Determination of optimal build orientation for additive manufacturing using Muirhead mean and prioritised average operators
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Abstract
Build orientation determination is one of the essential process planning tasks in additive manufacturing since it has crucial effects on the part quality, post-processing, build time and cost, etc. This paper introduces a method based on fuzzy multi-attribute decision making to determine the optimal build orientation from a finite set of alternatives. The determination process includes two major steps. In the first step, attributes that are considered in the determination and heterogeneous relationships of which are firstly identified. A fuzzy decision matrix is then constructed and normalised based on the values of the identified attributes, which are quantified by a set of fuzzy numbers. In the second step, two fuzzy number aggregation operators are developed to aggregate the fuzzy information in the normalised matrix. By comparing the aggregation results, a ranking of all alternative build orientations can then be generated. Two determination examples are used to demonstrate the working process of the proposed method. Qualitative and quantitative comparisons between the proposed method and other methods are carried out to demonstrate its feasibility, effectiveness, and advantages.
Keywords
Additive manufacturing Build orientation determination Optimal build orientation Correlative relationship Priority relationship Fuzzy multi-attribute decision makingList of symbols
- Ξ_{i}
A fuzzy number
- μ_{i}
The degree of membership of a fuzzy number
- <μ_{i}>
A fuzzy number whose degree of membership is μ_{i}
- S
An ordered set of fuzzy numbers
- S_{k}
The k-th partition of S
- |S_{k}|
The cardinality of S_{k}
- Ø
Empty set
- δ_{|Sk|}
The |S_{k}|-th real number corresponding to Ξ_{|Sk|}
- δ_{k}
A collection of δ_{1}, δ_{2}, …, δ_{|Sk|}
- Δ
A collection of δ_{1}, δ_{2}, …, δ_{N}
- p(i_{k})
A permutation of (1, 2, …, |S_{k}|)
- P_{|Sk|}
A set of all permutations of (1, 2, …, |S_{k}|)
- \( \Xi_{i} \oplus \Xi_{j} \)
The sum operation of Ξ_{i} and Ξ_{j}
- \( \Xi_{i} \otimes \Xi_{j} \)
The product operation of Ξ_{i} and Ξ_{j}
- aΞ_{r}
The multiplication operation of Ξ_{r}
- Ξ_{s}^{b}
The power operation of Ξ_{s}
- w_{i}
The weight of Ξ_{i}
- TT(Ξ_{i})
The TT value of a fuzzy number
- ξ_{p(ik)}
The power weight of p(i_{k})
- DIST(Ξ_{i}, Ξ_{j})
The distance between Ξ_{i} and Ξ_{j}
- SUPP(Ξ_{i}, Ξ_{j})
The support degree for Ξ_{i} from Ξ_{j}
- S_{i} ≻ S_{j}
S_{i} is preferred over S_{j}
- S_{0}
The fuzzy number <1>
- S_{k}
The fuzzy number <max(μ_{1}, μ_{2}, …, μ_{|Sk|})>
- T_{k}
The fuzzy number \( S_{0} \otimes S_{1} \otimes \cdots \otimes S_{k - 1} \)
- ξ_{ik}
The power weight of i_{k}
- O_{i}
An option or an orientation
- O
A set of options or orientations
- A_{j}
An attribute
- A
A set of attributes
- w
A set of weights
- M
A decision matrix
- v_{i,j}
The value of the j-th attribute of the i-th option
- r_{i,j}
The ratio of v_{i,j}
- M_{F}
A fuzzy decision matrix
- M_{N}
A normalised fuzzy decision matrix
- A_{i} ≻ A_{j}
A_{i} is preferred over A_{j}
- O_{i} ≻ O_{j}
O_{j} ranks behind O_{i}
Introduction
Additive manufacturing (AM), which is commonly known as three-dimensional (3D) printing, refers to the processes of accumulating materials layer upon layer to 3D objects from 3D model data (ISO/ASTM 52900 2015). AM processes provide better flexibility in 3D model design, generate fewer waste byproducts in part manufacturing, and require less time and cost in product development, over conventional subtractive manufacturing processes. In addition, AM processes enables the fabrication of components with complex geometries, heterogeneous materials, and user-customisable properties (Gibson et al. 2015). Some have anticipated that AM processes would bring revolutionary changes to the industry (Gao et al. 2015). Despite this, ensuring the repeatability of AM processes and the reproducibility of AM products is still considered as one of the biggest challenges to make the processes widely applied in the real world industry (Kim et al. 2017; Qin et al. 2019).
To tackle this challenge, substantial work needs to be carried out at a number of aspects (e.g. design and simulation for AM, process planning for AM, qualification and certification of AM parts, standardisation of AM processes, AM materials, and AM digital thread) (Kim et al. 2015, 2017), in which process planning is one of the most important aspects. Process planning for AM is the use of specific techniques to generate the process plans for building a part, which mainly include build orientation, support structure, 3D model slicing, and tool-path, according to the 3D model data of the part (Kulkarni et al. 2000; Ahsan et al. 2015; Liang 2018). It consists of a set of successive preparation steps before building the part, and determine the build orientation is the first step. Build orientation has direct influence on the subsequent preparation steps, namely support structure generation (Jiang et al. 2018a, b, 2019a, b, c), 3D model slicing (Xu et al. 2018), and tool-path planning (Xiao and Joshi 2018).
In the process of AM, the build orientation is the accumulating orientation of materials when building the part. The quality of a build orientation has important influence on the part quality, post-processing, build time and cost, etc. (Taufik and Jain 2013). Build orientation determination refers to the use of specific techniques to identify a proper orientation to build an AM part from an infinite number of theoretical orientations. It consists of two major tasks. One is to generate a certain number of alternative build orientations (ABOs) from the infinite set. The other is to determine the optimal build orientation (OBO) from the generated ABOs (Kulkarni et al. 2000). At present, build orientations are manually specified by AM machine operators according to their expertise. Under the same circumstances, different operators may be in favour of different build orientations, which may greatly increase the uncertainty of process planning and directly affect the build cost, build time, accuracy, and quality of the parts (Zhang et al. 2018).
To provide an effective tool for automatic determination of build orientations, a method based on fuzzy multi-attribute decision making is proposed in this paper. The method focuses on the second task in build orientation determination and assumes that the ABOs are known. It is carried out in two steps. In the first step, attributes that are considered in the determination and their relationships are identified. Values of the identified attributes will be obtained from vendor documents, benchmark data, experiments, or expert evaluation. The values are then fuzzified using a ratio model as a membership function. As such each attribute value is quantified by a fuzzy number (FN), which enables the use of aggregation operators upon the fuzzified values. In the second step, a fuzzy weighted power partitioned Muirhead mean (FWPPMM) operator and a fuzzy weighted power prioritised average (FWPPA) operator, are developed to aggregate all FNs of each ABO. The overall attribute value of each ABO can then be finally quantified by a single FN. By comparing these FNs (from high to low), a ranking of all ABOs can then be generated, and the one which is ranked first is then proposed as the OBO.
The remainder of the paper is organised as follows. An overview of related work is provided in section “Related work”. “OBO determination method” section explains the details of the proposed OBO determination method. Two examples and qualitative and quantitative comparisons are presented to illustrate and demonstrate the method in “Examples and comparisons” section. “Conclusion” section ends the paper with a conclusion.
