Exact and heuristics algorithms

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Exact and heuristics algorithms
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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

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Introduction

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

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Introduction

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

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Introduction

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

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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.

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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.

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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
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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?

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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.

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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
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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

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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

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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?
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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)

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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.

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Complexity of a problem

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

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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

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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

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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.

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

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