A Mathematical Analysis of Workforce Reduction and Completion Time
Imagine a scenario where a group of men are tasked with completing a project. The project can be achieved by 80 men working for 40 days. However, as the project progresses, the workforce diminishes by 5 men every 10 days. How can we calculate the exact time when the project will be completed? This article provides a step-by-step mathematical solution with the help of man-days and labor productivity concepts. Let's dive into the analysis.
Understanding the Problem
We start by examining the initial conditions. If 80 men can complete the work in 40 days, the total work required can be expressed as:
total_work 80 men * 40 days 3200 man-days
Here, 3200 man-days represent the total effort required to complete the project.
Step 1: Calculating Total Work
The total amount of work that needs to be done is 3200 man-days. This is calculated by the number of men available and the number of days they will work together initially.
Step 2: Work Done in the First 10 Days
During the initial 10 days, all 80 men are working. The work done in these 10 days is:
work_done_first_10_days 80 men * 10 days 800 man-days
Step 3: Remaining Work
After the first 10 days, the remaining work is calculated by subtracting the work done from the total work:
remaining_work 3200 man-days - 800 man-days 2400 man-days
Step 4: Workforce Dynamics
As the project progresses, the workforce decreases by 5 men every 10 days. The workforce changes as follows:
Days 0-10: 80 men Days 10-20: 75 men (80 - 5) Days 20-30: 70 men (75 - 5) Days 30-40: 65 men (70 - 5) Days 40-50: 60 men (65 - 5)Step 5: Calculating Work Done in Each 10-Day Interval
We now calculate the work done in each of these subsequent 10-day intervals:
Days 10-20: 75 men * 10 days 750 man-days Days 20-30: 70 men * 10 days 700 man-days Days 30-40: 65 men * 10 days 650 man-days Days 40-50: 60 men * 10 days 600 man-daysStep 6: Cumulative Work Done
Summing up the work done in these intervals, we get:
Work done in Days 0-10: 800 man-days Work done in Days 10-20: 750 man-days Work done in Days 20-30: 700 man-days Work done in Days 30-40: 650 man-days Work done in Days 40-50: 600 man-daysThe total work done after 50 days is:
total_work_done 800 750 700 650 600 3500 man-days
At the end of 50 days, the total work done is 3500 man-days, which surpasses the original 3200 man-days required to complete the project.
Step 7: Determining the Completion Time
With the workforce continuously reducing, we see that the work done after 50 days is more than enough to complete the project. Therefore, the project is completed in 50 days.
Conclusion
The mathematical analysis in this article showcases how a consistent calculation of man-days and workforce dynamics can help us predict and manage project completion time in scenarios where labor productivity fluctuates over time. By understanding these concepts, managers can better plan and allocate resources, ensuring timely completion of projects.