\r\n\tThe role of geodetic surveys in the realization of useful geophysical results is an aspect that aim to be emphasized in this book. The status of modern space geodetic techniques and some recent observations and the interpretation of these observations in terms of the Earth system are welcomed. Interesting topics from space geodesy that deals with the measurement and precise monitoring of small changes of the parameters they observe with high spatial-temporal resolution are important in this book. Even traditional geodetic techniques such as zenith or vertical angle measurements, traversing, levelling, arch measurements and triangulations are welcome.
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:null,priceUsd:null,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"e0e89f185fea8189b148b668cf7afa2e",bookSignature:"Prof. Anthony Afam Okiwelu",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8376.jpg",keywords:"Plate Tectonics, Basement Reactivation, Equatorial Atlantic Tectonics, Subduction, Beniof Zone, Seismic Moho, Isostatic Model, Plate Motion, Magnetometers, Relative Gravity, Geomagnetism, Equipotential Surfaces, Geodetic Reference Systems, Rock Magnetism, Geomagnetic reversal, Geoid, Dipole Field, Secular Variation, Lacoste-Romberg, Seismic Waves, Wavelets, Seismic Data Processing",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 10th 2018",dateEndSecondStepPublish:"July 31st 2018",dateEndThirdStepPublish:"September 29th 2018",dateEndFourthStepPublish:"December 18th 2018",dateEndFifthStepPublish:"February 16th 2019",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,editors:[{id:"139812",title:"Prof.",name:"Anthony",middleName:"Afam",surname:"Okiwelu",slug:"anthony-okiwelu",fullName:"Anthony Okiwelu",profilePictureURL:"https://mts.intechopen.com/storage/users/139812/images/system/139812.jpg",biography:"Dr. Anthony Okiwelu holds a Ph.D degree in Geophysics from the University of Calabar, Nigeria and M.Sc. Degree in Geophysics and B.Sc. Degree in Geology from the Universities of Ibadan and Calabar respectively. He is a Professor of Geophysics in Geophysics Unit, Physics Department, University of Calabar, Nigeria where he teaches magnetic prospecting, gravimetry, geomathematics and Earth Physics. His main research interest is in magnetics, gravimetry, tectonics and geopotential field models. He also has some publications in the areas of seismic and electrical methods. He is an active member of International body, SEG (Society of Exploration Geophysicists), NIP (Nigerian Institute of Physics), NMGS (Nigerian Mining and Geoscience Society) and Nigerian Environmental Society (NES).",institutionString:"University of Calabar",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"University of Calabar",institutionURL:null,country:{name:"Nigeria"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"10",title:"Earth and Planetary Sciences",slug:"earth-and-planetary-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"220813",firstName:"Danijela",lastName:"Sakic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/220813/images/6092_n.jpg",email:"danijela.s@intechopen.com",biography:"As Author Services Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As a PPM I am also involved in the acquisition of editors. 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Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3794",title:"Swarm Intelligence",subtitle:"Focus on Ant and Particle Swarm Optimization",isOpenForSubmission:!1,hash:"5332a71035a274ecbf1c308df633a8ed",slug:"swarm_intelligence_focus_on_ant_and_particle_swarm_optimization",bookSignature:"Felix T.S. Chan and Manoj Kumar Tiwari",coverURL:"https://cdn.intechopen.com/books/images_new/3794.jpg",editedByType:"Edited by",editors:[{id:"252210",title:"Dr.",name:"Felix",surname:"Chan",slug:"felix-chan",fullName:"Felix Chan"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"62907",title:"HAZ Phase Transformation and Thermal Damage for Laser Remanufacturing a High-Strength Stainless Steel",doi:"10.5772/intechopen.79910",slug:"haz-phase-transformation-and-thermal-damage-for-laser-remanufacturing-a-high-strength-stainless-stee",body:'Conventional high strength steel (CHSS) has been defined in advanced high-strength steels (AHSS) Application Guideline 3.0 by World Steel Association (IISI) as steel material with high strength (yield strength ≥ 210 MPa and tensile strength ≥ 370 MPa). In recent years, light alloy like aluminum and magnesium alloys were widely used in automobile lightening and it also promotes the emergence and development of the AHSS. As a typical example of high strength steels, martensitic stainless steel with good comprehensive mechanical properties, casting performance and excellent corrosion resistance was used in various large-scale and complex operating conditions equipment, such as large-scale compressor rotors, air-compressor blade and structural components of the nuclear reactor. These high performance parts with complicated structure and high manufacturing cost had great potential to remanufacture. The suitable material testing technique and perfect remanufacturing process were desperately needed for the remanufacturing of damaged equipment and it is environmentally friendly, more economical and efficient.
The laser cladding processing is an advanced high-energy beam remanufacturing techniques. As the light source, laser beam had high energy-density, high-precision and high flexible processing to achieve net shaping forming for manufacturing those structural parts or components with complex inner structure [1]. Taking full advantages of laser beam and based on the laser cladding, laser remanufacturing technology was used for remanufacturing the disabled parts of the damaged equipment and the original shape or external dimensions recovered accurately after proper subsequent machining. Meanwhile, the alloy powder material and the matrix of the work-piece produce high quality metallurgically bonding under heating of powerful laser during the remanufacturing processing and the overall mechanical properties of the remanufacturing parts possibly can meet or exceed that of new one by the regulation of the microstructure and the mechanical properties of heat affected zone and cladding. In general, as an advanced refabrication technology, laser remanufacturing was widely applied in repairing and regeneration of many high value-added equipment [2, 3, 4, 5, 6].
Although laser remanufacturing technology had been widely applied in many industries like aerospace, oil drilling, electricity, automobile and machine, there exist some unsolved problems causing seriously restricted the development and use of this technology. Quite different with the laser additive manufacturing process, the deposition of the alloy powder material was on the surface of the work-piece during the laser remanufacturing and therefore the interface between the cladding and the matrix cannot be neglected. Two specific aspects of the work were completed during the laser remanufacturing. Firstly, the high-density and high mechanical properties cladding part was obtained by the design and control of the laser cladding alloy powder material, the process optimization and the proper postprocessing like stress-relief heat treatment, which is similar to the laser additive manufacturing process. However, the regions adjacent to the surface of the work-piece (the depth of the region was about 100–1000 micrometers according to many studies) experienced different short multi-cycle with superhigh heating and cooling rate and moreover a series of nonequilibrium solid state phase transformation occurred during the interaction between laser beam and alloy powder material. Consequently, the microstructure of the heat affected zone was different with that of the work-piece before remanufacturing and the mechanical performance may deteriorate, namely thermal damage during the laser remanufacturing. Recently, the 3D print technology had drawn a lot of attention of large enterprises and academic institutions. Studies on the laser additive manufacturing processing optimizing were done by the combination of numerical simulation and experimental methods. Some mechanism of the grain morphology control was investigated by basic study on solidification nucleation and growth mechanisms of the local melting pool. Most of the studies focused on the pretreatment and heat-treatment during the laser cladding process and it turned out that the microstructure and mechanical properties improved significantly. Consequently, heat affected zone (HAZ) was the weak region for the laser remanufacturing parts and the relevant issues had deep effects on the development of this technology [7, 8, 9, 10, 11].
