An approach to budget allocation for an aerospace company—Fuzzy analytic hierarchy process and artificial neural network

ARTICLE IN PRESS Neurocomputing 72 (2009) 3477–3489

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

An approach to budget allocation for an aerospace company—Fuzzy analytic hierarchy process and artificial neural network Yu-Cheng Tang  Department of Accounting, National Changhua University of Education, No. 2, Shi-Da Road, Changhua 500, Taiwan

a r t i c l e in fo

abstract

Available online 12 June 2009

Budgetary allocations of resources are made in all businesses, but their volume and composition vary; and efficient budget allocation is fundamental to flow in businesses. The objective of the allocation problem is to determine the required budget for each department (or section) of a company so as to maximize the sum of the company’s benefits. The purpose of this paper is to find a suitable degree of fuzziness for preference rankings and to demonstrate an example of budget allocation using artificial intelligence programming, and fuzzy analytic hierarchy process (FAHP). An efficient budget allocation method using FAHP will be provided for businesses. This method is suitable for use in evaluating proposed policies (including tangible and intangible information). A comparison between FAHP and artificial neural network (ANN) will be also made in this paper. An aerospace company’s budget allocation problem is investigated as a case study in this research, which will illustrate how to solve this problem. The case study utilizes a two-stage interview (semistructured interview and in-depth interview) to select their budget allocations given a number of tangible and intangible criteria. The results from the case study are pertinent to other real-world allocation problems that share many of the characteristics of problems, such as decision makers’ subjective opinions. & 2009 Elsevier B.V. All rights reserved.

Keywords: Artificial intelligence Artificial neural network Budget allocation Fuzzy analytic hierarchy process Tangible criteria Intangible criteria Sensitivity analysis

1. Introduction Budgetary allocations of resources are made in all businesses, but their volume and composition vary; and efficient budget allocation is fundamental to flow in businesses. The central focus of budgeting is the allocation of financial resources between alternative uses over time with the aim of achieving some specified rates of return on current investments in the future [19]. Budget allocation is crucial to all manufacturing industries, especially hightech ones [34]. This problem is complicated owing to competing tasks and tight deadline requirements. Budget allocation typically begins by considering the investments that each facility would like to make in order to meet its needs and objectives [30,32]. However, budgets are usually limited; these potential investments are prioritized by evaluating them against a set of criteria. In the complex world of business, decisions made by decision makers (DMs) tend to deal with subjective interpretations of uncertain quantitative and qualitative information [6]. Moreover, DMs are usually willing or able to provide only incomplete information because of time pressure, lack of knowledge or data, and/or due to their limited expertise related to the problem

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situation [21]. This kind of decision-making and uncertainty in preference judgements can be modeled using the artificial intelligence (AI). AI is an advanced branch of science that studies the process of human thinking and attempts to apply this knowledge to simulate the same process in machines. There has been much research and development in the field of AI. AI consists of several branches, such as artificial neural networks (ANNs), genetic algorithms (GAs), fuzzy logic (FL), expert systems, problem solving and planning, nonmonotonic reasoning, logic programming, natural language processing, computer vision, robotics, learning, planning and various hybrid systems, which are combinations of two or more of the branches, etc., refer to [42–45]. The area of research includes speech and pattern recognition [4,20,24], natural language processing [20], learning from previous experiences [10], reasoning under situations providing limited or incomplete information, etc. AI is practically applied in the field of computer games, expert systems, neural networks, robotics, resource (budget) allocation [1,5,17,18,27,29,31,55] and many other fields of science and technology, especially utilized in elicited experts’ opinions. The methods for elicited experts’ opinions, such as Delphi method [3,17] and analytic hierarchy process (AHP) [28] are well known. Delphi method [3] was developed in the 1960s by the Rand Corporation at Santa Monica, CA. This technique attempts to

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develop forecasts through group consensus [13]. Delphi is a very flexible methodology as it combines both quantitative and qualitative elements and can be adapted to suit different research objectives [2]. However, it is a decision facilitation tool rather than a decision-making tool, which investigates policy issues and contributes to informed decision-making [26,33]. The AHP is to determine the relative ranking or preferences of decision alternatives. The fuzzy set theory can address imprecision in the data presented in the AHP, as well as offer a more linguistic interpretation of the preference judgements made [39]. The concepts of the fuzzy set theory have been integrated with the AHP, in the method referred to as fuzzy AHP (FAHP) [33]. FAHP is being used more and more frequently in multi-criteria decisionmaking because of its simplicity and similarity to human reasoning [38]. This method is suitable for use in evaluating proposed policies (including tangible and intangible information). Furthermore the FAHP also allows group decision-making [7] to derive priorities based on sets of pairwise comparisons [40]. FAHP can also be applied to the budget allocation problem in other industries to achieve a desired achievement level in a fuzzy environment. For more details about FAHP, refer to [41]. One of the powerful techniques of AI, artificial neural network (ANN), is the use of nonlinear statistical paradigms for recognizing complex patters. The development of ANNs usually generates a huge amount of data because it should represent the system being modeled as accurately as possible. ANN models have self-learning ability, by adjusting their parameters to reduce the error of estimation [58]. They are able to maintain accuracy when some data required for complete network function are missing [48]. The greatest advantage of ANNs over other modeling techniques is their capability to model complex, nonlinear processes without having to assume the form of the relationship between input and output variables [46]. The case study, the aviation industry, utilized in this paper is a unique field from a business, legal and regulatory perspective. The objective is to provide an efficient and fair method for the aviation industry. This paper analyses budget allocation in the aviation industry of Taiwan using the FAHP method and also compares this with the ANN technique. The remainder of the paper is organized as follows. Section 2 describes the techniques of the ANN, and the FAHP. Section 3 illustrates the case study of a Taiwanese manufacturing company. Section 4 presents the results and demonstrates the comparisons between the ANN techniques and the FAHP. Finally, conclusions will be presented in Section 5.