Related work
An ideal method for OBO determination is preferably a standardised method for practical applications. The latest AM process standards (ISO 17296-2 2015; ISO 17296-3 2014) provide a classification of AM processes, an introduction to the principle of each AM process, and a description to the performance test methods of AM processes, but have not yet enclosed a practical method for OBO determination.
To provide an effective tool for OBO determination, a number of methods have been presented during the past two decades. These methods can be classified into the following two categories on the basis of their used techniques:
A brief summarisation of some typical MOO methods
MOO method | Used MOO techniques | Optimised objectives |
---|---|---|
Cheng et al. (1995) | Weighted sum function | Part accuracy; build time |
Lan et al. (1997) | Self-developed algorithm | Surface quality; build time; complexity of support structure |
Alexander et al. (1998) | Self-developed algorithm | Cost; support volume; contact area with support; surface accuracy |
McClurkin and Rosen (1998) | Self-developed algorithm | Build time; accuracy; surface finish |
Hur and Lee (1998) | Genetic algorithm | Part accuracy; build time; support structure volume |
Xu et al. (1999) | Self-developed algorithm | Build cost; build time; building inaccuracy; surface finish |
Hur et al. (2001) | Genetic algorithm | Build time; surface quality |
Masood et al. (2003) | Generic mathematical algorithm | Volumetric error |
Thrimurthulu et al. (2004) | Genetic algorithm | Surface finish; build time |
Pandey et al. (2004) | Genetic algorithm | Surface roughness; build time |
Kim and Lee (2005) | Genetic algorithm | Post-processing time and cost |
Ahn et al. (2007) | Genetic algorithm | Post-machining time |
Canellidis et al. (2009) | Genetic algorithm | Build time; surface roughness; post-processing time |
Padhye and Deb (2011) | Genetic algorithm and particle swarm algorithm | Surface roughness; build time |
Strano et al. (2011) | Genetic algorithm | Surface roughness; energy consumption |
Zhang and Li (2013) | Genetic algorithm | Volumetric error |
Paul and Anand (2015) | Genetic algorithm | Cylindricity and flatness errors; support structure volume |
Ezair et al. (2015) | Self-developed algorithm | Support structure volume |
Delfs et al. (2016) | Self-developed algorithm | Surface quality; build time |
Luo and Wang (2016) | Principal component analysis | Volumetric error |
Zhang et al. (2017) | Genetic algorithm | Build time; build cost; production quality |
Brika et al. (2017) | Genetic algorithm | Surface roughness; build time; build cost; yield strength; tensile strength; elongation; vickers hardness; amount of support structure |
Chowdhury et al. (2018) | Genetic algorithm | Support structure volume; support structure accessibility; surface area contacting support; number of build layers; number of small openings; number of sharp corners; mean cusp height |
Mi et al. (2018) | Point clustering algorithm | Number of material changes |
Jaiswal et al. (2018) | Surrogate model | Material error; geometric error |
Al-Ahmari et al. (2018) | Weighted sum function | Geometric dimensioning and tolerancing value; production time |
Huang et al. (2018) | Genetic algorithm | Adaptive feature roughness; build time |
Raju et al. (2018) | Hybrid PSO-BFO algorithm | Hardness; flexural modulus; tensile strength; surface roughness |
Golmohammadi (2019) | Taguchi method | Amount of support material; mean roughness of the product |
A brief summarisation of some typical MADM methods
MADM method | Used MADM technique | Considered attributes | Importance of attributes | Relationships of attributes | Reducing of deviation effect |
---|---|---|---|---|---|
Pham et al. (1999) | Weighted sum score function | Cost; time; problematic features; optimally orientated features; overhanging area; support volume | Weights | Independent | No |
West et al. (2001) | Deviation function | Surface finish; accuracy; build time | Weights | Independent | No |
Byun and Lee (2006) | Simple weighted average operator | Surface roughness; build time; build cost | Weights | Independent | No |
Chen et al. (2008) | Fuzzy synthetic evaluation method | Base plane size; skewness of centre of gravity; height of centre of gravity; inaccessible volume; support-needed area; number of stock layers; removed material volume | Weights | Independent | No |
Zhang et al. (2016) | Deviation-similarity model in Zhang and Bernard (2014) | Surface quality; part accuracy; support structure volume; mechanical properties; build time; build cost; post-processing; favourableness of AM feature | Weights | Correlative | No |
Ransikarbum (2017) | Analytic hierarchy process | Build time; build cost; surface quality; part accuracy; mechanical properties; support volume | Weights | Independent | No |
Qie et al. (2018) | Ordered weighted averaging operator | Surface roughness; support volume; build time | Weights | Independent | No |
Zhang et al. (2018) | Statistical evaluation method | Surface quality; support structure | Weights | Independent | No |
Aiming at the two issues in the existing MADM methods for OBO determination, FS (Dubois and Prade 1980), the MM operator (Muirhead 1902), the PA operator (Yager 2008), the power average operator (Yager 2001), and the partitioned average operator (Dutta and Guha 2015) are introduced in this paper to construct a FWPPMM operator and a FWPPA operator to aggregate the values of the attributes of ABOs. An OBO determination method based on fuzzy MADM and the constructed operators is proposed. FS is a mathematical tool that can normalise the values of attributes into the numbers in [0, 1] to make them easy to handle. The MM operator is an aggregation operator that is applicable whenever all attributes are independent of each other, or there are correlative relationships between any two attributes, or any three or more attributes. The PA operator can capture priority relationships among attributes. The power average operator is an aggregation operator that can reduce the negative effect of the deviation of one or several argument values on the aggregation result. The partitioned average operator can handle the heterogeneous relationships among the aggregated arguments. Because of the combination of the MM, PA, power average, and partitioned average operators in the constructed operators, the proposed method can deal with the heterogeneous correlative relationship (HCR) and heterogeneous priority relationship (HPR) among the attributes of ABOs, while can also reduce the negative influence of the deviation of attribute values. Here HCR (or HPR) means that the attributes are partitioned into different partitions and the attributes in the same partitions are correlative with each other (have the same priority) whereas the attributes in different partitions are independent of each other (have different priority).
OBO determination method
Based on the description above, the present section is divided into three subsections, which explain the details of the FWPPMM and FWPPA operators, quantification of the attributes of ABOs, and generation of the OBO, respectively.
FWPPMM and FWPPA operators
Preliminaries
(1) Definition of FS In mathematics, a FS is a set whose elements have membership degrees, which indicate the degrees to which the elements belong to the set (Zadeh 1965). It can be formally defined as follow.
Definition 1
A FS S in a finite universe of discourse X is: S = {<x, µ_{S}(x)> | x∈X}, where µ_{S} : X → [0, 1] denotes the membership degree of x∈X to S, with the condition that 0 ≤ µ_{S}(x) ≤ 1.
(2) Definition of FN Generally, the values of the membership function µ_{S}(x) are called as FNs. The definition of a FN can be naturally obtained as follow.
Definition 2
A FN Ξ on a FS S = {<x, µ_{S}(x)> | x∈X} is: Ξ = <µ_{S}(x)>. For the sake of simplicity, the FN is usually denoted as Ξ = <µ> .
