Chaotic Simulator for Bilevel Optimization of Virtual Machine Placements in Cloud Computing


The drastic increase in engineering system complexity has spurred the development of highly efficient optimization techniques. Many real-world optimization problems have been identified as bilevel/multilevel as well as multiobjective. The primary aim of this work is to present a framework to tackle the bilevel virtual machine (VM) placement problem in cloud systems. This is done using the coupled map lattice (CML) approach in conjunction with the Stackelberg game theory and weighted-sum frameworks. The VM placement problem was modified from the original multiobjective (MO) problem to an MO bilevel formulation to make it more realistic albeit more complicated. Additionally comparative analysis on the performance of the CML approach was carried out against the particle swarm optimization method. A new bilevel metric called the cascaded hypervolume indicator is introduced and applied to measure the dominance of the solutions produced by both methods. Detailed analysis on the computational results is presented.

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Fig. 1
Fig. 2
Fig. 3


P j :

Power consumption of the jth server

y j :

Indicates whether server j is in use or not

x ij :

Indicates if VM i is assigned to server j

R pi :

CPU demand of each VM

T pj :

Threshold of CPU utilization associated with each server

T mj :

Threshold of memory utilization associated with each server

R mi :

Memory demand of each VM

W j :

Resource wastage of the jth server

ɛ o :

Positive real constant parameter

F 1 :

Total power consumption

F 2 :

Total resource wastage

f o :

Total VM utilization

P idle j :

Power consumption of the jth server when it is idle

P busy j :

Power consumption of the jth server when it is busy


Cumulative hypervolume indicator

(x *1 , x *2 , x *o ) :

Optimal candidate solution

(z 1 , z 2 , z o):

Nadir point

w 1, w 2 :

Weights for the weighted-sum approach

V ref :

Normalized hypervolume

K :

Scaling factor

I :

Total number of VMs

J :

Total number of servers

w, c 1 , c 2 , r 1 , r 2 :

Constant Parameters

v i (t):

Velocity of each particle

x i (t):

Position of each particle

T max :

Maximum limit of function evaluations

s i :

Initial social influence

p i :

Initial individual influence

ɛ :

Coupling parameter

j :

Iteration index

θ :

Logistic function constant

m :

Maximum number of iteration

s :

Index for the vertices on the lattice

y :

Mapping variable


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The authors thank the organizers of ICO’2018, 4–5th October 2018, Hard Rock Hotel, Pattaya, Thailand, for the opportunity to present their research results.

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Corresponding author

Correspondence to Timothy Ganesan.



A.1 xij for the best individual solution obtained using the CML approach

A.2 xij for the best individual solution obtained using the PSO approach

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Ganesan, T., Vasant, P. & Litvinchev, I. Chaotic Simulator for Bilevel Optimization of Virtual Machine Placements in Cloud Computing. J. Oper. Res. Soc. China (2021).

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  • Bilevel multiobjective
  • Coupled map lattices (CML)
  • Stackelberg game theory
  • Particle swarm optimization (PSO)
  • Cascaded hypervolume indicator (cHVI)
  • Virtual machine (VM) placement

Mathematics Subject Classification

  • 65K05
  • 90B50
  • 90B99
  • 91A65
  • 65P20
  • 68W50