2. Artificial intelligence (AI) techniques AI could be defined as the ability of computer software and hardware to do those things that we, as humans, recognize as intelligent behavior [54]. There are several branches within it. Nevertheless only the ANN and the FAHP techniques will be detailed in this paper.

these layers. To perform, preprocessed data are fed to the various neurons in the first layer—the input layer. The range of outputs from each neuron becomes the domain for one or more neurons in the next layer via synapses (or connections). A backpropagation neural (BPN) [59] network architecture is chosen for constructing each ANN model. BPN training method was applied in the ANN model training. In the BPN training process, the gradient of the performance function was used to determine how to adjust the weights to minimize the mean square error (MSE), and the average square error between the actual output and the desired output [58]. The following subsections illustrate the basic concepts of ANN with in this paper. 2.1.1. Layer The architecture of a typical ANN comprises three layers of interconnected nodes or neurons, each of which is connected to all the neurons in the successive layer (Fig. 1). An input layer is the layer where data are presented to the neural network whilst an output layer holds the response of the network to the input [45,46]. The layers between the input and output layers are the hidden layers. There may be one or more hidden layers. Neurons in each layer are fully or partially interconnected to preceding and subsequent layer neurons with each interconnection having an associated connection weight. 2.1.2. Processing element (PE) A neural network is a collection of small individually interconnected processing units, namely, nodes (called variously ‘‘neurons’’, ‘‘PEs’’ (‘‘processing elements’’) or ‘‘units’’) are connected together to form a network of neurons—hence the term ‘‘neural network.’’ Information is passed through these nodes along interconnections. An incoming connection has two values associated with it, an input value and a weight. The output of the unit is a function of the summed value [42]. Usually the sums of each neuron are weighted, and the sum is passed through a linear or nonlinear function known as an activation function (or transfer function). The different activation function will produce different output. The basic model of a neuron is illustrated in Fig. 2. In Fig. 2 the neuron model consists of two parts: the net function and the activation function [47]. The net function copes with how the network inputs {xi; 1rirN} are combined inside the neuron. An adder sums up all the inputs modified by their respective weights. In Fig. 2 the activity is referred to as linear combination. The output of the neuron, Oi, is related to the network input xi via a linear or nonlinear transformation called the activation function; y ¼ f(z). Finally, an activation function controls the amplitude of the output of the neuron. The most commonly used activation functions are sigmoid, hyperbolic tangent and linear [42,47]. Based on the above description, each neuron in the output layer determines its activity by following two step procedures.

2.1. Artificial neural networks (ANNs) ANN often called a ‘‘neural network’’ (NN), is a mathematical model or computational model based on biological neural networks. Much of the research on ANN have focused on accounting and finance problems [64], with special attention to bankruptcy prediction [60], credit evaluation [61], insolvency prediction [62], fraud detection [63], etc. The traditional ANN is a set of connected processing elements. Each of these processing elements (or neurons) performs a mathematic function. These neurons are then grouped together into layers. A complete network is made up of two or more of

Fig. 1. General configuration of an artificial neural network.

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x1

w1

Inputs . . xi

Σ wi

Activation Function Oi z θ

the delta rule for nonlinear activation functional and multilayer networks.

y Output

Weights Threshold summation or net function Fig. 2. Basic model of artificial neuron [69].