(3) Comparison rules of FNs Any two FNs can be compared via comparing their membership degrees. For example, suppose Ξ_{1} = <µ_{1}> and Ξ_{2} = <µ_{2}> are two arbitrary FNs, then Ξ_{1} < Ξ_{2} if and only if µ_{1} < µ_{2}; Ξ_{1} > Ξ_{2} if and only if µ_{1} > µ_{2}; and Ξ_{1} = Ξ_{2} if and only if µ_{1} = µ_{2}.
(6) MM operator The MM operator was firstly introduced to aggregate crisp numbers by Muirhead (1902). It is an all-in-one aggregation operator for capturing the relationships among multiple aggregated arguments, since it is applicable in the cases where all arguments are independent of each other, there are interrelationships between any two arguments, and there are interrelationships among any three or more arguments. The formal definition of the MM operator is as follow.
Definition 3
(7) PA (prioritised average) operator The PA operator, introduced by Yager (2008), has the capability of dealing with the HPR among the aggregated arguments. Its formal definition is as follow.
Definition 4
(8) Power average operator The power average operator, introduced by Yager (2001), can assign weights to the aggregated arguments via computing the support degrees between these arguments. This makes it possible to reduce the negative influence of the unduly high or unduly low argument values on the aggregation results. The formal definition of the power average operator is as follow.
Definition 5
(9) Partitioned average operator The partitioned average operator (Dutta and Guha 2015) can aggregate the arguments in different partitions using the same aggregation operator and aggregate the aggregation results of different partitions using the ordinary arithmetic mean operator. By this way, the HCR among arguments can be considered. The formal definition of the partitioned average operator is as follow.
Definition 6
FWPPMM operator
To handle the HCR among the attributes of ABOs and meanwhile reduce the negative effect of the deviation of attribute values on the OBO determination result, weights and the MM, power average, and partitioned average operators of FNs are combined to construct a FWPPMM operator. Based on the definitions of these operators (see Definitions 3, 5, and 6), the FWPPMM operator can be defined as follow.
Definition 7
It is easy to prove that the aggregation result of Eq. (12) [i.e. FWPPMM^{Δ}(Ξ_{1}, Ξ_{2}, …, Ξ_{n})] is still a FN. Thus if the aggregation results of different ABOs are computed, the comparison rules of FNs can be applied to generate a ranking of the ABOs. The determination of the OBO is naturally carried out on the basis of the ranking.
FWPPA operator
To deal with the HPR among the attributes of ABOs and meanwhile reduce the negative influence of the deviation of attribute values on the OBO determination result, weights and the PA and power average operators of FNs are combined to construct a FWPPA operator. Based on the definitions of these operators (see Definitions 4 and 5), the FWPPA operator can be defined as follow.
Definition 8
Let (Ξ_{1}, Ξ_{2}, …, Ξ_{n}) (Ξ_{i} = <μ_{i}>, i = 1, 2, …, n) be a collection of n FNs, S = {Ξ_{1}, Ξ_{2}, …, Ξ_{n}} be a set, S_{k} = {Ξ_{1}, Ξ_{2}, …, Ξ_{|Sk|}} (k = 1, 2, …, N) be N partitions of S (i.e. S_{1} ∪ S_{2} ∪ ··· ∪ S_{N} = S and S_{1} ∩ S_{2} ∩ ··· ∩ S_{N} = Ø), S_{1} ≻ S_{2} ≻ ··· ≻ S_{N} be the priority relationship among S_{1}, S_{2}, …, S_{N}, S_{0} = <1> and S_{k} = <max(μ_{1}, μ_{2}, …, μ_{|Sk|})> (k = 1, 2, …, N), \( T_{k} \, = \,S_{0} \otimes S_{1} \otimes \cdots \otimes S_{k - 1} \), and w_{1}, w_{2}, …, w_{n} be respectively the weights of Ξ_{1}, Ξ_{2}, …, Ξ_{n} such that 0 ≤ w_{1}, w_{2}, …, w_{n} ≤ 1 and w_{1} + w_{2} + ··· + w_{n} = 1. Then the aggregation function
It is easy to prove that the aggregation result of Eq. (19) [i.e. FWPPA(Ξ_{1}, Ξ_{2}, …, Ξ_{n})] is still a FN. Therefore, if the aggregation results of different ABOs are computed, the comparison rules of FNs can be used to generate a ranking of the ABOs. The determination of the OBO is naturally carried out based on the ranking.
Quantification of attributes of ABOs
Quantification of the attributes of ABOs aims to quantify the values of the attributes of ABOs in FNs. It takes a certain number of ABOs as input and output a normalised fuzzy decision matrix that quantifies the attributes of these ABOs. The quantification of the attributes of ABOs mainly consists of three tasks. The first one is identification of the attributes of ABOs and their relationships. Acquisition of the values of the identified attributes is the second task. The third task is transformation of the acquired values.
For the identification of the attributes of ABOs, there are already a number of research results available for reference. In the existing OBO determination methods in Tables 1 and 2, a variety of attributes of ABOs that are affected by build orientations has been identified, which mainly include cost, time, surface quality, accuracy, mechanical properties, support structure, and geometric features. Further, ISO 17296-3 (2014) has identified surface quality, dimensional and geometrical tolerance, mechanical properties, and build material requirements as the attributes that are affected by AM process. Based on these research results, the attributes affected by build orientation mainly include the followings (Taufik and Jain 2013; Zhang et al. 2016):
(1) Surface roughness. Build orientation has an important effect on surface roughness because of the layer upon layer building manner of AM processes. For example, planes or surfaces that are parallel or perpendicular to build orientation would have smaller roughness than those that have an angle with build orientation. Declining faces would be more seriously affected by stair-step effect (Zhang et al. 2016). When determining the build orientation, surface roughness should always be considered as a key factor. Most of the methods listed in Table 1 and Table 2 have taken into account this factor.
(2) Part accuracy. Part accuracy mainly contains dimensional error, geometric error, and volumetric error. Build orientation affects shrinkage, curling, and distortion, which are the major factors influencing part accuracy (Zhang et al. 2016). Thus, part accuracy is another important attribute for build orientation determination. Many existing orientation methods, which mainly include the methods of Cheng et al. (1995), McClurkin and Rosen (1998), Hur and Lee (1998), West et al. (2001), Masood et al. (2003), Zhang and Li (2013), Paul and Anand (2015), Luo and Wang (2016), Zhang et al. (2016), Ransikarbum and Kim (2017), Jaiswal et al. (2018), and Al-Ahmari et al. (2018), have studied the relationship between this attribute and build orientation.
(3) Volume of support structure. In general, support structure is needed when there are overhanging areas or declining faces. It is quite intuitive that different build orientations require different amount of supports. An ideal orientation should require as little support as possible, since support would affect build time, build cost, post-processing, and surface roughness (Jiang et al. 2018a, b, 2019a, b, c). In the orientation methods of Alexander et al. (1998), Hur and Lee (1998), Pham et al. (1999), Chen et al. (2008), Paul and Anand (2015), Ezair et al. (2015), Zhang et al. (2016), Brika et al. (2017), Ransikarbum and Kim (2017), Chowdhury et al. (2018), Qie et al. (2018), Zhang et al. (2018), and Golmohammadi and Khodaygan (2019), volume of support structure is considered as a key attribute.
