Post on 19-Mar-2016
description
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