Particle Swarm and Ant Colony Algorithms Hybridized for Multi-mode Resource-constrained

Project Scheduling Problem with Minimum Time Lag

Shan Miyuan, Wu Juan, Peng Danni School of Business Administration, Hunan University, Changsha,Hunan, 410082

Abstract—MMRCPSP/MIN has been proved to be a NP-hard problem. Based on the introduction of model of project scheduling problem with minimum time lag developed by Roland Heilmann, the paper analyzed two swarm intelligence algorithms which have good performance in solving combinatorial optimization problem—ACO and PSO algorithms. Based on analysis of these algorithms’ defects, the paper proposed that the defects of ACO algorithm such as hard to converge, performance significantly affected by parameters, high requirement for parameters and so on, can be made up by the high convergence of PSO algorithm. The parameters of ACO algorithm were set in terms of the solution of PSO algorithm. Through particle swarm iteration and making convergence effect and better objective solution of ACO algorithm as fitness value in order to lead the optimization of particle swarm, we can obtain the optimal solution as well as better convergence speed and effect. Finally, the model and problem solving process were programmed in the C++ language. Intensive computational experiments were done on cases in PSPLIB. The result shows that with the iteration of PSO algorithm, both the performance and convergence of ACO algorithm are improved.

Keywords—resource-constrained; project scheduling problem; Ant colony optimization; Particle swarm optimization

I. INTRODUCTION Since early 1960s, lots of research has been done on

RCPSP problem by researchers. The research includes 3 major branches: 1. Research on problem. Through it the model can close to practice constantly so that practical scheduling problem in project management can be solved by solving the model. 2. Generation methods of various problem cases set. There are 3 influential test cases base [1-3]: Patterson (1984), kolisch et al (1995), kolisch and sprecher (1996).3. Algorithms for solving the model. The algorithms includes accurate methods such as CPM, PERT, branch and bound method to get a near optimal, and various searching algorithms. However, heuristic is the most popular method so far in which genetic algorithm is a typical one. With the appearance of new algorithms, scholars because of its good performance, easy to understand and successful applications in other combinatorial optimization field applied a new swarm intelligence algorithm to RCPSP field since 1990s.

The swarm intelligence algorithms include PSO (particle swarm optimization) algorithm and ACO (ant colony optimization) algorithm. They both have advantages. But they also have some defects because it is not long since they were proposed, for example, the performance of PSO algorithm is susceptible to initialization, and the convergence of ACO algorithm is mainly determined by its parameters. In order to make up these defects to make the algorithm play a more important role in RCPSP field, this paper proposes particle swarm and ant colony algorithms hybridized which improves the performance of ACO algorithm through using PSO algorithm to optimize and adjust parameters of ACO algorithm. We apply this method to a difficult type in RCPSP problem set—MMRCPSP/MIN.

II. PROBLEM DESCRIPTION Multi-mode resource-constrained project scheduling

problem with minimum time lag (MMRCPSP/MIN) is a generalized combinatorial optimization problem. It requires the determination of execution mode of each activity in project and corresponding starting date and due date under the project precedence, resource and time lag constraints, so that the total project duration can be minimized. To the best of our knowledge, apart from two tabu search procedures (cf. De Reyck, 1998[4]; Franck, 1999[5]) and a branch-and bound procedure (Roland Heilmann, 2003) [6] the literature on MMRCPSP/max is completely void.

III. MODEL Let a project consist of a finite set 1,,,1,0 += nnV of

activities. Activity 0 represents the start and activity 1+n the completion of the project. Both are dummy activities. For each activity Vj ∈ a set ,,1 jj MM = of modes is available. Activities 0 and 1+n can be executed in only one mode:

1:: 10 == +nMM . Each activity Vj ∈ has to be performed in exactly one mode jM∈µ .

The processing time of activity j executed in mode µ is denoted by 0≥∈ Zp jµ . The processing time of activities 0 and

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1+n equals 0: 0::1101 == +npp . jS and jC stand for the start

time and the completion time ( of the performance) of activity j , respectively. If we define 10 ,0: += nSS stands for the project

duration. If activity j starts in mode µ at time jS , it is being

executed at each point in time ],[ µjjj pSSt +∈ .

Between the start time jS of activity j , which is performed in mode jM∈µ , and the start time lS of activity )( jll ≠ , which is performed in mode lM∈λ , a minimum time lag 0min ≥∈ Zd lj λµ can be given.

