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, x4≥0, x4 integer, … (note that there is sign-constraint on x2, sometimes we say “x2 is a free variable”)

7. If statements (Part 2) 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)

8. 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)

9. revenue or profit quantity cost quantity Diseconomy of Scale mathematically equivalent

10. revenue or profit cost quantity 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)

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

13. 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?

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

15. Sequential Decision Process • Discretize Time • Variables for each period • for example: #workers Wk, inventory level Ik period k=1 2 3 4 5 …

16. 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

17. 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

18. 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

20. 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

21. 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

22. 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

23. 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

24. 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

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

26. Objective • Hiring / Firing costs $3000*(H1+…+H7) $7000*(F1+…+F7) • Compensation $2600*(W2+…+W7) $2600*1.5*(O1+…+O7)/18 • Inventory $40*(I1+…+I7)

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