Optimizing Workforce in Project Management: A Mathematical Analysis

Project management involves effectively organizing and utilizing resources to achieve specific goals within a fixed time frame. One critical aspect is efficiently allocating personnel to complete tasks. This article uses a practical problem to illustrate the process of workforce optimization through mathematical calculations. Let's explore a scenario where work output is calculated and optimized between men and boys, reflecting real-world application in project management.

Problem Statement

The problem at hand states: “2 men or 3 boys can do a piece of work in 14 days working 6 hours each day. In how many days 6 men and 9 boys will complete a work twice as large working together 2 hours each day”. This scenario is a common challenge in project management, where hiring the correct number of workers to complete a project within a given timeframe is crucial.

Workforce Conversion

The first step in solving this problem is to establish the equivalence between men and boys in terms of work output. Given that 9 boys are equivalent to 6 men, we can infer that the work output of one boy is 0.667 (or 2/3) of the work output of one man. This relationship is pivotal in simplifying the calculations.

Step-by-Step Analysis

Step 1: Establishing Man-Hour and Boy-Hour Equivalents

First, let's calculate the man-hour and boy-hour equivalents for completing the original work:

2 men can do the work in 14 days working 6 hours each day, totaling 168 man-hour-days (2 x 14 x 6). 3 boys can do the same work in the same period, totaling 252 boy-hour-days (3 x 14 x 6).

In other words, 1 man-hour-day is equivalent to 1.5 boy-hour-days (252 / 168 1.5).

Step 2: Applying the Equivalence to Larger Workloads

When the work is doubled, the total man-hour-days required will also double to 336. Using the relationship established, 336 man-hour-days is equivalent to 224 boy-hour-days (336 / 1.5).

Step 3: Evaluating the New Workforce

Given that 6 men working 2 hours a day provide 12 man-hour (6 x 2) output, this translates to 18 boy-hour (12 x 1.5) output. With 9 boys, the total boy-hour output is 18 (9 x 2).

Combining both, the total output is 36 boy-hour (18 18). To complete the 224 boy-hour work, the time required is 224 / 36 6.22 days, approximately 28 days when rounded to the nearest whole number.

Conclusion

In summary, the problem illustrates the importance of understanding workforce dynamics and converting units to optimize project management. By weighing the work output of different types of workers, project managers can make informed decisions to meet project timelines effectively. This analysis not only highlights the mathematical approach to solving workforce optimization problems but also underlines the practical implications in real-life project scenarios.

Keywords

workforce optimization, project management, man-hour calculations