(4) Mechanical strength. The mechanical strength of a part manufactured by AM process is generally anisotropic. The tensile strength and yield strength in horizontal orientation are often greater than they in vertical orientation (Zhang et al. 2016). The KARMA knowledge base (http://karma.aimme.es) constructed based on a large number of practical experiments suggests that build orientation has direct impact on mechanical strength. The orientation methods presented by Zhang et al. (2016), Brika et al. (2017), Ransikarbum and Kim (2017), and Raju et al. (2018) have identified mechanical strength as an important attribute when dealing with the orientation problem.
(5) Build time. Build time mainly consists of layer scanning time, layer preparation time, and recoating time. It is not difficult to imagine that build orientation directly affects build time due to the layer by layer nature of AM technologies. Most of the orientation methods listed in Tables 1 and 2 have considered build time.
(6) Build cost. Build cost calculates all resources required in building a part, such as materials, machine, energy, and labour. It is also influenced by build orientation because different orientations would cause different support volume, material consumption, build time, and post-processing. Build cost has also been regarded as an important factor in most of the orientation methods listed in Tables 1 and 2.
(7) Post-processing. Post-processing is an important AM process activity. One of its purposes is to improve the surface quality and mechanical properties of the manufactured part. Build orientation has direct or indirect effect on this activity since it directly affects surface quality and mechanical properties. However, only a few orientation methods, such as the methods of Kim and Lee (2005), Canellidis et al. (2009), and Zhang et al. (2016), have considered post-processing. The reason could be the difficulty in establishing computable post-processing models or representing related experience.
(8) Geometric features. Geometric features can be simply understood as the geometric shapes that constitutes a part. As investigated by Pham et al. (1999) and Zhang et al. (2016), geometric features are also affected by build orientation. But very limited number of orientation methods have investigated the relationship between them.
(9) Need for filler material. Apart from the above attributes, need for filler material can also be selected as attributes when determining build orientation for specific AM processes (Singh et al. 2017). However, few orientation methods have taken into account this attribute separately. The reason could be the combination of this factor into build cost, since the needed filler material can be seen as one types of consumption materials.
In practice, the attributes of ABOs in specific situations can be identified via experiments, simulation, or expert experience. The complex relationships among the attributes of ABOs generally include independent relationship, correlative relationship, and priority relationship, where each relationship could possibly be heterogeneous. The independent and correlative relationships can be identified by expert evaluation and actual experiments. The priority relationship is the reflection of the preference of users and is generally specified by users.
For the acquisition of the values of attributes, various ways have been presented in the existing methods in Tables 1 and 2. They can be classified into acquisition from vendor documents and benchmark data, estimation based on experiments, prediction based on simulation, and evaluation based on expert experience according to their nature. In the proposed OBO determination method, the values of the attributes of ABOs can also be obtained by these ways.
For the transformation of the values of attributes into FNs, a specific membership function is required since the values of attributes are generally in crisp numbers. Brauers et al. (2008) tested the total ratio, Scharlig ratio, Weitendorf ratio, Van Delft and Nijkamp ratio, Juttler ratio, Stopp ratio, Korth ratio, and Peldschus ratio models and found that the best ratio model for transforming crisp numbers into FNs in MADM is the square root of the sum of squares of each option per attribute for such denominator. This ratio model is used as the membership function to transform attribute values into FNs in the proposed method.
On the basis of the description above and the basic principle of MADM, the quantification of attributes of ABOs in the proposed method is carried out according to the following steps:
(1) Identify the attributes of ABOs and their relationships.
(2) Obtain the values of the attributes of ABOs.
(3) Describe an OBO determination problem formally. In general, a MCDM problem can be described by a set of options O = {O_{1}, O_{2}, …, O_{m}}, a set of attributes A = {A_{1}, A_{2}, …, A_{n}}, a vector of weights w = [w_{1}, w_{2}, …, w_{n}], and a decision matrix M = [v_{i,j}]_{m×n} (i = 1, 2, …, m; j = 1, 2, …, n), where O_{1}, O_{2}, …, O_{m} are m different options, A_{1}, A_{2}, …, A_{n} are n different attributes, w_{1}, w_{2}, …, w_{n} are respectively the weights of A_{1}, A_{2}, …, A_{n} such that 0 ≤ w_{1}, w_{2}, …, w_{n} ≤ 1 and w_{1} + w_{2} +… + w_{n} = 1, and v_{i,j} is the numerical value of the j-th attribute of the i-th option. According to the set O, the set A, and the matrix M, the MADM problem can be described as: Making a decision with the help of a ranking of the elements of O based on A, w, and M. For an OBO determination problem, the ABOs, the attributes of ABOs, the relative importance of the attributes of ABOs, and the values of the attributes of ABOs can be respectively regarded as the options (O_{1}, O_{2}, …, O_{m}), the attributes (A_{1}, A_{2}, …, A_{n}), the weights of attributes (w_{1}, w_{2}, …, w_{n}), and the values of attributes (v_{i,j}) of a MADM problem. Based on this, the OBO determination problem is formally described as: Determining the OBO with the help of a ranking of the elements of O based on A, w, and M.
(5) Construct a fuzzy decision matrix. According to the transformation results, a fuzzy decision matrix is constructed as M_{F} = [r_{i,j}]_{m×n} (i = 1, 2, …, m; j = 1, 2, …, n).
(6) Normalise the fuzzy decision matrix. A MADM problem may have two types of attributes, i.e. benefit attributes and cost attributes. They respectively have positive and negative influences on decision making. To remove the influences of different types of attributes, the fuzzy decision matrix M_{F} is normalised as M_{N} = [Ξ_{i,j}]_{m×n} (i = 1, 2, …, m; j = 1, 2, …, n), where Ξ_{i,j} is <r_{i,j}> for benefit attribute A_{j} and is <1 − r_{i,j}> for cost attribute A_{j}.
Generation of the OBO
Generation of the OBO aims to generate the OBO from a certain number of ABOs. It takes the normalised fuzzy decision matrix and the identified relationships among the attributes of ABOs as input and output the OBO among the given ABOs. Based on the constructed FWPPMM and FWPPA operators, the generation of the OBO can be carried out according to the following steps:
(4) Calculate the overall attribute values of the given ABOs. If the relationships among attributes belong to HCR, then the FWPPMM operator in Eq. (12) is used to compute the overall attribute value Ξ_{i}. If the relationships among attributes belong to HPR, then the FWPPA operator in Eq. (19) is used to compute the overall attribute value Ξ_{i}.
(5) Rank the given ABOs. On the basis of the calculated overall attribute values Ξ_{i}, the given ABOs are ranked from high membership degree to low membership degree using the comparison rules of FNs.
(6) Determine the OBO. The OBO is determined as the ABO ranked first.
Examples and comparisons
In this section, two examples are firstly leveraged to illustrate the working procedure of the proposed OBO determination method. Then the effectiveness of the method is demonstrated via qualitative and quantitative comparisons to the existing methods for OBO determination.
