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Exact and heuristics algorithms

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Title: Exact and heuristics algorithms


1
Exact and heuristics algorithms
2
Introduction
  • An optimization problem is the problem of
    finding the best solution from all feasible
    solutions
  • Example we want the minimum cost path from s to
    a goal t

3
Introduction
  • Navigation
  • path-cost distance to node in miles
  • minimum gt minimum time, least fuel

4
Introduction
  • VLSI Design (Very-large-scale integration)
  • path-cost length of wires between chips
  • minimum gt least clock/signal delay

5
Introduction
  • Puzzle
  • path-cost number of pieces moved
  • minimum gt least time to solve the puzzle

6
Combinatorial optimization problem
  • A set of solution for combinatory optimization
    problem can be mathematically modeled using
  • Variables vector x (x1, x2, ..., xn),
  • Variable domaine D (D1, D2, ..., Dn), or (Di)
    i1,...,n finite sets,
  • Constraints set,
  • F Objective function to minimize or to
    maximize,
  • A set of all feasible solutions
  • S x (x1, x2, , xn) ? D / x satisfies all
    the constraints S is also called search space.

7
Combinatorial optimization problem
  • Facilities layout problem is consisted of a
    variety of problems. The main problems are
  • Storage (warehouses)
  • architectural design and general layout problem,
  • picking,
  • response time for the order processing,
  • minimization of travel distances in the
    warehouse, routing of pickers or automated guided
    vehicles,
  • personnel and machine Scheduling.

8
Combinatorial optimization problem
  • Example Warehouse location problem
  • Warehouses location problem It aims to
    compute optimal location for the warehouses in
    given area based on location of factories with
    their production capacities, the location of
    clients with their demands, warehouse storage
    capacities.

Math Model
Inputs
Solution
Outputs
9
Combinatorial optimization problem
  • Example Warehouse location problem
  • We need to answer these questions
  • how many warehouse are needed in a given area?
  • Where can we deploy them?
  • How to assign clients to the warehouse and
    respect constraints?

10
Combinatorial optimization problem
  • So, we can model this problem as follows
  • Let I1,,m be the set of possible locations to
    establish a warehouse,
  • J 1,,n be the set of customers,
  • Cij denoting the amount of transportation from
    warehouse i to customer j,
  • dj be the demand of customer j.
  • ai be the opening cost of warehouse i.
  • Let Yi be a decision variable that is not null if
    the warehouse i is opened and Xij a binary
    variable not null if the client j is assigned to
    warehouse i, and Wi is the ith warehouse
    capacity. These variables are summarized in the
    following table.

11
Combinatorial optimization problem
Variable Notation
Investment cost to build warehouse i ai
ith warehouse capacity Wi
Binary decision variable of affecting client j to warehouse i Xij
Decision variable to open or not warehouse i yi
Transportation Cost of client j toward the warehouse i Cij
Client j demand dj
Max number of initial warehouse m
Client number n
12
Combinatorial optimization problem
  • Objectives functions
  • F1 minimizes the investment cost ,
  • F2 minimizes the transportation cost,
  • We combine the two functions into a single
    objective function F
  • With

13
Combinatorial optimization problem
  • Constraints
  • Ensures that client j is affected only to one
    warehouse
  • guarantees that the sum of the demand dj is
    smaller than the warehouse capacity
  • Integrity constraints

14
Exact and heuristic solution
Exact Heuristic
An exact algorithm is typically deterministic and proven to yield an optimal result. A heuristic has no proof of correctness, often involves random elements, and may not yield optimal results.
Lot of iterations, lot of constraints ? Big computation resources ? Long time Does not explore all possible states of the problem ? short time
Exact solution Optimal (Good solution)
What to use? When?
15
Complexity of a problem
  • The theory of classifying problems based on how
    difficult they are to solve
  • P-problem (polynomial-time)
  • NP-problem (nondeterministic polynomial-time)

16
Complexity of a problem
  • The theory of classifying problems based on how
    difficult they are to solve.
  • A problem is assigned to the P-problem
    (polynomial-time) class if the number of steps
    needed to solve it is bounded by some power of
    the problem's size.
  • A problem is assigned to the NP-problem
    (nondeterministic polynomial-time) class if it
    permits a nondeterministic solution and the
    number of steps to verify the solution is bounded
    by some power of the problem's size.

17
Complexity of a problem
  • Problem complexity
  • We measure the time to solve a problem of input
    size n by a function T(n).
  • Example

18
Complexity of a problem
  • Problem complexity
  • Algorithm complexity can be expressed in Order
    notation, e.g. at what rate does work grow with
    N?
  • O(1) Constant
  • O(logN) Sub-linear
  • O(N) Linear
  • O(NlogN) Nearly linear
  • O(N2) Quadratic
  • O(XN) Exponential

19
Solution
  • Solution for combinatorial optimization problem
    includes different types of algorithms such as
  • Algorithms based on geometry
  • cut trees algorithms
  • Genetic Algorithms
  • Neighborhood search algorithms
  • Dynamic programming
  • Linear and non-linear programming
  • Mixed integer programming
  • Particle swarm optimization
  • Simulated annealing algorithms

20
Genetic algorithm introduction
  • Genetic Algorithms (GAs) are adaptive heuristic
    search algorithm premised on the evolutionary
    ideas of natural selection and genetic. The basic
    concept of GAs is designed to simulate processes
    in natural system necessary for evolution,
    specifically those that follow the principles
    first laid down by Charles Darwin of survival of
    the fittest. As such they represent an
    intelligent exploitation of a random search
    within a defined search space to solve a problem.

21
Genetic algorithm Chromosomes
  • Chromosomes are used to code information.
  • Example 3 warehouses, 5 clients

W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 1 2
22
Genetic algorithm Operators
Population
Select
Crossover
Mutation
No
Final iteration
Recombination
Yes
Best solution
23
Genetic algorithm Operators
Population
1- Randomly generate an initial population
(random chromosomes)
2 -Compute and save the fitness (Objective
function F) for each individual (chromosomes) in
the current population
Select
  • 3-Select some chromosomes from the population as
    an offspring individual
  • Randomly
  • using stochastic method

24
Genetic algorithm Operators
  • The crossover is done on a selected part of
    population (offspring) to create the basis of the
    next generation (exchange information).
  • This operator is applied with propability Pc

Crossover
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 1 2
Father
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 1 2
Mother
25
Genetic algorithm Operators
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 1 2
Father
Crossover
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 3 1 2 2
Mother
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 3 1 1 2
Child 1
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 2 2
Child 2
26
Genetic algorithm Operators
  • This operation is a random change in the
    population. It modifies one or
  • more gene values in a chromosome to have a new
    chromosom value in the pool.
  • This operator is applied with propability Pm

Mutation
W1 W2 W3 C1 C2 C3 C4 C5
1 0 1 3 2 1 1 2
Current
New
W1 W2 W3 C1 C2 C3 C4 C5
0 0 1 3 2 1 1 2
27
Genetic algorithm Operators
  • Recombination combines the chromosomes from the
    initial population and the new offspring
    chromosomes.

Recombination
  • Repeat a fixed number of iteration or until the
    solution converge to one solution (always with
    the best fitness) .

Final iteration
28
Genetic algorithm Operators
Population
Select
Crossover
Mutation
No
Final iteration
Recombination
Yes
Best solution
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