Latest Space News
Mon, 25 Sep 2017 10:36:00 +0100
NASA’s asteroid sample return spacecraft successfully used Earth’s gravity on Friday to slingshot itself on a path toward the asteroid Bennu, for a rendezvous next August. At 12:52 p.m. EDT on Sept. 22, the OSIRIS-REx (Origins, Spectral Interpretation, Resource Identification, and Security – Regolith Explorer) spacecraft came within 10,711 miles (17,237 km) of Antarctica, just south of Cape Horn, Chile, before following a route north over the Pacific Ocean. OSIRIS-R...
Mon, 25 Sep 2017 09:48:00 +0100
NASA’s Hubble Space Telescope helped an international team of astronomers find that an unusual object in the asteroid belt is, in fact, two asteroids orbiting each other that have comet-like features. These include a bright halo of material, called a coma, and a long tail of dust. Hubble was used to image the asteroid, designated 300163 (2006 VW139), in September 2016 just before the asteroid made its closest approach to the Sun. Hubble’s crisp images revealed that it was actual...
Mon, 25 Sep 2017 09:02:00 +0100
Arianespace’s ninth launch of 2017 is now scheduled for September 29 on a dual-payload Ariane 5 mission to geostationary transfer orbit with Intelsat 37e (for the global operator Intelsat) and BSAT-4a (for Space Systems Loral in the framework of a turnkey contract for the Japanese operator B-SAT). Ariane 5 in the Final Assembly BuildingDesignated Flight VA239 in Arianespace’s mission numbering system for its launcher family, the liftoff originally had been set for September 5, b...
Sun, 24 Sep 2017 08:00:00 +0100
A United Launch Alliance (ULA) Atlas V rocket carrying a payload for the National Reconnaissance Office lifted off from Space Launch Complex-3 on Sept. 23, at 10:49:47 p.m. PDT. Designated NROL-42, the mission is in support of national security. "Congratulations to the entire team for overcoming multiple challenges throughout this launch campaign. From Hurricane Irma schedule impacts to replacing to a first stage battery this week - the team maintained a clear focus on mission success," said...
Thu, 21 Sep 2017 08:38:00 +0100
For the first time in its 19-year history, SMi Group’s Global MilSatCom Conference & Exhibition will be featuring South Korea in its speaker line-up for this year’s event. With recent threats surging within the region, South Korea has instilled urgency in the race to perfect milsatcom systems to ensure overall robust defence capability. Commander of Defence Communication Command, Brigadier General Nag Jung Choi, will be presenting a keynote address on Day 3 of the conference...
Wed, 20 Sep 2017 16:45:00 +0100
In the first act of lunar exploration, Neil Armstrong and Buzz Aldrin were major characters. In setting its sights on the Moon, ESA hopes to bring many more actors to this off-world stage. By testing the market for transport services to the Moon, ESA aims to push the limits of technology and create new models of space business. Touching down on the Moon was a monumental moment in human history. Eight short years and enormous resources took humankind to the lunar surface, initially for le...
Wed, 20 Sep 2017 12:00:00 +0100
Astronomers have used ALMA to capture a strikingly beautiful view of a delicate bubble of expelled material around the exotic red star U Antliae. These observations will help astronomers to better understand how stars evolve during the later stages of their life-cycles. In the faint southern constellation of Antlia (The Air Pump) the careful observer with binoculars will spot a very red star, which varies slightly in brightness from week to week. This very unusual star is called U Antliae and...
Wed, 20 Sep 2017 11:03:00 +0100
SES, EchoStar Corp., and Airbus Defence and Space announced today that the new EchoStar 105/SES-11 spacecraft, built by Airbus, has been shipped from the Airbus facilities in Toulouse, France, to Cape Canaveral, Florida, for its forthcoming launch by SpaceX in October. The first satellite that EchoStar has constructed with manufacturer Airbus, EchoStar 105/SES-11 will be launched by SpaceX on a flight-proven Falcon 9 rocket from Launch Complex 39A at Kennedy Space Center, Florida. EchoStar 10...
Tue, 19 Sep 2017 19:30:00 +0100
Today Lockheed Martin (NYSE: LMT) debuted a new family of satellite buses that form the core of nearly every space mission. From nanosats to high-powered satellites, the versatile lineup incorporates dozens of major enhancements and hundreds of common components that speed production and reduce cost. The company announced the lineup at the Air Force Association Air, Space & Cyber Conference. "We've invested $300 million in revamping our satellite solutions from top to bottom, applying what we...
Tue, 19 Sep 2017 15:54:00 +0100
FLYHT Aerospace Solutions Ltd. (TSX VENTURE: FLY) (OTCQX: FLYLF) (the "Company" or "FLYHT") is pleased to announce a flight trial with Inmarsat, and would like to bring attention to an Inmarsat press release titled Inmarsat to demonstrate advanced SwiftBroadband-Safety capabilities in flight trial with FLYHT on September 19, 2017. This trial will demonstrate the use of FLYHT's Automated Flight Information Reporting System (AFIRS(tm)) to send data to the UpTime(tm) Cloud management platform via I...
Tue, 19 Sep 2017 14:53:00 +0100
