Time-Cost-Quality Tradeoff Modeling based on Resource Allocation

Project time-cost-quality tradeoff problem (PTCQTP) can be defined as follows: a project is represented by an activity-on-node network with $n$ activities that is an acyclic digraph $G=(A)$, where $A=\{0, \ldots, n+1\}$ is the set of nodes (construction activities). In the network both node $(0)$ and node $(n+1)$ are dummy activities. $P$ is the set of all paths in the activity-on-node network, starting from activity $(0)$ and ending at activity $(n+1)$ and $P_{l}$ is the set of activities contained in path $l \in P$.

Each activity $i \in A$ is associated with its time $T_{i}$, cost $C_{i}$, and quality $Q_{i} .$ The earliest/latest starting times $\left(\mathrm{EST}_{i}^{s} / \mathrm{LST}_{i}^{s}\right)$ for each activity $i$ are easily calculated using the forward-backward passes. Each activity $i$ can be decomposed into four resources of $L_{i}$ (construction labor), $M_{i}$ (construction material and machine), $E_{i}$ (construction equipment), and $A_{i}$ (construction administration). Construction labor $L_{i}$ is associated with labor productivity $\mathrm{LP}_{i}$, labor cost $\mathrm{LC}_{i}$, labor amount $\mathrm{LA}_{i}$, and labor quality $\mathrm{LQ}_{i}$. Construction material $M_{i}$ is associated with material and machine cost $\mathrm{MC}_{i}$ and material quality $\mathrm{MQ}_{i} .$ Construction equipment $E_{i}$ is associated with equipment productivity $\mathrm{EP}_{i}$, equipment cost $\mathrm{EC}_{i}$, equipment amount $\mathrm{EA}_{i}$, and equipment quality $\mathrm{EQ}_{i}$. Construction administration $A_{i}$ is associated with administration cost $\mathrm{AC}_{i}$ and administration quality $\mathrm{AQ}_{i}$.

In order to guarantee public safety and interest, local governments would supervise and secure all construction projects to be above a minimum quality level $\left(Q^{\mathrm{min}}\right).$ If any part of a construction project fails to conform with the minimum construction quality standards, the project could not be delivered properly, and the unqualified parts (LQ $_{i}$, $\mathrm{MQ}_{i}, \mathrm{EQ}_{i}$ ) should be replaced or reworked until this quality conforms to the minimum requirements such as the minimum labor quality $\left(\mathrm{LQ}_{i}^{\mathrm{min}}\right)$, the minimum material quality $\left(\mathrm{MQ}_{i}^{\min }\right)$, the minimum equipment quality $\left(\mathrm{EQ}_{i}^{\min }\right)$, and the minimum administration quality $\left(\mathrm{AQ}_{i}^{\mathrm{min}}\right) .$ The reworks or replacement of construction parts obviously increase cost and delay schedule if the parts of inferior quality are detected by supervisors according to regulations and codes. Research on rework or replacement is so complicated that it would mislead this paper into game method rather than optimization analysis, so this model assumes that any part of a construction part could not be below its minimum standard.

Since the project delivery time defined clearly in construction agreements is a crucial factor for project owner and contractors, contractors should complete and deliver the project to the owner in time. Otherwise the contractors will pay a certain penalty because of delay delivery. This kind of contract conditions will encourage the contractor to set up a project time plan in advance. Contractors are naturally interested in controlling cost actively and minimize the project total cost ($C$).

As discussed previously in aspects of local government’s regulations in securing construction quality, project owner’s stimulus to shorten construction time, and contractor’s intrinsic motivation to reduce construction cost, PTCQTP can be formally stated as follows: given a network with a lot of nodes, that is, activities by their sequences, durations, costs, and qualities, a general status is determined by each activity according to at least one of the following objectives: minimize the project duration, maximize the requirements of construction quality codes and standards, and minimize budget.

