The Use Of Grid Computing

Published Date: 02 Nov 2017

Solving NP-hard combinatorial optimisation problems requires large hardware resources mainly for large size data instances.

There is a lot progress and evolution in parallel environments and architectures able to offer a wide variety and quantity of resources (processors and memory space) which contribute to resolve many time-consuming problems.

Today we are faced with a spectacular evolution Incentives and distributed architectures such as parallel machines, grids, clouds and skis which opens full horizon in particular in the field of research and especially in combinatorial problems.

The strong competition in the global market requires companies to improve the quality of their products especially for industrial companies such as the printing industry, the wood manufacturing industry, the textile industry and many other production or service environments.

Scheduling, in general speaking, concerns the allocation of limited resources to

tasks over time. It is a decision-making process that has as a goal the optimization

of one or more objectives (Pinedo 2008). The resources and the activities may vary a lot. We can think of resources as being machines in a production environment and the activities being the operations that have to be performed on these machines; or the resources may be processors in a computing environment and the activities executions of computer programs. Other examples can be found in the areas of personnel scheduling, maintenance scheduling, airlines, hospitals, classrooms, project scheduling, and other types of services. Traditionally, the production environment is the most common and probably the most appealing application area. Its simplified setting makes the area of machine scheduling one of the most active research branches of scheduling.

Scheduling theory has been extensively studied in the literature and has many

applications in real life situations. Scheduling problems arise in many areas such as manufacturing, computer processing, transportation, production planning and so on.

Though, the flow shop problem is still considered as a very hard nut to crack.

Despite 20-year attempts, some of the instances remain unsolved. Thus, flowshop became a benchmark problem in combinatorial optimization, and deterministic scheduling. Though many methods have been proposed to solve combinatorial optimization problems, Branch-and-Bound (BB) remains the main algorithm delivering guaranteed optimum solutions. BB partially enumerates the solutions, and this process often can be envisioned as searching a tree. Various approaches are used to prune the tree, but still search spaces of combinatorial problems remain huge.

Therefore, parallel branch-and-bound (PBB) is used to reduce the running time [1, 3]. BB parallelization introduces a host of new complications [3, 7]. Overcoming

them requires making design decisions which influence performance of PBB.

By combining computational resources of many institutions Grid environments

provide computational power not available to any single institution separately.

Therefore, grid is a very attractive computing platform for combinatorial

optimization applications. Yet, grid has a number of inherent peculiarities such

as resource heterogeneity and volatility which must be dealt with when designing

a robust application. Overall, the three elements: benchmark hard combinatorial problem, parallel branch-and-bound, the grid environment make the implementation practically hard. Therefore, these three elements can serve as a benchmark of maturity and robustness in parallel processing. With such a goal 2nd Grid Plugtests Flowshop Challenge was organized by European Telecommunications Standards Institute in 2005 [4]. A testbed for solving benchmark hard problems on the grid was provided to verify usability and maturity of the computational platform and the programming environment. The code presented in this paper scored 1st prize in the above competition. This paper is dedicated to the grid parallel processing, and the challenges which must be faced by application designers, rather than to solving flowshop problem itself. Flowshop serves as a benchmark here.

Search trees for hard combinatorial problems have exponential size in the length of the input (here the number of jobs n). Since it is not desirable, various methods are applied to limit the search tree.

The more the search tree is cut, the smaller the set of visited nodes and the faster the algorithm is. Yet, sometimes shallower cuts but faster to calculate give faster BB than deep time consuming cuts. Parallelization is a natural step to speed up BB. Yet, PBB has to deal with several problems. A common approach is to distribute the search tree between computers. Since the search tree is instance-dependent, its structure is unknown before the runtime.

Consequently, the tree cannot be partitioned statically because some computers would quickly run out of work, while the other wouldbe overloaded. However, an optimistic scenario is also possible. When several computers search the solution space simultaneously, then a good upper bound UB may be found earlier than in the sequential run. As a result, some parts of the search tree which would have been visited in a sequential run, may be pruned in a parallel run. Such phenomena are known as performance anomalies in PBB [11, 12].

II. Problem statement : PFSP-makespan

The permutation flow-shop scheduling problem minimizing makespan denoted (PFSP-makespan) Using the notation specified in [22], this problem is denoted F|prmu |Cmax.