With the ''Space Strategy for Europe'' from the European Commission and the new ESA roadmap on the table, the European space stakeholders will have the great opportunity to debate and influence, at the highest level, the medium and long term development of European space programmes and of the spread of space services in many economic and societal activities as well as in a wide range of public fields, including security and defence. Download Conference Agenda | Register to a...
Tue, 19 Sep 2017 14:00:00 +0100
Kratos Defense & Security Solutions, Inc. (NASDAQ:KTOS), a leading National Security Solutions provider, announced today that it has recently received approximately $8.2 million in orders or specialized products in support of satellite global positioning system protection and communication systems. Work on these recent orders will be performed in secure Kratos manufacturing facilities. Due to customer related, competitive and other considerations, no additional information will be provided rel...
Tue, 19 Sep 2017 11:51:00 +0100
The Sun's recent activity has caught the interest of scientists and space weather forecasters worldwide, highlighting the need to keep a watchful eye on our star and its awesome power. On 6 and 10 September, our Sun produced a pair of solar flares, the strongest observed in over 10 years. They were accompanied by huge eruptions of billions of tonnes of matter into space. While many such eruptions fall back onto the hot surface, these two did not and became 'coronal mass ejections' (CMEs) -...
Tue, 19 Sep 2017 08:19:00 +0100
NASA's Hubble Space Telescope has observed a planet outside our solar system that looks as black as fresh asphalt because it eats light rather than reflecting it back into space. This light-eating prowess is due to the planet's unique capability to trap at least 94 percent of the visible starlight falling into its atmosphere. The oddball exoplanet, called WASP-12b, is one of a class of so-called "hot Jupiters," gigantic, gaseous planets that orbit very close to their host ...
Fri, 15 Sep 2017 18:13:00 +0100
Raytheon Company (NYSE: RTN) will continue its support for NOAA's Advanced Weather Interactive Processing System program that ingests and analyzes weather data, creating useful visualizations for forecasters and distributing time-sensitive weather statements such as watches and warnings across the United States. Used by more than 140 National Weather Service field offices, the AWIPS program informs decisions that save lives and protect property during natural disasters. Under the $269 mill...
Fri, 15 Sep 2017 15:00:00 +0100
The international Cassini mission has concluded its remarkable exploration of the Saturnian system in spectacular style, by plunging into the gas planet's atmosphere. Confirmation of the end of mission arrived at NASA's Jet Propulsion Laboratory at 11:55 GMT/13:55 CEST with the loss of the spacecraft's signal having occurred 83 minutes earlier at Saturn, some 1.4 billion km from Earth. With the rocket propellant for manoeuvering the spacecraft fully expended as planned touring Saturn and i...
Fri, 15 Sep 2017 14:17:00 +0100
Four of the latest set of Galileo navigation satellites will be launched on Ariane 6 rockets - ESA's first contract to use Europe's new vehicle. The launches are scheduled between the end of 2020 and mid-2021, using two Ariane 62 rockets - the configuration of Europe's next-generation launch vehicle that is best suited to haul the two 750 kg navigation satellites into their orbits at 23 222 km altitude. Under development, Ariane 6 is Europe's newest launcher, designed to extend guaranteed ...
Fri, 15 Sep 2017 14:00:00 +0100
The Space Alliance, formed by Thales Alenia Space (Thales 67%, Leonardo 33%) and Telespazio (Leonardo 67%, Thales 33%), today signed a partnership with the US-based company Spaceflight Industries which includes the following elements: A minority investment in Spaceflight Industries, which through its BlackSky business, has developed a geospatial platform and plans to build and operate a constellation of 60 small high-resolution observation satellites featuring very short revisit times; ...
Tue, 12 Sep 2017 16:33:00 +0100
Lockheed Martin (LM) unveiled a new small unmanned aircraft system (UAS) that has been designed and developed in the UK at the Defence and Security Equipment International (DSEI). OUTRIDER is a lightweight, canister launched UAS that has been designed by engineers at Lockheed Martin UK's Havant facility in partnership with Wirth Research, an engineering company with specialisation in aerodynamics and composite materials. At only four inches wide and weighing only 1.7 kilograms, it is designed...
Mon, 11 Sep 2017 13:00:00 +0100
Orbital ATK, a global leader in aerospace and defense technologies, today announced significant progress on the industry's first commercial in-space satellite servicing system. The Mission Extension Vehicle-1 (MEV-1) spacecraft successfully completed its critical design review earlier this year and is now in production with about 75% of the platform and payload components already delivered to the company's Satellite Manufacturing Facility in Virginia. The spacecraft will begin system-level testi...
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    Personal information
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    Candidate Profile
     Date Submitted:20-03-2014
     Last Modified:26-03-2014 (02:49)
    Job information
     Current job:Senior Research Engineer
     Employment Term:Permanent
     Job location:Anywhere
     Date available:within 2 months
     Industry:,
     KeywordsAstrodynamics, Optimal Control, Trajectory Optimization, Nonlinear Dynamical Systems, Vibrations
    CV

