Efficient Greedy Algorithm with Hill Climbing for Technicians and Interventions Scheduling Problem
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An Iterated Greedy Metaheuristic for the Blocking Job Shop Scheduling Problem
Dipartimento di Ingegneria Via della Vasca Navale, 79 00146 Roma, Italy An Iterated Greedy Metaheuristic for the Blocking Job Shop Scheduling Problem Marco Pranzo1, Dario Pacciarelli2 RT-DIA-208-2013 Novembre 2013 (1) Universit`adegli Studi Roma Tre, Via della Vasca Navale, 79 00146 Roma, Italy (2) Universit`adegli Studi di Siena Via Roma 56 53100 Siena, Italy. ABSTRACT In this paper we consider a job shop scheduling problem with blocking (zero buffer) con- straints. Blocking constraints model the absence of buffers, whereas in the traditional job shop scheduling model buffers have infinite capacity. There are two known variants of this problem, namely the blocking job shop scheduling (BJSS) with swap allowed and the BJSS with no-swap. A swap is needed when there is a cycle of two or more jobs each waiting for the machine occupied by the next job in the cycle. Such a cycle represents a deadlock in no-swap BJSS, while with a swap all the jobs in the cycle move simulta- neously to their subsequent machine. This scheduling problem is receiving an increasing interest in the recent literature, and we propose an Iterated Greedy (IG) algorithm to solve both variants of the problem. The IG is a metaheuristic based on the repetition of a destruction phase, which removes part of the solution, and a construction phase, in which a new solution is obtained by applying an underlying greedy algorithm starting from the partial solution. Comparison with recent published results shows that the iterated greedy outperforms other state-of-the-art algorithms on benchmark instances. -
GREEDY ALGORITHMS Greedy Algorithms Solve Problems by Making the Choice That Seems Best at the Particular Moment
GREEDY ALGORITHMS Greedy algorithms solve problems by making the choice that seems best at the particular moment. Many optimization problems can be solved using a greedy algorithm. Some problems have no efficient solution, but a greedy algorithm may provide a solution that is close to optimal. A greedy algorithm works if a problem exhibits the following two properties: 1. Greedy choice property: A globally optimal solution can be arrived at by making a locally optimal solution. In other words, an optimal solution can be obtained by making "greedy" choices. 2. optimal substructure. Optimal solutions contain optimal sub solutions. In other Words, solutions to sub problems of an optimal solution are optimal. Difference Between Greedy and Dynamic Programming The most difference between greedy algorithms and dynamic programming is that we don’t solve every optimal sub-problem with greedy algorithms. In some cases, greedy algorithms can be used to produce sub-optimal solutions. That is, solutions which aren't necessarily optimal, but are perhaps very dose. In dynamic programming, we make a choice at each step, but the choice may depend on the solutions to subproblems. In a greedy algorithm, we make whatever choice seems best at 'the moment and then solve the sub-problems arising after the choice is made. The choice made by a greedy algorithm may depend on choices so far, but it cannot depend on any future choices or on the solutions to sub- problems. Thus, unlike dynamic programming, which solves the sub-problems bottom up, a greedy strategy usually progresses in a top-down fashion, making one greedy choice after another, interactively reducing each given problem instance to a smaller one. -
AI-KI-B Heuristic Search
AI-KI-B Heuristic Search Ute Schmid & Diedrich Wolter Practice: Johannes Rabold Cognitive Systems and Smart Environments Applied Computer Science, University of Bamberg last change: 3. Juni 2019, 10:05 Outline Introduction of heuristic search algorithms, based on foundations of problem spaces and search • Uniformed Systematic Search • Depth-First Search (DFS) • Breadth-First Search (BFS) • Complexity of Blocks-World • Cost-based Optimal Search • Uniform Cost Search • Cost Estimation (Heuristic Function) • Heuristic Search Algorithms • Hill Climbing (Depth-First, Greedy) • Branch and Bound Algorithms (BFS-based) • Best First Search • A* • Designing Heuristic Functions • Problem Types Cost and