2. MADM Methods in AM
2.1. AOs in AM
Aggregation operators (AOs), also known as aggregation functions, are mathematical functions for grouping together multiple values to obtain a single summary value 
. The most well-known AO is the weighted averaging operator (i.e., a simple additive weighting method or weighted sum model). Other AOs commonly used in MADM include the ordered weighted averaging operator, power averaging operator, weighted Heronian mean operator, weighted Bonferroni mean operator, weighted Maclaurin symmetric mean operator, and weighted Muirhead mean operator. An important feature of AOs is that they can generate summary attribute values and a ranking of alternatives, which greatly facilitate the final decision. Another important feature is that each AO has its specific capability in solving MADM problems. For example, the ordered weighted averaging operator can capture the ordered positions of attribute values; the power averaging operator can reduce the influence of extreme attribute values on the aggregation results; the weighted Heronian mean operator and weighted Bonferroni mean operator can capture the interactions between two attributes; the weighted Maclaurin symmetric mean operator and weighted Muirhead mean operator can capture the interactions among multiple attributes. Because of such feature and capabilities, AOs have been widely applied to solve MADM problems in many fields. In the field of additive manufacturing (AM), AOs have been applied to tackle the problems below.
2.1.1. AOs in Part Orientation
Pham et al. 
developed a decision support system based on the weighted averaging operator to solve the part orientation problem in stereolithography (SLA), where alternative orientations are obtained via feature recognition and the best orientation is selected via considering the overhanging region area, support volume, build time, build cost, and problematic features. Byun and Lee 
presented an approach based on the weighted averaging operator to determine the optimal orientation to build an AM part. In this approach, the alternative orientations of an AM part are generated according to the surfaces of the convex hull of its three-dimensional (3D) model and the optimal orientation for building the part is determined under the consideration of surface roughness, build time, and manufacturing cost. Al-Ahmari et al. 
developed a decision system based on the weighted averaging operator for the automatic generation of part build orientations in selective laser melting (SLM), where feasible orientations are obtained via feature recognition and the maximisation of the tolerances of all features and the final orientation is generated based on the predicted part accuracy and build time.
2.1.2. AOs in AM-Related Assessment
Moreno-Cabezali and Fernandez-Crehuet 
carried out a risk assessment on AM research and development projects based on a fuzzy weighted triangular averaging operator. A set of risks with negative influence on project objectives are identified from the literature and assessed via a survey answered by ninety experts. The responses of the experts are aggregated using the AO. The aggregation results show that defects occurring during the building of a part presents the most critical risk.
2.1.3. AOs in Process Selection
Qin et al. 
presented a generic approach based on two fuzzy Archimedean power-weighted Bonferroni mean operators for the selection of AM processes. This approach has characteristics in providing good versatility and flexibility, capturing the interrelationships of performance parameter types and the risk attitudes of decision makers, and reducing the influence of extreme performance parameter values on the aggregation results. Qin et al. 
constructed two linguistic interval-valued intuitionistic fuzzy Archimedean prioritised operators and applied them in AM machine selection. These two AOs can process the situation where the attribute weights are unknown and the attributes are in different priority levels. Qin et al. 
presented a linguistic interval-valued intuitionistic fuzzy Archimedean power weighted Muirhead mean operator and studied its application in 3D printer selection. Compared to the AOs in 
, this AO can also capture the interrelationships among multiple attributes.
2.1.4. AOs in Multiple Problems
Huang et al. 
developed a generic approach based on a fuzzy Hamacher power weighted Maclaurin symmetric mean operator for MADM problems in design for AM. This approach was demonstrated using an AM machine and material selection example and an optimal build orientation determination example. Compared to the AO in 
, the developed approach also has the capability to capture the risk attitudes of decision makers.
2.2. AHP in AM
Analytical hierarchy process (AHP) is a structured MADM method based on mathematics and psychology for organising and analysing complex decisions 
. In this method, an MADM problem is first decomposed into different hierarchies in the order of the overall objective, the sub-objective in each level, the attributes, and the alternatives. Then, the priority of each element in each level to an element of the previous level is calculated via solving the eigenvector of a positive reciprocal pairwise comparison matrix. Finally, the weighted averaging operator is used to calculate the final priorities of all alternatives to the overall objective, and the alternative with the highest final priority is determined as the best alternative. AHP provides a systematic, simple, and practical method for structuring an MADM problem, quantifying the elements, relating the elements to the overall objective, and evaluating the alternatives. It has been extensively studied, refined, and applied in many areas since its introduction. In the area of AM, AHP has been applied to solve the problems below.