For a given job sequence, we define the makespan as the time to complete the schedule. The PFSP is to find a job sequence with minimum makespan.

More precisely, the PFSP can be stated mathematically as follows: if we have a permutation (sequence) =((1), (2),…, (n)) of the set J of jobs, then we define the completion time, Ci( j) of job ( j) on machine Mi as follows:

C1(1) = p1(1)

Ci(1)= Ci_1,(1)+ pi(1), i= 2, . . . ,m

C1 (j)= C1j_1+ p1( j), j = 2, . . . , n

Ci( j)= max(Ci(j-1), Ci-1,(j)) + pi( j), i= 2, . . . ,m and J= 2, . . . ,n

The makespan is defined as

Cmax() = Cm(n)

The PFSP is to find a permutation s* in the set S of all permutations of the set {1, 2, …, n} such that

Cmax(*) =

The first study of the PFSP goes back to 1954 when Johnson published his seminal paper [28] showing that, for the two-machine case, the following property holds:

Job i precedes job j in an optimal sequence if:

min{p1i, p2j} ≤ min{p1j , p2i}

A nice consequence of this result is that the two-machine flow shop problem can be solved in polynomial time [29]. However, for m≥3, the PFSP is NP-hard.

In this work, we consider the mono-objective case, which aims to minimize the overall completion time of all jobs, i.e. the makespan. The PFSP can be solved by two broad classes of methods: Exact methods and Heuristic methods. In this paper we are interested only on the exact methods to solve the PFSP with makespan criterion using parallel B&B algorithm.

is one of the most widely studied scheduling problems in literature. Indeed, this interest is due to the practical aspects of this problem and its simple formulation that hides its hardness. Though, the flow shop problem is still considered a very difficult problem to solve. The PFSP with two machines (m = 2) is polynomially solved using Johnson’s rule [14] which generates an optimal schedule in O(n log n) time. However, for m > 3 the problem is shown to be strongly NP-hard [17].

The main objective of the PFSP is to insure processing N jobs on M machines placed in series while maintaining the same order of all jobs on all machines.

The PFSP is defined as follows. We consider a set {J1,…, Jn} of n jobs to be executed on a set {M1,…,Mm} of m dedicated machines.. Execution time of job Ji on machine Mj is a non-negative integer pij. The problem consists in finding the shortest schedule denoted by Cmax.

consists of N jobs{J1, J2…..Jn}which must be processed in M machines{M1, M2…,Mm} placed in series

A Flow shop problem

Each job is processed first in M1 ,next in M2, and so on, until it’s processed in Mm .

All jobs are available at time zero.

At any time, each machine can process at most one job

At any time, each job can be processed on at most one machine

Once the processing of a job on a machine has started, it must be not interrupted(no pr-emption is allowed)

III. The Sequential B&B algorithm solving the PFSP with makespan criterion

In this paper, we investigate the permutation flow shop problem. This problem can be described as follows. Each job from the job set J = {1, 2… n} has to be processed on m machines M1, M2… Mm, in this order. The processing time of job j on machine Mi is pij. All jobs are ready for processing at time zero. The machines are continuously available from time zero onwards (no breakdowns). At any time, each machine can process at most one job and each job can be processed on at most one machine. No pre-emption is allowed (that is, once the processing of a job on a machine has started, it must be completed without interruption). Only permutation schedules are allowed (i.e. all jobs have the same ordering sequence on all machines). The problem is to find a processing order of the n jobs, such that the time Cmax (makespan) is minimized. Based on the notation presented in [7], this problem is denoted F | prmu |Cmax.

It is well-known that the case of two machines (F2 | |Cmax), could be easily solved using Johnson’s rule which generates an optimal schedule in O(n log n) time [18][20]. For m ≥ 3, however, the problem is shown to be strongly NP-hard [8].

To solve such kind of problems (hard combinatorial optimization problem), there exist mainly two classes of methods; exact methods and approximate methods.

Indeed, the B&B algorithm is an exact method applied in order to obtain the optimal solution to a wide class of combinatorial optimization and scheduling problems. It was introduced by Ignall and Shrage [19]. The principle of the B&B method is to make an implicit search through the space of all possible feasible solutions of the problem. These enumerative optimization methods require a substantial computing time and can subsequently only solve relatively small sized instance problems. In this subsection, we try to briefly describe the basic components of a classical B&B algorithm. Suppose that we have a discrete set S, and an optimal solution x*  S, such that F(x*) = min F(x) x  S, where F is called the objective function, S is called feasible region or domain and x feasible solution.

