Work and Labor Optimization in Construction Projects: A Mathematical Analysis

Construction projects often face unexpected challenges, such as reduced labor force due to workers dropping out, which can impact their completion timeline. This article delves into a mathematical problem rooted in real-world scenarios to understand the optimal number of days required to finish a project given the workers' productivity and unexpected drops.

Problem Introduction

A construction project was planned to be completed in a specific number of days with 150 workers. However, due to various unforeseen circumstances, 4 workers dropped out from the second day onwards. If the project took 8 more days than initially planned, how many days were originally planned to complete this work?

Mathematical Breakdown

To solve this problem, let's define some key variables:

D: The original number of days planned to finish the work. Total work: The total work required to complete the project, expressed in terms of worker-days.

Total Work Calculation

The total work can be expressed as 150D worker-days, where 150 is the number of workers and D is the number of days. This is straightforward since the project is initially planned for 150 workers for D days.

Worker Dynamics

The number of workers decreases by 4 each day starting from the second day, as workers drop out:

Day 1: 150 workers Day 2: 146 workers Day 3: 142 workers ... and so on

Work Done Over Extended Days

The work done in the extended days can be calculated as an arithmetic series:

Where the number of workers follows the pattern 150 - 4n, and the total work can be expressed as:

[ 150D sum_{n0}^{D-8} (150 - 4n) ]

The sum of this arithmetic series can be calculated using the formula:

[ S frac{n}{2}(a l) ]

Where:

a 150 l 150 - 4(D - 8) n D - 8

Solving the Equation

By equating the total work done to the original planned work, we get:

[ 150D frac{D - 8}{2} (150 150 - 4(D - 8)) ]

After simplifying, the equation becomes:

[ 150D (D - 8) (272 - 4D) ]

Rearranging terms, we get a quadratic equation:

[ 2D^2 - 30D - 1088 0 ]

Solving this quadratic equation using the quadratic formula:

[ D frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

With a 2, b 30, and c -1088, we can calculate:

[ D frac{-30 pm sqrt{30^2 - 4 cdot 2 cdot (-1088)}}{2 cdot 2} ]

And further simplifying, we obtain two potential solutions:

[ D frac{68}{4} 17 quad (valid) ]

[ D frac{-128}{4} -32 quad (not valid) ]

Therefore, the original number of days planned to complete the work is 17 days.

Conclusion

This mathematical problem highlights the importance of understanding the dynamics of labor productivity in construction projects. By carefully analyzing the reduction in the number of workers and its impact on the completion time, we can optimize project planning and resource allocation to meet deadlines effectively.