## Time-Cost-Quality Tradeoff Modeling based on Resource Allocation

Read this article. The paper presents an optimization model that enables managers to effectively evaluate trade-offs related to time, cost, and other competing priorities. Pay particular attention to Section 6 as it provides an illustrated example of building a home.

### Decision Variables and Assumptions

#### Relationship among Equipment Productivity, Quality, and Cost

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 $20 \%$.