    EDUCATION
    University of Southern California – Los Angeles, CA 90089-1453, USA
    ▪ Ph.D. in Aerospace Engineering (December 2012)
    ◦ Supervisor: Prof. Firdaus E. Udwadia
    ◦ Dissertation: New Approaches to Satellite Formation-Keeping and the Inverse Problem of the Calculus of Variations
    Yonsei University – Seoul 120-749, Republic of Korea
    ▪ M.S. in Astronomy  and Space Science (February 2008)
    ◦ Supervisor: Prof. Sang -Young Park
    ◦ Thesis: Analytic Solutions to Fuel-Optimal Reconfiguration of Satellite Formation Flying
    ▪ B.S. in Astronomy and Space Science (major) and Physics (minor) (February  2006)

    WORK EXPERIENCE
    Samsung Techwin Co., Ltd., Power Systems Division,  Control Group           (March 2013 – )
    –  Gyeonggi-do 463-400, Republic of Korea
    ▪  Works  as  a  senior  research  engineer.  Develops  control  algorithms  for  propulsion  control  and monitoring systems of MUAVs (Medium-altitude Unmanned Aerial Vehicles) together with Korean  Agency  for  Defense  Development.  Main  work  includes  1)  analysis  of  the  control  requirements,  2)  design  of  controllers  using  Matlab/Simulink,  and  3)  verification  of  the designed controllers via software testing and HILS (Hardware-In-the-Loop Simulations).