Examples
Example 1
The values of the five attributes of the six ABOs of the part in Example 1
Source: The values of surface roughness, build time, build cost, and favourableness of AM features were cited from Zhang et al. (2016)
ABO | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|
O_{1} | 18.34 | 2.88 | 105.33 | 1588.82 | 2.72 |
O_{2} | 21.31 | 3.09 | 143.08 | 2119.19 | 1.65 |
O_{3} | 21.31 | 3.09 | 143.08 | 3646.78 | 1.65 |
O_{4} | 20.02 | 5.11 | 271.47 | 1889.14 | 1.65 |
O_{5} | 20.97 | 4.63 | 231.40 | 1474.98 | 1.00 |
O_{6} | 21.90 | 4.82 | 250.10 | 745.22 | 1.00 |
With the given five attributes of ABOs and the relationships among them, the values of the five attributes in Table 3 and the weights of the five attributes in w, determination of the OBO among the six ABOs can be carried out using the proposed method. The procedure of the determination consists of the following twelve steps:
(1) Identify the attributes of ABOs and their relationships. The attributes of the six ABOs are identified as surface roughness (A_{1}), build time (A_{2}), build cost (A_{3}), volume of support structure (A_{4}), and favourableness of AM features (A_{5}). The four attributes are divided into two partitions {A_{1}, A_{5}} and {A_{2}, A_{3}, A_{4}}, where A_{1} and A_{5} are correlative, A_{2}, A_{3}, and A_{4} are correlative, and {A_{1}, A_{5}} and {A_{2}, A_{3}, A_{4}} are independent of each other.
(2) Obtain the values of the attributes of ABOs. The values of the surface roughness, build time, build cost, and favourableness of AM features of each ABO are cited from Zhang et al. (2016), which were estimated using the prediction models in an AM process planning platform named KARMA Tool (http://karma.aimme.es/app). The values of the volume of support structure of each ABO were obtained via using Meshmixer to generate the support structure of each ABO and using MeshLab to compute the volume of the generated support structure of each ABO.
(3) Describe an OBO determination problem formally. Based on the given O = {O_{1}, O_{2}, O_{3}, O_{4}, O_{5}, O_{6}}, A = {A_{1}, A_{2}, A_{3}, A_{4}, A_{5}}, w = [0.2, 0.2, 0.2, 0.2, 0.2], and M = [v_{i,j}]_{6×5} (i = 1, 2, 3, 4, 5, 6; j = 1, 2, 3, 4, 5; the values of v_{i,j} are listed in Table 3), the OBO determination problem in the example can be formally described as: Determining the OBO with the help of a ranking of the elements of O based on A, w, and M.
The FNs that stand for the values of the five attributes of the six ABOs of the part in Example 1
ABO | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|
O_{1} | <0.3621> | <0.2907> | <0.2142> | <0.3080> | <0.6490> |
O_{2} | <0.4208> | <0.3119> | <0.2910> | <0.4108> | <0.3937> |
O_{3} | <0.4208> | <0.3119> | <0.2910> | <0.7069> | <0.3937> |
O_{4} | <0.3953> | <0.5157> | <0.5520> | <0.3662> | <0.3937> |
O_{5} | <0.4141> | <0.4673> | <0.4706> | <0.2859> | <0.2386> |
O_{6} | <0.4324> | <0.4865> | <0.5086> | <0.1445> | <0.2386> |
(5) Construct a fuzzy decision matrix. According to the transformation results, a fuzzy decision matrix is constructed as:
where the values of r_{i,j} are listed in Table 4.
(10) Calculate the overall attribute values of the given ABOs. Because the relationships among A_{1}, A_{2}, A_{3}, A_{4}, and A_{5} belong to HCR, the FWPPMM operator in Eq. (12) is used to compute the overall attribute values of the six ABOs [When adapting the FWPPMM operator, Δ = (δ_{1}, δ_{2}), δ_{1} = (1, 2), and δ_{2} = (1, 2, 3)]. The computed results are as follows:
Ξ_{1} = <0.4503>; Ξ_{2} = <0.3691>; Ξ_{3} = <0.3141>; Ξ_{4} = <0.3010>; Ξ_{5} = <0.3120>; Ξ_{6} = <0.3246>
(11) Rank the given ABOs. According to the comparison rules of FNs, the six ABOs are ranked from high membership degree to low membership degree as: O_{1} ≻ O_{2} ≻ O_{6} ≻ O_{3} ≻ O_{5} ≻ O_{4}
(12) Determine the OBO. Based on the ranking, the OBO is determined as O_{1}.
Example 2
The values of the five attributes of the seven ABOs of the part in Example 2
Source: The values were cited from Zhang et al. (2016)
ABO | A_{1} | A_{2} | A_{3} | A_{4} | A_{5} |
---|---|---|---|---|---|
O_{1} | 5.40 | 4.32 | 57 | 510 | 1.65 |
O_{2} | 3.56 | 4.27 | 56 | 650 | 2.72 |
O_{3} | 5.76 | 4.32 | 57 | 712 | 1.65 |
O_{4} | 3.85 | 4.27 | 56 | 702 | 2.72 |
O_{5} | 8.32 | 6.83 | 84 | 395 | 0.85 |
O_{6} | 5.29 | 7.56 | 92 | 652 | 1.95 |
O_{7} | 8.35 | 6.79 | 83 | 522 | 0.85 |
With the given five attributes of ABOs and the relationships among them, the values of the five attributes in Table 5 and the weights of the five attributes in w, determination of the OBO among the seven ABOs can be carried out using the proposed method. The procedure of the determination consists of the following twelve steps:
(1) Identify the attributes of ABOs and their relationships. The attributes of the seven ABOs are identified as surface roughness (A_{1}), build time (A_{2}), build cost (A_{3}), volume of support structure (A_{4}), and favourableness of AM features (A_{5}). The five attributes are divided into three partitions {A_{1}, A_{5}}, {A_{2}, A_{3}} and {A_{4}}, where the relationship among these partitions is the heterogeneous priority relationship {A_{4}} ≻ {A_{2}, A_{3}} ≻ {A_{1}, A_{5}}.
(10) Calculate the overall attribute values of the given ABOs. Because the relationships among A_{1}, A_{2}, A_{3}, A_{4}, and A_{5} belong to HPR, the FWPPA operator in Eq. (19) is used to compute the overall attribute values of the seven ABOs:
Ξ_{1} = <0.5812>; Ξ_{2} = <0.4951>; Ξ_{3} = <0.4694>; Ξ_{4} = <0.4889>; Ξ_{5} = <0.5818>; Ξ_{6} = <0.4546>; Ξ_{7} = <0.5180>
(11) Rank the given ABOs. According to the comparison rules of FNs, the seven ABOs are ranked from high membership degree to low membership degree as: O_{5} ≻ O_{1} ≻ O_{7} ≻ O_{2} ≻ O_{4} ≻ O_{3} ≻ O_{6}
(12) Determine the OBO. Based on the ranking, the OBO is determined as O_{5}.