Step 1: Compute the total weighted input xj X xj ¼ zi wij

(1)

i

where zi is the activity level of the jth unit in the previous layer and wij is the weight of the connection between the ith and the jth unit. Step 2: Calculate the activity zj using activation function of the total weighted input. The sigmoid function, one of the activation functions, as an example here. zj ¼

1 1 þ exj

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2.1.5. Backpropagation neural BPN was first described by Werbos in 1974 [49] and recognized through the work of Rumelhart, Hinton and Williams [50] in 1986, and it led to a ‘‘renaissance’’ in the field of artificial neural network research. BPNs are one of the most common neural network structures [51], as they are simple and effective, and have found a home in a wide assortment of machine learning applications, such as character recognition. The central idea behind neruo network is that the errors for the units of the hidden layer are determined by BPN. To train the neural network, we must try to minimize this error. To minimize the error the neuron connection weights and biases must be modified. The error function is defined to calculate the error of the neural network. The most popular way to minimize the error function is the gradient descent method. The error function E is defined by E¼

1X ðz  Oj Þ2 2 j j

(3)

(2)

where Oj is the desired output of the jth unit. Unit j is a typical unit in the output layer and unit i is a typical unit in previous layer.

2.1.3. Learning function Learning function is mathematical procedure used to automatically adjust the network’s weights and biases. The learning function can be applied to individual weights and biases within a network. The cost function C is an important concept in learning, as it is a measure of how far away we are from an optimal solution to the problem that we want to solve. Learning algorithms search through the solution space in order to find a function that has the smallest possible cost. There are three major learning paradigms, each corresponding to a particular abstract learning task. These are supervised learning, unsupervised learning and reinforcement learning. Usually any given type of network architecture can be employed in any of those tasks. In supervised training, both the inputs and the outputs are provided. The network then processes the inputs and compares its resulting outputs against the desired outputs. Errors are then propagated back through the system, causing the system to adjust the weights which control the network. A commonly used cost is the MSE which tries to minimize the average error between the network’s output, f(x), and the target value y over all the example pairs. In unsupervised training, the network is provided with inputs but not with desired outputs. The system itself must then decide what features it will use to group the input data. This is often referred to as self-organization or adaption.

2.1.6. Four steps of BPN algorithm Step 1: Compute how fast the error changes as the activity of an output unit is changed.

2.1.4. Multilayer perceptron Each neuron in one layer connects with a certain weight to every other neuron in the following layer. These networks are commonly referred to as multilayer perceptron (feed-forward) which consists of an input and an output layer with one or more hidden layers of nonlinearly activating neurons. To train a neural network to perform some task, we must adjust the weights of each unit in such a way that the error between the desired output and the actual output is reduced. This process requires that the neural network compute the error derivative of the weights (EW). In other words, it must calculate how the error changes as each weight is increased or decreased slightly. The BPN [52,53] is the most widely used method for determining EW. BPN can also be considered as a generalization of

EAj ¼

@E ¼ zj  Oj @zj

(4)

The error derivative (EA) is the difference between the actual and the desired output. Step 2: Compute how fast the error changes as the total input received by an output unit is changed. This quantity (EI) is the answer from step 1 multiplied by the rate at which the output of a unit changes as its total input is changed. EIj ¼

@E @E @zj ¼ ¼ EAj zj ð1  zj Þ @xj @zj @xj

(5)

Step 3: Compute how fast the error changes as a weight on the connection into an output unit is changed. EW is the error derivative of the weights. EW ij ¼

@E @E @xj ¼ ¼ EIj zi @wij @xj @wij

(6)

Step 4: Compute how fast the error changes as the activity of a unit in the previous layer is changed. To compute the overall effect on the error, all these separate effects on output units should be added together. It is the answer in step 2 multiplied by the weight on the connection to that output unit. EAi ¼

@E X dE dxj X ¼  ¼ EIj wij @zi dxj dzi j j

(7)

In this paper, the gradient descent BPN algorithm is utilized for training data. Training of the ANNs was done using the commercial software NeuroSolutions [56]. 2.2. Triangular fuzzy numbers (TFNs) Central to the fuzzy set theory is the notion of the fuzzy number. In applications it is often convenient to work with TFNs because of their computational simplicity [16,25], and they are useful in promoting representation and information processing in

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a fuzzy environment [24]. In addition, TFNs are the most utilized in FAHP studies (e.g., [11,14,22,36]). A TFN A can be defined by a triplet (l, m, u) and the membership function mA ðxÞ can be defined by [14,27] 8 xl > > ; lxm > > > >u  m > : 0 otherwise where x is the mean value of A and l, m, u are real numbers. In this paper only the three relevant algebraic operations are illustrated. Define two TFNs A and B by the triplets A ¼ (l1, m1, u1) and B ¼ (l2, m2, u2). Then (i) addition: A(+)B ¼ (l1, m1, u1)+(l2, m2, u2) ¼ (l1+l2, m1+m2, u1+u2), (ii) multiplication: A  B ¼ (l1, m1, u1)  (l2, m2, u2) ¼ (l1l2, m1m2, u1u2), (iii) inverse: (l1, m1, u1)1E(1/u1, 1/m1, 1/l1), where E represents approximately equal to. 2.3. Construction of the FAHP comparison matrices In this paper the modified synthetic extent FAHP is utilized, which was originally introduced in [11] and developed in [36]. One reason for its employment is that it allows for incompleteness of the pairwise judgements made, though it is not the only FAHP approach to allow this (see Interval Probability Theory in [15]). This feature reflects its suitability in decision problems where uncertainty exists in the judgement-making process. The aim of any FAHP method is to place an order of preference on a number of decision alternatives, i.e., a prioritized ranking of decision alternatives. Central to this method is a series of pairwise comparisons, indicating the relative preferences between pairs of decision alternatives in the same hierarchy. The linguistic variables used to make the pairwise comparisons are those associated with the standard 9-unit scale [28]; see Table 1. It is difficult to map qualitative preferences to point estimates, and hence a degree of uncertainty is associated with some or all pairwise comparison values in an FAHP problem [57]. Using TFNs with the pairwise comparisons made, the fuzzy comparison matrix X ¼ (xij)n  n can be formed as shown below: 3 2 x11 x12 x13    x1n 7 6x 6 21 x22 x23    x2n 7 7 6 6 x31 x32 x33    x3n 7 X¼6 7 6 . .. 7 .. .. 6 .. . 7 . . 5 4 xn1 xn2 xn3    xnn

