## 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.

### Problem Definition

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.