2.2.1. AHP in Process Selection
Braglia and Petroni 
proposed a management support approach based on AHP for the selection of AM technologies. This approach decomposes the selection problem into a four-level structure. The first level represents the overall objective of AM technology selection. The overall objectives are divided into five sub-objectives: price, friendliness, characteristics, cost, and time, which are included at the second level. The sub-objectives of friendliness, characteristics, and cost make up the third level. The last level consists of alternative AM machines. Armillotta 
presented an AM technique selection approach based on an adaptive AHP decision model. This approach also decomposes the selection problem into a four-level structure: application, categories of AM techniques, attributes, and alternative AM techniques. It is improved with a procedure that adapts the parameters of the AHP model to AM technique specifications. Lokesh and Jain 
developed a systematic approach based on AHP for the selection of AM technologies. This approach also solves the selection problem via a four-level structure. The first level stands for the overall issue. The product requirement issues, process requirement issues, social and environmental issues, and user- and company-related issues constitute the second level. Their sub-issues are included at the third level. The last level consists of alternative AM technologies.
2.2.2. AHP in Adhesive Selection
Arenas et al. 
presented an approach based on AHP to select a structural adhesive that best conjugates mechanical benefits and adaptation to FDM from five different types of structural adhesives, including cyanoacrylate, polyurethane, epoxy, acrylic, and silicone. This approach tackles the selection problem through four levels. The first level represents the overall objective and the last one includes the five alternative adhesives. The technological criterion, adjustment to FDM, and economic criterion form the second level. Their sub-criteria are included at the third level. The experimental results suggest that polyurethane is the best adhesive to bond ABS parts fabricated by FDM.
2.2.3. AHP in Part Selection
Knofius et al. 
developed an approach based on AHP for the selection of parts for AM in service logistics. The overall objective is divided into the securing supply, reducing the downtime, and reducing the cost. Both securing supply and reducing downtime are further decomposed into supply options and supply risk, while reducing the cost is further divided into the remaining usage period, manufacturing and order costs, and supply options. Muvunzi et al. 
constructed an evaluation model based on AHP for the selection of part candidates for AM in transport sector. Six selection criteria including geometric complexity, production volume, function, opportunity for design improvement, time to manufacture, and material removal are considered in the model. These criteria are ranked according to the requirements of the transport equipment manufacturing industry via the use of AHP. Foshammer et al. 
presented a knowledge management-based approach to identify the aftermarket and legacy parts suitable for AM. The part identification entailed AHP, semi-structured interviews, and workshops. Compared to the existing approaches, the presented approach integrates knowledge management-based part identification with current operations and supply chains.
2.2.4. AHP in Multiple Problems
Uz Zaman et al. 
introduced a generic decision methodology based on AHP for the selection of AM processes and materials. The criteria for AM process selection include geometry complexity, minimum layer thickness, accuracy, build volume, build speed, machine cost, and labour cost. The criteria for AM material selection come from nine material indices. The introduced methodology provides a guideline for designers to achieve a strong foothold in the AM industry. Hodonou et al. 
presented an integrated material-design-process selection methodology based on AHP for aircraft structural components manufactured by subtractive and additive processes. This methodology solves the selection problem via a three-level structure. The first level is the overall objective of the material-design-process selection. The second level consists of three selection attributes including the mass, buy-to-fly ratio, and manufacturing cost. Alternatives are included at the third level. The methodology is adopted to redesign an aircraft component for machined Al7075-T6 and for SLM AlSi10Mg powders. Kadkhoda-Ahmadi et al. 
developed a multi-criterion evaluation system based on AHP to solve the process and resource selection problem in design for AM. The working process of this system mainly includes a screening step, a comparative assessment step, and a ranking step. In the first step, the AM process, machine, and material are screened according to some technical and economic evaluation criteria. The second and third steps are carried out using AHP where the sub-criteria including the build time, accuracy performance, and cost are considered.
