1. Principles of Teamwork
Learning how to function as a team member, and make your team succeed is one of the key objectives of this course. You can not get a good grade without it.
Team work requires work. Not a free lunch. Pays off though.

2. The indisputable laws of teamwork
The Law of Significance. People try to achieve great things by themselves mainly because of the size of their ego, their level of insecurity, or simple naiveté and temperament.
The Law of the Big Picture. The goal is more important than the role. Members must be willing to subordinate their roles and personal agendas to support the team vision.
The Law of the Niche. All players have a place where they add the most value. Essentially, when the right team member is in the right place, everyone benefits.
The Law of The Bad Apple. Rotten attitudes ruin a team. The first place to start is with yourself. Are you at our best?
The Law of Countability. Teammates must be able to count on each other when it counts. Are you dedicated to the team’s success? Can people depend on you?
The Law of Communication. Interaction fuels action. Effective teams have teammates who are constantly talking, and listening to each other.
The Law of Dividends. Investing in the team compounds over time. Make the decision to build a team

3. Principles of Teamwork
An important heuristic: Set up regular meeting times, rather than planning ad-hoc meetings. At least once a week + a back-up time. Tie to HW deadlines. You can not skip a meeting, unless the whole team decides it is not needed.

4. Now, a real life problem.
Break out in groups. Each group will earn activity points for the in-class work.
SET-UP: A sports equipment company manufactures 3 types of
balls: golf, ping-pong, and tennis. The sizes (radii) of the balls
are 2, 3, and 5 cm respectively. The company ships each ball type
separately, but uses only one size of container for all its shipping needs.
The container is a perfect cube made out of padded, thermally insulating
novel material (nano-carbon + trace amounts of gold) manufactured by
just one company in the World. The boxes are loaded on the trucks by
a single worker, using only her sheer muscle power. A lion’s share of the
shipping cost is in the cost of getting the truck from A to B, so sending
trucks partially filled with the merchandize is guaranteed to be non-optimal.
PROBLEM: What is the optimal box size that will
minimize the shipping costs? How many balls of each type will fit into
such box (to within 10 % relative error)?

5. The shipping problem cont’d
Initial brainstorming:
Boxes are expensive.
Cost = (cost per box) x( number of boxes per truck).
Each truck must be filled with boxes,
so we really need to minimize cost
per truck.

6. The shipping problem cont’d
So it looks like we need to minimize the amount of
expensive carton used per box,
while preserving the total volume of all the boxes
that fill up the truck.
Is this it? (don’t focus on the “distilled” problem yet!)
Strategy for complex problems: Is this it?
Is this “distilled” problem the only one we need to solve? Have we left anything out?

7. The shipping problem cont’d
Remember, we need to maximize
the amount of merchandize (balls) per truck. Are we going to have more
of them in larger boxes or smaller boxes?
Second problem: packing of hard spheres.
What is the best packing?

8. The shipping problem cont’d
Remember, we need to maximize
the amount of merchandize (balls) per truck. Are we going to have more
of them in larger boxes or smaller boxes?
Second problem: packing of hard spheres.
What is the best packing (that is the one that minimizes empty space)? Is the best packing achieved in larger or smaller
boxes?

9. The shipping problem cont’d
Strategy: divide and conquer. Complex problem divided into simpler parts.
The problem-solving group may break down into two parts now.

10. The shipping problem cont’d
Problem I: Do we need large boxes or small boxes?
Box sides that touch each other are “wasted”, so need fewest touching sides, that is largest boxes within the other given constraints.

11. The shipping problem cont’d
Problem II: Spheres packed inside the
box. What box size will optimize the packing, that is minimize
the “wasted” empty space?
Heuristic: simplify.

12. The shipping problem cont’d
Problem II: Spheres packed inside the
box. What box size will optimize the packing, that is minimize
the “wasted” empty space?
Heuristic: simplify.
3D -> 2D (circles in a square)
Get a feel for it. One circle.

13. The shipping problem cont’d
Problem II: Spheres packed inside the
box. Heuristic: circles in a sphere. Maximize
packing density = (volume of spheres)/(volume of box)

14. The shipping problem cont’d
What have we learned from the 2D case?
The dependence of the packing ratio on the box size is not monotonic, but appears to reach an asymptotic value of ~ 0.78 for large size box.
Seems to occur at box size a = k^2
We want the box size so that integer # balls of 2,3,5 (diameters 4,6,10) fit. a=60*m. Thus, 60*m = k^2. Looks like this may lead to very large m, and inadmissibly large box size. Need to
Get away from the exact packing.
Fortunately, for large box size a >> ball size the surface/volume arguments suggest that we can achieve a close to optimal packing even without being perfect.

15. The shipping problem cont’d
Now back to the spheres:
More complex, but essentially the same thing conceptually. Most tight packing of identical spheres fills ~ 3/4 of the total volume. Given the size of the box of ~1x1x1 m, its volume is 1m^3. The balls will take up 3/4 of it, from which we can compute the number of balls of each type.

16. The shipping problem cont’d
Our tactical steps (heurtsics ) from the beginning:
Brainstorm
Divide and Conquer
Simplify
Get hands dirty. Explore trends.