Previously in chapter 4
Sponsored Links
This presentation is the property of its rightful owner.
1 / 27

Previously in Chapter 4 PowerPoint PPT Presentation


  • 87 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Previously in Chapter 4

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation 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 Analysis

If 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

minf(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 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)

  • Binary variables are hard for solvers

    • though better than if statements

  • Sometimes can be avoided

    • for example: diseconomies of scale(certain piecewise linear functions)


revenue

or profit

quantity

cost

quantity

Diseconomy of Scale

mathematically equivalent


revenue

or profit

cost

quantity

Economy of Scale

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?


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)

  • 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

  • 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)

  • 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 (4.12)

  • 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

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)

  • 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

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 (4.12)

  • 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

  • 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 (4.12)

  • 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

  • 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


  • Login