1

δ

0 vk−1

Numerical value

Definition

1 3

Equally preferred 2, 4, 6, 8 Moderately Intermediate values between the two adjacent preferred judgements Strongly preferred Very strongly preferred Extremely preferred

5 7 9

lij

vk+1 uij

Fig. 3. Description of d degree of fuzziness.

where xij is an element of the comparison matrix. The reciprocal property of the comparison matrix is that: xji ¼ 1=xij ; i, j ¼ 1,y,n; and the subscripts i and j refer to the row and column, respectively, where any entry is located, and n is the number of rows and columns. The pairwise comparisons are described by values taken from a pre-defined set of ratio scale values as presented in Table 1. The ratio comparison between the relative preference of elements indexed i and j on a criterion can be modeled through a fuzzy scale value associated with a degree of fuzziness. Then, for an element of X, xij is a fuzzy number defined as xij ¼ (lij, mij, uij), where lij, mij, uij are the lower bound, modal and upper bound values for xij, respectively. In general, given the entry mij in a fuzzy scale value has the kth value vk, then lij and uij have values either side of the vk scale value, describing the fuzziness of the judgement given in xij. In [36], this fuzziness is influenced by a degree of fuzziness d, where mijlij ¼ uijmij ¼ d. That is, the value of d is a constant and is considered an absolute distance from the lower bound value (lij) to the modal value (mij) or the modal value (mij) to the upper bound value (uij), see Fig. 3. From Fig. 3, the fuzzy number representing the fuzzy judgement made is defined by (mijd, mij, mij+d), with its associated inverse fuzzy number described by (1=mij þ d, 1=mij , 1=mij  d). In the case of mij given a value of one (mij ¼ 1) off the leading diagonal (iaj), the general form of its associated fuzzy scale value is defined as (1=1 þ d, 1, 1+d).1 For example, with mij ¼ 1, the fuzzy scale value will be (0.6667, 1, 1.5) when d ¼ 0.5.

2.4. Value of fuzzy synthetic extent Let C ¼ {C1,C2,y,Cn} be a criteria set, where n is the number of criteria and D ¼ {D1, D2,y,Dm} is a decision alternative set with m the number of decision alternatives. Let M 1C i ; M 2C i ; . . . ; Mm C i be values of extent analysis of the ith criteria for m decision alternatives. Here i ¼ 1,2,y,n and all the M jC i ðj ¼ 1; 2; . . . ; mÞ are TFNs. The value of fuzzy synthetic extent Si with respect to the ith criteria is defined as

Si ¼

m X j¼1

Table 1 Scale of relative preference based on [37].

δ vk mij

2 M jC i

4

n Y m X

31 MjC i 5

(9)

i¼1 j¼1

where  represents fuzzy multiplication and the superscript 1 represents the fuzzy inverse. The concepts of synthetic extent can also be found in [8,12,23]. To obtain the estimates for the sets of weight values under each criterion, it is necessary to consider a principle of comparison for fuzzy numbers. For example, for two fuzzy numbers M1 and M2, the degree of possibility of M1ZM2 is defined as VðM 2  M 1 Þ ¼ hgtðM 1 \ M 2 Þ ¼ mM1 ðxd Þ

(10)

1 The expression for (1=1 þ d, 1, 1+d) is supported by [37] who for the traditional AHP state that the distribution of the scale value above and below one are analogous.

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Budget Allocation

Capital Budgets C2

Operating Budgets C1 The relations between government’s and company’s strategy C11 The development ability of section

C12

Co-ordination with time table based on before’s experiences

C21

Exploitation ability of new products

C22 C23

The expenditure of materials

C13

The degree of environment pollution

The expenditure of salary

C14

The complexity of business execution C24

The expenditure of services

C15

Purchasing fixed assets

D1

D2

D3

D4

D5

D6

C25

D7

Fig. 4. The budget allocation structure of the department FSD.