Until now, studies on the mechanism of the microstructure evolution and mechanical properties distribution were sorely lacking. The key parts of the issues were the nonequilibrium phase transformation and the evaluations of the mechanical properties of HAZ. First of all, the regions in different depth in HAZ experienced different thermal cycles and eventually the microstructure and the microhardness of HAZ show a stepwise characteristic. Besides, the microstructure of HAZ may have further changes as the following and continuous heat cycles. For some material systems like martensite stainless steel, a set of characteristic parameters (maximum temperature, heating rate and cooling rate) of the heat cycles in HAZ can be acquired by methods like computer numerical simulation and temperature-measuring technology. To some extent, the formation mechanism of the final microstructure in HAZ can be explained and predicted by the classic phase transformation theory. However, some none-equilibrium microstructure frequently appear in HAZ, which make controlling the microstructure and the mechanical properties challenging. In addition, the mechanical properties evaluation of HAZ was challenging but significant. According to previous studies, by prefabricated delimitation the butt joint specimens were obtained as the mechanical properties tests samples or obtained directly from the laser cladding part along the cladding direction.
Martensite phase transformation was a displacive phase transformation. During the martensite phase transformation, lattice type were changed by shear of material atoms, as well as the mechanical, thermodynamic properties and specific volume of the material. The thermodynamic and latent heat of phase transformation changes will have massive effects on the evolution of the temperature field.
During the laser cladding, material will experience a complete cycle from room temperature up to melting points. The physical properties of material changed as the occurrence of the solid state phase transformation and then eventually affected the evolution of the temperature field and stress field. The studies on the temperature and stress field evolution during the laser remanufacturing were based on the reveal of physical properties evolution. Furthermore, the heat cycles of different regions in HAZ can be obtained from the temperature field evolution and then the mechanism of the microstructure and mechanical properties distribution in HAZ may be revealed. Last but not the least, we hope the final microstructure and mechanical properties of HAZ can be accurately predicted and well controlled. However, the solid state phase transformation during the laser cladding was a non-equilibrium phase transformation and the kinetics of phase transformation were obviously affected by the superhigh heating rate and cooling rate [12, 13].
In the case that no phase transformation occurs for some generic solid material, the temperature rises will increase as the internal energy rises, so is the volume. On the contrary, the temperature and the volume decrease as the internal energy fall.
The specimens of the thermal expansion coefficient measurement were shown as Figure 1. The dilatometry experiment was processed in DIL801 dilatometer (Figure 2). In the temperature range (−150~1500°C), the maximum heating rate was 50°C/min and the maximum cooling rate was 25°C/min at low temperatures. In addition, the maximum heating rate rise to 100°C/min and the maximum cooling rate reached to 180°C/min as well. The errors of linear expansion coefficient was set as 0.03*10–6/°C, the gradient of temperature was 10°C/min. Using nitrogen as carrier gas, the flow rate was set as 20 ml/°C. The ambient temperature was 25°C and humidity was 35%.
Specimens for the free dilatometry test.
DIL801 single-sample dilatometer.
The radial strain of the specimens for the thermal expansion coefficient experiments were used as measurement results. The strain was decomposed into two parts in the Ti (Range from room temperature to Ms): One part were the strain variations of martensite and austenite as temperature changes. The rest were caused by the volume effect of martensite phase transformation. It could be described as follows:
dealed as:
where
For FV520B high-strength stainless steel, the free expansion curve was shown as Figure 3. The critical temperature of transformation from pearlite to austenite (Ac1) was 600°C and the complete transformation temperature of austenite (Ac3) was 900°C. The starting temperature of the martensite phase transformation (Ms) was 130°C.
The free dilatometry test of FV520B steel.
Shown as Figure 4, in the temperature between 130 and 600°C, the strain of martensite was 0.0057 and that of austenite was 0.088. The expansion coefficient of austenite was 18.72 and that of martensite was 12.13. In high temperature, the expansion coefficient of austenite was 21.5 and that of martensite reached up to nearly 19 at temperature over 600°C.
Temperature range of two phase regions during the free dilatometry test.
Figure 5 shows the temperature range of martensite transformation during the free dilatometry test. In the martensite phase transformation period, the strain caused by volume effect during phase transformation was 0.0067249 and the real strain should be 0.00328 considering the thermal contraction. Ideally, the material volume increased by 0.97% compared with that at Ms. In addition, the material volume at 320°C was equivalent to that at room temperature.
Temperature range of martensite transformation during the free dilatometry test.
For materials in the temperature between Ms and Ac line, it could be martensite, austenite or dual-phase structure. The lower was the temperature, the higher was the density contrast of the two phase states. Consequently, the volume effect during phase transformation was more significant and the volume changes increased as the temperature drop. For the solid state phase transformation of the common steel, the transformation temperature of pearlite was the highest, next was bainite and that of martensite was the lowest [9]. In general, the volume effect of these phase transformations were corresponding to the transformation temperatures.
It had been noted above that superhigh heating rate and cooling rate had magnificent impact on the physical parameters of material during the laser remanufacturing. Studies indicates that the heating rate could reached over 500°C by methods of computer numerical simulation and temperature measurements. The simulation experiments were designed and conducted to study the effect of heating rate on the solid state phase transformation. Limited to the instrument condition, the heating rates were set as 0.25, 1, 5, 10, 30, 80 and 100°C/s. The results revealed that the rise of heating rates of each thermal cycles significantly increased Ac1 and Ac3 of FV520B (as shown in Figure 6). The expansion coefficient of martensite at 640°C increased apparently as the heating rate increased from 5 to 30°C/s. In general, heating rates had a great effect on the kinetics parameters of the austenite phase transformation.
The free dilatometry test of FV520B steel in different level of heating rate.
In recent years, many studies were concerned about the effects of solid state phase transformation on the temperature and stress field and the mechanisms were revealed by experimental methods. However, the theoretical analysis were not enough in conditions of the complex projects like laser remanufacturing, which involved process control with many parameters. Therefore, computer numerical simulation was considered as a practical method to obtain precise, comprehensive and quantitative results, which had engineering meanings.