2.2.5. AHP in Material Selection
Alghamdy et al. 
presented a material selection methodology based on AHP for AM applications. This methodology consists of two steps, including the screening of available materials and the ranking of alternative materials. The first step was implemented using a heuristic and analytical algorithm based on AHP, where the performance, physical, and thermal requirements are considered. The second step is carried out via a decision matrix, where the alternative materials are ranked based on the cost and best performance.
2.2.6. AHP in AM-Related Assessment
Foteinopoulos et al. 
proposed an approach for the evaluation of the key performance indicators in the construction sector AM. This approach is based on a modified version of AHP, which was found to greatly decrease the needed comparisons and minimise the preparation time when the number of variables is large. Sonar et al. 
developed an approach based on AHP to identify and prioritise AM implementation factors. In this approach, a total of eleven AM implementation factors, including AM technology, top management commitment, technological awareness, information sharing, organisation capability and human resource, education and training, supply chain coordination, market support, customer and service management, process improvement practices, and financial capability are identified and ranked via AHP. Bappy et al. 
constructed a model based on AHP and evidential reasoning to assess the social impacts of AM technologies. In this model, AHP is used to rate and structure the relevant attributes of social impacts, while evidential reasoning is applied to aggregate the subjective judgmental belief structure data. The model output includes the average state of the social impact together with the uncertainty for each attribute.
2.2.7. AHP in Production Scheduling
Ransikarbum et al. 
developed a decision support model based on multi-objective optimisation and AHP for production and distribution planning in material extrusion, SLA, and SLS. In this model, a multi-objective optimisation technique is applied to schedule component batches to a network of AM machines, and AHP is adopted to analyse the trade-offs among conflicting objectives.
2.2.8. AHP in Design Selection
Rochman et al. 
presented an approach based on AHP for the selection of 3D printing COVID-19 mask design. The selection criteria are identified from the aspects of customer, production, and cost, which include the usefulness, easy to use, material selection, print time, print cost, and additional material cost.
2.3. TOPSIS in AM
Technique for order of preference by similarity to an ideal solution (TOPSIS) is an MADM method that ranks the alternatives according to their geometric distances from the best solution and from the worst solution 
. This method has an assumption that the attributes are monotonically increasing or decreasing. The normalisation of attribute measures is usually needed in it since they are often of incongruous dimensions. The best alternative determined by the method should have the shortest geometric distance from the best solution and the longest geometric distance from the worst solution. TOPSIS is a method of compensatory aggregation that allows trade-offs between attributes, where a bad result in one attribute can be negated by a good result in another one. This provides a more realistic modelling approach than non-compensatory methods. TOPSIS has been widely applied to a variety of fields since its introduction. In the field of AM, TOPSIS has been used in the problems below.
2.3.1. TOPSIS in Process Selection
Vahdani et al. 
presented a group decision-making method based on a fuzzy modified TOPSIS. In this method, the performance measures of the alternatives with respect to the attributes as well as the weights of the attributes are quantified via linguistic variables and are converted into triangular fuzzy numbers. To differentiate between alternatives in the evaluation process, a collective index is introduced. The method is demonstrated via an industrial robot selection example and an AM process selection example. Ic 
proposed an experimental design approach based on TOPSIS for the selection of computer-integrated manufacturing technologies. This approach combines TOPSIS and experiment design, which greatly reduces the computation cost and time in TOPSIS. The approach was validated using four computer-integrated manufacturing technology selection problems, including industrial robot selection, AM process selection, machine tool selection, and plant layout design. Yildiz and Ugur 
evaluated 3D printers used in AM by using interval type-2 fuzzy TOPSIS. In the evaluation process, the max printing speed, max build volume, layer resolution, price, and positioning precision are identified as attributes, and the performance measures of the alternative 3D printers with respect to these attributes are described using interval type-2 fuzzy numbers.
2.3.2. TOPSIS in Part Orientation
Yu et al. 
studied the personalised design of part build orientation in AM. TOPSIS is used to calculate a score for each orientation during the rotation of a part. The proportional–integral–derivative controller rotates the part according to the error between the target and score. A suitable orientation is determined when the error is eliminated.