Cost Estimation • \Real" cost is known for each operator. • Accumulated cost g(n) for a leaf node n on a partially expanded path can be calculated. • For problems where each operator has the same cost or where no information about costs is available, all operator applications have equal cost values. For cost values of 1, accumulated costs g(n) are equal to path-length d. • Sometimes available: Heuristics for estimating the remaining costs to reach the final state. • h^(n): estimated costs to reach a goal state from node n • \bad" heuristics can misguide search! Cost and Cost Estimation cont. Evaluation Function: f^(n) = g(n) + h^(n) Cost and Cost Estimation cont. • \True costs" of an optimal path from an initial state s to a final state: f (s). • For a node n on this path, f can be decomposed in the already performed steps with cost g(n) and the yet to perform steps with true cost h(n). • h^(n) can be an estimation which is greater or smaller than the true costs. -
CAP 4630 Artificial Intelligence
CAP 4630 Artificial Intelligence Instructor: Sam Ganzfried [email protected] 1 • http://www.ultimateaiclass.com/ • https://moodle.cis.fiu.edu/ • HW1 out 9/5 today, due 9/28 (maybe 10/3) – Remember that you have up to 4 late days to use throughout the semester. – https://www.cs.cmu.edu/~sganzfri/HW1_AI.pdf – http://ai.berkeley.edu/search.html • Office hours 2 • https://www.cs.cmu.edu/~sganzfri/Calendar_AI.docx • Midterm exam: on 10/19 • Final exam pushed back (likely on 12/12 instead of 12/5) • Extra lecture on NLP on 11/16 • Will likely only cover 1-1.5 lectures on logic and on planning • Only 4 homework assignments instead of 5 3 Heuristic functions 4 8-puzzle • The object of the puzzle is to slide the tiles horizontally or vertically into the empty space until the configuration matches the goal configuration. • The average solution cost for a randomly generated 8-puzzle instance is about 22 steps. The branching factor is about 3 (when the empty tile is in the middle, four moves are possible; when it is in a corner, two; and when it is along an edge, three). This means that an exhaustive tree search to depth 22 would look at about 3^22 = 3.1*10^10 states. A graph search would cut this down by a factor of about 170,000 because only 181,440 distinct states are reachable (homework exercise). This is a manageable number, but for a 15-puzzle, it would be 10^13, so we will need a good heuristic function. -
Greedy Estimation of Distributed Algorithm to Solve Bounded Knapsack Problem
Shweta Gupta et al, / (IJCSIT) International Journal of Computer Science and Information Technologies, Vol. 5 (3) , 2014, 4313-4316 Greedy Estimation of Distributed Algorithm to Solve Bounded knapsack Problem Shweta Gupta#1, Devesh Batra#2, Pragya Verma#3 #1Indian Institute of Technology (IIT) Roorkee, India #2Stanford University CA-94305, USA #3IEEE, USA Abstract— This paper develops a new approach to find called probabilistic model-building genetic algorithms, solution to the Bounded Knapsack problem (BKP).The are stochastic optimization methods that guide the search Knapsack problem is a combinatorial optimization problem for the optimum by building probabilistic models of where the aim is to maximize the profits of objects in a favourable candidate solutions. The main difference knapsack without exceeding its capacity. BKP is a between EDAs and evolutionary algorithms is that generalization of 0/1 knapsack problem in which multiple instances of distinct items but a single knapsack is taken. Real evolutionary algorithms produces new candidate solutions life applications of this problem include cryptography, using an implicit distribution defined by variation finance, etc. This paper proposes an estimation of distribution operators like crossover, mutation, whereas EDAs use algorithm (EDA) using greedy operator approach to find an explicit probability distribution model such as Bayesian solution to the BKP. EDA is stochastic optimization technique network, a multivariate normal distribution etc. EDAs can that explores the space of potential solutions by modelling be used to solve optimization problems like other probabilistic models of promising candidate solutions. conventional evolutionary algorithms. The rest of the paper is organized as follows: next section Index Terms— Bounded Knapsack, Greedy Algorithm, deals with the related work followed by the section III Estimation of Distribution Algorithm, Combinatorial Problem, which contains theoretical procedure of EDA. -