VðM2  M 1 Þ ¼ hgtðM 1 \ M2 Þ ¼

l1  u2 ¼d ðm2  u2 Þ  ðm1  l1 Þ

(11)

The degree of possibility for a convex fuzzy number M to be greater than the number of k convex fuzzy numbers Mi (i ¼ 1,2,y,k) can be given by the use of the operations max and min [9] and can be defined by VðM  M 1 ; M 2 ; . . . ; Mk Þ ¼ V½ðM  M 1 Þ and ðM  M 2 Þ and . . . and ðM  M k Þ ¼ minVðM  M i Þ; 0

0

i ¼ 1; 2; . . . ; k 0

W 0 ¼ ðd ðD1 Þ; d ðD2 Þ; . . . ; d ðDm ÞÞ

(12)

0

The vector W is normalized and denoted: W ¼ ðdðD1 Þ; dðD2 Þ; . . . ; dðDm ÞÞ

(13)

When two elements (fuzzy numbers), say M1 ¼ (l1, m1, u1) and M2 ¼ (l2, m2, u2) in a fuzzy comparison matrix satisfy l1u240 then VðM 2  M 1 Þ ¼ hgtðM 1 \ M 2 Þ ¼ mM2 ðxd Þ, with mM2 ðxd Þ given by [58] 8 l1  u2 < ; l  u2 (14) mM2 ðxd Þ ¼ ðm2  u2 Þ  ðm1  l1 Þ 1 : 0 otherwise 3. Case study This section presents the descriptions of the case study Aerospace Company (referred to as AS) and the details of the budget allocation investigated throughout this study. The problem concerns an aerospace company and its allocation of company’s annual budget. Effective allocation is conducive to regular business operations and fairness of budgets allocation. The case study utilized a two-stage interview (semi-structured interview and in-depth interview) to provide a number of tangible and intangible criteria.

national aerospace development objectives, AS was transformed from a military establishment into a government-owned company under the authority of the Ministry of Economic Affairs of Taiwan. As a market-oriented commercial entity, AS is working in tandem towards commercialization, privatization and globalization. Pursuant with the new company objectives, AS’s business strategy has turned from products and services for solely military applications to a well-balanced diversified provider to both military and commercial markets. To meet the increasingly competitive environment of the commercial markets, AS underwent an organizational restructure in July 2000. This restructuring program divided AS into four business departments, namely Business Management Department, Business Development Department, Flight Services Department and Engineering Department. Functions are clearly defined, business is strategically oriented, and the overall work force is consolidated. Based on the exceptional professional standing of AS, only the vice-director of Flight Services Department (FSD) was willing to help with this case study. Therefore, the budget allocation of FSD is investigated in this case study. The FSD is currently performing tasks for the ROC Air Force and anticipates the opportunity to provide airborne test bed services to both the military as well as research organizations. The FSD has seven sections: Flight Operations Section, Engineering Section, Business Section, Flight Support Section, Field Operation Inspection Section, Flight Safety Section, and Planning and Administration Section (including Secretariat Section and the Office of Director) which are denoted as D1,D2,y,D7, respectively. These are the decision alternatives in this case study. The budgets of AS are provided every year. The allocation method depends on the department. For the FSD, budget allocation decisions are normally made by three senior superintendents, who are the vice-director of FSD, the head of Planning and Administration Section and the head of Secretariat Section (hereafter referred to as decision makers—DMs). 3.2. The details of the budget allocation problem

3.1. Descriptions of the AS company AS, was established in 1969 under the authority of Republic of China Air Force, and was later transferred to the Chung-Shan Institute of Science and Technology in 1983. In 1996 in support of

The first stage that the company engaged in was the identification of the necessary criteria to be considered, which was obtained through a semi-structured interview with the DMs who are three senior superintendents of FSD. For the allocation

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Low Low Low Low Office furniture: 0.1 Medium Low Low Medium Office furniture: 0.1 Medium Low Medium Medium Overhaul equipment: 0.5 High Medium High Medium Replace a new model equipment: 1 High High Low High Office furniture: 0.2 Medium Medium High High Replace a new model equipment: 1 C2 C21 C22 C23 C24 C25

High High Medium High New machine: 10

Low Medium None 3 0.2 Low Medium None 5 0.5 Low Medium 60 30 1.5 High High None 11 1.5 Medium High 60 30 5 Medium High None 13 1.5

D6 D5 D4 D3 D2 Section

C1 C11 C12 C13 C14 C15

2 For the two criteria, the DMs all agree to give 0.4 for C1 weight and 0.6 for C2. Therefore, it is not necessary to compare these two criteria with each other.

Criteria

Full information about these criteria and their sub-criteria is presented in Table 2. Assume AS is going to allocate a billion to FSD. Given the necessary details of the criteria, sub-criteria and decision alternatives, three DMs were asked to indicate preferences between pairs of sub-criteria2 and then between pairs of alternatives over the different sub-criteria through the structured interview. The results of the pairwise comparisons made by the DMs of FSD are illustrated in Table 3 for the comparisons between 10 sub-criteria and Tables 4 and 5 for the comparisons on subcriteria of operating budgets and capital budgets between

Table 2 Information table of the budget allocation for those seven sections (unit: million).