Comparisons
Qualitative comparison
In general, a qualitative comparison among different OBO determination methods can be made through comparing their characteristics. Section 2 classified the existing OBO determination methods into MOO and MADM methods and briefly described the major characteristics of each category of methods. In a nutshell, the computation cost of most MOO methods would be expensive as the number of ABOs or objectives increases. MOO algorithms may encounter difficulty in forming the Pareto front when dealing with some optimisation cases with more than three objectives, which are common in most OBO determination problems. In addition, large number of optimal solutions on the Pareto front would bring difficulty to later determination (Zhang et al. 2018). MADM methods avoid these issues to some extent. In a MADM method, the computation time generally increases in a polynomial relation with the number of ABOs or attributes, and the number of the optimal solutions is usually one or several as the number of ABOs or attributes increases. However, the existing MADM methods have not handled the complex relationships among attributes comprehensively and nor taken into account the reducing of the negative effect of the deviation of attribute values on the OBO determination result.
Compared to the existing MOO methods, the proposed method does not need expensive computation cost [It is not difficult to obtain that the time complexity of the proposed method is polynomial from Eqs. (12) and (19)] and will not generate those Pareto-optimal solutions that make determination difficult when the number of attributes is more than three. Compared to the existing MADM methods, the proposed method not only can deal with the complex relationships among the attributes of ABOs, but also reduce the influence of the deviation of attribute values on the OBO determination result.
Quantitative comparison
The details and results of the quantitative comparison
Method | Benchmark | Relationships | Ranking | OBO |
---|---|---|---|---|
Cheng et al. (1995) | Example 1 | Independent | O_{1} ≻ O_{3} ≻ O_{2} ≻ O_{6} ≻ O_{5} | O_{1} |
Pham et al. (1999) | Example 1 | Independent | – | O_{1} |
Pandey et al. (2004) | Example 1 | Independent | – | O_{1} |
Byun and Lee (2006) | Example 1 | Independent | – | O_{1} |
Canellidis et al. (2009) | Example 1 | Independent | – | O_{1} |
Zhang et al. (2016) | Example 1 | Correlative | O_{1} ≻ O_{2} = O_{3} ≻ O_{5} ≻ O_{6} ≻ O_{4} | O_{1} |
Ransikarbum and Kim (2017) | Example 1 | Independent | O_{1} ≻ O_{3} (O_{2}) ≻ O_{2} (O_{3}) ≻ O_{5} ≻ O_{6} | O_{1} |
Qie et al. (2018) | Example 1 | Independent | O_{6} ≻ O_{2} ≻ O_{5} ≻ O_{4} ≻ O_{1} ≻ O_{3} | O_{6} |
The proposed method | Example 1 | HC | O_{1} ≻ O_{2} ≻ O_{6} ≻ O_{3} ≻ O_{5} ≻ O_{4} | O_{1} |
Zhang et al. (2016) | Example 2 | Correlative | O_{5} ≻ O_{1} ≻ O_{7} ≻ O_{2} ≻ O_{4} ≻ O_{3} ≻ O_{6} | O_{5} |
The proposed method | Example 2 | HP | O_{5} ≻ O_{1} ≻ O_{7} ≻ O_{2} ≻ O_{4} ≻ O_{3} ≻ O_{6} | O_{5} |
As can be seen from Table 6, the OBO of the proposed method is the same as the OBOs of the methods of Cheng et al. (1995), Pham et al. (1999), Pandey et al. (2004), Byun and Lee (2006), Canellidis et al. (2009), Zhang et al. (2016), and Ransikarbum and Kim (2017) when the used benchmark is Example 1. These indicate its feasibility and effectiveness. In addition, the ranking of the proposed method is exactly the same as the ranking of the method of Zhang et al. (2016) when the used benchmark is Example 2, which also indicates the feasibility and effectiveness of the proposed method.
The details of the two additional comparison experiments
Experiment | Purpose | Used operators | Benchmark | Adjusted element | Results |
---|---|---|---|---|---|
Experiment 1 | Independent and ignore deviation effect versus deal with HCR and reduce deviation effect | SWA vs. FWPPMM | Example 1 | Ξ_{1,2} and Ξ_{1,3} | Figure 6 |
Experiment 2 | Independent and Ignore deviation effect versus deal with HPR and reduce deviation effect | SWA vs. FWPPA | Example 2 | Ξ_{5,4} | Figure 7 |
The values of the adjusted elements in the two additional comparison experiments
Adjusted element | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|
Ξ_{1,2} of M_{N} in Example 1 | 0.7093 | 0.6093 | 0.5093 | 0.4093 | 0.3093 | 0.2093 | 0.1093 | 0.0093 |
Ξ_{1,3} of M_{N} in Example 1 | 0.7858 | 0.6858 | 0.5858 | 0.4858 | 0.3858 | 0.2858 | 0.1858 | 0.0858 |
Ξ_{5,4} of M_{N} in Example 2 | 0.7520 | 0.7220 | 0.6920 | 0.6620 | 0.6320 | 0.6020 | 0.5720 | 0.5420 |
As can be summarised from the quantitative comparison and the two experiments, the proposed method is feasible and effective for solving the OBO determination problems, and has distinctive capabilities of dealing with the HCR and HPR among attributes and reducing the effect of the deviation of attribute values compared to the existing methods.
Conclusion
In this paper, a method for determining the optimal build orientations from a finite set of alternatives is proposed based on fuzzy multi-attribute decision making. This method firstly quantifies the attributes of all alternative build orientations, and then generates the optimal build orientations from the alternatives. The quantification of attributes transforms the values of attributes into fuzzy numbers using a ratio model. A fuzzy decision matrix is constructed and normalised based on the fuzzy numbers. Later, two fuzzy number aggregation operators (i.e. FWPPMM and FWPPA) have been constructed to aggregate the fuzzy information in the normalised matrix. All attribute values of each alternative are aggregated into a fuzzy number using these aggregation operators. By comparing the fuzzy numbers, the optimal solution is determined.
The major contribution of the paper is the development of a multi-attribute decision making based method for determination of the optimal build orientation from a set of alternatives. Compared with the existing methods, the developed method does not need expensive computation cost, can deal with complex relationships among the attributes of alternative build orientations, and at the same time can reduce negative influence of the distortion of attribute values. To the best of the knowledge, the method is the first multi-attribute decision making based build orientation determination method that considers both the handling of the complex relationships of build orientation attributes and the reducing of the effect of attribute value deviation.
As the present paper addresses only the determination of the optimal build orientation, future work will aim especially at the study of the generation of alternative build orientations. In process planning for AM, there are four basic tasks, which are build orientation determination, support structure generation, 3D model slicing, and tool-path planning. To develop a complete build orientation determination method for AM, the method would be extended to include support structure generation, 3D model slicing, and tool-path planning.
Supplementary material
The authors are releasing the Java implementation code of the proposed OBO determination method at GitHub. Every reader can freely access https://github.com/YuchuChingQin/OptimalBODetermination to download the code. The authors hope that the implemented method will provide a reliable basis for other decision-making tasks in manufacturing environment and foster further research in the development of other new OBO determination methods.
Notes
Acknowledgements
The authors are very grateful to Dr. Yicha Zhang at the Department of Mechanical Engineering and Design, University of Technology of Belfort-Montbéliard, France for his help in providing the STL files of the 3D models of the two parts in this paper and Dr. Peizhi Shi at the School of Computer Science, The University of Manchester, UK for his help in writing the implementation code of the constructed FWPPMM operator. The authors also would like to acknowledge the insightful comments from the two anonymous reviewers for the improvement of the paper and the financial supports by the EPSRC UKRI Innovation Fellowship (Ref. EP/S001328/1), the EPSRC Fellowship in Manufacturing (Ref. EP/R024162/1), and the EPSRC Future Advanced Metrology Hub (Ref. EP/P006930/1).