ρR and vR denote the set of renewable resources and the set of nonrenewable resources, respectively. 0>∈ ZRk

ρ stands for the capacity of renewable resource )( ρRkk ∈ which is

available at each point in time. 0>∈ ZR vk stands for the

capacity of nonrenewable resource )( vRkk ∈ which is available in total. Provided that activity j is performed in mode µ , kjr µ

ρ units of renewable resource )( ρRkk ∈ are consumed in total. For activities 0 and 1+n , we set

)(0:: 11 10ρρρ Rkrr knk ∈== + and )(0:: 11 10

vkn

vk

v Rkrr ∈== + So, the MMRCPSP/min can be stated as follows [6]: Objective: 1+nSMin (1) s.t.

lj lmjmjl SS δ≥− ),( Elj >∈< (2)

)0;(),,(

≥∈≤∑∈

tRkRrtSMAj

kkjm jρρρ (3)

∑∈

∈≤Vj

vk

vkjm

v RkRr j )( (4)

)( VjMm jj ∈∈ 0≥jS )( Vj ∈ 00 =S

Where, Activities and time lags are represented by an activity-on-node (or briefly AON) network );,( δEVN = with node setV , arc set E , and arc weight functionδ . Each element of node set V represents and activity.

The objective is to determine a schedule ),( SM such that the time lags (2) are observed, the constraints w.r.t. the renewable resources (3) and nonrenewable resources (4) are met, and the project duration is minimized (1). Such a schedule is called optimal . A schedule ),( SM obeying (2)-(4) is called feasible .

IV. PARTICLE SWARM ALGORITHMS AND ANT COLONY

ALGORITHMS Swarm intelligence meta-heuristics, namely, particle

swarm optimization (PSO) and ant colony optimization (ACO) are recently proved to be successful approaches to solve complex optimization problems.

A. Particle Swarm Optimization Algorithm The particle swarm optimization (PSO) algorithm was first

proposed by Kennedy and Eberhart [7], which is inspired by the observations for bird flocking and fish schooling. A number of birds/fishes flock synchronously, change direction suddenly, and scatter and regroup together. Each individual, called a particle, benefits from the historical experience of its own and that of the other members of the swarm during the search for food.

The PSO proceeds as follows. Given an optimization function )(Pf where P is a vector of n real-valued random variables; a swarm of particles is generated at random for targeting the optimum solution *P . Each particle is represented as ),,,( 21 iniii pppP = , Si ,,2,1= , where S is the swarm size. The particle is a candidate solution in the n-dimensional real number space and iteratively moves in the problem space. The PSO enriches the swarm intelligence by storing the best solutions seen by every particle. In particular, particle i remembers the best position it visited so far, referred to as ipbest , and the best position by its neighbors. There are two versions for defining the neighbors’ best position, namely lbest and gbest . In the local version, each particle keeps track of the best position lbest attained by the particles within its topological neighborhood. For the global version, the best position gbest is determined by any particles in the entire swarm. Hence, the gbest model is a special case of the lbest model. The PSO is an iterative evolutionary algorithm. At each iteration, particle i adjusts its velocity ijv and position ijp through each dimension j by referring to the personal best position ( ijpbest ) and the swarm’s best position ( jgbest , if the global version is adopted) using Esq. (5) and (6) as follows [8]:

)]()([ 2211 ijjijijijij pgbestrcppbestrcvKv −+−+= (5)

and ijijij vpp += (6)

Where 1c and 2c are the acceleration constants, 1r and 2r are random real numbers drawn from )1,0(U , and K is the constriction factor. Clerc and Kennedy [9] has pointed out that the use of a constriction factor is needed to insure convergence of the PSO, and it is determined by:

ϕϕϕ 42

22 −−−

=K

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Where 21 cc +=ϕ , 4>ϕ . Typically, ϕ is set to 4.1 and K is thus 0.729.

B. Ant Colony Optimization Algorithm In the early 1990s, ant colony optimization (ACO) [10-12]

was introduced by M. Dorigo and colleagues as a novel nature-inspired metaheuristic for the solution of hard combinatorial optimization (CO) problems. The inspiring source of ACO is the foraging behavior of real ants.

ACO algorithms are stochastic search procedures. Their central component is the pheromone model, which is used to probabilistically sample the search space. The pheromone model can be derived from a model of the tackled CO problem, defined as follows:

Definition 1. A model ),,( fSP Ω= of a CO problem consists of:

(1) a search (or solution) space S defined over a finite set of discrete decision variables and a set Ω of constraints among the variables;

(2) an objective function +→ RSf : to be minimized.

The search space S is defined as follows: Given is a set of n discrete variables iX with values iD

jjiij vvDv , 1=∈ ,

ni ,1= . A variable instantiation, that is the assignment of a value i

jv to a variable iX , is denoted by ij

i vX = . A feasible solution Ss ∈ is a complete assignment that satisfies the constraints. A feasible solution Ss ∈* is called a globally optimal solution (or global optimum), if )()( * sfsf ≤ Ss ∈∀ . The set of globally optimal solutions is denoted by SS ∈* . To solve a CO problem one has to find a solution ** Ss ∈ .

