9 research outputs found
Mathematical models and decomposition methods for the multiple knapsack problem
We consider the multiple knapsack problem, that calls for the optimal assignment of a set of items, each having a profit and a weight, to a set of knapsacks, each having a maximum capacity. The problem has relevant managerial implications and is known to be very difficult to solve in practice for instances of realistic size. We review the main results from the literature, including a classical mathematical model and a number of improvement techniques. We then present two new pseudo-polynomial formulations, together with specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. Extensive computational experiments show the effectiveness of the proposed approaches
A PTAS for the Multiple Knapsack Problem
The Multiple Knapsack problem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the Generalized Assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation for it is implicit in the work of Shmoys and Tardos [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme for MKP.
Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a ptas-preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits
A multiperiod optimization model to schedule large-scale petroleum development projects
This dissertation solves an optimization problem in the area of scheduling large-scale
petroleum development projects under several resources constraints. The dissertation
focuses on the application of a metaheuristic search Genetic Algorithm (GA) in solving
the problem. The GA is a global search method inspired by natural evolution. The
method is widely applied to solve complex and sizable problems that are difficult to
solve using exact optimization methods. A classical resource allocation problem in
operations research known under Knapsack Problems (KP) is considered for the
formulation of the problem.
Motivation of the present work was initiated by certain petroleum development
scheduling problem in which large-scale investment projects are to be selected subject to
a number of resources constraints in several periods. The constraints may occur from
limitations in various resources such as capital budgets, operating budgets, and drilling
rigs. The model also accounts for a number of assumptions and business rules encountered in the application that motivated this work. The model uses an economic
performance objective to maximize the sum of Net Present Value (NPV) of selected
projects over a planning horizon subject to constraints involving discrete time dependent
variables.
Computational experiments of 30 projects illustrate the performance of the model.
The application example is only illustrative of the model and does not reveal real data. A
Greedy algorithm was first utilized to construct an initial estimate of the objective
function. GA was implemented to improve the solution and investigate resources
constraints and their effect on the assets value.
The timing and order of investment decisions under constraints have the prominent
effect on the economic performance of the assets. The application of an integrated
optimization model provides means to maximize the financial value of the assets,
efficiently allocate limited resources and to analyze more scheduling alternatives in less
time
The 0 -1 multiple knapsack problem
In operation research, the Multiple Knapsack Problem (MKP) is classified as a
combinatorial optimization problem. It is a particular case of the Generalized Assignment
Problem. The MKP has been applied to many applications in naval as well as financial
management. There are several methods to solve the Knapsack Problem (KP) and
Multiple Knapsack Problem (MKP); in particular the Bound and Bound Algorithm
(B&B). The bound and bound method is a modification of the Branch and Bound
Algorithm which is defined as a particular tree-search technique for the integer linear
programming. It has been used to obtain an optimal solution. In this research, we provide
a new approach called the Adapted Transportation Algorithm (ATA) to solve the KP and
MKP. The solution results of these methods are presented in this thesis. The Adapted
Transportation Algorithm is applied to solve the Multiple Knapsack Problem where the unit profit of the items is dependent on the knapsack. In addition, we will show the link
between the Multiple Knapsack Problem (MKP) and the multiple Assignment Problem
(MAP). These results open a new field of research in order to solve KP and MKP by
using the algorithms developed in transportation.Master of Science (MSc) in Computational Scienc
Contribution à la résolution de problèmes d'optimisation combinatoire : méthodes séquentielles et parallèles
