# Lecture 1

## 1. Computational Models

- Optimization models
- Statistical models
- Simulation models

### 1.1. Optimization Models

- An objective function that is to be maximized or minimized, e.g.,
- Minimize time spent traveling from New York to Boston

- A set of constraints (possibly empty) that must be honored, e.g.,
- Cannot spend more than $100
- Must be in Boston before
`5:00PM`

### 1.2. Knapsack Problem

- You have limited strength, so there is a maximum weight knapsack that you can carry
- You would like to take more stuff than you can carry
- How do you choose which stuff to take and which to leave behind?
- Two variants
- 0/1 knapsack problem
- Continuous or fractional knapsack problem

- Each item is represented by a pair,
`<value, weight>`

- The knapsack can accommodate items with a total weight of no more than
`w`

- A vector,
`L`

, of length n, represents the set of available items. Each element of the vector is an item
- A vector,
`V`

, of length n, is used to indicate whether or not items are taken. If **V[i] = 1**, item **I[i]** is taken. If **V[i] = 0**, item **I[i]** is not taken
- Find a
`V`

that Maximizes:
- $\displaystyle\sum_{i=0}^{n-1}V[i]*I[i].\text{value}$
- Subject to the constraint that:
- $\displaystyle\sum_{i=0}^{n-1}V[i]*I[i].\text{weight} \le w$

## 2. Brute Force Algorithm

- Procedure:
- Enumerate all possible combinations of items. That is to say, generate all subsets of the set of subjects. This is called the
**power set**.

- Remove all of the combinations whose total units exceeds the allowed weight.

- From the remaining combinations choose any one whose value is the largest.

- Dark Side:
- Will take a considerable time to get the power set.
- There will lots of
`V`

s to indicate whether or not items are taken.

## 3. Greedy Algorithm

- Put “
**best**” available item in knapsack
- Procedure:
- Define what the best means.
- Sort the items.
- Take the items from the best to the worst, stop when it hit the maximum.

- Dark Side:
- Sequence of locally “optimal” choices don’t always yield a globally optimal solution