    JOURNAL PUBLICATIONS
    ▪  Udwadia, F. E., Wanichanon, T., and  Cho, H., “Methodology for Satellite Formation-Keeping in the  Presence  of  System  Uncertainties,”  Journal  of  Guidance,  Control,  and  Dynamics.
    (Accepted for publication)
    ▪  Udwadia,  F.  E.  and  Cho,  H.,  “First  Integrals  and  Solutions  of  Duffing-Van  der  Pol  Type Equations,” Journal of Applied Mechanics, Vol. 81, No. 3, 2014, pp. 034501-1 – 034501-4.
    DOI: 10.1115/1.4024673
    ▪  Cho,  H.  and  Udwadia,  F.  E. ,  “Explicit  Control  Force  and  Torque  Determination  for  Satellite Formation-Keeping  with  Attitude  Requirements,”  Journal  of  Guidance,  Control,  and Dynamics, Vol. 36, No.2, 2013, pp. 589-605.
    DOI: 10.2514/1.55873
    ▪  Udwadia,  F.  E.  and  Cho,  H.,  “Lagrangians  for  Damped  Linear  Multi-Degree-of-Freedom Systems,” Journal of Applied Mechanics, Vol. 80, No.4, 2013, pp. 041023-1 – 041023-10.
    DOI: 10.1115/1.4023019
    ▪  Cho, H., Park, S. –Y., Yoo, S. –M., and Choi, K. –H., “Analytical Solution to Optimal Relocation of  Satellite  Formation  Flying  in  Arbitrary  Elliptic  Orbits,”  Aerospace  Science  and Technology, Vol. 25, No.1, 2013, pp. 161-176.
    DOI: 10.1016/j.ast.2012.01.005
    ▪  Cho,  H.,  Park,  S.  –Y.,  Park,  H.  –E,  and  Choi,  K.  –H.,  “Analytic  Solution  to  Optimal Reconfigurations  of  Satellite  Formation  Flying  in  Circular  Orbit  under  J2  Perturbation,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 48, No. 3, 2012, pp.  2180-2197.
    DOI: 10.1109/TAES.2012.6237587
    ▪  Udwadia,  F. E.,  Leitmann,  G.,  and  Cho,  H.,  “Some  Further  Remarks  on  Hamilton’s  Principle,” Journal of Applied Mechanics, Vol. 78, No. 1, 2011, pp. 011014-1 – 011014-4.
    DOI: 10.1115/1.4002435
    ▪  Cho,  H.  and  Udwadia,  F.  E.,  “Explicit  Solution  to  the  Full  Nonlinear  Problem  for  Satellite Formation-Keeping,” Acta Astronautica, Vol. 67, No. 3-4, 2010, pp. 369-387.
    DOI: 10.1016/j.actaastro.2010.02.010
    ▪  Cho, H.  and Yu, A., “New Approach to Satellite Formation-Keeping: Exact Solution to the Full Nonlinear Problem,” Journal of Aerospace Engineering, Vol. 22, No. 4, 2009, pp. 445-455.
    DOI: 10.1061/(ASCE)AS.1943-5525.0000013
    ▪  Cho, H.  –C.  and  Park, S.  –Y.,  “Analytic Solution for Fuel-Optimal Reconfiguration in Relative Motion,”  Journal of Optimization Theory and Applications, Vol. 141, No. 3, 2009, pp. 495-512.
    DOI: 10.1007/s10957-008-9482-3
    ▪  Cho, H., Park, S.  –Y., and Choi, K. –H., “Application of Analytic Solution in Relative Motion to Spacecraft Formation Flying in Elliptic Orbit,”  Journal of Astronomy and Space Sciences, Vol. 25, No. 3, 2008, pp. 255-266.
    DOI: 10.5140/JASS.2008.25.3.255

    CONFERENCE PRESENTATIONS
    ▪  Cho,  H.,  and  Udwadia,  F.  E.,  (ASME  2012  International  Mechanical  Engineering  Congress  & Exposition,  IMECE  2012,  Houston,  Texas,  November  2012)  Inverse  Problem  for Lagrangian Dynamics for Multi-Degree-of-Freedom Systems with Linear Damping.
    ▪  Udwadia,  F.  E.,  and  Cho,  H.  (The  13 th ASCE  Earth  and  Space  2012  Conference,  Pasadena, California, April 2012) New Solutions to the Exact Formation-keeping Control of Satellites with Attitude Constraints.
    ▪  Cho,  H.,  and  Udwadia,  F.  E.,  (AIAA  SPACE  2011  Conference  &  Exposition,  Long  Beach, California,  September  2011)  New  Approach  to  Formation-Keeping  Problem  of  Multiple Satellites with Attitude Requirements.
    ▪  Wanichanon,  T.,  Cho,  H.,  and  Udwadia,  F.  E., (AIAA  SPACE  2011  Conference  &  Exposition, Long  Beach,  California,  September  2011)  Satellite  Formation-Keeping  Using  the Fundamental Equation in the Presence of Uncertainties in the System.
    ▪  Cho,  H.,  and  Udwadia,  F.  E.,  (Engineering  Mechanics  Institute  (EMI)  2010  Conference,  Los Angeles,  California,  August  2010)  Explicit  Solution  to  Satellite  Formation-Keeping Problem with Attitude Constraints.
    ▪  Cho,  H.,  and  Park,  S.  –Y.,  (The  16th Mediterranean  Conference  on  Control  and  Automation, Corsica,  France,  June  2008)  Simple  Analytic  Solution  to  the  Optimal  Reconfigurations  in General Relative Motions.
    ▪  Cho, H.,  Park,  S.  –Y.,  and  Choi,  K.  –H.,  (The  Annual  Spring  Conference  of  the  Korean  Space Science  Society,  Cheongju,  Rep.  of  Korea,  April  2008)  Simple  Analytic  Solutions  for Optimal Reconfigurations in General Relative Motion.
    ▪  Cho,  H.,  Park,  S.  –Y.,  and  Choi,  K.  –H.,  (The  Annual  Fall  Conference  of  the  Korean  Space Science  Society,  Daegu,  Rep.  of  Korea,  October  2007)  An  Analytic  Solution  to  FuelOptimal  Reconfiguration  of  Satellite  Formation  Flying  in  the  Presence  of  J2  Geopotential Disturbance.
    ▪  Cho, H. –C., Yoo, S. –M., Park, S. –Y., and Choi, K. –H., (The Annual Spring Conference of the Korean  Space  Science  Society,  Busan,  Rep.  of  Korea,  April  2007)  Applications  of Analytical Solutions into Reorienting, Resizing, and Retargeting of Formation Flying.
    ▪  Yoo, S. –M., Cho, H. –C., Park, S. –Y., and Choi, K. –H., (The Annual Spring Conference of the Korean Space Science Society, Busan, Rep. of Korea, April 2007) Numerical Validation of Analytical  Solution to  Optimal  Reconfiguration  of  Satellite  Formation  Flying  in  Arbitrary Elliptical Orbits.
    ▪  Cho,  H.  –C.,  Park,  S.  –Y.,  Yoo,  S.  –M.,  and  Choi,  K.  –H.,  (The  17th AAS/AIAA  Space  Flight Mechanics Meeting, Sedona, AZ, January 2007) Analytical Solution to Optimal Relocation of Satellite Formation Flying in  Arbitrary Elliptical Orbits.