Les problèmes d'optimisation combinatoire sont souvent des problèmes très difficiles dont la résolution par des méthodes exactes peut s'avérer très longue ou peu réaliste. L'utilisation de méthodes heuristiques permet d'obtenir des solutions de bonne qualité en un temps de résolution raisonnable. Les heuristiques sont aussi très utiles pour le développement de méthodes exactes fondées sur des techniques d'évaluation et de séparation. Nous nous sommes intéressés dans un premier temps à proposer une méthode heuristique pour le problème du sac à dos multiple MKP. L'approche proposée est comparée à l'heuristique MTHM et au solveur CPLEX. Dans un deuxième temps nous présentons la mise en oeuvre parallèle d'une méthode exacte de résolution de problèmes d'optimisation combinatoire de type sac à dos sur architecture GPU. La mise en oeuvre CPU-GPU de la méthode de Branch and Bound pour la résolution de problèmes de sac à dos a montré une accélération de 51 sur une carte graphique Nvidia Tesla C2050. Nous présentons aussi une mise en oeuvre CPU-GPU de la méthode du Simplexe pour la
résolution de problèmes de programmation linéaire. Cette dernière offre une accélération de 12.7 sur une carte graphique Nvidia Tesla C2050. Enfin, nous proposons une mise en oeuvre multi-GPU de l'algorithme du Simplexe, mettant à contribution plusieurs cartes graphiques présentes dans une même machine (2 cartes Nvidia Tesla C2050 dans notre cas). Outre l'accélération obtenue par rapport à la mise en oeuvre séquentielle de la méthode du Simplexe, une efficacité de 96.5 % est obtenue, en passant
d'une carte à deux cartes graphiques.Combinatorial optimization problems are difficult problems whose solution by exact methods can be time consuming or not realistic. The use of heuristics permits one to obtain good quality solutions in a reasonable time. Heuristics are also very useful for the development of exact methods based on branch and bound techniques. The first part of this thesis concerns the Multiple Knapsack Problem (MKP). We propose here a heuristic called RCH which yields a good solution for the MKP problem. This approach is compared to the MTHM heuristic and CPLEX solver. The second part of this thesis concerns parallel implementation of an exact method for solving combinatorial optimization problems like knapsack problems on GPU architecture. The parallel implementation of the Branch and Bound method via CUDA for knapsack problems is proposed. Experimental results show a speedup of 51 for difficult problems using a Nvidia Tesla C2050 (448 cores). A CPU-GPU implementation of the simplex method for solving linear programming problems is also proposed. This implementation offers a speedup around 12.7 on a Tesla C2050 board. Finally, we propose a multi-GPU implementation of the simplex algorithm via CUDA. An
efficiency of 96.5% is obtained when passing from one GPU to two GPUs
Knapsack Problems with Side Constraints
The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once.
The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem.
These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts
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Network models with generalized upper bound side constraints
The objective of this thesis is to develop and
computationally test a new algorithm for the class of
network models with generalized upper bound (GUB) side
constraints. Various algorithms have been developed to
solve the network with arbitrary side constraints problem;
however, no algorithm that exploits the special structure
of the GUB side constraints previously existed. The
proposed algorithm solves the network with GUB side
constraints problem using two sequences of problems. One
sequence yields a lower bound on the optimal value for the
problem by using a Lagrangean relaxation based on relaxing
copies of some subset of the original variables. This is
achieved by first solving a pure network subproblem and
then solving a set of single constraint bounded variable
linear programs. Because only the cost coefficients
change from one pure network subproblem to another, the
optimal solution for one subproblem is at least feasible,
if not optimal, for the next pure network subproblem. The
second sequence yields an upper bound on the optimal value
by using a decomposition of the problem based on changes
in the capacity vector. Solving for the decomposed
problem corresponds to solving for pure network
subproblems that have undergone changes in lower and/or
upper bounds. Recently developed reoptimization
techniques are incorporated in the algorithm to find an
initial (artificial) feasible solution to the pure network
subproblem.
A program is developed for solving the network with
GUB side constraints problem by using the relaxation and
decomposition techniques. The algorithm has been tested
on problems with up to 200 nodes, 2000 arcs and 100 GUB
constraints. Computational experience indicates that the
upper bound procedure seems to perform well; however, the
lower bound procedure has a fairly slow convergence rate.
It also indicates that the lower bound step size, the
initial lower bound value, and the lower and upper bound
iteration strategies have a significant effect on the
convergence rate of the lower bound algorithm