Previously in Chapter 4

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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
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
Sensitivity Analysis

If you are missing these columns

Sensitivity Analysis

make sure it is checked

If statements (Part 1)
• 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”)

If statements (Part 2)

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)

If statements (Part 3)
• 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

Diseconomy of Scale

mathematically equivalent

revenue

or profit

cost

quantity

Economy of Scale

quantity

mathematically equivalent

Projects

(worth a couple of homeworks)

• Groups of up to 3
• Topic areas:
• optimization (should start around now)
• stochastic models (later)
Optimization Projects
• airline scheduling
• asset allocation
• production planning
• class scheduling
• tournament setup
• design optimization
• comparing algorithms

I will post more details online

Examples
• Airline scheduling
• Virgin America network
• How many planes are needed?
• Asset Allocation
• July ‘08 Northwestern endowment at \$8b
• How would you have invested it?
Todo

Group should meet me

• discuss project
• negotiate deliverables
• earlier for optimization topics
Sequential Decision Process
• Discretize Time
• Variables for each period
• for example: #workers Wk, inventory level Ik

period k=1

2

3

4

5

Production Planning (4.12)
• 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
Problem Summary
• Producing snow tires
• Monthly demand: Oct-March
• Goal: cheaply meet demand
• Decisions:
• hire or fire, overtime, production quantity
• Inventory cost, trainees are less productive
Production Planning (4.12)
• 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 Planning (4.12)
• 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
Variables

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
Production Planning (4.12)
• 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
Timeline

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

Production Planning (4.12)
• 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
Constraints
• 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
Production Planning (4.12)
• 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
Objective
• Hiring / Firing costs

\$3000*(H1+…+H7)

\$7000*(F1+…+F7)

• Compensation

\$2600*(W2+…+W7)

\$2600*1.5*(O1+…+O7)/18

• Inventory

\$40*(I1+…+I7)

Variations and Extensions
• Transportation Problem with delays
• Multiple products
• Multiple production steps
• Warehouses
• Everything combined