Assign a construction team to excavate earth, to erect formwork, to mix concrete, or to do other jobs; their working quality will decline if they intend to increase productivity, and an approximate linear relationship between labor productivity and labor quality is observed here. A more complex relationship function between labor productivity and labor quality is also considered in later case study:

$\mathrm{LPRD}_{(i)}=\mathrm{LPRD}_{i}^{\max }-\mathrm{LQK}_{i} \times\left(\mathrm{LQ}_{(i)}-\mathrm{LQ}_{i}^{\min }\right),$           (1)

where $L Q_{(i)}=$ actual quality level of construction labor $(i)$ working in activity $(i)$, $\mathrm{LQ}_{(i)} \in\left(\mathrm{LQ}_{i}^{\min }, \mathrm{LQ}_{i}^{\max }\right) ; \mathrm{LQ}_{i}^{\max }=$ maximum quality level of construction labor $(i)$ working in activity $(i) ; \quad \mathrm{LQK}_{i}=\left(\mathrm{LPRD}_{i}^{\max }-\mathrm{LPRD}_{i}^{\min }\right) /\left(\mathrm{LQ}_{i}^{\max }-\mathrm{LQ}_{i}^{\min }\right)$; LPRD $_{i}^{\min }=$ minimum productivity level of construction labor $(i)$ working in activity (i); LPRD $_{i}^{\max }=$ maximum productivity level of construction labor (i) working in activity $(i)$; LPRD $_{(i)}=$ actual productivity level of construction labor $(i)$ working in activity $(i), \operatorname{LPRD}_{(i)} \in\left(\operatorname{LPRD}_{i}^{\min }, \operatorname{LPRD}_{i}^{\max }\right)$.

The time of a construction activity is mainly determined by its job quantities and productivities rather than its material or machine quality, and the material can hardly interfere with the activity's time, either. Thereafter an approximate linear relationship between material quality and material cost is determined.

The relationship between quality and quality cost for a manufacturing company

$\mathrm{MC}_{(i)}=\mathrm{MC}_{i}^{\mathrm{min}}+\mathrm{MQK}_{i} \times\left(\mathrm{MQ}_{(i)}-\mathrm{MQ}_{i}^{\mathrm{min}}\right)$          (2)

where $\mathrm{MQ}_{(i)}=$ actual quality level of construction material in activity (i), $\mathrm{MQ}_{(i)} \in\left(\mathrm{MQ}_{i}^{\min }, \mathrm{MQ}_{i}^{\max }\right) ; \mathrm{MQ}_{i}^{\min }=$ minimum quality level of construction material in activity $(i) ; \mathrm{MQ}_{i}^{\max }=$ maximum quality level of construction material in activity $(i)$; $\mathrm{MQK}_{i}=\left(\mathrm{MC}_{i}^{\max }-\mathrm{MC}_{i}^{\min }\right) /\left(\mathrm{MQ}_{i}^{\max }-\mathrm{MQ}_{i}^{\min }\right) ; \quad \mathrm{MC}_{i}^{\min }=\operatorname{minimum} \quad$ cost of construction material in activity $(i) ; \mathrm{MC}_{i}^{\max }=$ maximum cost of construction material in activity $(i) ; \mathrm{MC}_{(i)}=a$ $\mathrm{MC}_{(i)} \in\left(\mathrm{MC}_{i}^{\min }, \mathrm{MC}_{i}^{\max }\right)$

Construction equipment is a crucial factor of construction techniques to increase construction quality, to reduce cost, and to shorten time. In order to calculate construction time variation impacted by equipment, a modified factor to labor productivity caused by equipment $(i)$ is introduced:

$\operatorname{PRD}_{(i)}=\operatorname{LPRD}_{(i)} \times \operatorname{DEK}_{(i)} \text {, }$       (3)

where $\operatorname{PRD}_{(i)}$ is the actual productivity in activity $(i) ; \operatorname{DEK}_{(i)}$ is a modified factor to labor $(i)$ productivity by changes of construction equipment parameters; $\operatorname{LPRD}_{(i)}$ is labor productivity in activity $(i)$.