1. Operating budgets C1: Operating budget contains estimates of the total value of resources required for the performance of the operation including reimbursable work or services for others. It also includes estimates of workload in terms of total work units identified by cost accounts. 2. Capital budgets C2: Capital budgeting (or investment appraisal) is the planning process used to determine FSD’s long term investments such as new machinery, new aircrafts, replacement machinery, new plants, new products and research and development projects. 3. The relations between government and company strategy C11: The budgets of FSD sections will depend on how far their strategy relates to government strategy. The stronger the relationship, the higher the budget. 4. The development ability of section C12: The DMs will give a larger budget to the sections which have the potential for development and products which can meet market demand. 5. The expenditure on materials C13: If the section needs more expenditure on material or if the cost of production increases then it will be allocated a larger budget. 6. The expenditure on salary C14: A section with a large number of employees requires more expenditure on pay, and therefore, a larger budget allocation. 7. The expenditure on services C15: All expenditure other than materials and salary belongs to service expenditure. 8. Co-ordination with other organizations’ timetables based on past experiences C21: Based on previous experience, if the section has higher co-ordination with other organizations’ time table then it will be allocated a higher budget. 9. Exploitation ability of new products C22: If the section has higher ability to exploit new products then it will have higher need to invest heavily in purchasing new equipment. In this situation, the section should have a larger budget than others. 10. The degree of environment pollution C23: If the section produces more pollution then it also needs a larger budget to purchase equipment to prevent pollution. 11. The complexity of section affairs C24: If the section has more complex affairs then it might have more projects than others. In this situation, this section will need to have a larger budget than others. 12. Purchasing fixed assets C25: Apart from the basic requirements, if the section needs more fixed assets then it will be allocated a larger budget.

D7

process of the sections, two criteria and 10 sub-criteria are defined under the FSD requirements by the DMs. The two criteria are operating and capital budgets. These two criteria have their own sub-criteria constructed them in Fig. 4. The descriptions of these criteria and sub-criteria are shown below:

High Low None 10 20

Y.-C. Tang / Neurocomputing 72 (2009) 3477–3489

D1

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Table 3 Fuzzy comparison matrix based on the pairwise comparison between sub-criteria. (a) C1

C11

C12

C13

C14

C15

(b) C2

C21

C22

C23

C24

C25

C11

1

5 7 7

9 1/6 5

1 1/9 5

9 1/5 5

C21

1

5 5 3

5 6 5

5 4 1/7

3 1 3

C12

1/5 1/7 1/7

1

1/5 1/7 1/7

1/5 1/9 1/3

1/3 1/6 1/3

C22

1/5 1/5 1/3

1

5 1 5

5 1/3 1/5

1 1/4 3

C13

1/9 6 1/5

5 7 7

1

1/5 1/9 1

5 4 1

C23

1/5 1/6 1/5

1/5 1 1/5

1

1 1/4 1/7

1 1/4 1/3

C14

1 9 1/5

5 9 3

5 9 1

1

9 9 1

C24

1/5 1/4 7

1/5 3 5

1 4 7

1

1/5 1 7

C15

1/9 5 1/5

3 6 3

1/5 1/4 1

1/9 1/9 1

1

C25

1/3 1 1/3

1 4 1/3

1 4 3

5 1 1/7

1

decision alternatives. Following on from the comparison between sub-criteria, 10 further fuzzy comparison matrices are constructed to represent comparisons between decision alternatives on each of the sub-criteria.

4. Results of the FAHP and ANN analysis on the budget allocation problem 4.1. Results from the FAHP analysis 4.1.1. Determining the degrees of fuzziness Utilizing the expressions described in Sections 2.3 and 2.4, I apply the modified synthetic extent FAHP to the data on budget allocation case study also previously described. Firstly, Fuzzy numbers require a suitable degree of fuzziness. The use of sensitivity analysis in this research to determine the degrees of fuzziness will be explained. The objective of a typical sensitivity analysis is to find out how the ranking of the sub-criteria or decision alternatives will be changed when the input data (preference judgements and degrees of fuzziness) are changed into new values. To illustrate this, I first consider the comparisons made between the sub-criteria based on the DMs, listed in Table 3. There are five lines in Figs. 5a and b, respectively. Those represent the weight values associated with the different subcriteria. In Fig. 5a the numbers (with sub-criteria) on the d-axis represent the degree of fuzziness appearance points with respect to each sub-criterion. For example the degree of fuzziness d up to 0.25 (on Fig. 5a d-axis), shows that ‘‘the expenditure of salary’’—C14 has the absolute dominant preference. This means that C14 is an important sub-criterion to be considered when the DMs make decisions, so the weight value is 1. After d reaches 0.25, the sub-criterion C11 has the next priority weight. The next criterion is C13 which has a priority weight as d approaches 0.35, etc. The values of d at which the sub-criteria have positive weight values (non-zero) are hereafter referred to as appearance points. For the fuzzy comparison matrix with judgements between sub-criteria (see Fig. 5a), all the sub-criteria have positive weights when d is greater than 2.72. This means that if d is less than 2.72 some sub-criteria have no positive weights [35]. Discussion of this aspect within traditional AHP suggests that the DM does not favor

