Previously in Chapter 4

1
Previously in Chapter 4

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

2
Agenda

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

3
Sensitivity Analysis
If you are missing these columns
4
Sensitivity Analysis
5
Sensitivity Analysis
make sure it is checked
6
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,
x40, x4 integer,
(note that there is sign-constraint on
x2, sometimes we say x2 is a free variable)
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If statements (Part 2)

• 0 x and
• If xb, then yc, else yd
• create binary 0/1 variable z
• add the constraints
• (b-x)/b z (if xb, then z1)
• z1(b-x)/b (if xgtb, then z0)
• yczd(1-z) (if z1, then yc else yd)

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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)

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Diseconomy of Scale
mathematically equivalent
10
Economy of Scale
quantity
mathematically equivalent
11
Projects

• 10 of final grade
• (worth a couple of homeworks)
• Groups of up to 3
• Topic areas
• optimization (should start around now)
• stochastic models (later)

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Optimization Projects

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

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Examples

• 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?

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Todo

• Group should meet me
• discuss project
• negotiate deliverables
• and deadlines
• earlier for optimization topics

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Sequential Decision Process

• Discretize Time
• Variables for each period
• for example workers Wk, inventory level Ik

period k1
2
3
4
5

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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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

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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

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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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

19
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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

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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

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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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

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Timeline
Production Decision Rk units with regular
time Ok units with overtime
Period k
Ik units inventory
next period
prev. period
Dk units shipped
Hk hired Fk fired
Wk workers
Tk trained workers
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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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

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Constraints

• Inventory
• I10, Ik1IkRkOk-Dk
• Meeting Demand
• Ik1 0
• Workforce
• W190, Wk1WkHk-Fk
• TkWk-Fk, T7100
• Capacity
• Rk18Tk8Hk
• Ok (18/4)Tk
• Nonnegativity

25
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
• Add constraints
• laws of motion constraints connecting a period
  to the next
• Add objective
• Solve

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Objective

• Hiring / Firing costs
• 3000(H1H7)
• 7000(F1F7)
• Compensation
• 2600(W2W7)
• 26001.5(O1O7)/18
• Inventory
• 40(I1I7)

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Variations and Extensions

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