IE 312 Review 1. The Process 2 Problem Model Conclusions Problem Formulation Analysis.
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Transcript of IE 312 Review 1. The Process 2 Problem Model Conclusions Problem Formulation Analysis.
![Page 1: IE 312 Review 1. The Process 2 Problem Model Conclusions Problem Formulation Analysis.](https://reader030.fdocuments.in/reader030/viewer/2022012914/5a4d1b6e7f8b9ab0599b438a/html5/thumbnails/1.jpg)
IE 312 Review
1
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The Process
2
Problem
Model
Conclusions
Problem Formulation
Analysis
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Problem Formulation
3
What is the objective? Maximize profit, Minimize inventory, ...
What are the decision variables? Capacity, routing, production and stock levels
What are the constraints? Capacity is limited by capital Production is limited by capacity
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Analysis
4
Optimization Algorithm Computer Implementation
Excel (or other spreadsheet)
Optimization software (e.g., LINDO)
Modeling software (e.g., LINGO)
IncreasingComplexity
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Optimization Algorithms
5
Find an initial solution Loop:
Look at “neighbors” of current solution Select one of those neighbors Decide if to move to selected solution Check stopping criterion
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Tractability and Validity
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Is a model tractable? Can it be solved, or is it too complex and/or large?
Is a model valid? Do we reach the same conclusions experimenting
with the model as we would experimenting with the real system?
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Mathematical Programming
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General Model Minimize or maximize some objective function Subject to some constraints
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Linear versus Nonlinear
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A function is linear if it is a weighted sum of the decision variables, otherwise nonlinear
A linear program (LP) has a linear objective function f and constraint functions g1,…,gm
A nonlinear program has at least one nonlinear function
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Integer Programs
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A variable is discrete if it can only take a limited or countable number of values
A variable is continuous if it can take values in a specific interval
Mathematical programs can be continuous discrete (integer/combinatorial) mixed
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Classification Summary
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Linear Program (LP) ILP
Nonlinear Program (NLP) INLP
Integer Program (IP)
Increased difficulty
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Solution Techniques
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Models Thousands (or millions) of variables Thousands (or millions) of constraints
Complex models tend to be valid Is the model tractable?
Solution techniques
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Improving Search
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Begin at feasible solution Advance along a search path Ever-improving objective function value
Neighborhood: points within small positive distance of current solution
Stopping criterion: no improvement
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Local Optima
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Improving search finds a local optimum
May not be a global optimum(only a heuristic solution)
Tractability: for some models there is only one local optimum (which hence is global)
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Tractability
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The most tractable models for improving search are models with unimodal objective function (linear special case) convex feasible region (linear special case)
Here every local optimum is global
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Solving LP Models
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Improving Search Unimodal Convex feasible region Should be successful!
Special Form of Improving Search Simplex method
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Optimal Solutions
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Every optimal solution is a boundary point We can find an improving direction whenever
we are at an interior point If optimum unique the it must be an
extreme point of the feasible region If optimal solution exist, an optimal
extreme point exists
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LP Standard Form
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Easier if we agree on exactly what a LP should look like
Standard form only equality main constraints only nonnegative variables variables appear at most once in left-hand-side
and objective function all constants appear on right hand side
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Extreme Points
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Know that an extreme point optimum exists Will search trough extreme points
An extreme point is define by a set of constraints that are active simultaneously
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Improving Search
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Move from one extreme point to a neighboring extreme point This defines the directions Simplex is a special case of improving search
that only uses these directions Extreme points are adjacent if they are
defined by sets of active constraints that differ by only one element
An edge is a line segment determined by a set of active constraints
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Basic Solutions
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An easy way of keeping track of the Simplex directions is by using some linear algebra
Extreme points are defined by set of active nonnegativity constraints
A basic solution is a solution that is obtained by fixing enough variable to be equal to zero, so that the equality constraints have a unique solution
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Simplex Algorithm
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Starting point A basic feasible solution (extreme point)
Direction Follow an edge to adjacent extreme point:
Increase one nonbasic variable Compute changes needed to preserve equality
constraints One direction for each nonbasic variable
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Sensitivity Analysis
22
Basic Question: How does our solution change as the input parameters change? The objective function?