B&B is characterized by the three following basic components:

A branching strategy splits the region of feasible solutions into subregions or subproblems through the addition of constraints. Convergence of B&B is assured if the size of each new generated subproblem is smaller than the original problem, and the number of feasible solutions to the original problem is finite.

The lower bounding is an algorithm which provides a lower bound (LB), on the value of each solution in a subproblem generated by the branching scheme. A stronger bound eliminates relatively more nodes of the search tree. But if its computational requirements turn excessively large, it may become advantageous to search through larger parts of the tree, using a weaker but more quickly computable bound.

A search strategy or exploration strategy selects a node from which to branch. For instance, there is the depth first strategy, the large first strategy and the best first strategy.

The following algorithm describes the ideas presented above:

LIST={S};

UB: value of some heuristic solutions. CurrentBest: heuristic solution

While LIST≠ Ø Do

Choose a branching node k from LIST

Remove k from LIST

Generate children child(i) for i=1,…,nk and compute the corresponding lower bounds LBi.

For i=1 To nk DO

IF LBi<UB Then

IF child(i) consists of a single solution THEN

UB= LBi

CurrentBest= solution corresponding with child(i)

ELSE add child(i) to LIST

The figure (Fig.1) shows an example for the resolution of the PFSP using the B&B algorithm, in a single processor.

Fig. . Sequential resolution steps of the PFSP with B&B

4.2 Branching Scheme

4.3 Bounding Techniques

4.4 Parallelizing for the Grid

IV The use of Grid computing

IV.1 Introduction

In the past few years, new approaches of distributed and parallel architectures have emerged in order to coordinate the use of geographically dispersed underused resources and/or to solve hard and complex problems which require a large number of resources (processors).

The implementation of a parallel and distributed algorithm for solving a combinatorial optimization problem depends strictly on the hardware architecture on which the application will be deployed. In this context, in order to use the grid computing for solving the PFSP, we have to make some readjustment of our parallel algorithm that we presented in [4].

We have chosen the grid Grid5000 [6] for several reasons: Grid’5000 is suitable for complex scientific applications, large-scale simulations and data analysis, also it is a safe environment based on a high speed dedicated network.

IV.2 Grid’5000 architecture

Grid'5000 is a French research grid offering a large scale nationwide infrastructure for large scale parallel and distributed computing research. Seventeen laboratories are involved in France with the objective of providing the community a test bed allowing experiments in all the software layers between the network protocols up to the applications [6]. Fig. 2 shows clearly the hardware architecture of the Grid’5000, and its principal components.

Fig. 2. The Grid’5000 architecture [10]

IV.3 Use of grids to solve combinatorial optimization problems: review of literature

The use of large scale architectures such as Grid computing in many scientific purposes is to satisfy the requirement of applications which are very time consuming and require a large number of processors to be solved. For example, in the combinatorial optimization field, and in particular for NP-hard problems, the use of grids is the best cure to decrease running times of some problems, and even to solve some problems not yet solved. In this context, several studies have been interested in Grid computing, which continues to attract more researchers.

In the following, we will present a review of state of the art related to the use of grids to solve combinatorial optimization problems.

In [11], authors were interested in the particular optimization problem; the quadratic assignment, in their paper they showed clearly that grid and cluster computing provide tremendous power to optimization methods, and especially for large instances of data.

Moreover, several researchers from the combinatorial optimization community have focused on the programming environment of large scale. In [12], M. Mezmaz, N. Melab and E.-G. Talbi present a parallel B&B algorithm using Grid’5000, they resolved the tail056 Taillard’s benchmark in 25 days. In fact, although this may be considered as a promising result, more investments have to be investigated to reduce the running time which is still huge.

In [13], authors propose a dynamic load balancing approach for the parallel Branch and Bound algorithm on the computational grid. They perform their experimentations on the bi-objective permutation flow shop problem.

In [9], Bendjoudi, A., Melab, N., Talbi, E.-G propose a P2P design and implementation of a parallel B&B algorithm on the ProActive grid middleware. They applied their application to the flow shop scheduling problem and experimented it on a computational pool of 1500 CPUs from the GRID'5000. Remark that the authors used other benchmarks than we used in our study.