2.3.3. TOPSIS in AM-Related Assessment
Priarone et al. 
assessed the environmental and economic impact of wire arc AM, where TOPSIS is applied to generate high-resolution maps of the results within the decision-making space. Alsaadi 
studied the prioritisation of challenges for the effectuation of sustainable AM via grey TOPSIS. In this study, fifteen sustainable AM challenges were identified from the literature and ranked using the TOPSIS in grey environment. The ranking results show that training towards sustainable AM benefits and limited materials recycling potential are significant challenges. Agrawal 
presented an approach to analyse the sustainable design guidelines for AM applications. In this approach, twenty-six guidelines are identified and divided into four groups. Grey axiomatic design is used to determine the weight of each group and grey TOPSIS is applied to prioritise the guidelines. The prioritisation results suggest that the design for the reusability and optimisation of build orientation to reduce the build time and surface roughness are the top identified guidelines.
2.3.4. TOPSIS in Parameter Optimisation
Kamaal et al. 
studied the influence of build orientation, infill percentage, and layer height on the tensile strength and impact strength of FDM carbon fibre–polylactic acid composite. TOPSIS is used to find the best set of build orientation, infill percentage, and layer height that would obtain the maximum strength using minimum material. Sugavaneswaran et al. 
studied the combined effect of FDM and vapour smoothening process parameters on part quality. The process parameters include build the orientation angle, build surface normal, and exposure time, whilst the part quality indicators are the surface roughness and dimensional error percentage. TOPSIS is applied in this study to determine the optimal set of process parameters. Kumar et al. 
presented a hybrid approach for twin screw extrusion parametric optimisation. In this approach, the analysis of variance is used to produce alternative sets of process parameters, and TOPSIS is applied to carry out multi-objective selection.
2.3.5. TOPSIS in Material Selection
Jha et al. 
developed a material selection approach based on TOPSIS for the biomedical application of AM. In this approach, the materials for the biomedical application of AM are first identified according to survey of the literature and the experience of the experts, and TOPSIS is applied to prioritise the identified materials.
2.4. Other Single Methods in AM
In addition to AOs, AHP, and TOPSIS, other single MADM methods, including deviation function (DF) 
, fuzzy synthetic evaluation (FSE) 
, graph theory and matrix approach (GTMA) 
, multi-objective optimisation by ratio analysis (MOORA) 
, knowledge value measuring (KVM) 
, vlsekriterijuska optimizacija i komoromisno resenje (VIKOR) 
, complex proportional assessment (COPRAS) 
, analytical network process (ANP) 
, axiomatic design (AD 
, elimination et choix traduisant la réalité (ELECTRE) 
, preference selection index (PSI) 
, best–worst method (BWM) 
, and three-way decision model (3WDM) 
, have also been applied to address the following problems in the field of AM.
2.4.1. Other Single Methods in Multiple Problems
West et al. 
presented a process planning approach to aid SLA users in the selection of proper process variables to obtain the desired part performance. This approach achieves a balance of objectives described by geometric tolerances, surface finishes, and build time. The process variables to be determined include build orientation, layer thicknesses, z-level wait time, sweep period, hatch overcure, and fill overcure. The determination process is carried out using an MADM method based on the deviation function. Palanisamy et al. 
applied BWM to select a suitable AM machine and material for a product. The selection of a suitable machine is based on the criteria including the cost, accuracy, variety of materials, and material wastage. The selection of the best material is based on respondent requirement, in which the criteria that affect the overall cost of the product are considered.
2.4.2. Other Single Methods in Process Selection
Lan et al. 
developed a decision support system for AM process selection, where the alternative AM processes are determined via a knowledge-based expert system, and the most suitable AM process is selected using an FSE model. Rao and Padm 
presented a GTMA-based methodology for the selection of an AM process that best suits the end use of a given product, where a selection index obtained from a digraph of selection attributes is introduced to evaluate and rank the alternative AM processes. Chakraborty 
studied the application of MOORA for MADM in a manufacturing environment, where the selection of an industrial robot, a manufacturing system, a numerical control machine, a machining process, an AM process, and an inspection system are addressed.