All the involved parameters were listed below: laser power, scanning rate, laser spot size and lapping rate were 1.5KW, 10 mm/s, 3 mm, 50% respectively. Using the technique of deactivate and reactivate element, the cladding material were activated by passes and layers. By building the coupling thermo-mechanical model, the evolution of the temperature field during the cladding was revealed. This work focused on the short and varying thermal cycles in positions with different depth in HAZ during the single-pass and multi-layer claddings. Figure 7 was the geometries of the laser cladding samples during the single-pass and multi-layer processes [12, 14, 15].
Geometry of laser clad sample: (a) single-layer; (b) multilayer.
Figure 8 was the temperature evolution at midpoint of single-pass cladding sample. The temperature of the melting pool reached 1833°C after 3.5 s. Meanwhile, the temperature measurement experiment at the midpoint was conducted by using RaytekMM infrared thermometer. The results showed that the laser beam scan midpoint of the single track after 3.7 s and the temperature of the melting pool was 2175°C, which was consistent with the simulation results.
Temperature variation at midpoint of single-layer laser clad sample.
Figure 9 was the thermal cycles in HAZ during the single-pass cladding. The maximum temperature in regions close to surface of the work-piece reached over 1000°C, which is higher than the complete austenitic temperature Ac3. In addition, the measurement results of the heating rate was around 500°C/s and that of the cooling rate was over 100°C. For regions in different depth of HAZ, the maximum temperatures of the thermal cycles were the main difference, which could directly affected the final microstructure in HAZ. Meanwhile, the simulation results could be provided as evidence for the studies on the mechanism of the microstructure evolution.
The thermal cycles in HAZ.
Figure 10 was the thermal cycles in HAZ during the multi-layer cladding. In multi-layer laser remanufacturing process, the coating is deposited layer by layer and HAZ was created repeatedly [16]. In addition, the repeatedly thermal effect on microstructure in HAZ was hard to control and predict. Some studies pointed that the microstructure in HAZ experienced approximate tempering during the repeatedly thermal cycles and toughening effect on microstructure was observed. The numerical simulation results showed that microstructure of HAZ experienced thermal cycles with over 600°C maximum temperature during the subsequent deposition process after the first layer cladding. It was considered that the multi-cycle heating and cooling process could affect the final microstructure and mechanical properties in HAZ.
The temperature at the midpoint of the five pass of the first layer, with and without preheating.
Figure 11 was the schematic of phase transformation for the martensite stainless steel during the laser remanufacturing process. Austenite was assumed to be the initial phase during solidification process. In addition, the martensite transformation occurred during the cooling process due to the high cooling rates. As temperature decreased, the martensite transformation started at Ms and finished at Mf. When a new layer of FV520B was deposited, the previously deposited material experienced a new thermal cycle. Once the rising temperature was higher than Ac1, the transformation of martensite to austenite occurred. The percentage of austenite phase increased linearly as temperature rise.
Evolution of volume fraction of austenite during phase transformation during Laser cladding directed shaping.
As the physical properties of material depended on temperature and phases, the accuracy of the physical properties based on the appropriate description on the evolution of phase. In laser cladding process, different regions had different thermal cycles and external load, which could contribute to the change of the kinetics and effect of solid state phase transformation. Therefore it was necessary to carry out systematical research in the influence of cooling condition on the solid state phase transformation during the cooling process [17].
A combined experiment by methods of dilatometry and metallography was conducted and the CCT curve was obtained by the L78 RITA transformation measuring apparatus. According to the thermal cycles of the laser cladding processes, two austenitic procedures were designed to study the effect of the austenitization condition on the microstructure evolution. The process of case 1 was designed as below: the sample heated up to 1000°C at heating rate of 300°C/s and kept the temperature for 5 s. In the cooling period, the sample was cooled to 810°C at the cooling rate of 100°C/s, kept the temperature for 5 mins and then was cooled at cooling rate of 0.02, 0.08, 0.2, 0.5, 1 and 50°C/s respectively. For case 2, the sample heated up to 1300°C at the same heating rate, was kept the temperature for 5 s and cooled to 810°C at the same cooling rate. Then, the sample was cooled at cooling rate of 0.02, 0.08, 0.2, 0.5, 1°C/s respectively. The phase transformation temperatures were obtained using tangent method.
Figure 12 was the continuous cooling transformation of FV520B stainless steel. Figures 13 and 14 were metallograph of microstructures in different austenitic procedures. The results showed that the microstructures were all martensite in the designed cooling parameters due to the hardenability of FV520B stainless steel. For case 1, the microstructure was lath martensite with finely distributed precipitates. In addition, the proportion of martensite with sparse laths increased with the increasing cooling rate in the process. For case 2, the microstructure was lath martensite as well. However, the microstructure of martensite in case 2 was coarser than that of case 1 and dispersedly distributed precipitates were barely observed. As shown in Figures 15 and 16, the phase transformation temperature of case 2 was 20°C lower than that of case 1 and both increased as the cooling rate became higher; the microhardness of case 2 was 60 HV higher than that of case 1 and both decreased as the increasing cooling rate. In general, the maximum temperature of thermal cycle had a dominating effect on the microstructure, microhardness and phase transformation temperature rather than cooling rate.
Welding continuous cooling transformation of FV520B stainless steel: (a) Case I; (b) Case II.
OM images of specimen Case I under different cooling rates. (a) 0.02°C/s; (b) 0.2°C/s; (c) 0.5°C/s; (d) 1°C/s.
OM images of specimen Case II under different cooling rates. (a) 0.02°C/s; (b) 0.2°C/s; (c) 0.5°C/s; (d) 1°C/s.
It was considered that the maximum temperature was 1000°C with short heat preservation in case 1 and the sample experienced complete austenitic process with precipitates partly dissolving in the matrix. In addition, the precipitates further grew coarse and solubility in the matrix decreased, which eventually contributed to the decreasing microhardness. However, the maximum temperature of case 2 reached 1250°C and no ferrite existed at high temperature. In general, the sample of case 2 experienced complete austenitic process with precipitates completely dissolving in the matrix. The increasing solid solubility of alloy elements contributed to the higher microhardness than that of case 1. Meanwhile, phase transformation temperature of case 1 and case 2 were both low and it could caused by the strengthen effect of austenite and growth of precipitates during the cooling period at low cooling rate.
The microhardness of laser remanufacturing specimens was higher than that from CCT process due to the superhigh heating rate in the actual laser cladding process. According to the numerical simulation results, the temperature gradient in high temperature period reached to 1000°C/s during the laser cladding process and the maximum heating rate could be set as 500°C/s due to the restrictions of instrument conditionFigures 15 and 16.