Greedy Algorithms Greedy Strategy: Make a Locally Optimal Choice, Or Simply, What Appears Best at the Moment R
Overview Kruskal Dijkstra Human Compression Overview Kruskal Dijkstra Human Compression Overview One of the strategies used to solve optimization problems CSE 548: (Design and) Analysis of Algorithms Multiple solutions exist; pick one of low (or least) cost Greedy Algorithms Greedy strategy: make a locally optimal choice, or simply, what appears best at the moment R. Sekar Often, locally optimality 6) global optimality So, use with a great deal of care Always need to prove optimality If it is unpredictable, why use it? It simplifies the task! 1 / 35 2 / 35 Overview Kruskal Dijkstra Human Compression Overview Kruskal Dijkstra Human Compression Making change When does a Greedy algorithm work? Given coins of denominations 25cj, 10cj, 5cj and 1cj, make change for x cents (0 < x < 100) using minimum number of coins. Greedy choice property Greedy solution The greedy (i.e., locally optimal) choice is always consistent with makeChange(x) some (globally) optimal solution if (x = 0) return What does this mean for the coin change problem? Let y be the largest denomination that satisfies y ≤ x Optimal substructure Issue bx=yc coins of denomination y The optimal solution contains optimal solutions to subproblems. makeChange(x mod y) Implies that a greedy algorithm can invoke itself recursively after Show that it is optimal making a greedy choice. Is it optimal for arbitrary denominations? 3 / 35 4 / 35 Chapter 5 Overview Kruskal Dijkstra Human Compression Overview Kruskal Dijkstra Human Compression Knapsack Problem Greedy algorithmsFractional Knapsack A sack that can hold a maximum of x lbs Greedy choice property Proof by contradiction: Start with the assumption that there is an You have a choice of items you can pack in the sack optimal solution that does not include the greedy choice, and show a A game like chess can be won only by thinking ahead: a player who is focused entirely on Maximize the combined “value” of items in the sack contradiction. -
Types of Algorithms
Types of Algorithms Material adapted – courtesy of Prof. Dave Matuszek at UPENN Algorithm classification Algorithms that use a similar problem-solving approach can be grouped together This classification scheme is neither exhaustive nor disjoint The purpose is not to be able to classify an algorithm as one type or another, but to highlight the various ways in which a problem can be attacked 2 1 A short list of categories Algorithm types we will consider include: Simple recursive algorithms Divide and conquer algorithms Dynamic programming algorithms Greedy algorithms Brute force algorithms Randomized algorithms Note: we may even classify algorithms based on the type of problem they are trying to solve – example sorting algorithms, searching algorithms etc. 3 Simple recursive algorithms I A simple recursive algorithm: Solves the base cases directly Recurs with a simpler subproblem Does some extra work to convert the solution to the simpler subproblem into a solution to the given problem We call these “simple” because several of the other algorithm types are inherently recursive 4 2 Example recursive algorithms To count the number of elements in a list: If the list is empty, return zero; otherwise, Step past the first element, and count the remaining elements in the list Add one to the result To test if a value occurs in a list: If the list is empty, return false; otherwise, If the first thing in the list is the given value, return true; otherwise Step past the first element, and test whether the value occurs in the remainder of the list 5 Backtracking algorithms Backtracking algorithms are based on a depth-first* recursive search A backtracking algorithm: Tests to see if a solution has been found, and if so, returns it; otherwise For each choice that can be made at this point, Make that choice Recur If the recursion returns a solution, return it If no choices remain, return failure *We will cover depth-first search very soon (once we get to tree-traversal and trees/graphs). -
Fast and Efficient Black Box Optimization Using the Parameter-Less Population Pyramid