References
- Ahn, D., Kim, H., & Lee, S. (2007). Fabrication direction optimisation to minimise post-machining in layered manufacturing. International Journal of Machine Tools and Manufacture,47(3–4), 593–606.Google Scholar
- Ahsan, A. N., Habib, M. A., & Khoda, B. (2015). Resource based process planning for additive manufacturing. Computer-Aided Design,69, 112–125.Google Scholar
- Al-Ahmari, A. M., Abdulhameed, O., & Khan, A. A. (2018). An automatic and optimal selection of parts orientation in additive manufacturing. Rapid Prototyping Journal,24(4), 698–708.Google Scholar
- Alexander, P., Allen, S., & Dutta, D. (1998). Part orientation and build cost determination in layered manufacturing. Computer-Aided Design,30(5), 343–356.Google Scholar
- Ancău, M., & Caizar, C. (2010). The computation of Pareto-optimal set in multicriterial optimisation of rapid prototyping processes. Computers & Industrial Engineering,58(4), 696–708.Google Scholar
- Brauers, W. K. M., Zavadskas, E. K., Peldschus, F., & Turskis, Z. (2008). Multi-objective decision-making for road design. Transport,23(3), 183–193.Google Scholar
- Brika, S. E., Zhao, Y. F., Brochu, M., & Mezzetta, J. (2017). Multi-objective build orientation optimisation for powder bed fusion by laser. Journal of Manufacturing Science and Engineering,139(11), 111011.Google Scholar
- Byun, H. S., & Lee, K. H. (2006). Determination of the optimal build direction for different rapid proto-typing processes using multi-criterion decision making. Robotics and Computer-Integrated Manufacturing,22(1), 69–80.Google Scholar
- Canellidis, V., Giannatsis, J., & Dedoussis, V. (2009). Genetic-algorithm-based multi-objective optimisation of the build orientation in stereolithography. International Journal of Advanced Manufacturing Technology,45(7–8), 714–730.Google Scholar
- Chen, Y. H., Yang, Z. Y., & Ye, R. H. (2008). A fuzzy decision making approach to determine build orientation in automated layer-based machining. In Proceedings of the 2008 IEEE international conference on automation and logistics (pp. 1–6).Google Scholar
- Cheng, W., Fuh, J. Y., Nee, A. Y., Wong, Y. S., Loh, H. T., & Miyazawa, T. (1995). Multi-objective optimisation of part-building orientation in stereolithography. Rapid Prototyping Journal,1(4), 12–23.Google Scholar
- Chowdhury, S., Mhapsekar, K., & Anand, S. (2018). Part build orientation optimisation and neural network-based geometry compensation for additive manufacturing process. Journal of Manufacturing Science and Engineering,140(3), 031009.Google Scholar
- Delfs, P., Tows, M., & Schmid, H. J. (2016). Optimised build orientation of additive manufactured parts for improved surface quality and build time. Additive Manufacturing,12, 314–320.Google Scholar
- Dubois, D., & Prade, H. (1980). Fuzzy sets and systems: Theory and applications. New York: Academic Press.Google Scholar
- Dutta, B., & Guha, D. (2015). Partitioned Bonferroni mean based on linguistic 2-tuple for dealing with multi-attribute group decision making. Applied Soft Computing,37, 166–179.Google Scholar
- Ezair, B., Massarwi, F., & Elber, G. (2015). Orientation analysis of 3D objects toward minimal support volume in 3D printing. Computers & Graphics,51, 117–124.Google Scholar
- Gao, W., Zhang, Y., Ramanujan, D., Ramani, K., Chen, Y., Williams, C. B., et al. (2015). The status, challenges, and future of additive manufacturing in engineering. Computer-Aided Design,69(12), 65–89.Google Scholar
- Gibson, I., Rosen, D. W., & Stucker, B. (2015). Additive manufacturing technologies: 3D printing, rapid prototyping, and direct digital manufacturing (2nd ed.). New York: Springer.Google Scholar
- Golmohammadi, A. H., & Khodaygan, S. (2019). A framework for multi-objective optimisation of 3D part-build orientation with a desired angular resolution in additive manufacturing processes. Virtual and Physical Prototyping,14(1), 19–36.Google Scholar
- Huang, R., Dai, N., Li, D., Cheng, X., Liu, H., & Sun, D. (2018). Parallel non-dominated sorting genetic algorithm-II for optimal part deposition orientation in additive manufacturing based on functional features. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science,232(19), 3384–3395.Google Scholar
- Hur, S. M., Choi, K. H., Lee, S. H., & Chang, P. K. (2001). Determination of fabricating orientation and packing in SLS process. Journal of Materials Processing Technology,112(2–3), 236–243.Google Scholar
- Hur, J., & Lee, K. (1998). The development of a CAD environment to determine the preferred build-up direction for layered manufacturing. International Journal of Advanced Manufacturing Technology,14(4), 247–254.Google Scholar
- ISO 17296-2. (2015). Additive manufacturing—General principles—Part 2: Overview of process categories and feedstock. Geneva: International Organisation for Standardisation.Google Scholar
- ISO 17296-3. (2014). Additive manufacturing—General principles—Part 3: Main characteristics and corresponding test methods. Geneva: International Organisation for Standardisation.Google Scholar
- ISO/ASTM 52900. (2015). Additive manufacturing—General principles—Terminology. Geneva: International Organisation for Standardisation.Google Scholar
- Jaiswal, P., Patel, J., & Rai, R. (2018). Build orientation optimisation for additive manufacturing of functionally graded material objects. International Journal of Advanced Manufacturing Technology,96(1–4), 223–235.Google Scholar
- Jiang, J., Lou, J., & Hu, G. (2019a). Effect of support on printed properties in fused deposition modelling processes. Virtual and Physical Prototyping. https://doi.org/10.1080/17452759.2019.1568835.CrossRefGoogle Scholar
- Jiang, J., Stringer, J., & Xu, X. (2019b). Support optimization for flat features via path planning in additive manufacturing. 3D Printing and Additive Manufacturing. https://doi.org/10.1089/3dp.2017.0124.CrossRefGoogle Scholar
- Jiang, J., Stringer, J., Xu, X., & Zhong, R. (2018a). Investigation of printable threshold overhang angle in extrusion-based additive manufacturing for reducing support waste. International Journal of Computer Integrated Manufacturing,31(10), 961–969.Google Scholar
- Jiang, J., Xu, X., & Stringer, J. (2018b). Support structures for additive manufacturing: A review. Journal of Manufacturing and Materials Processing,2(4), 64.Google Scholar
- Jiang, J., Xu, X., & Stringer, J. (2019c). Optimisation of multi-part production in additive manufacturing for reducing support waste. Virtual and Physical Prototyping. https://doi.org/10.1080/17452759.2019.1585555.CrossRefGoogle Scholar
- Kim, H. C., & Lee, S. H. (2005). Reduction of post-processing for stereolithography systems by fabrication direction optimisation. Computer-Aided Design,37(7), 711–725.Google Scholar
