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# Previously in Chapter 4 - PowerPoint PPT Presentation

Previously in Chapter 4. Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems. Agenda. Sensitivity Analysis Optimization tricks: If statements Diseconomy of Scale Projects Sequential Decision Processes a.k.a. Production Planning.

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Presentation Transcript

• 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

min f(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

(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)

or profit

quantity

cost

quantity

Diseconomy of Scale

mathematically equivalent

or profit

cost

quantity

Economy of Scale

quantity

mathematically equivalent

(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

• 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

• 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

• “laws of motion”: constraints connecting a period to the next

• 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

• “laws of motion”: constraints connecting a period to the next

• 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

• “laws of motion”: constraints connecting a period to the next

• 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

• “laws of motion”: constraints connecting a period to the next

• 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

• “laws of motion”: constraints connecting a period to the next

• 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

• “laws of motion”: constraints connecting a period to the next

• 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