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one criterion and ignore all others, but rather places the criteria at various levels. Besides, when pairwise comparisons are made between criteria, it is expected that all weights should have positive values. It is suggested therefore that it is useful to choose a minimum workable degree of fuzziness. The expression ‘‘minimum workable degree of fuzziness’’ is defined as the largest of the values of d at the various appearance points of sub-criteria on the d-axis. When considering the final results, the domain of workable d is expressed as dT , and is defined by the maximum of the various minimum workable degrees of fuzziness throughout the problem, that is dT ¼ maxðdT C ; dT 1 ; dT 2 ; . . . ; dT n1 ; dT n Þ where the subscript T is the maximum of the minimum workable d values in the n+1 (Tc matrix) fuzzy comparison matrices. For the comparisons between decision alternatives with respect to individual sub-criteria fuzzy comparison matrices (on C11,C12,y,C15, and C21,C22,y,C25) their minimum workable d values are (0.91, 1.43, 1.82, 2.05, 3.65) and (1.7, 2.35, 1.605, 1.98, 3.65), respectively (see Figs. 6a–e and 7a–e, respectively). When considering the final results, the domain of workable d is expressed as dT , and is defined by the maximum of the various minimum workable degrees of fuzziness throughout the problem, that is dT ¼ maxð2:72; 1:76; 0:91; 1:43; 1:82; 2:05; 3:65; 1:7; 2:35; 1:605; 1:98; 3:65Þ ¼ 3:65 where the subscript T is the maximum of the minimum workable d values in the 12 fuzzy comparison matrices. It follows that for this problem the workable region of d is d43.65 and the results on weights should possibly only be considered in the workable d region. The use of the minimum workable degree of fuzziness is intended to exclude values of d at which there are no positive weights for the decision alternatives. However the use of a workable value of d is not to be strictly enforced. When d ¼ 3.65, the weight values for sub-criteria based on each criterion are derived from Tables 4 and 5 and the weight values are listed in Table 6. For example, for C1, the weight values for the comparisons among C11 to C15 are (0.2496, 0.0531, 0.2409, 0.2680, 0.1883). The last column presented in Table 6 is the weight values for decision alternatives over the different sub-criteria. For instance, for C11, the weight values for the comparisons among the three DMs are (0.1554, 0.1527, 0.1661, 0.1364, 0.1143, 0.1161, 0.1591). In Table 6, the final results show that the Engineering Section can have a larger budget allocation than the rest of sections. The next section is the D4, and D7 is the section with the least expenditure in the DMs’ minds. From the comparison between the sub-criteria based on two criteria in Table 6, the most preferred sub-criterion out of the five sub-criteria in operating budget is C14 ‘‘the expenditure of salary’’ and in capital budget, C21 ‘‘coordination with other organizations’ timetables based on previous experiences’’. This means for the operating budget in FSD, salaries take up the greater part of the budget. It shows the importance of human capital. For the capital budget in FSD, co-ordination with other organizations’ timetables is more important than other subcriteria. FSD uses previous work experiences to decide which section can have a larger budget allocation. 4.1.2. Examination of interview data The final results presented in Table 6 also can be examined in both Figs. 5–7 and Tables 3–5. Figs. 5–7 also can be compared with Tables 3–5. For example, Fig. 5a, the sub-criterion ‘‘the expenditure on salary’’ (C14) is the key factor when DMs make their judgments, as also demonstrated in Table 6. Examining from Table 3a, in row C14, DM2 places ‘‘extremely preferred’’3 when comparing C14 with C11, 3

The scale of relative preference based on Table 1.

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Table 4 Pairwise comparisons for sub-criteria of operating budgets between decision alternatives. D2