- Kim, D. B., Witherell, P., Lipman, R., & Feng, S. C. (2015). Streamlining the additive manufacturing digital spectrum: A systems approach. Additive Manufacturing,5(1), 20–30.Google Scholar
- Kim, D. B., Witherell, P., Lu, Y., & Feng, S. C. (2017). Toward a digital thread and data package for metals-additive manufacturing. Smart and Sustainable Manufacturing Systems,1(1), 75–99.Google Scholar
- Kulkarni, P., Marsan, A., & Dutta, D. (2000). A review of process planning techniques in layered manufacturing. Rapid Prototyping Journal,6(1), 18–35.Google Scholar
- Lan, P. T., Chou, S. Y., Chen, L. L., & Gemmill, D. (1997). Determining fabrication orientations for rapid prototyping with stereolithography apparatus. Computer-Aided Design,29(1), 53–62.Google Scholar
- Liang, J. S. (2018). An ontology-oriented knowledge methodology for process planning in additive layer manufacturing. Robotics and Computer-Integrated Manufacturing,53(10), 28–44.Google Scholar
- Luo, N., & Wang, Q. (2016). Fast slicing orientation determining and optimising algorithm for least volumetric error in rapid prototyping. International Journal of Advanced Manufacturing Technology,83(5–8), 1297–1313.Google Scholar
- Masood, S. H., Rattanawong, W., & Iovenitti, P. (2003). A generic algorithm for a best part orientation system for complex parts in rapid prototyping. Journal of Materials Processing Technology,139(1–3), 110–116.Google Scholar
- McClurkin, J. E., & Rosen, D. W. (1998). Computer-aided build style decision support for stereolithography. Rapid Prototyping Journal,4(1), 4–13.Google Scholar
- Mi, S., Wu, X., & Zeng, L. (2018). Optimal build orientation based on material changes for FGM parts. International Journal of Advanced Manufacturing Technology,94(5–8), 1933–1946.Google Scholar
- Muirhead, R. F. (1902). Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proceedings of the Edinburgh Mathematical Society,21, 144–162.Google Scholar
- Padhye, N., & Deb, K. (2011). Multi-objective optimisation and multi-criteria decision making in SLS using evolutionary approaches. Rapid Prototyping Journal,17(6), 458–478.Google Scholar
- Pandey, P. M., Thrimurthulu, K., & Reddy, N. V. (2004). Optimal part deposition orientation in FDM by using a multicriteria genetic algorithm. International Journal of Production Research,42(19), 4069–4089.Google Scholar
- Paul, R., & Anand, S. (2015). Optimisation of layered manufacturing process for reducing form errors with minimal support structures. Journal of Manufacturing Systems,36(7), 231–243.Google Scholar
- Pham, D. T., Dimov, S. S., & Gault, R. S. (1999). Part orientation in stereolithography. International Journal of Advanced Manufacturing Technology,15(9), 674–682.Google Scholar
- Qie, L., Jing, S., Lian, R., Chen, Y., & Liu, J. (2018). Quantitative suggestions for build orientation selection. International Journal of Advanced Manufacturing Technology,98(5–8), 1831–1845.Google Scholar
- Qin, Y., Qi, Q., Scott, P. J., & Jiang, X. (2019). Status, comparison, and future of the representations of additive manufacturing data. Computer-Aided Design,111(6), 44–64.Google Scholar
- Raju, M., Gupta, M. K., Bhanot, N., & Sharma, V. S. (2018). A hybrid PSO–BFO evolutionary algorithm for optimisation of fused deposition modelling process parameters. Journal of Intelligent Manufacturing. https://doi.org/10.1007/s10845-018-1420-0.CrossRefGoogle Scholar
- Ransikarbum, K. & Kim, N. (2017). Multi-criteria selection problem of part orientation in 3D fused deposition modelling based on analytic hierarchy process model: A case study. In Proceedings of the 2017 IEEE international conference on industrial engineering and engineering management (pp. 1455–1459).Google Scholar
- Singh, S., Ramakrishna, S., & Singh, R. (2017). Material issues in additive manufacturing: A review. Journal of Manufacturing Processes,25, 185–200.Google Scholar
- Strano, G., Hao, L., Everson, R. M., & Evans, K. E. (2011). Multi-objective optimisation of selective laser sintering processes for surface quality and energy saving. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture,225(9), 1673–1682.Google Scholar
- Taufik, M., & Jain, P. K. (2013). Role of build orientation in layered manufacturing: A review. International Journal of Manufacturing Technology and Management,27(1–3), 47–73.Google Scholar
- Thrimurthulu, K. P., Pandey, P. M., & Reddy, N. V. (2004). Optimum part deposition orientation in fused deposition modelling. International Journal of Machine Tools and Manufacture,44(6), 585–594.Google Scholar
- West, A. P., Sambu, S. P., & Rosen, D. W. (2001). A process planning method for improving build performance in stereolithography. Computer-Aided Design,33(1), 65–79.Google Scholar
- Xiao, X., & Joshi, S. (2018). Automatic toolpath generation for heterogeneous objects manufactured by directed energy deposition additive manufacturing process. Journal of Manufacturing Science and Engineering,140(7), 071005.Google Scholar
- Xu, J., Gu, X., Ding, D., Pan, Z., & Chen, K. (2018). A review of slicing methods for directed energy deposition based additive manufacturing. Rapid Prototyping Journal,24(6), 1012–1025.Google Scholar
- Xu, F., Loh, H. T., & Wong, Y. S. (1999). Considerations and selection of optimal orientation for different rapid prototyping systems. Rapid Prototyping Journal,5(2), 54–60.Google Scholar
- Yager, R. R. (2001). The power average operator. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans,31(6), 724–731.Google Scholar
- Yager, R. R. (2008). Prioritised aggregation operators. International Journal of Approximate Reasoning,48(1), 263–274.Google Scholar
- Zadeh, L. A. (1965). Fuzzy sets. Information and Control,8, 338–353.Google Scholar
- Zhang, Y., & Bernard, A. (2014). An integrated decision-making model for multi-attributes decision-making (MADM) problems in additive manufacturing process planning. Rapid Prototyping Journal,20(5), 377–389.Google Scholar
- Zhang, Y., Bernard, A., Gupta, R. K., & Harik, R. (2016). Feature based building orientation optimisation for additive manufacturing. Rapid Prototyping Journal,22(2), 358–376.Google Scholar
- Zhang, Y., Bernard, A., Harik, R., & Karunakaran, K. P. (2017). Build orientation optimisation for multi-part production in additive manufacturing. Journal of Intelligent Manufacturing,28(6), 1393–1407.Google Scholar
- Zhang, Y., Harik, R., Fadel, G., & Bernard, A. (2018). A statistical method for build orientation determination in additive manufacturing. Rapid Prototyping Journal,25(1), 187–207.Google Scholar
- Zhang, J., & Li, Y. (2013). A unit sphere discretisation and search approach to optimise building direction with minimised volumetric error for rapid prototyping. International Journal of Advanced Manufacturing Technology,67(1–4), 733–743.Google Scholar
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