A better equipment quality performance will improve construction productivity, so the modified factor $\mathrm{DEK}_{(i)}$ could be derived from the equipment quality $\mathrm{EQ}_{(i)}$ :

$\mathrm{DEK}_{(i)}=\mathrm{DEK}_{i}^{\min }+\mathrm{DQK}_{i} \times\left(\mathrm{EQ}_{(i)}-\mathrm{EQ}_{i}^{\min }\right)$      (4)

where $\mathrm{DQK}_{i}=\left(\mathrm{DEK}_{i}^{\max }-\mathrm{DEK}_{i}^{\min }\right) /\left(\mathrm{EQ}_{i}^{\max }-\mathrm{EQ}_{i}^{\min }\right) .$

Construction equipment quality and equipment cost is also assumed as an approximate linear function just like construction material:

$\mathrm{EC}_{(i)}=\left[\mathrm{EC}_{i}^{\min }+\mathrm{EQK}_{i} \times\left(\mathrm{EQ}_{(i)}-\mathrm{EQ}_{i}^{\min }\right)\right]$       (5)

where $\mathrm{EQ}_{(i)}=$ actual quality level of construction equipment $(i)$ in activity $(i)$, $\mathrm{EQ}_{(i)} \in\left(\mathrm{EQ}_{i}^{\min }, \mathrm{EQ}_{i}^{\max }\right) ; \mathrm{EQ}_{i}^{\min }=$ minimum quality level of construction equipment (i) in activity $(i) ; \mathrm{EQ}_{i}^{\max }=$ maximum quality level of construction equipment $(i)$ in activity $(i) ; \mathrm{EQK}_{i}=\left(\mathrm{EC}_{i}^{\max }-\mathrm{EC}_{i}^{\min }\right) /\left(\mathrm{EQ}_{i}^{\max }-\mathrm{EQ}_{i}^{\min }\right) ; \mathrm{EC}_{i}^{\min }=$ minimum cost of construction equipment $(i)$ in activity $(i) ; \mathrm{EC}_{i}^{\max }=$ maximum cost of construction equipment $(i)$ in activity $(i) ; \mathrm{EC}_{(i)}=$ actual cost of construction equipment in activity (i), $\mathrm{EC}_{(i)} \in\left(\mathrm{EC}_{i}^{\min }, \mathrm{EC}_{i}^{\max }\right)$.

Work overtime usually decreases construction productivity and increases hourly cost rate. Then construction equipment cost $\mathrm{EC}_{(i)}$ will be modified by factor $\alpha_{i}$ :

$\begin{aligned} \mathrm{EC}_{(i)}=& {\left[\mathrm{EC}_{i}^{\min }+\mathrm{EQK}_{i} \times\left(\mathrm{EQ}_{(i)}-\mathrm{EQ}_{i}^{\min }\right)\right] \times \alpha_{i} } \\ =& {\left[\mathrm{EC}_{i}^{\min }+\mathrm{EQK}_{i} \times\left(\mathrm{EQ}_{(i)}-\mathrm{EQ}_{i}^{\min }\right)\right] } \\ & \times\left[1+\left(\mathrm{DPK}_{(i)}-1\right) \times \mathrm{EOK}_{i}\right], \end{aligned}$        (6)  

where $\alpha_{i}=$ construction equipment cost modification factor during overtime because of extra or additional construction equipment, $\alpha_{i}=1+\left(\mathrm{DPK}_{(i)}-1\right) \times \mathrm{EOK}_{i}$; $\mathrm{EOK}_{i}=$ productivity decreased rate during overtime per unit time (e.g., hour), normally $20 \%$.

A construction team consisting of sufficient crew members could improve construction quality and consume a reasonable cost, but the construction team hardly impacts on construction productivities. Therefore it is assumed that administration cost and administration quality are an approximate linear function:

$\mathrm{AC}_{(i)}=\mathrm{AC}_{i}^{\min }+\mathrm{AQK}_{i} \times\left(\mathrm{AQ}_{(i)}-\mathrm{AQ}_{i}^{\min }\right)$        (7)

where $\mathrm{AQ}_{(i)}=$ actual quality level of construction administration $(i)$ in activity $(i)$, $\mathrm{AQ}_{(i)} \in\left(\mathrm{AQ}_{i}^{\min }, \mathrm{AQ}_{i}^{\max }\right) ; \quad \mathrm{AQ}_{i}^{\min }=$ minimum quality level of construction administration $(i)$ in activity $(i) ; \mathrm{AQ}_{i}^{\max }=$ maximum quality level of construction administration $(i)$ in activity $(i) ; \mathrm{AQK}_{i}=\left(\mathrm{AC}_{i}^{\max }-\mathrm{AC}_{i}^{\min }\right) /\left(\mathrm{AQ}_{i}^{\max }-\mathrm{AQ}_{i}^{\min }\right) ;$

$\mathrm{AC}_{i}^{\min }=$ minimum cost of construction administration $(i)$ in activity $(i) ;$

$\mathrm{AC}_{i}^{\max }=$ maximum cost of construction administration $(i)$ in activity $(i) ;$

$\mathrm{AC}_{(i)}=$ actual cost of construction administration $(i)$ in activity,

$\mathrm{AC}_{(i)} \in\left(\mathrm{AC}_{i}^{\min }, \mathrm{AC}_{i}^{\max }\right) .$

Since work overtime might increase administration cost, the construction administration cost will be modified by factor $\beta_{i}$:

$\begin{aligned} \mathrm{AC}_{(i)}=& {\left[\mathrm{AC}_{i}^{\min }+\mathrm{AQK}_{i} \times\left(\mathrm{AQ}_{i}-\mathrm{AQ}_{i}^{\min }\right)\right] \times \beta_{i} } \\=& {\left[\mathrm{AC}_{(i)}=\mathrm{AC}_{i}^{\min }+\mathrm{AQK}_{i} \times\left(\mathrm{AQ}_{(i)}-\mathrm{AQ}_{i}^{\min }\right)\right] } \\ & \times\left[\mathrm{ACRK}_{i}+\frac{1-\mathrm{ACRK}_{i}}{\mathrm{DPK}_{(i)}}\right], \end{aligned}$   (8)

where $\beta_{i}=$ administration cost modification factor during work overtime because of extra or additional construction equipment, $\beta_{i}=\mathrm{ACRK}_{i}+\left(1-\mathrm{ACRK}_{i}\right) / \mathrm{DPK}_{(i)}$; $\mathrm{ACRK}_{i}=$ administration hourly cost rate factors in activity $(i)$ when overtime working is applicable, usually $2.0$.

When labors are working together as a construction crew in a construction activity $(i)$, their working duration is the time of the construction activity, which could be estimated by the construction quantities (QNT) and the actual productivity, overtime factors:

$\operatorname{DUR}_{(i)}=\frac{\mathrm{QNT}_{(i)}}{\operatorname{PRD}_{(i)} \times \mathrm{DPK}_{(i)}},$           (9)

where $\mathrm{DUR}_{(i)}=$ duration (time) of construction activity $(i) ; \mathrm{QNT}_{(i)}=$ quantities of construction activity $(i) ; \quad \mathrm{PRD}_{(i)}=$ actual $\quad$ productivity in activity $\quad(i)$; $\mathrm{DPK}_{(i)}=$ overtime factors if accelerated construction speed is desired, $\mathrm{DPK}_{(i)} \in[1.0,1.5]$ when overtime varies from 0 to 4 hours per day since standard working time is eight hours per day. DPK $_{(i)}=1.0$ means no overtime work assigned.

Labor cost in construction activity $(i)$ would be determined by standard daily cost and overtime work cost [18]. It is assumed that work overtime is paid in an hourly cost rate comparing with standard labor cost:

$\begin{aligned} \mathrm{LC}_{(i)}=& \mathrm{LCD}_{i}^{S} \times \mathrm{DUR}_{(i)}+\left(\mathrm{DPK}_{(i)}-1\right) \\ & \times\left(\mathrm{LCRK}_{(i)} \times \mathrm{LCD}_{i}^{S}\right) \times \mathrm{DUR}_{(i)} \\=& \mathrm{LCD}_{i}^{S} \times \mathrm{DUR}_{(i)} \times\left[1+\left(\mathrm{DPK}_{(i)}-1\right) \times \mathrm{LCRK}_{(i)}\right] \\=&\left(\mathrm{LCD}_{i}^{S} \times \mathrm{QNT}_{(i)}\right) \\ & \times\left(\left[\mathrm{LPRD}_{i}^{\max }-\mathrm{LQK}_{i} \times\left(\mathrm{LQ}_{(i)}-\mathrm{LQ}_{i}^{\min }\right)\right]\right.\\ &\left.\times\left[\mathrm{DEK}_{i}^{\min }+\mathrm{DQK}_{i} \times\left(\mathrm{EQ}_{(i)}-\mathrm{EQ}_{i}^{\min }\right)\right]\right)^{-1} \\ & \times\left(\mathrm{LCRK}_{i}+\frac{1-\mathrm{LCRK}_{i}}{\mathrm{DPK}_{(i)}}\right) \end{aligned}$         (10)

where $\mathrm{LC}_{(i)}=$ labor cost in construction activity $(i) ; \mathrm{LCD}_{i}^{s}=$ standard labor cost per unit time (e.g., day) in construction activity $(i) ; \mathrm{LCRK}_{i}=$ labor cost rate factors in activity $(i)$ when overtime work is applicable, usually $2.0.$

Since a construction project comprises various resources such as materials, machines, method, labors, and even management, the overall quality $Q$ of a construction project is calculated by each activity's quality $\mathrm{AQP}_{(i)}$ and its quality weight $\mathrm{WT}_{i}$ :

$\mathrm{Q}=\sum\limits_{i=1}^{n}\left(\mathrm{WT}_{i} \times \mathrm{AQP}_{(i)}\right),$ (11)

where $Q=$ the general quality of a construction project $; n=$ number of activities in a construction project; $\mathrm{WT}_{i}=$ quality weight indicator of each construction activity $(i)$, and $\sum_{i=1}^{n} \mathrm{WT}_{i}=1.0 ; \mathrm{AQP}_{(i)}=$ quality performance of construction activity (i) calculated by its labor quality, material quality, construction equipment quality, and administration $\mathrm{AQP}_{(0)}=\mathrm{LWT}_{i} \times \mathrm{LQ}_{(i)}+\mathrm{MWT}_{i} \times \mathrm{MQ}_{(i)}+\mathrm{EWT}_{i} \times \mathrm{EQ}_{(i)}+\mathrm{AWT}_{2} \times \mathrm{AQ}_{(i)} ; \mathrm{LWT}_{i}$ $\mathrm{AQP}_{(i)}=\mathrm{LWT}_{i} \times \mathrm{LQ}_{(i)}+\mathrm{MWT}_{i} \times \mathrm{MQ}_{(i)}+\mathrm{EWT}_{i} \times \mathrm{EQ}_{(i)}+\mathrm{AWT}_{i} \times \mathrm{AQ}_{(i)} ; \mathrm{LWT}_{i}$, $\mathrm{MWT}_{i}$, EWT $_{i}, \mathrm{AWT}_{i}=$ weight indicators of construction labor, material, equipment, and administration in activity $(i)$ respectfully; $\mathrm{LQ}_{(i)}, \mathrm{MQ}_{(i)}, \mathrm{EQ}_{(i)}, \mathrm{AQ}_{(i)}=$ quality of construction labor, material, equipment, and administration in activity $(i)$ respectfully.

The overall cost of a construction project is added up with each construction activity's cost and its administration cost $\mathrm{AC}_{(i)}$. Thereby the overall cost $\mathrm{C}$ is calculated as follows:

$C=\sum\limits_{i=1}^{n}\left(\mathrm{LC}_{(i)}+\mathrm{MC}_{(i)}+\mathrm{EC}_{(i)}+\mathrm{AC}_{(i)}\right),$                (12)

where $C=$ the overall cost of a construction project; $\mathrm{LC}_{(i)}, \mathrm{MC}_{(i)}, \mathrm{EC}_{(i)}, \mathrm{AC}_{(i)}=$ the cost of labor, materials, equipment, and administration in construction activity (i) respectfully; $n=$ number of all construction activities.