Martensite transformation start temperature under Case I and Case II.
Hardness under the Case I and Case II.
As above, austenitic process and cooling rate had a remarkable impact on the phase transformation temperature and it could be interpreted as the effect of thermal cycle on the valid elements of material composition. It was assumed that the maximum temperature and heating rate during the heating period in the thermal cycle had great influence on the martensitic phase transformation temperature. Moreover, the strengthen effect and reducing of Ms took place due to the low cooling rate during the same austenitic process. In general, the thermal cycles had a significant effect on the metallographic transformation [18, 19].
Figure 17 were microstructure distribution of HAZ. It could be seen that the HAZ microstructure showed a stepwise characteristic. The microstructure of regions adjacent to interface was coarse martensite and no ferrite existed in high temperature. In thermal cycles of these regions, it experienced complete austenitic phase transformation and precipitates fully dissolved in the matrix and consequently the final microstructure of the regions were martensite with less lath characteristics. In the middle area of HAZ where the maximum temperature of the thermal cycle was close to Ac3 and the matrix underwent the complete austenitic as well. For the partially austenization zone, the laser energy input was relatively low and the matrix experienced partially austenization due to a thermal cycle with a maximum temperature between Ac1 and Ac3. Therefore, the microstructure was lath martensite with more dispersedly distributed precipitate and particles. The microstructure of regions in the bottom of HAZ was similar to the original microstructure of FV520B and it was considered to undergo the thermal cycle with a maximum temperature near Ac1.
OM images of laser RM FV520B: (a) Macrostructure of laser RM FV520B (b) region close to the interface (c) partially molten zone (d) completely austenization zone (e) partially austenization zone (f) FV520B substrate.
Figure 18 was the microhardness distribution of cladding layer, HAZ and substrate. The microhardness distribution was considered consistent with that of microstructure. In the complete melting zone where the maximum temperature of thermal cycle was higher than AC3, it showed high hardness due to the solution strengthen. In the partially austenization zone, the microhardness was lower than other regions.
Microhardness distribution of HAZ-substrate.
Figure 19 was the tensile test results of FV520B steel and cladding layer of laser remanufacturing FV520B. It could be seen that the yield stress, tensile stress and elongation of FV520B steel were respectively 830 Mpa, 970 Mpa and 23% and that of FV520B laser cladding layer were 920 Mpa, 1280 Mpa and 11% respectively. The results indicated that the strength of alloy increased and ductility decreased after laser remanufacturing process.
Comparison of tensile test curves between FV520B steel and as-deposited component.
Figure 20 was the tensile test curves of laser remanufacturing butt samples and FV520B steel samples. The results showed that two samples had similar stress–strain curve. It could be predicted that the fracture section of butt samples was the bottom of HAZ due to the strengthen and descend of ductility of HAZ after laser remanufacturing process. In this research, FV520B work-piece underwent aging treatment after forging process and showed relative low strength but high ductility. In general, laser remanufacturing FV520B had high strength and ductility, which reached the combination properties of forgings.
Comparison of tensile test curves between FV520B steel and butt sample.
Figure 20 was the fracture surface morphologies of laser remanufacturing FV520B samples. Ductile characteristics were observed without any apparent flaws or inclusion. In addition, High density and tiny dimples were observed at high magnification, where nanoscale spherical particles were founded. In general, the results showed that the material had good ductility with uniform distribution of nanoscale particles and no segregation at grain boundaries Figure 21, crack or inclusion were founded in the fracture surface [20].
Fracture surface morphologies of FV520B steel fabricated by laser cladding. (a) Macro morphology of fracture surface; (b) fracture surface by 5000 times; (c) fracture surface by 10,000 times; (d) fracture surface by 20,000 times.
Laser remanufacturing is an advanced repairing method to remanufacture damaged parts based on laser cladding processing. To reveal the mechanism of solid state phase transformation and the microstructure evolution during the laser cladding processing, it is necessary to study the effect of solid state phase transformation on physical properties and temperature filed by numerical simulation and experimental method. For FV520B stainless steel, the result of free dilatometry test showed that heating rates had a great effect on the kinetics parameters of the austenite phase transformation and the volume effect of phase transformations were corresponding to the phase transformation temperatures. According to the results of thermal simulation experiment and metallographic observation, it was considered that the multi-cycle heating and cooling process affected the final microstructure and mechanical properties in HAZ. The maximum temperature of thermal cycle had a dominating effect on the microstructure, microhardness and phase transformation temperature rather than cooling rate. Thermal cycle influenced significantly the metallographic transformation and consequently decided the final mechanical performance. By using the optimal process, laser remanufacturing FV520B had high strength and ductility, which reached the combination properties of forgings.
The work was supported by the National Key Research and Development of China (Grant No. 2016YFB1100205), and 973 Project of China (Grant No. 2011CB013403).
The necessity to assign resources such as machines to jobs that need to be performed without interruption, where the time required for a machine to perform a certain job is known in advance, is a widely encountered problem. It can occur for example in production planning or when assigning landing and take-off stripes to planes in airports. Sometimes these resources become unavailable for predetermined periods of time, for example because of necessary maintenance. Minimizing the maximum completion time of all tasks is often considered as a goal, for example such that the workers who operate the machines can undertake other activities afterward or go home early. As a consequence, the problem of scheduling on multiple machines with predefined unavailability periods (downtimes) to minimize the maximum completion time, that is, the latest completion time of a job in a schedule, has been considered. A closely related problem, of scheduling with fixed jobs, where on each machine certain jobs have to be performed at predefined times, has also been considered. The difference between these two problems is in the meaning of the objective function: when scheduling with fixed jobs, the maximum completion time of the jobs must be at least the latest completion time of a fixed job, whereas the maximum completion time when scheduling in the presence of unavailability periods can occur before the end of an unavailability period. We consider the static nonresumable variant of the problem of scheduling with unavailability periods, where the downtimes are known for each machine before the schedule needs to be made, and where jobs that start executing before a downtime cannot resume execution after it.
\nIn these problems, the job execution times are usually assumed to be given as an integer number of computing units such as clock cycles or of other suitable units such as time units. Similarly, the starttimes and endtimes of unavailability periods or of fixed jobs are assumed to be given as integer multiples of adequately chosen time units. We note that any problem with rational numbers as job durations and starttimes and endtimes of downtimes or of fixed jobs can be transformed into an equivalent problem where these entities are integers by multiplying them with an adequate factor, thus there is arguably no loss of generality in this assumption when considering the representation of any practical problem.
