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

make sure it is checked

If statements (Part 1) Not in typical optimization formulation Harder for solvers minf(x 1,x 2,…,x n ) s.t.g 1 (x 1,x 2,…,x n ) ≤ b 1 g 2 (x 1,x 2,…,x n ) = b 2 … x 1 ≤0, x 3 binary, x 4 ≥0, x 4 integer, … (note that there is sign-constraint on x 2, sometimes we say “x 2 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)

Diseconomy of Scale quantity revenue or profit quantity cost mathematically equivalent

Economy of Scale revenue or profit 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 W k, inventory level I k period k=12345…

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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Variables For each period # hired H k, #fired F k #trained and trainee workers –total #workers W k, #trained workers T k units produced overtime used –R k units produced with regular hours, –O k units produced with overtime inventory I k

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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Timeline Period k W k #workers H k #hired F k #fired Production Decision R k #units with regular time O k #units with overtime T k #trained workers I k #units inventory D k #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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Constraints Inventory: I 1 =0, I k+1 =I k +R k +O k -D k Meeting Demand: I k+1 ≥ 0 Workforce W 1 =90, W k+1 =W k +H k -F k T k =W k -F k, T 7 =100 Capacity R k ≤18T k +8H k O k ≤(18/4)T k 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 period 4.Add constraints –“laws of motion”: constraints connecting a period to the next 5.Add objective 6.Solve

Objective Hiring / Firing costs $3000*(H 1 +…+H 7 ) $7000*(F 1 +…+F 7 ) Compensation $2600*(W 2 +…+W 7 ) $2600*1.5*(O 1 +…+O 7 )/18 Inventory $40*(I 1 +…+I 7 )

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