After that, in 2009, D. Caromel, A. di Costanzo, L. Baduel and S. Matsuoka propose in [14], a framework built over the master-workers approach that helps programmers to distribute problems over grids. They performed their experimental results using the flow shop scheduling problem, with randomly generated instances.

In [15], authors propose a parallel B&B algorithm which runs on the Grid. The proposed algorithm is parallelized with the hierarchical master-worker paradigm in order to efficiently compute fine-grain tasks on the Grid. The algorithm is implemented on the Grid testbed by using GridRPC middleware, Ninf-G and Ninf.

In this context, ETSI and INRIA at Sophia Antipolis France organized in 2005 the 2nd Grid PLUGTESTS Flow Shop Challenge Contest [16]. The goal of this contest is focused on solving the well known benchmarks problems of Taillard [5]. Indeed, in 2007, a challenge was launched in order to solve some of the instances of combinatorial optimization problem, for instance the PFSP, and a call for competition has been announced in the web [17].

5. Experiments

A. Bendjoudi, N. Melab, and E.-G Talbi, "P2P design and implementation of a parallel branch and bound algorithm for grids," , International Journal of Grid and Utility Computing, vol. 1, pp. 159-168, 2009.

D. Caromel, A. di Costanzo, L. Baduel, and S. Matsuoka, " Grid’BnB: A parallel Branch and Bound Framework for Grids," International conference on high performance computing (HIPC), 2007. 

E.Taillard, "Benchmarks for basic scheduling problems," European Journal of Operational Research, vol. 64, pp. 278-285, 1993.

G. B. McMahon, and P. G. Burton, "Flow-Shop Scheduling with the Branch-and-Bound Method," OPERATIONS RESEARCH, Vol. 15, No. 3, May-June 1967, pp. 473-481.

J. Carlier, and I. Rebai, "Two branch-and-bound algorithms for the permutation flowshop problem," European Journal of Operational Research, vol. 90, n° 2, pp. 238-251, 1996.

K. Aida, K., Y. Yoshiaki Futakata, and T. Osumin, "Parallel Branch and Bound Algorithm with the Hierarchical Master-Worker Paradigm on the Grid Information and Media Technologies," 2(1), 17–30 (2007).

K. Fujisawa, M. Kojima, A. Takeda, and M. Yamashita, "Solving large scale optimization problems via Grid and Cluster Computing," Research Report B400 (December 2003).

M. Mezmaz, N. Melab, and E.-G.Talbi, "An efficient load balancing strategy for grid based branch and bound algorithm," Parallel Computing, Large Scale Grids 33(4-5), 302–313 (2007).

M. Mezmaz, N. Melab, and E.-G.Talbi, "A Grid-enabled B&B Algorithm for Solving Challenging Combinatorial Optimization Problems," IEEE International Parallel and Distributed Processing Symposium, pp. 1–9 (March 2007).

M. Pinedo, "Scheduling: theory, algorithms, and systems," Englewood CliKs, NJ: Prentice-Hall, 1995.

S. Kouki, M. Jemni, and T. Ladhari, "Deployment of Solving Permutation Flow Shop Scheduling Problem on the Grid," Conference on Grid and Distributed Computing (GDC) 2010, LNCS, Springer, December 13-15, 2010, Jeju Island, Korea.

S. Kouki, M. Jemni, and T. Ladhari, "Design of parallel distributed algorithm for the Permutation Flow Shop Problem," Conférence internationale sur les NOuvelles TEchnologies de la REpartition (NOTERE ) IEEE, pp : 65-72, 31 May-2 June, 2010, Tozeur, Tunisia.

S. Kouki, T. Ladhari, and M. Jemni, "A Parallel Distributed Algorithm for the Permutation Flow Shop Scheduling Problem," International Conference on Algorithms and Architectures for Parallel Processing (ICA3PP), LNCS, Springer, pp 328-337, May 21-23, 2010, Busan, Korea.

S. M. Johnson, "Optimal two- and three-stage production schedules with setup times included," Naval Research Logistics Quarterly, vol. 8, pp. 1-61, 1954.

www.grid5000.fr/

www2.lifl.fr/~talbi/challenge2007/.

www-sop.inria.fr/oasis/plugtest2005/2ndGridPlugtestsReport/

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