left(\mathrm{L}\right)=\mathrm{ }\frac{\mathrm{NQUAL}\left(\mathrm{L}\right)–\mathrm{CRQUAL}\left(\mathrm{L}\right)}{\mathrm{NTIME}\left(\mathrm{L}\right)–\mathrm{CRTIME}\left(\mathrm{L}\right)}$

         (3) 

$\mathrm{INTERCPTQ}\left(\mathrm{L}\right)=\mathrm{CRQUAL}\left(\mathrm{L}\right)–\mathrm{SQ}\left(\mathrm{L}\right)\times \mathrm{CRTIME}\left(\mathrm{L}\right)$

     (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:

$\mathrm{OBJFNT}:\mathrm{Y}\left(‘{\mathrm{N}}^{‘}\right)=\mathit{OBJT}$

         (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)

$\mathrm{ETIME}:\mathrm{Y}\left(‘{\mathrm{I}}^{‘}\right)+X\left(L\right)–Y(‘J^{‘})\le 0$

        (8)

$\mathrm{LBTACT}:\mathrm{X}\left(‘{\mathrm{L}}^{‘}\right)\ge \mathit{CRTIME}\left(L\right)$

        (9)

$\mathrm{UBTACT}:\mathrm{X}(‘{\mathrm{L}}^{‘})\le \mathrm{NTIME}\left(\mathrm{L}\right)$

        (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 time-cost-quality tradeoff based on the above formulation are:

1. 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.
2. 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.
3. 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 time-cost-quality of construction project which may not be appropriate. However, the linearity of relationships makes computation easy. Also, this model can be used to accommodate non-linear relationships which makes it a foundation for the models developed in the future.

Multi-Objective 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 time-cost-quality trade-off 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 three-dimensional representation of the trade-offs 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 two-folds:

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 Time-cost-quality trade-off 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:

1. $\mathit{Project\; Time}=\sum _{i=1}^l{T}_i^n$

           (14)

Where,

$T_i^n$

: duration of activity (i) on the critical path using resource utilization n

2. $\mathit{Project\; Cost}=\sum _{i=1}^l{(M}_i^n+D_i^n\times R_i^n+B_i^n)$

        (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)

3. $\mathit{Project\; Quality}=\sum _{i=1}^l{\mathit{wt}}_i\sum _{k=1}^K{\mathit{wt}}_{i,k}\times Q_{i,k}^n$

        (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:

1. Initialization
2. Fitness Functions Evaluation
3. 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
• Cross-over rate

The mutation and cross-over 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:

1. 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.
2. A new Child population is generated by selection, cross-over 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 cross-over 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.
3. 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.
4. 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.
5. The combined population is then sorted based on their Pareto optimal rank. As a tie-breaker, the solution with higher crowding distance is given preference.
6. 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 Time-Cost-Quality Trade-off Problem ^[5]

Fuzzy-Multi-Objective Particle Swarm Optimization

This method incorporates the uncertainty in time-cost-quality relationship associated with various construction methods for each activity by considering the traditional time-cost-quality problem as a fuzzy time-cost-quality problem and solving it using a fuzzy-multi-objective particle swarm optimization algorithm. The time, cost and quality are considered as fuzzy numbers. This approach of fuzzy time-cost-quality analysis also takes into account the goodness-of-fit 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)

$u_T=\frac{T^{+}–T}{T^{+}–T^{–}}$

           (18) 

$u_C=\frac{C^{+}–C}{C^{+}–C^{–}}$

           (19)

$u_Q=\frac{{Q–Q}^{–}}{Q^{+}–Q^{–}}$

           (20)

Where

$w_T{,w}_C,w_Q$

: Weights of criteria

$T^{+}{,C}^{+},Q^{+}$

: Maximum time cost and quality of each activity

$T^{–}{,C}^{–},Q^{–}$

: 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 time-cost-quality 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

$\mathit{U}$

 into a closed unit interval [0,1], where for 

$\mathit{x}\mathit{\in }\mathit{U}$

,