Discrete Optimization [Chen, Batson, Dang: Applied Integer programming]

Chapter 1-2 - Tomas Lidén, Marcus Posada (ITN/KTS) Seminar #1, 2015-03-24

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 [-”-]

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)

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)

”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

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

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

Production planning problems (2.4)

• Lot sizing

Big ”M”: Capacitated:

Note: single product, state (secondary) variables s_t

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)

Workforce scheduling• Full time workers

Fractional values can be used for part timers

Note: same model as Capital budgeting..

Workforce scheduling• Part time additions

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𝑦𝑖𝑗 = 1 𝑥𝑖𝑗 > 0

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𝑀 𝑢𝑖

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