\nBoth the problem of multiprocessor scheduling on fixed jobs and that of multiprocessor scheduling with unavailability periods are strongly NP-hard as they are more general than the strongly NP-hard multiprocessor scheduling problem (MSP), which has no downtimes or fixed jobs.
\nFor scheduling with downtimes, it is NP-hard to find a schedule that ends within a given constant multiple of an optimal schedule even when scheduling on identical machines with at most one downtime on each machine. We discuss this in more detail in Section 4.2. To obtain approximation results for scheduling with unavailability periods in this context, assumptions about the downtimes were made such as the assumption that only a fraction of the processors can be unavailable at the same time [1, 2], or by comparing the generated schedule to the latest among the end of an optimal schedule or the latest end of a downtime, thus essentially considering scheduling with fixed jobs [3, 4].
\nTo describe the performance of an approximation algorithm, we use the notion of a worst-case approximation bound. In this work, we call worst-case approximation bound of an algorithm A when applied to a scheduling problem SP the largest ratio between the maximum completion time of a schedule produced by A and the maximum completion time of an optimal schedule for a problem instance of SP.
\nFor the problem of multiprocessor scheduling with fixed jobs to minimize the maximum completion time, even in the case where there is at most one fixed job on each machine, it has been shown in [5] that no polynomial algorithm can achieve a worst-case approximation bound that is less than 1.5 unless \n
The case where all downtimes are at the beginning of the processing time on all machines is called scheduling with nonsimultaneous machine available times, as the machines start processing jobs nonsimultaneously. For this problem, polynomial-time algorithms with constant worstcase approximation bounds exist.
\nFor scheduling on identical nonsimultaneous parallel machines, MULTIFIT achieves a tight worst-case approximation bound of 24/19 (~1.2632) [9] and another algorithm achieves a bound of 5/4 [10], while in the case of scheduling on uniform nonsimultaneous parallel machines, a MULTIFIT variant has a worst-case approximation bound of 1.382 [11], which was shown by generalizing the bound obtained for MULTIFIT when scheduling on uniform processors in [12]. Experimental results suggest that for scheduling on nonsimultaneous uniform machines, the MULTIFIT variant from [11] is adequate for use in practice, as we discuss in Section [4].
\nIn Section 2, we describe the ways in which the content of this work can be used. In Section 3, we introduce the algorithms LPT and MULTIFIT. In Section 4, we consider scheduling with unavailability periods, while first focussing on scheduling with nonsimultaneous machine available times in Section 4.1, and on the more general case where downtimes do not have to occur at the beginning of the schedule in Section 4.2. In Section 5, we present results on scheduling with fixed jobs. Section 6 contains the description of main techniques used in the worst-case approximation bound proofs and Section 7 contains concluding remarks and a discussion of some of the challenges in this area.
\nWe next present ways to use the content of this work.
\nThis work aims to provide a deeper understanding of multiple related problems that involve scheduling on parallel machines with fixed jobs or unavailability periods to minimize the maximum completion time. We explain why multiprocessor scheduling with at most one unavailability period on each machine does not have a polynomial-time approximation algorithm with a constant worst-case approximation bound unless P = NP, which is the main reason why results on this topic are hard to obtain.
\nFurthermore, we observe that most results in this area involve variants of LPT and MULTIFIT, and comment on the other results obtained. We also hope that this work will increase awareness of these results and of how they relate to each other.
\nThe heuristics presented and referenced in this work can be used directly in practice or for research purposes to solve the problems they address. The heuristics based on LPT and MULTIFIT are fast and easy to implement and some of them have best possible worst-case approximation bounds in the class of polynomial algorithms unless P = NP for the problems they address. In addition to worst-case approximation results, this work also highlights for some cases experimental insights into how the heuristics would perform in practice based on how they perform for randomly generated instances. As expected, they perform much better for such instances than in the worst case. Also, for some cases, references to more complex methods are provided in case the user prefers to use those.
\nThis work presents the main proof techniques used to obtain worst-case approximation bounds for LPT- and MULTIFIT-like heuristics when the aim is to minimize the maximum completion time. Thus, the interested reader is provided with a concise description of the tools that can be used to develop such proofs, and he or she may not have to read hundreds of pages in order to become aware of all of them or work with an expert in the area when developing such a proof. Even for people with expertise in the area, one or more of the ideas presented may be new and helpful.
\nThe algorithms LPT and MULTIFIT are among the most studied approximation algorithms for multiprocessor scheduling with or without unavailability periods or fixed jobs. In this section, we describe the basic versions of these algorithms for MSP, which can be stated as follows: given a set of m machines P and \n
The algorithm LPT [7] works as follows:
\nAlgorithm 1 The largest processing time algorithm (LPT) | \n
1: Order jobs in nonincreasing order of their processing time. 2: In this order assign each job at the earliest possible time allowed by the schedule that exists when the job is assigned. \n \n \n \n \n \n \n | \n
The algorithm MULTIFIT was first introduced by Coffmann Garey and Johnson in 1978 [8]. It uses binary search for the end of its resulting schedule and receives as input an accuracy ε with which it determines this schedule end. In each iteration it assigns a deadline and attempts to create a schedule that contains all tasks that ends at or before that deadline by using the bin packing algorithm first fit decreasing (FFD). If a feasible schedule is created, it decreases the deadline and otherwise it increases the deadline. This process is repeated until the difference between the current deadline and the previously considered deadline is less than ε. More formally, the algorithm is described as Algorithm 2.
\nThe MULTIFIT algorithm results in a schedule with a maximum completion time that is within \n
An example of a LPT-schedule and a MULTIFIT schedule for the same problem instance are presented in Figures 1 and 2 respectively.
\nAlgorithm 2 The algorithm MULTIFIT | \n
1: Order the jobs in non-increasing order of their duration. 2: Assign upper bound (ub) and lower bound (lb) for the end of schedule; (for example, \n 3: Assign \n 4: FFD: Assign tasks in non-increasing order on the first processor on which they fit while respecting the deadline (the processors are considered in each iteration in the same order). 5: If all tasks are successfully scheduled decrease the upper bound: \n 6: Else increase the lower bound: \n 7: If \n \n \n | \n
A LPT schedule. The jobs are numbered according to the order in which they are considered. At start, when all processors are available at the same time, they are considered in the order p1, p2, p3 in this example.
A MULTIFIT schedule together with a possible deadline. The jobs are numbered according to the order in which they are considered. The processors are considered in the order p1, p2, p3 when generating the schedule.