V. ANT COLONY AND PARTICLE SWARM HYBRID OPTIMIZATION ALGORITHM

Through the introduction about ACO above, we can know that pheromone updating equation and visit probability equation are the most important in ACO algorithm. Therefore, the values of related parameters have great impact on the performance of ACO algorithm. The paper will adjust related parameters in ACO algorithm by using PSO algorithm to make ACO algorithm achieve the best performance. The paper will adopt method proposed in reference [13] which embeds PSO to ACO to search the best parameters and make ACO algorithm achieve the best performance. The content of hybrid algorithm is as follows:

A. PSO Parameters,Iinitial Particles and Fitness Value The algorithm is run each time on a MMRCPSP/MAX

instance. The set of parameters of ACO that define a point in

the PSO domain are αβϕ ,, and the numbers of ants ( m ). Where βϕ,,m is integer, otherwise α is real value. All of the parameters explained follow:

According to the introduction in section 3, we know that 3 most important equations in ACO are:

Path-selecting probability formula:

∈

= ∑∈

otherwise

Njift

t

tpk

i

Nlilil

ijij

ijk

ki

0

][)]([

][)]([

)( βϕ

βϕ

ητ

ητ

where ijj Ri /1=η is a priori available heuristic information, βϕ , are two parameters which determine the relative influence of pheromone trail and heuristic information, and k

iN is the feasible neighborhood of ant k ,that is ,the set of activities which is the successor of processing activity.

The solution construction ends after each ant has completed a scheduling, that is, after each ant has constructed a sequence of activity in some process model. Next, the pheromone trails are updated. In ACO this is done by first lowering the pheromone trails by a constant factor (this is pheromone evaporation) and then allowing each ant to deposit pheromone on the arcs that belong to its scheduling:

),()()()1()1(1

jitttm

k

kijijij ∀∆+−=+ ∑

=ττατ

Where 10 << α is the pheromone trail evaporation rate and m is the number of ants. )(tk

ijτ∆ is the amount of pheromone ant k deposits on the arcs; it is defined as

=∆otherwise

kantbyusedisjiarciftDtk

kij 0

),()(/1)(τ

Where )(tD k is the length of the kth ant’s scheduling.

Meanwhile, each parameter pertains to its related [Minimum, Maximum].

PSO’s fitness value is the makespan of the best scheduling computed by an ACO using the related parameters in the given instance. If comparing two different positions they have the same makespan value then computing time is considered. We consider better of those parameters that minimize the makespan of the scheduling and secondly the time of computing. For computing the fitness of a given position, first integer parameters ),,( βϕm are rounded up, secondly the algorithm runs five trials of the ACO algorithm using the rounded parameters in the MMRCPSP/max instance and returns the best value obtained from the trial.

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For each particle of the population its initial velocity is set randomly. For half of the population the initial position is set using predefined parameters assuring that for every parameter there will be a particle containing a value covering the full range. The positions of the other half of the initial population are set randomly.

Parameters for the PSO have been set following: c1=c2=2, the inertia weight is set initially to 1 and gradually decreasing from 1 to 0.1 (at each PSO iteration ωω 99.0= ). Maximum number of iterations is set to 200 due to computing time constraints.

B. Algorithm Flow Charts The algorithm PSO-ACO flow chart is as follows:

Initialize particlesiteration(P)=1:maxiteration(P)

initialize fitness

iteration(A)=1:maxiteration(A)Perform a trial ACO with

particle parameters

w=0.99w

Update fitness if needand iteration(A)=iteration(A)+1

Update the best parameters byeach particle and best

parameters by the population

if iteration>=maxiterationoutput bestfitness and best

particle,elseiteration(P)=iteration(P)+1

and return to first step

Figure 1. Flow chart of PSO-ACO algorithm

VI. COMPUTATIONAL

A. Test Instances A test set applied PSPLIB’s instances --- MMRCPSP/max

(kolisch, 1995) [3]with the essential exceptions that the maximum number of nondummy activities was increased from 10 to 50 and the number of (non)renewable resources from 2 to 3 and the number of mode from 3 to 5. But PSPLIB’s instances are all MMRCPSP/max cases while algorithm in this paper only suits for cases with minimum time lag. Thus we removed all cases with max time lag and only solved the minimum duration of project that minimum time lag exists between multi-mode activities.

The performance analysis has been done on an Intel Pentium Ⅳ 1.73GHz personal computer under the operating system Windows XP. All algorithms have been coded in ANSI C using the MS Visual C++ 6.0 Developer Studio

B. Analysis of Result Because of the limit of text size, the paper just report the

experiment result of one type from the test set. Table 1 listed the parameters PSO algorithm selected in the iteration. Figure 2

presented the final results of various cases. The result improvement in PSO iteration was shown from top to bottom. From figure 2 we can see that:

• When ACO algorithm is applied to different projects, the parameters that make ACO algorithm achieve the best performance differ with the network complexity, resource strength, work contents and activity number of project.

