discrete optimization - liu
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Discrete Optimization [Chen, Batson, Dang: Applied Integer programming]
Chapter 1-2 - Tomas Lidén, Marcus Posada (ITN/KTS) Seminar #1, 2015-03-24
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Outline• Ch 1: Introduction, classification [Tomas]
• Ch 2: Modeling and Models • 2.1: Assumptions [Marcus] • 2.2: Modeling process [-”-] • 2.3: Project selection [Tomas] • 2.4: Production planing [-”-] • 2.5: Workforce scheduling [-”-] • 2.6: Transportation and distribution [Marcus] • 2.7: Multi-commodity network flow [-”-] • 2.8: Network optimization [-”-] • 2.9: Supply chain planning [-”-]
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Ch 1: Introduction and classification
Mathematical programming (problem) = constrained optimization (problem)
• ”programming”: planning activities that consume resources and/or meet requirements, expressed as constraints
• Not coding a computer program!
• Short hand: ”program”LP = linear programming / program (linear optimization)
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Problem classesMIP: Mixed Integer Program (also MILP: Mixed Integer Linear Program)
LP: Linear Program (no y)
IP: (pure) Integer Program (no x)
• One constraint: Integer knapsack program (|b| = 1)
BIP: Binary (Integer) Program (y \in {0, 1})
• 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)
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”Standard form”• Here:
Maximize,≤ constraints,non-negative variables (also called canonical form)
• Bertsimas, Lundgren:Minimize, = constraints,non-negative variables (also called augmented or slack form)
• Easy to transform(negations, extra variables)
• Single constraints for bounds
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Combinatorial optimization problems (COP)
• A finite set of solutions, often representable by graph structures
• Classical examples:
• Assignment problem
• Traveling salesman problem (TSP)
• Vehicle routing problem
• Constraint satisfaction (sudoku, game-of-life, queens etc)
• Can be formulated as BIP, one variable per possible solution
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Successful applications• Transportation and distribution
• Manufacturing
• Communications
• Military and government
• Finance
• Energy
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Modeling tips, blogs etc• Paul Rubin - http://orinanobworld.blogspot.se/
• Formulating optimization problems
• Branching and integer/binary variables
• Other bloggers: Laura McLay, Jean-Francois Puget, Marco Lübbecke, Michael Trick
• Organizations:
• Informs - https://www.informs.org/
• 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
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Production planning problems (2.4)
• Lot sizing
Big ”M”: Capacitated:
Note: single product, state (secondary) variables s_t
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Just-in-time production• Minimize inventory cost for multi-product
production
• Somewhat strange formulation in text book: variables x_jt and s_jt unnecessary; ”pair of inequality constraints”? Index error in objective.
• Could use surplus ( ) and shortage ( ) Inventory balance: Objective: (note: only state variables in objective function)
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Workforce scheduling• Full time workers
Fractional values can be used for part timers
Note: same model as Capital budgeting..
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Workforce scheduling• Part time additions
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𝑦𝑖𝑗 = 1 𝑥𝑖𝑗 > 0
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𝑀 𝑢𝑖
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