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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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Previously in chapter 4
Previously in Chapter 4

  • Assignment Problems

  • Network Flow Problems

  • Vehicle Routing Problems

  • Transportation Problems

  • Staffing Problems


Agenda
Agenda

  • Sensitivity Analysis

  • Optimization tricks: If statements

  • Diseconomy of Scale

  • Projects

  • Sequential Decision Processes

    • a.k.a. Production Planning


Sensitivity analysis
Sensitivity Analysis

If you are missing these columns



Sensitivity analysis2
Sensitivity Analysis

make sure it is checked


If statements part 1
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
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)


If statements part 3
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

revenue

or profit

quantity

cost

quantity

Diseconomy of Scale

mathematically equivalent


Economy of scale

revenue

or profit

cost

quantity

Economy of Scale

quantity

mathematically equivalent


Projects
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
Optimization Projects

  • airline scheduling

  • asset allocation

  • production planning

  • class scheduling

  • tournament setup

  • design optimization

  • comparing algorithms

    I will post more details online


Examples
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?


Previously in chapter 4
Todo

Group should meet me

  • discuss project

  • negotiate deliverables

  • and deadlines

    • earlier for optimization topics


Sequential decision process
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
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


Problem summary
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 4 121
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


Production planning 4 122
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


Variables
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 4 123
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


Timeline
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 4 124
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


Constraints
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 4 125
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


Objective
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
Variations and Extensions

  • Transportation Problem with delays

  • Multiple products

  • Multiple production steps

  • Warehouses

  • Everything combined