• After the use of PSO algorithm, ACO algorithm can adjust the value of parameters automatically. Therefore, with the iteration of PSO algorithm, both the performance and convergence of ACO algorithm are improved.

• When apply ACO algorithm to solve project scheduling problem, heuristic pheromone can produce significant effect on the performance of ACO algorithm. So selecting a good prior rule is an important factor of obtaining the optimal solution. The experiment results indicate that the shortest activity first rule prior to other rules.

TABLE I. PARAMETERS IN THE PSO ITERATION

EIM WE WP SMLV GPER 1 3 3 0．1 0．8 0 3 -1 0．5 0．8 1 1 0 0．22 0．59 0 4 1 0．77 0．49 0 4 3 0．8 0．48

Where:

EIM: enlightenment information mode; WE: weight of enlightenment; WP: weight of information; GPER: global pheromone evaporation rate

Figure 2. Experiment result example

VII. CONCLUSION Resource constrained project scheduling problem is a

classic NP-hard problem. The processing difficulty of multi-mode resource-constrained project scheduling problem with

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minimum and maximum time lag is enhanced because of the complexity of time lag and project logical relationship. Based on the introduction of multi-mode resource-constrained project scheduling problem model with minimum time lag developed by Roland Heilmann, the paper analyzes two swarm intelligence algorithms which have good performance in solving combinatorial optimization problem—ACO and PSO algorithms. Based on analysis of defects of these algorithms, the paper proposes that the defects of ACO algorithm such as hard to converge, performance significantly affected by parameters and high requirement for parameters and so on, can be made up by the high convergence of PSO algorithm. We set the parameters of ACO algorithm as the solution representation of PSO algorithm. Through particle swarm iteration and making convergence effect and better solution of ACO algorithm as fitness value in order to lead the optimization of particle swarm, we can obtain the optimal solution and better convergence speed and effect. Meanwhile, through a number of experiments, we show that by using PSO algorithm to optimize ACO algorithm, ACO algorithm can get better performance.

REFERENCES [1] Patterson, James H. “A Comparison of Exact Approaches for solving the

Multiple Constrained Resource. Project Scheduling Problem.” Management Science, 1984(30):854-867.

[2] Kolisch Rainer, Arno Sprecher, and Andreas Drexl. Characterization and generation of a general Class of Resource-Constrained Project Scheduling Problems. Management Science. 1995(41):1693-1703.

[3] Rainer Kolisch, Arno sprecher. PSPLIB – A project scheduling problem library. European Journal of operational Research 96(1996):205-216.

[4] Roland Heilmann. Abranch-and –bound procedure for the multi-mode resource-constrained project scheduling problem with minimum and maximum time lags. European Journal of Operational Research. 2003(144):348-365.

[5] De Reyck, B., Scheduling projects with generalized precedence relations: Exact and heuristic procedures. Dissertation, Katholieke Universiteit Levuven, Belgium. 1998.

[6] Franck,B.,Prioritatsregelverfahren for die resource cenbesch rankte project planning mit und ohne Kalender. Doctoral Dissertation. Shaker, Aachen. 1999.

[7] J Kennedy, R.C. Eberhart, Particle swarm optimization, in : Proceedings of the IEEE International Conference on Neural Networks, vol. IV, 1995:1942-1948.

[8] Peng-Yeng Yin, Jing-Yu wang. A particle swarm optimization approach to the nonlinear resource allocation problem. Applied Mathematics and Computation. 2006: 1-11.

[9] M. clerc, J. Kennedy. The particle swarm explosion, stability, and convergence in a multidimensional comples space, IEEE Transaction on Evolutionary Computation 6(2002):58-73.

[10] M. dorigo, Optimization, learning and natural algorithms (in Itqlian), Ph.D. thesis, di;partmento di Elettronica, Politecnico di Milano, Italy, 1992.

[11] N. Dorigo, V. Maniezzo, A. Colorni: Positive feedback as a search strategy, Tech. Report 91-016, Dipartimento di Elettronica, Politecnico di Milano, Italy, 1991.

[12] M. dorigo, V. Maniezzo, A. colorni, Ant system: optimization by a colony of cooperating agent, IEEE Trans. Systems, Man, Cybernet. Part B 26(1)(1996)29.

[13] Gomez-Cabrero, D., Ranasinghe, D.N., Fine-Tuning the Ant Colony System Algorithm Through Particle Swarm Optimization, Technical Report TR07-2005, Departamento de Estadistica e Investigacion Operativa, Universitat de Valencia, Burjassot, Spain, 2005.

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