Math 5593 Linear Programming Week 2

29 Pages·2013·1.73 MB·English

WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Math 5593 Linear Programming Week 2
2. Intro to AMPL
University of Colorado Denver, Fall 2013, Prof. Engau
1 What is AMPL, and What is it Not?
2 Production Models: Maximizing Profits
3 Diet and Other Input Models: Minimizing Costs
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
AMPL Website, Software, and Book (free downloads)
AMPL Website (provides downloads, examples, and more)
http://www.ampl.com
Software: academic or 30-day trial (unrestricted); permanent
student version (limited to 300 variables and 300 constraints)
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
What is AMPL, and What is it Not?
Recallthefollowinggeneralstepsforsolvingareal-lifeproblem:
1 Model Formulation: the representation of real-life decision
problems as abstract system of variables, objectives, and
constraints that represent the general form of the problem
2 Data Collection (Statistics): the use of statistical methods
to collect available data and estimate unknown parameters
thatdefinethespecificinstanceoftheproblemtobesolved
3 Problem Generation: the formulation of a specific objective
function and constraint equations from the model and data
4 Computational Optimization: the execution of a program or
solver that applies a numerical algorithm to find an optimal
solution and assign optimal values to all decision variables
5 Decision Analysis: the interpretation of this solution for the
original problem and evaluation of its practical relevance
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
AMPL - A Modeling Language for Math Programming
AMPLisamodelinglanguage(“matrixgenerator”),notasolver!
enables the formulation of models using “natural” language
“understands” certain conditional or logical statements
uses additionally specified data to translate these
statements into algebraic form as input to a solver
AMPL supports many commercial and open-source solvers:
CPLEX (linear and convex quadratic with mixed-integer)
GUROBI (linear and convex quadratic with mixed-integer)
LPSOLVE (linear with mixed-integer)
MINOS (linear and general nonlinear)
SNOPT (general nonlinear)
For a full list, see http://www.ampl.com/solvers.html
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Other Online Sources for Optimization Software
COmputational INfrastructure for Operations Research
http://www.coin-or.org
NEOS (Network-Enabled Optimization System) Server
http://neos.mcs.anl.gov/neos
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
A Mythical Steelmaking Operation (AMPL Book Ch. 1)
Problem: A steel company must decide how to allocate next
week’s time on a rolling mill. The mill takes unfinished slabs of
steel as input and can produce two semi-finished products:
bands or coils. The mill’s two products come off the rolling line
at different rates and have different profitabilities. To further
complicate matters, weekly production amounts must be limited
to not exceed currently booked orders. How many bands and
coils should be produced to bring in the greatest total profit?
Solution: We will use the five-step process outlined above:
1 Model Formulation: define variables, constraints, objective
2 Data Collection: find actual values for problem parameters
3 Problem Generation: convert problem into algorithmic form
4 Computational Optimization: run solver and get solution
5 Decision Analysis: does the solution make sense?
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Mythical Steelmaking Step 1. Model Formulation
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Mythical Steelmaking Step 1. A First AMPL Model
set (index set, can be used in sums/other model entities)
param (data parameter, can be indexed, must get value)
var (decision variables, can be indexed, can be fixed)
max(imize)/min(imize) (objective, must be named)
subject to (constraint, can be indexed, must be named)
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Mythical Steelmaking Step 2. Data Collection (AMPL)
Tons per hour: Bands 200
Coils 140
Profit per ton: Bands $25
Coils $30
Maximum tons: Bands 6,000
Coils 4,000
Available time on mill in hours: 40
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL
WhatisAMPL,andWhatisitNot? ProductionModels:MaximizingProfits DietandOtherInputModels:MinimizingCosts
Steps 3 (and 4): Problem Generation (and Solution)
maximize 25x +30x
B C
1 1
subject to x + x ≤ 40
B C
200 140
0 ≤ x ≤ 6000
B
0 ≤ x ≤ 4000
C
Math5593LinearProgramming-Fall2013-Week2-UnitI:Models&Applications 2.IntrotoAMPL