In Khrais et al. 
, an FSE approach to select an AM process for producing prototypes was presented. In this approach, fuzzy if–then rules and fuzzy sets are used to translate the appropriateness of each process into each attribute, and the best process is identified according to the overall efficiencies of alternative processes. In Roberson et al. 
, a model for evaluating and prioritising 3D printing technologies based on some purchasing considerations was constructed, in which the prioritisation was implemented using an MADM method based on DF. In Vinodh et al. 
, the application of fuzzy VIKOR for the selection of AM processes in an agile environment was studied. It was found that FDM is the most suitable process for fabricating the prototypes of pump impeller.
2.4.3. Other Single Methods in Parameter Optimisation
Patel et al. 
studied the application of PSI to select the optimal process parameters for FDM polylactic acid, where five attributes including the tensile strength, tensile module, surface roughness, compressive strength, and compressive module are considered. Raykar and D’Addona 
applied VIKOR to select the optimal set of layer thickness, bed temperature, printing speed, and infill percentage for FDM, where the responses include the material weight, material length, and total time. Deomore and Raykar 
applied VIKOR to select the best set of the layer thickness, infill percentage, bed temperature, printing speed, infill pattern, build orientation, air gap, and raster angle for FDM, where the responses are also the material weight, material length, and total time.
2.4.4. Other Single Methods in Material Selection
Exconde et al. 
studied the selection of virgin polymer resins and recycled post-consumer plastics for use in 3D printer filaments. The ELECTRE method was used to select the best material. The study suggests that the virgin low-density polyethylene is the best alternative filament. Qin et al. 
proposed a 3WDM-based approach for the selection of materials in metal AM. The effectiveness of the approach is demonstrated via a quantitative comparison with several existing approaches. The demonstration results show that the proposed approach is as effective as the existing approaches and is more flexible and advantageous than them.
2.4.5. Other Single Methods in AM-Related Assessment
Agrawal and Vinodh 
studied the prioritisation of drivers of sustainable AM. Forty drivers were analysed and rated from eight perspectives using BWM. The rating results show that the key drivers are eco-design, green innovation, and energy conservation.
2.5. Hybrid Methods in AM
2.5.1. Hybrid Methods in Process Selection
Byun and Lee 
developed a decision support system for the selection of an AM process. In this system, the performance measures of the alternatives with respect to the attributes are quantified by either numerical values or linguistic variables, the weights of the attributes are determined via AHP, and the alternatives are ranked by a modified TOPSIS method. Borille et al. 
applied six MADM methods including TOPSIS, GTMA, AHP, multiplicative AHP, SPA, and VDI to select AM technologies. It is demonstrated that not all methods produce the same ranking of AM technologies. Rao and Patel 
presented a hybrid method to solve MADM problems in the manufacturing environment. This method is obtained by integrating PROMETHEE with AHP and FSE. It is demonstrated via four examples about cutting fluid selection, manufacturing program selection, end-of-life scenario selection, and AM process selection.
In Mahapatra and Panda 
, GRA and fuzzy TOPSIS are applied to select AM processes. The results of GRA are compared with that of fuzzy TOPSIS. It is concluded that SLS is the process for the best dimensional accuracy and surface quality. In Paul et al. 
, a comparison of three MADM methods considering a case of selection of 3D printers is carried out. The methods are TOPSIS, SIMA, and PROMETHEE, in which the weights are determined using ANP. In Vimal et al. 
, a decision support system for AM process selection considering environmental criteria was developed. The ranking mechanism of this system is based on a hybrid of ANP and TOPSIS. The ranking results show that SLA is the best process based on environmental considerations.
2.5.2. Hybrid Methods in AM-Related Assessment
Liao et al. 
established a hybrid MADM framework for evaluating and enhancing 3D printing service providers. This framework was realised using the DEMATEL-based network process and VIKOR. Cruz and Borille 
applied three MADM methods to compare AM with the machining process of a titanium part used in the aerospace industry. The three methods are AHP, SPA, and VDI. It was found from the comparison results that topology optimisation-SLM is a strong candidate for manufacturing titanium parts for aerospace application. Wang et al. 
presented a fuzzy systematic approach to assess the factors critical to the applicability of AM technologies in the aircraft industry. This approach combines a fuzzy weighted geometric operator with fuzzy AHP.