    INVITED TALK
    ▪  Cho,  H.,  and  Udwadia,  F.  E.,  (The  12th ASCE  Earth  and  Space  2010  Conference,  Honolulu, Hawaii,  March  2010)  Explicit  Analytic  Solution  to  the  Satellite  Formation-Keeping Problem.

    COMPUTER SKILLS
    Matlab/Simulink, C/C++, Fortran, LabVIEW, Maple, Mathematica, SolidWorks, Solid Edge, PLC

    FELLOWSHIPS
    ▪ Brain Korea 21 Fellowship, Korean Science and Engineering Foundation, 2006  – 2008
    ▪ National Research Lab Fellowship, Korean Science and Engineering Foundation, 2006 – 2008
    ▪ Academic Excellence Scholarship, Yonsei University, 2005
    ▪ Academic Excellence Scholarship, Yonsei University, 2004

    EXTERNAL REVIEWER  FOR:
    ▪ Journal of Aerospace Engineering (8 papers)
    ▪ Applied Mathematics and Computation (5 papers)
    ▪ Journal of Optimization Theory and Applications (2 papers)
    ▪ ASME Applied Mechanics Reviews (1 paper)
    ▪ Journal of Sound and Vibration (1 paper)

    RESEARCH INTERESTS
    Astrodynamics,  Optimal  Control,  Trajectory  Optimization,  Satellite  Formation  Flying,  Nonlinear Dynamical Systems, Multi-body Dynamics and Control, Linear, Nonlinear, and Random Vibrations, Numerical Integration of Ordinary/Partial Differential Equations

    TEACHING EXPERIENCES
    ▪  Teaching Assistant  for the junior-level  Computer-Aided  Analyses  for Aero-Mechanical Design.
    (Instructor:  Dr.  Yann  Staelens):  Duties  included  grading ,  hands-on  training  of  SolidWorks  and Solid Edge, and holding office hours                                
    (Spring 2011, Spring 2012)
    ▪ Teaching Assistant  for the junior-level Dynamics. (Instructor: Prof. Roger G. Ghanem, Dr. David C. Wilcox): Duties included lectures and holding office hours
    (Spring 2010, Fall 2010, Fall 2011)
    ▪  Teaching  Assistant  for  the  graduate-level  Linear  Control  Systems  II.  (Instructor:  Prof.  Henryk Flashner): Duties included grading, lectures, and holding office hours                      
    (Fall 2009, Fall 2010)
    ▪  Teaching  Assistant  for  the  graduate-level  Engineering  Analytical  Methods.  (Instructor:  Prof. Satwindar S. Sadhal): Duties included grading and holding office hours            
    (Spring 2010)
    ▪  Teaching  Assistant  for  the  graduate-level  Engineering  Analysis.  (Instructor:  Prof.  Charlie Campbell): Duties included grading and holding office hours                       
    (Fall 2009)
    ▪ Teaching Assistant  for the senior-level  Control Systems Laboratory. (Instructor: Dr. Jerry Chen): Duties included grading and holding office hours                                
    (Spring 2009)
    ▪  Teaching  Assistant  for  the  senior-level  Space  Flight  Dynamics.  (Instructor:  Prof.  Sang-Young Park) Duties included grading, lectures, and hands-on training of STK (Satellite Tool Kit®)
    (Spring 2006)