The MULTIFIT algorithm tends to produce more balanced schedules than LPT, and, as a consequence it tends to perform better when the aim is to minimize the maximum completion time. It also has a higher time complexity, as it tries to make a schedule about \n
If the parameter \n
The binary search for the deadline within the MULTIFIT algorithm happens within \n
As a consequence, the number of times the MULTIFIT loop is called is polynomial in the size of the input, as any reasonable upper bound is at most the sum of the processing times of all jobs, which can be represented within at most the total number of bits used to represent all jobs. In [4], Grigoriu and Friesen also comment that if the upper bound is 2 years, the lower bound is 0 and the deadline is determined with an accuracy of 10−13 s, the MULTIFIT loop is called at most 40 times.
\nThe time complexity of MULTIFIT is thus \n
In this section, we first present results for the case where all unavailability periods are at the beginning of the schedule. Then, we present results for the more general case where the unavailability periods can occur anywhere in the schedule.
\nThis section addresses the case where the processors may have unavailability periods at the start of their processing time. This situation is more general than the multiprocessor scheduling problem (where there are no fixed jobs or downtimes) and less general than the problems of scheduling with fixed jobs or with unavailability periods. As the less general multiprocessor scheduling problem is NP-hard, so are the problems of scheduling on identical machines with nonsimultaneous machine available times (NMSP: nonsimultaneous multiprocessor scheduling problem) and scheduling on uniform processors with nonsimultaneous machine available times (UNMSP) when minimizing the maximum completion time. Due to the NP-hardness of these problems, polynomial-time approximation algorithms like LPT and MULTIFIT and their variants have been studied for their solution. As before, we will continue to denote with the number of processors in the problem instance being considered with m.
\nFor NMSP, worst-case approximation bounds for LPT of \n
When MULTIFIT is used for MSP, a deadline results in periods of equal duration in which jobs can be scheduled on each processor; thus the schedules resulting from using any ordering of processors in step (4) of MULTIFIT have the same maximum completion time. When considering NMSP, thus allowing for nonsimultaneous machines, the order in which processors are considered becomes relevant, as the period in which jobs can be executed on each processor corresponding to a deadline depends on the time the processor becomes available. MULTIFIT variants that address such situations usually order the processors in non-decreasing order of their periods in which jobs can be scheduled. Thus, in this case, the ordering is in non-increasing order of the times at which the processors become available.
\nA bound of 9/7 (about 1.2857) was obtained for MULTIFIT by Chang and Hwang [14]. In [10], Kellerer gives a problem instance of NMSP for which the approximation factor of its MULTIFIT schedule is 24/19 (about 1.2632). More recently, 24/19 was shown to be the exact worst-case approximation bound when using MULTIFIT for NMSP by Hwang and Lim [9]. By comparison, a tight worst-case approximation bound of 13/11 (about 1.18182) was shown by Yue [15] for MUTLIFIT when applied to MSP.
\nFor the uniform multiprocessor scheduling problem with simultaneous machine available times (UMSP), that is, where processors execute jobs at different speeds, the amount of jobs that fit on a processor corresponding to a given MULTIFIT deadline depends on the speed of that processor. Usually, the slowest processor is considered to have a speed of 1, and for each job j, the time it would take to process it on this processor \n
For UMSP, approximation bounds of 1.4 and 1.382 were obtained for MULTIFIT by Friesen and Langston [16] and by Chen [12] respectively. In [17], Burkard and He derive a worst-case approximation bound of \n
In [11], Grigoriu and Friesen show that bounds that apply to the MULTIFIT variants from previous work such as [12, 16, 17] where scheduling on two uniform processors is considered also apply to a slightly different proposed variant of MULTIFIT for UNMSP, LMULTIFIT, which was first proposed in [4] in a more general form. The difference between the MULTIFIT variants from [12, 16, 17] on the one hand and LMULTIFIT on the other hand is that in the latter, the choice of the initial upper and lower bounds is not given explicitly within the algorithm, and thus the worst-case approximation bound proofs are more general, as they work for any initial choices of upper and lower bounds for the duration of the resulting schedule. A first step in the proofs that the bounds hold in the more general case, where there are uniform nonsimultaneous parallel machines, was to show that LMULTIFIT obeys the bounds of the earlier MULTIFIT variants in the simultaneous machines case for the instances considered in those works.
\nUsing LPT for UNMSP has been considered in [18], where worst-case approximation bound of 5/3 was shown in the general case, as well as a better bound for the case where there are only two machines.
\nFor the case where the number of machines is constant, a PTAS exists for UNMSP [11], which was derived from a PTAS for scheduling on a constant number of uniform processors with fixed jobs from [5]. As the objective function for scheduling with fixed jobs that are at the beginning of the schedule and scheduling with unavailability periods that are at the beginning of the schedule differ, the PTAS from [5] can not be used unaltered to address UNMSP. To address UNMSP, the PTAS from [5] is first run for the transformed problem instance where the unavailability periods become fixed jobs, and then for all problem instances resulting from successively removing the machine with the latest end of a fixed job from the transformed instance [11]. This accounts for the cases where a number between 1 and \n
In [19], a lower bound is derived for the end of an optimal schedule of an UNMSP instance, and using that bound approximation factors for LMULTIFIT schedules of randomly generated instances are determined. The reasonably extensive experiments described in [19] suggest that LMULTIFIT is a good option for solving UNMSP in practice, not only because of being fast and easy to implement, but also because it has very good approximation factors (less than 1.03) for the generated instances with an average of at least five jobs for each machine. In order to obtain the approximation factors, a lower bound for the end of the optimal schedule that can be calculated directly from the problem instance was used.
\nIn this section, we consider the multiprocessor scheduling problem where downtimes can occur at any time during the scheduling horizon.
\nSurveys with focus on scheduling with availability constraints are given in [20, 21, 22, 23, 24]. Besides the makespan, the authors of these works survey work on various other objective functions such as total completion time, and also address additional variants of the problem, such as its resumable version.