In Zhang et al. 
, an optimal set algorithm based on AHP, TOPSIS, and the Baldwin effect was developed to evaluate the cloud 3D printing order task execution. This algorithm can perform the automatic matching and optimisation of alternative services. In Yoris-Nobile et al. 
, life cycle assessment and MADM analysis to determine the performance of 3D printed cement mortars and geopolymers were carried out, where the life cycle assessment is performed to study the environmental impact of materials, and the MADM analysis is based on AHP, IEM, TOPSIS, and WASPAS and applied to select the most suitable dosages.
2.5.3. Hybrid Methods in Multiple Problems
Zhang and Bernard 
constructed an integrated model for MADM problems in process planning for AM. This integrated model is an aggregation of a deviation model and a similarity model. The deviation model, which is inspired by TOPSIS, measures the deviation extent of each alternative to the aspired goal based on the geometric distance between them. The similarity model, which is based on GRA, measures the similarity between alternatives and the expected goal via analysing the curve shape of each alternative. Algunaid and Liu 
developed a decision support system for the selection of AM processes, machines, and materials from a large-scale option pool. This system is based on a hybrid MADM method that integrates DEMATEL, AHP, and a modified TOPSIS.
2.5.4. Hybrid Methods in Part Orientation
Zhang et al. 
presented a feature-based build orientation optimisation method for AM. In this method, alternative build orientations are obtained from shape feature recognition, and the best build orientation are selected from the alternatives using the integrated model in 
. Qin et al. 
developed an automatic determination approach of the build orientation for an SLM part. In this approach, the alternative build orientations for an SLM part are generated via facet clustering, while the optimal build orientation for the part is determined using AHP and the weighted averaging operator. Ransikarbum et al. 
established an integrated MADM framework for part build orientation analysis in AM. In this framework, the quantitative data are assessed using DEA, the preferences of decision makers are analysed using AHP, and a suitable build orientation is determined using LN.
2.5.5. Hybrid Methods in Material Selection
Zhang et al. 
studied the material selection of 3D printed continuous carbon fibre-reinforced composites. A systematic hierarchical structure of multiple criteria considering the environmental, economic, social, and physical impacts is established. An integrated MADM method containing fuzzy BWM, GRA, and fuzzy VIKOR is presented to solve the material selection problem. Agrawal 
carried out a critical analysis of the rank reversal approach for sustainable AM material selection. Four MADM methods including the weighted averaging operator, MOORA, TOPSIS, and VIKOR were applied to compare the materials. The comparison results show that Accura HPC, TPU Elastomer, and Duraform EX are, respectively, the best sustainable material for SLA, FDM, and SLS. Malaga et al. 
studied the material selection for metal AM process. IEM and CODAS are applied to determine the priority order of alternative materials. The results show that aluminium alloy AlSi12Cu2Fe, tool steel H13, and aluminium alloy AlSi10Mg are the top-ranking materials for metal AM.
In Mastura et al. 
, the concurrent material selection of a natural fibre filament for FDM was investigated. An integration of AHP and ANP is introduced to select suitable natural fibres for FDM.
2.5.6. Hybrid Methods in Parameter Optimisation
Sakthivel and Vinodh 
applied the grey-based Taguchi method to optimise the slice height, part fill style, and build orientation of FDM. The response parameters include the build time, surface roughness, and hardness. The optimisation results are verified using a hybrid approach based on AHP and TOPSIS. Koli et al. 
investigated the effect of the current speed, welding speed, and gas flow rate on the ultimate tensile strength, micro-hardness, compressive residual stress, and total elongation of SS308L samples fabricated by the wire arc AM-cold metal transfer process. The optimal set of the process parameters is determined via an integrated MADM method based on fuzzy AHP, fuzzy MARCOS, and the analysis of means. Patil et al. 
presented an MADM method based on AHP and VIKOR for the selection of the best process parameters for FDM. The layer thickness, printing speed, infill percentage, and zig–zag pattern in FDM are simultaneously optimised by the method.