Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems...

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 AnalysisIf 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”)

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)

• 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

mathematically equivalent

Economy of Scale

revenueor profit

quantity

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?

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

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

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

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

• CapacityRk≤18Tk+8Hk

Ok ≤(18/4)Tk

• Nonnegativity

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

Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems...

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