\nUnless assumptions about the unavailability periods are made or unless P = NP, there is no polynomial-time approximation algorithm with a constant worst-case approximation bound for the problem of scheduling with unavailability periods to minimize the maximum completion time, since there is a polynomial-time reduction from the NP-hard problem of 3-Partition to the problem of finding a schedule that has a maximum completion time that is at most \n
For resumable scheduling, where the execution of jobs may continue after a downtime that interrupted them, but where jobs cannot be preempted by the scheduling algorithm, and where one machine does not shut down and all other machines shut down at most once, Lee shows that the makespan of LPT is in the worst case \n
In [1], Hwang and Chang make the assumption that at most half of the machines are unavailable at any time, and show for this situation that the worst-case approximation bound of LPT is 2. In [3], it is shown that no polynomial algorithm can have a better bound than 2 for this problem unless \n
In [26], scheduling with at most one unavailability period on each machine is considered and exact algorithms are given for small problem instances. The authors consider separately the case of identical jobs, and also consider total completion time beside the makespan as an objective function. For larger instances, they propose an LPT-like algorithm, which assigns jobs in nonincreasing order of their processing time to the fastest machine on which they would finish being processed at the earliest time. They also do experiments on a total number of 68 generated instances where error margins of at most 5.6% are observed. They do not compare their heuristic to the previously proposed heuristic from [4], which we discuss in Section 5.1.2, which was also proposed for this problem, even though its worst-case approximation bound was shown for the objective function of scheduling with fixed jobs.
\nThe problem of scheduling with fixed jobs is given as a number of processors, where each processor may have jobs that must be executed during certain given periods on it, together with a number of other jobs which can be executed by any processor. As noted in Section 1, job durations or required execution times are expressed for example as a number of significant units such as clock cycles. For uniform processors this number represents the time needed by a job to be executed on the slowest processor. In case there are uniform processors, each processor also has a speed factor, by which the time needed by a job on the slowest processor is divided in order to obtain the time needed for the processor to execute the job.
\nAs noted before, the problem of scheduling with fixed jobs differs from the problem of scheduling with unavailability periods in that its maximum completion time cannot be earlier than the latest completion time of a fixed job.
\nIn [5], Scharbrodt et al. give a polynomial-time approximation scheme for scheduling on a constant number of uniform machines with fixed jobs. They also show that it is NP-hard to obtain a schedule that ends within a factor of less than 1.5 when scheduling on identical processors with at most one fixed job on each machine. Even though they do not specify their result in this way, their proof that no polynomial-time approximation algorithm can have a better worst-case approximation bound than 1.5 for multiprocessor scheduling with fixed jobs does not use the fact that there can be more than one fixed job on each machine, which implies the previous statement.
\nIf all fixed jobs are at the beginning of the schedule, the results presented in Section 4.1 apply, as the optimal schedule of each problem instance of scheduling on nonsimultaneous machines can only potentially get worse when scheduling with fixed jobs is considered instead, and since the resulting schedule of an approximation algorithm ends later for scheduling with fixed jobs only if its maximum completion time is the completion time of a fixed job, which the optimal schedule also needs to execute.
\nWe next consider the case where there is at most one fixed job on each machine in Section 5.1, and the case where there can be multiple fixed jobs on each machine in Section 5.2.
\nWhen scheduling on multiple processors with at most one fixed job on each machine, LPT and MULTIFIT variants have been shown to achieve a worst-case approximation bound of 1.5, which is best possible for this problem unless \n
For scheduling on identical machines with at most one fixed job on each machine, an LPT-like algorithm, LPTX, was given in [3], for which a worst-case approximation bound of 1.5 was shown. Before running LPT, LPTX creates an order of processors, which is then used by LPT to break ties in case two processors become available at the same time. The ordered list of processors created before applying LPT is built in two steps:
All processors that have an unavailability period that is not at the beginning of the schedule are assigned to this list in non-decreasing order of the start of their downtime.
All other processors (that is, those which have the downtimes at the beginning of the scheduling period or have no downtime at all) are appended to the list built in the previous step in non-decreasing order of the times at which they can start executing jobs.
In the more general case of scheduling on uniform machines with at most one fixed job on each machine, a MULTIFIT-like algorithm, LMULTIFIT, was given in [4], which achieves the worst-case approximation bound of 1.5. For a MULTIFIT variant to work in the presence of downtimes, it must be specified how it deals with the fact that there are more than one time interval in which jobs can be scheduled on one processor.
\nAfter a MULTIFIT deadline is assigned and before applying the bin packing algorithm FFD (see Section 3), LMULTIFIT orders all time intervals in which jobs can be scheduled in non-decreasing order of their length. Here, the length of a time interval is the duration of the time interval multiplied by the speed factor of the processor on which the interval occurs.
\nFor scheduling on identical processors with fixed jobs, where the number of fixed jobs on a machine can be arbitrary, approximation algorithms that are much more complex than LPT and MULTIFIT were given in [27], with a worst-case approximation bound of \n
In [28], a very long proof is outlined that LMULTIFIT achieves a worst-case approximation bound of 1.5 when scheduling on identical processors with at most two fixed jobs on each machine. In [29], an algorithm using two MULTIFIT-like algorithms is shown to have a worst-case approximation bound of 1.625, which likely can be improved to 1.6 without excessive effort.
\nThe time complexity of the MULTIFIT-like and LPT-like algorithms is significanlty lower than that of the algorithms from [6, 27], and they are also significantly easier to implement; however, there is little hope in our opinion that bounds better than 1.55 can be shown in the general case for such algorithms with proofs of reasonable length. Given such problems, the user must decide what is best suited for his or her needs.
\nWorst-case approximation results in this research area about LPT and MULTIFIT variants mainly use proofs by contradiction in which some proof techniques appear very often. We next describe two techniques that we consider to be the most relevant, and then comment upon some other methods that are used relatively often. In the following, we call job lengths the durations of the jobs on the slowest processor.
\nA very well-known proof method is that of assuming that there exists a minimal counterexample to a theorem T, that is, a problem instance for which the algorithm’s schedule does not obey that theorem, with a minimal number of processors, of jobs, of downtimes (or of fixed jobs) that do not start at the beginning of the schedule, and/or of other quantities that can be minimized which are chosen to fit the statements to prove. A minimal counterexample exists whenever there is a counterexample, and thus, showing that it does not exist proves T. In order to define minimality among counterexamples, the author of the proof first chooses a partial order among the problem instances. An instance which is minimal according to this partial order among the instances for which a theorem T does not apply is called a minimal counterexample.
\nThis method can be very powerful, because after assuming that a minimal counterexample does not obey a theorem T, many useful properties of a minimal counterexample can be derived from the fact that no lesser counterexample exists, which can ultimately lead to a contradiction.
\nHere, a lesser counterexample is a counterexample with less processors, or less tasks, or less downtimes, or with a job with a smaller length, depending on how the order of instances was defined. Showing that a minimal counterexample does not exist is usually significantly easier than developing a direct proof for T.
\nThe theorem to prove could be that the worst-case approximation bound holds, but it can also be an intermediary result that is later used to prove the worst-case approximation bound. One could address instances that have a certain property first, and then show that the worst-case approximation bound holds for these, for example by using a minimal counterexample among these instances, and then do the same for all other instances.
