Exact and heuristics algorithms

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 Concept of cost

Introduction Navigation path-cost = distance to node in miles minimum => minimum time, least fuel

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

Introduction Puzzle path-cost = number of pieces moved minimum => least time to solve the puzzle

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) i=1,...,n finite sets, Constraints set, F: Objective function to minimize or to maximize, A set of all feasible solutions S = {x = (x1, x2, , xn)} ϵ D / x satisfies all the constraints S is also called search space.

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. Common problem

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. Inputs Math Model Solution Outputs Common problem

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? Common problem

Combinatorial optimization problem So, we can model this problem as follows: Let I={1,…,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. Common problem

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 Common problem

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: Common problem

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 Common problem

Exact and heuristic solution 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?

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)

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. The class of P-problems is a subset of the class of NP-problems, but there also exist problems which are not NP.

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

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

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

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.

Genetic algorithm: Chromosomes Chromosomes are used to code information. Example: 3 warehouses, 5 clients W1 W2 W3 C1 C2 C3 C4 C5 1 3 2

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

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

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 3 2 Father W1 W2 W3 C1 C2 C3 C4 C5 1 3 2 Mother

Genetic algorithm: Operators W1 W2 W3 C1 C2 C3 C4 C5 1 3 2 Father Crossover W1 W2 W3 C1 C2 C3 C4 C5 1 3 2 Mother W1 W2 W3 C1 C2 C3 C4 C5 1 3 2 Child 1 W1 W2 W3 C1 C2 C3 C4 C5 1 3 2 Child 2

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 3 2 Current New W1 W2 W3 C1 C2 C3 C4 C5 1 3 2

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

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