Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems...
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Transcript of Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems...
![Page 1: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/1.jpg)
Previously in Chapter 4
• Assignment Problems
• Network Flow Problems
• Vehicle Routing Problems
• Transportation Problems
• Staffing Problems
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Agenda
• Sensitivity Analysis
• Optimization tricks: If statements
• Diseconomy of Scale
• Projects
• Sequential Decision Processes– a.k.a. Production Planning
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Sensitivity AnalysisIf you are missing these columns
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Sensitivity Analysis
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Sensitivity Analysis
make sure it is checked
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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”)
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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)
![Page 8: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/8.jpg)
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)
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Diseconomy of Scale
quantity
revenueor profit
quantity
cost
mathematically equivalent
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Economy of Scale
revenueor profit
quantity
quantity
cost
mathematically equivalent
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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)
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Optimization Projects
• airline scheduling• asset allocation• production planning• class scheduling• tournament setup• design optimization• comparing algorithms
I will post more details online
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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?
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Todo
Group should meet me
• discuss project
• negotiate deliverables
• and deadlines– earlier for optimization topics
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Sequential Decision Process
• Discretize Time
• Variables for each period– for example: #workers Wk, inventory level Ik
period k=1 2 3 4 5 …
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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
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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
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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
![Page 19: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/19.jpg)
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
![Page 20: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/20.jpg)
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
![Page 21: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/21.jpg)
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
![Page 22: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/22.jpg)
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
![Page 23: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/23.jpg)
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
![Page 24: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/24.jpg)
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
![Page 25: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/25.jpg)
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
![Page 26: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/26.jpg)
Objective
• Hiring / Firing costs$3000*(H1+…+H7)
$7000*(F1+…+F7)
• Compensation$2600*(W2+…+W7)
$2600*1.5*(O1+…+O7)/18
• Inventory$40*(I1+…+I7)
![Page 27: Previously in Chapter 4 Assignment Problems Network Flow Problems Vehicle Routing Problems Transportation Problems Staffing Problems.](https://reader031.fdocuments.in/reader031/viewer/2022032309/56649d405503460f94a1a247/html5/thumbnails/27.jpg)
Variations and Extensions
• Transportation Problem with delays
• Multiple products
• Multiple production steps
• Warehouses
• Everything combined