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Previously in Chapter 4

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- Assignment Problems
- Network Flow Problems
- Vehicle Routing Problems
- Transportation Problems
- Staffing Problems

- Sensitivity Analysis
- Optimization tricks: If statements
- Diseconomy of Scale
- Projects
- Sequential Decision Processes
- a.k.a. Production Planning

If you are missing these columns

make sure it is checked

- Not in typical optimization formulation
- Harder for solvers

minf(x1,x2,…,xn)

s.t.g1(x1,x2,…,xn) ≤ b1

g2(x1,x2,…,xn) = b2

…

x1 ≤0, x3 binary, x4≥0, x4 integer, …

(note that there is sign-constraint on x2,

sometimes we say “x2 is a free variable”)

0 ≤x and

If x≤b, then y=c, else y=d

- create binary 0/1 variable z
- add the constraints
(b-x)/b ≤ z(if x≤b, then z=1)

z≤1+(b-x)/b(if x>b, then z=0)

y=cz+d(1-z)(if z=1, then y=c else y=d)

- Binary variables are hard for solvers
- though better than if statements

- Sometimes can be avoided
- for example: diseconomies of scale(certain piecewise linear functions)

revenue

or profit

quantity

cost

quantity

mathematically equivalent

revenue

or profit

cost

quantity

quantity

mathematically equivalent

- 10% of final grade
(worth a couple of homeworks)

- Groups of up to 3
- Topic areas:
- optimization (should start around now)
- stochastic models (later)

- airline scheduling
- asset allocation
- production planning
- class scheduling
- tournament setup
- design optimization
- comparing algorithms
I will post more details online

- Airline scheduling
- Virgin America network
- 2 flight/day per link
- How many planes are needed?

- Asset Allocation
- July ‘08 Northwestern endowment at $8b
- How would you have invested it?

Group should meet me

- discuss project
- negotiate deliverables
- and deadlines
- earlier for optimization topics

- Discretize Time
- Variables for each period
- for example: #workers Wk, inventory level Ik

period k=1

2

3

4

5

…

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

- Producing snow tires
- Monthly demand: Oct-March
- Goal: cheaply meet demand
- Decisions:
- hire or fire, overtime, production quantity

- Inventory cost, trainees are less productive

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

For each period

- # hired Hk, #fired Fk
- #trained and trainee workers
- total #workers Wk, #trained workers Tk

- units produced
- overtime used
- Rk units produced with regular hours,
- Ok units produced with overtime

- inventory Ik

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

Production Decision

Rk #units with regular time

Ok #units with overtime

Period k

Ik #units inventory

prev. period

next period

Dk #units shipped

Hk #hired

Fk #fired

Wk #workers

Tk #trained workers

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

- Inventory:
I1=0, Ik+1=Ik+Rk+Ok-Dk

- Meeting Demand:
Ik+1 ≥ 0

- Workforce
W1=90, Wk+1=Wk+Hk-Fk

Tk=Wk-Fk, T7=100

- Capacity
Rk≤18Tk+8Hk

Ok ≤(18/4)Tk

- Nonnegativity

- List time periods
- maybe add an extra at beginning and end

- List variables (things to keep track of)
- states and actions

- Make timeline for a single period
- Add constraints
- “laws of motion”: constraints connecting a period to the next

- Add objective
- Solve

- Hiring / Firing costs
$3000*(H1+…+H7)

$7000*(F1+…+F7)

- Compensation
$2600*(W2+…+W7)

$2600*1.5*(O1+…+O7)/18

- Inventory
$40*(I1+…+I7)

- Transportation Problem with delays
- Multiple products
- Multiple production steps
- Warehouses
- Everything combined