The overall time $T$ of a construction project can be easily calculated by nodes as activities in an acyclic digraph network:

$T=\max _{i=1, n}\left(\mathrm{EST}_{(i)}+\operatorname{Dur}_{(i)}\right),$           (3)

where $T=$ overall time in construction project, $\operatorname{EST}_{(i)}=\max _{h=1, i-1}\left(\operatorname{EST}_{(h)}+\operatorname{Dur}_{(h)}\right)$, as well as the earliest starting time of activity $(i)$ derived by its predecessors, and the first $\mathrm{EST}_{(1)}=0$.

The PTCQTP model is to minimize the project overall time while conforming to the requirements of construction quality standards within a specified budget, which can be stated as follows:

$\begin{array}{ll}\min & Z_{1}=T, \\ \text { s.t. } & C \cdot X \leq A ; Q \cdot X \geq B, \end{array}$          (14)

where $A=$ the specified maximum cost; $B=$ the minimum requirements of construction quality;

$\mathrm{X}=\left[\mathrm{DPK}_{(1)} \cdots \mathrm{DPK}_{(n)} \mathrm{LQ}_{(1)} \cdots \mathrm{LQ}_{(n)} \mathrm{MQ}_{(1)} \cdots \mathrm{MQ}_{(n)} \mathrm{EQ}_{(1)} \cdots \mathrm{EQ}_{n} \mathrm{AQ}_{(1)} \cdots \mathrm{AQ}_{(n)}\right]_{1 \times 5 n}^{T}$,

a vector of all variables in the PTCQTP model.

In order to solve this comprehensive time-cost-quality tradeoff problem, a global optimization algorithm is necessary. The genetic algorithm is widely applied in optimization solution and pattern search is one of direct search methods, both methods can solve global optimization problems. Tests of different algorithms developed for this model reveal that the pattern search algorithm cannot solve this complex nonlinear programming problem by direct search which often falls into local optimization, but the genetic algorithm can find out a global optimization though solutions are not precise but acceptable.

The genetic algorithm has been widely applied in previous works of the literature; thereafter the development process of genetic algorithm for the PTCQTP model is not worthy of detailing here.

(1) If $Q=0.8$ or above and $C=\$ 350000$ or below.

If the overall quality objective is $0.8$ or above and the overall cost objective is $\$ 350000$ or below, the optimal overall time is 64 days suggested by the PTCQTP model.

All optimized construction arrangements and resource utilizations suggested by this model are available; for example, the optimal working hours of each activity are shown in Figure 4. The optimal working arrangement reveals that nearly all activities are carried out in overtime working hours since the construction time is limited. Although overtime working demands higher labor cost, overtime working shortens construction time and secures construction quality.

Figure 4 

Optimized working hours a day in twenty activities.

The optimized material quality performances in 20 construction activities vary much, as shown in Figure 5. For example, materials in the activity (6) "-Refill foundation earth" are required to maintain the highest quality level $(=0.95)$, while the activity (10) "Build block in 1st story" and (12) "-Install reinforcing for 2nd story" only need to be the lowest quality level $(=0.7)$. Obviously the lowest quality requirement would save cost.

Figure 5 

Optimized material quality performances in twenty activities.

(2) Optimized overall time-cost-quality.

A part of optimized results (shown in Table 3) are excerpted from numerous calculations when construction cost and quality are confined within potential solution boundaries. A visual tradeoff among construction time, cost, and quality is presented in Figure 6.

Table 3 

The minimum time of a building project while project cost and general quality are bounded.

Figure 6 

A visual three-dimensional tradeoff pattern among project time, cost, and quality.

When a moderate quality performance, for example, 0.86, is set and the overall cost is $320,000, the overall time is about 90 days by the tradeoff model. The overall cost will climb to $390,000 if the overall time is shortened to 54 days and the quality objective is unchanged.

Tradeoff curves between the overall cost and time under different quality objectives are shown in Figure 7, which proves that traditional linear assumptions between project cost and time are reasonable somehow.

Figure 7 

Tradeoff relationship between construction cost and time when the quality sets 0.80, 0.86, and 0.92.