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 AnalysisIf you are missing these columns

Sensitivity Analysis

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 andIf 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)

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)

Diseconomy of Scale

quantity

revenueor profit

quantity

cost

mathematically equivalent

Economy of Scale

revenueor profit

quantity

quantity

cost

mathematically equivalent

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)

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– 2 flight/day per link– 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• and deadlines

– 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)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. 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)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. Solve

Production Planning (4.12)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. 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)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. Solve

Timeline

Period k

Wk #workers

Hk #hiredFk #fired

Production DecisionRk #units with regular timeOk #units with overtime

Tk #trained workers

Ik #units inventory

Dk #units shipped

next periodprev. period

Production Planning (4.12)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. Solve

Constraints• Inventory:

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

• Meeting Demand:Ik+1 ≥ 0

• WorkforceW1=90, Wk+1=Wk+Hk-Fk

Tk=Wk-Fk, T7=100• Capacity

Rk≤18Tk+8Hk

Ok ≤(18/4)Tk

• Nonnegativity

Production Planning (4.12)

1. List time periods– maybe add an extra at beginning and end

2. List variables (things to keep track of)– states and actions

3. Make timeline for a single period4. Add constraints

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

5. Add objective6. 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