\nSometimes it is enough to define the partial order only using the number of processors [11], while in most cases, it is useful to include multiple characteristics of problem instances, such as all or a part of the characteristics enumerated above. In one situation, a minimal counterexample was defined to also have minimal job lengths, meaning that if in a minimal counterexample the length of one job is reduced, the resulting instance is not a counterexample [28].
\nFor a problem instance that does not obey a worst-case approximation bound, there is a job (\n
The schedule \n
Therefore, the optimal schedule has more total execution time than \n
There are infinite ways of assigning weights, and there is no unique strategy that leads to success. Usually, the weight function is monotonic with regard to job lengths, and, as \n
In order to reason easier about time, one can divide all durations of time intervals in which jobs can be scheduled by X if scheduling on identical processors. For uniform processors, such intervals can be divided by the time it would take \n
In the case of uniform processors, the time intervals can have unbounded lengths because the speed factors may be arbitrarily high. A way to describe the amount of jobs assigned within a schedule in such an interval is to use task densities, which can be defined for each task as being the ratio between its weight and its length. Also, a task type density can be defined as a lower bound for all possible task densities of tasks of that type. The concept of task density can be used in order to reason about time intervals that may be very long. For example, the total weight of all tasks in an interval of length t is at most t times the maximum task type density among all task types represented within that interval.
\nWe use the notations from the previous subsection. Since X is not contained in SA, there must be a processor p* in the optimal schedule that has a total execution time that is greater than that of SA on p*. Such a processor can be analyzed in detail and may be shown to have certain properties that, in conjunction with other methods, result in proofs of useful theorems. For example, the existence of p* may imply a certain minimum duration of the optimal schedule, in conjunction with the observation that X could not be fitted by A on p* without causing the maximum completion time of SA to exceed B.
\nIn this work, we considered worst-case approximation for scheduling on multiple machines with availability constraints or with fixed jobs in order to minimize the maximum completion time. We surveyed results obtained in this research area and commented upon the algorithms used.
\nProminent among these algorithms are LPT and MULTIFIT and their variants, whereas for multiprocessor scheduling with fixed jobs, more complicated algorithms were used to achieve best possible worst-case approximation bounds in the class of polynomial algorithms assuming that \n
The problem of scheduling with availability constraints cannot be approximated by a polynomial-time algorithm with a constant worst-case approximation bound, even if there is at most one downtime on each machine, unless assumptions about the downtimes are made. The results we presented in this area address the problem of scheduling on identical processors with at most one downtime on each machine, with various assumptions.
\nDue to its different objective function, the problem of scheduling on identical parallel machines with fixed jobs allowed for the development of a polynomial-time approximation algorithm with a worst-case approximation bound of 1.5, and the development of a PTAS for scheduling on a constant number of uniform machines with fixed jobs was also possible.
\nThe MULTIFIT and LPT variants developed for the discussed variants of these problems could be useful in practice, as their time complexity is low and thus they should be able to address very large problem instances, as they are easy to implement, and because in some cases their worst-case approximation bounds could be considered to be good enough. In the case of scheduling on uniform nonsimultaneous machines, the average performance of a MULTIFIT variant was studied, and shown to be very good, as the experiments suggest that in general, for instances that can be relevant in practice and for which exhaustive search is not an option, the algorithm returns schedules with a maximum completion time that is within 3% of that of an optimal schedule.
\nWe also elaborated on the most encountered proof techniques in worst-case approximation bound proofs for LPT and MULTIFIT variants.
\nThe limitations of the presented works result mainly from the difficulty of the problem of scheduling with unavailability periods when considering the subject of approximation. To assess how well the proposed heuristics for this problem perform under such conditions is difficult, as it is hard to have a good estimate of the optimal schedule unless it is computed by an exact algorithm as was done in [26]. This problem has therefore been addressed only by considering the special case where there is at most one downtime on each machine.
\nIn the future, it may be interesting to compare the heuristics proposed for the same problems experimentally.
\nAnother limitation of the discussed works and of many other research works on scheduling results from the fact they attempt to understand problems with one, two, or at most three aspects at one time, whereas in many practical problems such as some production planning problems, many aspects occur at once. For example, availability constraints can appear alongside a multitude of other constraints that have to be considered simultaneously. These can be precedence constraints, that certain jobs have to be assigned to certain machines, or preferences of the manufacturer that the machines should not have more than a 60% or another predefined load for example in order to leave room for unexpected events. Furthermore, orders often come online, and if an urgent order from an important client needs to be given priority, this can alter the delivery times of other orders. Also, delivery times and delays have a big relevance in practice, as not delivering on time can cause fines. Such practical problems can also have sequence-dependent setup times, the necessity for setup operators to be present to perform setups, the preference that setup times are kept low by putting jobs from the same family of types of jobs consecutively on machines whenever possible, the necessity for workers to attend to certain machines while production takes place, and worker breaks and holidays. The preexisting schedule also has to be kept unchanged for a predefined time period since materials are brought to the production place in preparation for the production process. In addition, orders may have priorities and deadlines. For such problems, given the time constraints in which the schedule needs to be generated and that there can be thousands of jobs, usually a heuristic is employed that first orders the jobs for example by using priorities assigned to them and/or their deadlines and then schedules them on the machines in that order while also obeying all constraints and attempting to fulfill all preferences. Difficulties in researching such problems include that probably for different sets of orders different scheduling strategies may be better, and that an optimal schedule may be very hard to find and thus it is hard to quantify how well a heuristic performs.
\nThis publication was supported by the Open Access Publication Fund of Technische Universität Berlin.
\nThe author declares no conflict of interest.
\nMSP | multiprocessor scheduling problem |
NMSP | nonsimultaneous multiprocessor scheduling problem |
UNMSP | uniform nonsimultaneous multiprocessor scheduling problem |
LPT | largest processing time |
FFD | first fit decreasing |
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\n\nFor a quote please contact us directly at orders@intechopen.com The quote will be sent to you within 1-2 business days.
\n\nAll of the books and chapters can be browsed online. To obtain InTechOpen's full book catalogue in PDF, please contact us.
\n\n\n\nIntechOpen works with award winning print-houses and we hold to the fact that all of our printed products are of the highest quality.
\n\nPrint copies of our publications are most often purchased as individual purchases by universities, libraries, institutions and academia personnel, hence increasing the visibility and outreach of our authors' published work among science communities and institutions. Our books are available at our direct Print Sales Department and through selected representatives throughout the world.
\n\nIndia - CBS Publishers & Distributors Pvt. Ltd.
\n\nASEAN - Books International
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\n\nFor partnership opportunities, please contact orders@intechopen.com.
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