    RESEARCH EXPERIENCES
    University of Southern California, Los Angeles, CA 90089-1453
    ▪  New  methodology  for  satellite  formation-keeping  with  attitude  requirements  in  the  presence  of system uncertainties                                          (August 2008 – May 2012)
    The  purpose  of  this  research  is  to  analytically  develop  exact  formation-keeping  guidance schemes  that  determine  control  inputs  and  state  trajectories  in  the  presence  of  both  orbital  and attitude requirements. The obtained solution is new in that 1) it is assumed that the leader satellite’s reference  orbit  can  be  any  arbitrary  one  and  the  follower  satellites  are  required  to  point  to  an arbitrarily  chosen  specific  spot  in  space  that  may  be  time-varying,  2)  a  method  to  correct  initialcondition  misalignment  problems  due  to  sensor  measurement  noise  is  developed  so  that  both  the orbital  and  attitude  control  requirements  are  asymptotically  satisfied  at  any  desired  rate  of convergence,  and  3)  the  effect  of  model  uncertainties  is  also  considered  by  augmenting  an additional additive controller  based on a generalization of the notion  of sliding surface control.  It should  be  stressed  that  nonlinear  relative  dynamics  is  considered  in  its  entirety  without  any linearizations/approximations  and  the  solution  is  obtained  in  an  explicit  form,  which  shows  the possibility of precisely controlling nonlinear multi-body systems.
    ▪ New approaches to the inverse problem of the calculus of variations                                         (August 2010 – December 2012)
    To develop the equations of motion for a mechanical system, it is usual to define a Lagrangian function first and then obtain the equations of motion by applying the Lagrange equation when the forces are derivable from  a potential function. However, it is interesting  and useful  to  investigate the types of forces that can be engendered when substituting the forces into the Lagrange equation guarantees the proper equations of motion even when the forces do not arise from a potential, that is,  when  they  are  nonconservative.  In  this  inverse  problem   of  the  calculus  of  variations,  by extending  the  existing  Bolza’s  approach,  a  more  general  and  systematic  way  of  handling  the problem  is  found  for  single-degree-of-freedom  (SDOF)  systems.  Then,  new  results  for  linearly damped  multi-degree-of-freedom  (MDOF)  systems  are  obtained  using  extensions  of  results  for SDOF  systems.  Conservation  laws  for  these  damped  MDOF  systems  are  also  found  using  the Lagrangians  obtained,  and  the  first  integrals  and  the  analytical  solutions  of  some  nonlinear oscillators (ex. Duffing-van der Pol oscillator) are newly proposed. These new findings can be used for development of  approximate solutions of nonlinear differential equations or various numerical techniques.
    Yonsei University, Seoul 120-749, Republic of Korea
    ▪ Analytical analysis of fuel-optimal reconfiguration of formation flying   (Sponsor: Korean Science and Engineering Foundation)                        (March 2006 – July 2008)
    The problem of relative motion, formation flying, and rendezvous has been studied.  The  most significant advantage of the research has been the development of completely analytical  results, by use  of  the  Fourier  series  and  the  Pontryagin’s  minimum  principle.  The  major  contributions  have been  the  solutions  to  the  fuel-optimal  rendezvous  problem  for  satellite.  The  results  are  also  valid when the orbits of the satellites are non-circular with arbitrary eccentricity. The  J2-perturbed case has been  also considered.  These findings allow  us to  predict the form  of the optimal solutions in advance  without having  to  solve  the  problem  and  significantly  reduce  the  amount  of  calculations required.


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