# Optimizing duration of activities

I would like to understand optimization through a simple application, and then progressing towards understanding more general concepts. My inquiry starts with its application: Optimizing duration of activities in a day.

The length of the day and duration of each activity are known. The problem I'm seeking to solve is optimizing the length of each activity. No priorities are considered, so lengths are to be adjusted proportionally.

Here's an example: Assuming the allotted time is 16 hours and the sum of the duration of activities is greater than 16 hours, time slots are squeezed proportionally like this:

(Activity, Desired length, computed optimum length)

A, 3h, 2h 50min

B, 2h, 1h 55min

C, 1h, 0h 54min

...

Lacking a math background, I would appreciate guidance on where to start tackling this.

1. What is this sort of optimization and related concepts called?
2. What are the pros and cons of possible approaches?
3. Any book recommendations?

Thanks!

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The time squeezing described is not proportional. – André Nicolas Jul 10 '11 at 21:35

I see no optimization in this example. To formulate a mathematical optimization problem, you need an objective: a function of certain variables which you want to maximize or minimize. You may also have constraints on those variables.

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If you just want to adjust the lengths of activities "proportionally", couldn't you just do:

computed optimum length = desired length * (allotted time / sum of the duration of the activities)

So if you have a 16 hour time slot and the following activities:
(Activity, Desired length, computed optimum length)

• A, 6h
• B, 5h
• C, 4h
• D, 3h30m
• E, 2h45m

Then the "computed optimum length" of activity A is

computed optimum length = desired length * (allotted time / sum of the duration of the activities)
= 5h * 16h / (6h+5h+4h+3h30m+2h45m)
= 5h * 16/21.25
= 3.764h
~= 3h46m

Is this simple formula sufficient for your "optimization"?

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