Discrete Optimization
[Chen, Batson, Dang: Applied Integer programming]
Chapter 1-2 - Tomas Lidén, Marcus Posada (ITN/KTS)
Seminar #1, 2015-03-24
Outline
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Ch 1: Introduction, classification
[Tomas]
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Ch 2: Modeling and Models
• 2.1: Assumptions
• 2.2: Modeling process
• 2.3: Project selection
• 2.4: Production planing
• 2.5: Workforce scheduling
• 2.6: Transportation and distribution
• 2.7: Multi-commodity network flow
• 2.8: Network optimization
• 2.9: Supply chain planning
[Marcus]
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[Tomas]
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[Marcus]
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Ch 1: Introduction and
classification
Mathematical programming (problem) = constrained
optimization (problem)
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”programming”: planning activities that consume
resources and/or meet requirements, expressed as
constraints
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Not coding a computer program!
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Short hand: ”program”
LP = linear programming / program
(linear optimization)
Problem classes
MIP: Mixed Integer Program (also
MILP: Mixed Integer Linear Program)
LP: Linear Program (no y)
IP: (pure) Integer Program (no x)
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One constraint: Integer knapsack
program (|b| = 1)
BIP: Binary (Integer) Program (y \in {0, 1})
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One constraint: Knapsack problem
(|b| = 1)
Notational conventions
real numbers
negative real (< 0)
positive real (> 0)
integer numbers (Zahlen)
negative integer
non-negative integer
positive integer (Natural)
”Standard form”
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Here:
Maximize,
≤ constraints,
non-negative variables
(also called canonical form)
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Bertsimas, Lundgren:
Minimize,
= constraints,
non-negative variables
(also called augmented or slack form)
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Easy to transform
(negations, extra variables)
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Single constraints for bounds
Combinatorial optimization
problems (COP)
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A finite set of solutions, often representable by graph structures
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Classical examples:
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Assignment problem
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Traveling salesman problem (TSP)
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Vehicle routing problem
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Constraint satisfaction (sudoku, game-of-life, queens etc)
Can be formulated as BIP, one variable per possible solution
Successful applications
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Transportation and distribution
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Manufacturing
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Communications
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Military and government
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Finance
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Energy
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Modeling tips, blogs etc
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Paul Rubin - http://orinanobworld.blogspot.se/
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Formulating optimization problems
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Branching and integer/binary variables
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Other bloggers: Laura McLay, Jean-Francois Puget, Marco Lübbecke,
Michael Trick
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Organizations:
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Informs - https://www.informs.org/
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SOAF - http://www.soaf.se/
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Project selection problems
(2.3)
Knapsack problem
• Maximize value of limited bag
Capital budgeting
• Multi-period
Parameters: number of projects (n), cost in period t (a_tj), net present
value (c_j), available budget in period t (b_t)
Variables: select project j or not (y_j)
Constraints: budget in each period
State variables: none
Objective: maximize net present value of selected projects
Production planning
problems (2.4)
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Lot sizing
Big ”M”:
Capacitated:
Note: single product, state (secondary) variables s_t
Just-in-time production
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Minimize inventory cost for multi-product
production
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Somewhat strange formulation in text book:
variables x_jt and s_jt unnecessary; ”pair of
inequality constraints”? Index error in objective.
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Could use surplus ( ) and shortage ( )
Inventory balance:
Objective: (note: only state variables in objective function)
Workforce scheduling
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Full time workers
Fractional values can be
used for part timers
Note: same model as
Capital budgeting..
Workforce scheduling
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Part time additions
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𝑦𝑖𝑗 = 1
𝑥𝑖𝑗 > 0
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𝑀
𝑢𝑖
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