Question

A citizen decides to campaign for the election of a candidate for city council. Her goal is to generate at least 279 votes by a combination of door-to-door canvassing, letter writing, and phone calls. She figures that each hour of door-to-door canvassing will generate five votes, each hour of letter writing will generate three votes, and each hour on the phone willl generate four votes. She would like to devote at least eight hours to phone calls and spend at most half her time at door-to-door canvassing. How much time should she allocate to each task in order to achieve her goal in the least amount of time? First identify the objective function to be minimized. Let x, y, and z be the number of hours spent door-to-door canvassing, letter writing, and making phone calls, respectively.

Answer 1

Solution:

Given this information, we know that the total time (in hours) is to be minimized. As denoted, the total time is (x + y + z), we can clearly tell that objective function is:

Minimizing x + y + z

Further, we have the following constraints:

If 1 hour of door harassing generates 5 votes, then x hours will generate 5x votes in total. Going this way, given y and z hours on letter writing and making phone calls respectively, total votes by letter writing is 3y and by phone calls is 4z.

Since at least 279 votes are required in total, we have following inequality:

5x + 3y + 4z >= 279

Also, at least 8 hours devotion to phone calls imply z >= 8

Finally, total time being (x + y + z), half of this time becomes: (x + y + z)/2 and door to door harassing should be atmost this value, so we have x <= (x + y + z)/2

Rewriting this: x - y - z <= 0

Thus, to conclude: the citizen wishes to minimize (x + y + z), subject to

1. 5x + 3y + 4z >= 279

2. z >= 8

3. x - y - z <= 0

We have to find optimal values for x, y and z

Solving as follows:

Using the excel solver and the above objective function and constraints, we get the solution as:

(x*, y*, z*) = (31, 0, 31)