Fast and Efficient Black Box Optimization Using the Parameter-less Population Pyramid B. W. Goldman [email protected] Department of Computer Science and Engineering, Michigan State University, East Lansing, 48823, USA W. F. Punch [email protected] Department of Computer Science and Engineering, Michigan State University, East Lansing, 48823, USA doi:10.1162/EVCO_a_00148 Abstract The parameter-less population pyramid (P3) is a recently introduced method for per- forming evolutionary optimization without requiring any user-specified parameters. P3’s primary innovation is to replace the generational model with a pyramid of mul- tiple populations that are iteratively created and expanded. In combination with local search and advanced crossover, P3 scales to problem difficulty, exploiting previously learned information before adding more diversity. Across seven problems, each tested using on average 18 problem sizes, P3 outperformed all five advanced comparison algorithms. This improvement includes requiring fewer evaluations to find the global optimum and better fitness when using the same number of evaluations. Using both algorithm analysis and comparison, we find P3’s effectiveness is due to its ability to properly maintain, add, and exploit diversity. Unlike the best comparison algorithms, P3 was able to achieve this quality without any problem-specific tuning. Thus, unlike previous parameter-less methods, P3 does not sacrifice quality for applicability. There- fore we conclude that P3 is an efficient, general, parameter-less approach to black box optimization which is more effective than existing state-of-the-art techniques. Keywords Genetic algorithms, linkage learning, local search, parameter-less. 1 Introduction A primary purpose of evolutionary optimization is to efficiently find good solutions to challenging real-world problems with minimal prior knowledge about the problem itself. -
Hybrid Metaheuristics Using Reinforcement Learning Applied to Salesman Traveling Problem
13 Hybrid Metaheuristics Using Reinforcement Learning Applied to Salesman Traveling Problem Francisco C. de Lima Junior1, Adrião D. Doria Neto2 and Jorge Dantas de Melo3 University of State of Rio Grande do Norte, Federal University of Rio Grande do Norte Natal – RN, Brazil 1. Introduction Techniques of optimization known as metaheuristics have achieved success in the resolution of many problems classified as NP-Hard. These methods use non deterministic approaches that reach very good solutions which, however, don’t guarantee the determination of the global optimum. Beyond the inherent difficulties related to the complexity that characterizes the optimization problems, the metaheuristics still face the dilemma of exploration/exploitation, which consists of choosing between a greedy search and a wider exploration of the solution space. A way to guide such algorithms during the searching of better solutions is supplying them with more knowledge of the problem through the use of a intelligent agent, able to recognize promising regions and also identify when they should diversify the direction of the search. This way, this work proposes the use of Reinforcement Learning technique – Q-learning Algorithm (Sutton & Barto, 1998) - as exploration/exploitation strategy for the metaheuristics GRASP (Greedy Randomized Adaptive Search Procedure) (Feo & Resende, 1995) and Genetic Algorithm (R. Haupt & S. E. Haupt, 2004). The GRASP metaheuristic uses Q-learning instead of the traditional greedy- random algorithm in the construction phase. This replacement has the purpose of improving the quality of the initial solutions that are used in the local search phase of the GRASP, and also provides for the metaheuristic an adaptive memory mechanism that allows the reuse of good previous decisions and also avoids the repetition of bad decisions. -
L09-Algorithm Design. Brute Force Algorithms. Greedy Algorithms