More/less profit or cost The optimal values of decision variables?
Make different decisions! Why?
Only have estimates of input parameters May want to change input parameters
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What We Know
23
Qualitative Answers for All Problems Quantitative Answers for Linear Programs
(LP) Dual program Same input parameters Decision variables give sensitivities Dual prices Easy to set up Theory is somewhat complicated
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Discrete Optimization
24
Wide range of problems LPs with additional integer constraints Knapsack & capital budgeting Set packing, covering, and partitioning Traveling salesman and routing
How do we solve these problems? Much more difficult that the
corresponding continuous problems
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Exponential Growth
25
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7
Number of variables
Num
ber o
f sol
utio
ns
kk22klog k
2k
k2
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Relaxations
26
Discrete problems are hard Relax them to an easier problem Our original Swedish Steel formulation was a
relaxation of the real problem Can always relax a zero-one problem by
allowing the variable to take any value on the interval [0,1]
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Continuous Relaxations
27
LP relaxation or continuous relaxation is when a model with discrete variables is assume to have only continuous variables
If a constraint relaxation is infeasible, the original model is infeasible as well
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Relationship with Relaxed Solution
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The optimal value of the relaxed problem is an upper/lower bound for the original maximization/minimization model
Feasible solutionsin relaxed model
Feasible solutionsin original model
Optimum
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Final Relationship
29
If an optimal solution to the relaxed model is feasible for the original model, it is also optimal for the original model.
Feasible solutionsin relaxed model
Feasible solutionsin original model
Optimum for the relaxed model
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Further Bound
30
Have that the optimal value of the relaxed problem is an upper/lower bound for the original maximization/minimization model
The objective value of any integer feasible solution to a maximization/minimization problem is a lower/upper bound on the integer optimal value
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Rounding Example
31
integer and 0,4232503150s.t.
64.0max
21
21
21
21
xxxx
xxxx
Solve LP relaxation for (x1,x2)=(376/193,950/193)
Integer Programming (IP) solution is (x1,x2)=(5,0)
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More Efficient Enumeration
32
Looking at every solution takes prohibitively long
Can we somehow account for every solution without actually looking at every solution?
Would like to be able to eliminate a bunch of solutions without evaluating them!
Branch and bound
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Using Relaxations
33
If a relaxation of a candidate problem is infeasible that branch can be fathomed
If optimal solution of a relaxed candidate problem is worse than incumbent then fathom branch
If optimal solution of a relaxed candidate problem is feasible for full candidate then fathom branch and update candidate if necessary
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Knapsack Model
34
Intuitive idea: what is the most valuable collection of items that can be fit into a backpack?
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Capital Budgeting
35
Multidimensional knapsack problems are often called capital budgeting problems
Idea: select collection of projects, investments, etc, so that the value is maximized (subject to some resource constraints)
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Assignment Problems
36
Assignment problems deal with optimal pairing or matching of objects in two distinct sets
Decision variable
Let A be the set of allowed assignments and cij be the cost of assigning i to j.
otherwise0
toassigned is if1 jixij
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Traveling Salesman Problem (TSP)
37
Ames
Fort Dodge
BooneCarrollMarshalltown
West Des Moines
Waterloo
What is the shortest route,starting in Ames, that visitseach city exactly ones?
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Solving TSP
38
We can use branch-and-bound to get an exact solution to the TSP problem
As always, the key to implementing branch-and-bound is to relax the problem so that we can easily solve the relaxed problem, but we still get good bounds
How can we relax the TSP?
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Single Machine Sequencing
39
A set of jobs Find the best order in which to sequence the
jobs Solution
Tabu search Branch-and-bound
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Neighborhood/Local Search
40
Find an initial solution Loop:
Look at “neighbors” of current solution Select one of those neighbors Decide if to move to selected solution Check stopping criterion
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Random Search Methods
41
Tabu Search Maintain a tabu list of solution changes
A move made entered at top of tabu list Fixed length (5-9) Neighbors restricted to solutions not requiring a tabu
move