Algorithm Design Techniques (II) Brute Force Algorithms. Greedy Algorithms. Backtracking DSA - lecture 9 - T.U.Cluj-Napoca - M. Joldos 1 Brute Force Algorithms Distinguished not by their structure or form, but by the way in which the problem to be solved is approached They solve a problem in the most simple, direct or obvious way • Can end up doing far more work to solve a given problem than a more clever or sophisticated algorithm might do • Often easier to implement than a more sophisticated one and, because of this simplicity, sometimes it can be more efficient. DSA - lecture 9 - T.U.Cluj-Napoca - M. Joldos 2 Example: Counting Change Problem: a cashier has at his/her disposal a collection of notes and coins of various denominations and is required to count out a specified sum using the smallest possible number of pieces Mathematical formulation: • Let there be n pieces of money (notes or coins), P={p1, p2, …, pn} • let di be the denomination of pi • To count out a given sum of money A we find the smallest ⊆ = subset of P, say S P, such that ∑di A ∈ pi S DSA - lecture 9 - T.U.Cluj-Napoca - M. Joldos 3 Example: Counting Change (2) Can represent S with n variables X={x1, x2, …, xn} such that = ∈ xi 1 pi S = ∉ xi 0 pi S Given {d1, d2, …, dn} our objective is: n ∑ xi i=1 Provided that: = ∑ di A ∈ pi S DSA - lecture 9 - T.U.Cluj-Napoca - M. Joldos 4 Counting Change: Brute-Force Algorithm ∀ ∈ n xi X is either a 0 or a 1 ⇒ 2 possible values for X. -
Artificial Intelligence 1: Informed Search
Informed Search II 1. When A* fails – Hill climbing, simulated annealing 2. Genetic algorithms When A* doesn’t work AIMA 4.1 A few slides adapted from CS 471, UBMC and Eric Eaton (in turn, adapted from slides by Charles R. Dyer, University of Wisconsin-Madison)...... Outline Local Search: Hill Climbing Escaping Local Maxima: Simulated Annealing Genetic Algorithms CIS 391 - Intro to AI 3 Local search and optimization Local search: • Use single current state and move to neighboring states. Idea: start with an initial guess at a solution and incrementally improve it until it is one Advantages: • Use very little memory • Find often reasonable solutions in large or infinite state spaces. Useful for pure optimization problems. • Find or approximate best state according to some objective function • Optimal if the space to be searched is convex CIS 391 - Intro to AI 4 Hill Climbing Hill climbing on a surface of states h(s): Estimate of distance from a peak (smaller is better) OR: Height Defined by Evaluation Function (greater is better) CIS 391 - Intro to AI 6 Hill-climbing search I. While ( uphill points): • Move in the direction of increasing evaluation function f II. Let snext = arg maxfs ( ) , s a successor state to the current state n s • If f(n) < f(s) then move to s • Otherwise halt at n Properties: • Terminates when a peak is reached. • Does not look ahead of the immediate neighbors of the current state. • Chooses randomly among the set of best successors, if there is more than one. • Doesn’t backtrack, since it doesn’t remember where it’s been a.k.a. -
A. Local Beam Search with K=1. A. Local Beam Search with K = 1 Is Hill-Climbing Search
4. Give the name that results from each of the following special cases: a. Local beam search with k=1. a. Local beam search with k = 1 is hill-climbing search. b. Local beam search with one initial state and no limit on the number of states retained. b. Local beam search with k = ∞: strictly speaking, this doesn’t make sense. The idea is that if every successor is retained (because k is unbounded), then the search resembles breadth-first search in that it adds one complete layer of nodes before adding the next layer. Starting from one state, the algorithm would be essentially identical to breadth-first search except that each layer is generated all at once. c. Simulated annealing with T=0 at all times (and omitting the termination test). c. Simulated annealing with T = 0 at all times: ignoring the fact that the termination step would be triggered immediately, the search would be identical to first-choice hill climbing because every downward successor would be rejected with probability 1. d. Simulated annealing with T=infinity at all times. d. Simulated annealing with T = infinity at all times: ignoring the fact that the termination step would never be triggered, the search would be identical to a random walk because every successor would be accepted with probability 1. Note that, in this case, a random walk is approximately equivalent to depth-first search. e. Genetic algorithm with population size N=1. e. Genetic algorithm with population size N = 1: if the population size is 1, then the two selected parents will be the same individual; crossover yields an exact copy of the individual; then there is a small chance of mutation.