D3

D4

D5

D6

D7

(b) C12

D1

D2

D3

D4

D5

D6

D7

(c) C13

D1

D2

D3

D4

D5

D6

D7

(d) C14

D1

D2

D3

D4

D5

D6

D7

(e) C15

D1

D2

D3

D4

D5

D6

D7

D1

1

1 1 1

1/3 1 1/5

3 1 3

3 3 3

3 3 3

1/5 5 1/7

D1

1

1/5 1 1

3 1 3

1 1 4

3 3 5

3 3 5

5 5 4

D1

1

1/5 3 1/5

3 5 3

1/5 1/3 1/5

1 3 2

3 3 1

3 7 2

D1

1

1/5 1/3 1/3

1 3 2

1/5 1/4 1/3

5 5 2

5 5 2

2 3 2

D1

1

1 1/5 1/3

1 1 1/2

1 1 2

5 5 2

5 7 2

7 1/9 1/5

D2

1 1 1

1

1/3 1 1/3

3 1 2

3 3 3

3 1 3

1/5 5 1/6

D2

5 1 1

1

7 1 1

3 1 3

5 3 2

5 3 2

7 5 3

D2

5 1/3 5

1

7 5 5

1 1 1/3

5 3 4

7 3 4

7 7 5

D2

5 3 3

1

5 5 3

1 1 1

7 5 4

7 5 5

5 5 3

D2

1 5 3

1

3 5 3

3 5 3

5 7 4

5 8 4

7 1/5 1/6

D3

3 1 5

3 1 3

1

5 1 4

7 1 4

7 1 5

1/3 5 1/3

D3

1/3 1 1/3

1/71

1

1/3 1 3

1 1 3

1 1 2

3 5 2

D3

1/3 1/5 1/3

1/7 1/5 1/5

1

1/7 1/5 1/5

1/3 1/3 1/2

1 1/3 1/2

1 1 1

D3

1 1/3 1/2

1/5 1/5 1/3

1

1/5 1/4 1/3

5 3 3

5 4 3

1 2 2

D3

1 1 2

1/3 1/5 1/3

1

7 1 2

5 4 3

5 6 3

1/7 1/9 1/5

D4

1/3 1 1/3

1/3 1 1/2

1/5 1 1/4

1

1 3 2

1 3 2

1/7 5 1/7

D4

1 1 1/4

1/3 1 1/3

3 1 1/3

1

3 2 3

3 2 2

3 5 3

D4

5 3 5

1 1 3

7 5 5

1

7 5 4

7 5 5

7 7 5

D4

5 4 3

1 1 1

5 4 3

1

7 5 2

7 5 2

3 5 2

D4

1 1 1/2

1/3 1/5 1/3

1/7 1 1/2

1

5 4 3

5 6 3

1/7 1/9 1/6

D5

1/3 1/3 1/3

1/3 1/3 1/3

1/7 1 1/4

1 1/3 1/2

1

1 1 1

1/7 5 1/7

D5

1/3 1/3 1/5

1/5 1/3 1/2

1 1 1/3

1/3 1/2 1/3

1

1 1 2

3 3 2

D5

1 1/3 1/2

1/5 1/3 1/4

3 3 2

1/7 1/5 1/4

1

1 1 2

1 3 2

D5

1/5 1/5 1/2

1/7 1/5 1/4

1/5 1/3 1/3

1/7 1/5 1/2

1

2 3 1

1/5 1/4 1/3

D5

1/5 1/5 1/2

1/5 1/7 1/4

1/5 1/4 1/3

1/5 1/4 1/3

1

2 4 2

1/9 1/9 1/6

D6

1/3 1/3 1/3

1/3 1 1/3

1/7 1 1/5

1 1/3 1/2

1 1 1

1

1/7 3 1/7

D6

1/3 1/3 1/5

1/5 1/3 1/2

1 1 1/2

1/3 1/2 1/2

1 1 1/2

1 1 1

3 3 2

D6

1/3 1/3 1

1/7 1/3 1/4

1 3 2

1/7 1/5 1/5

1 1 1/2

1

1 3 2

D6

1/5 1/5 1/2

1/7 1/5 1/5

1/5 1/4 1/3

1/7 1/5 1/2

1/2 1/3 1

1

1/7 1/5 1/3

D6

1/5 1/7 1/2

1/5 1/8 1/4

1/5 1/6 1/3

1/5 1/6 1/3

1/2 1/4 1/2

1

1/9 1/9 1/6

D7

5 1/5 7

5 1/5 6

3 1/5 3

7 1/5 7

7 1/5 7

7 1/3 7

1

D7

1/5 1/5 1/4

1/7 1/5 1/3

1/3 1/5 1/2

1/3 1/5 1/3

1/3 1/3 1/2

1/3 1/3 1/2

1

D7

1/3 1/7 1/2

1/7 1/7 1/5

1 1 1

1/7 1/7 1/5

1 1/3 1/2

1 1/3 1/2

1

D7

1/2 1/3 1/2

1/5 1/5 1/3

1 1/2 1/2

1/3 1/5 1/2

5 4 3

7 5 3

1

D7

1/7 9 5

1/7 5 6

7 9 5

7 9 6

9 9 6

9 9 6

1

1

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Y.-C. Tang / Neurocomputing 72 (2009) 3477–3489

(a) C11

Table 5 Pairwise Comparisons for sub-criteria of Capital Budgets between decision alternatives.

D1 1 3 3 2

1

5 3 3

1/3 1/3 1 2

3 3 2

1 3 1/3 1/3 1 1 1/3

D2

D3

D4

D3

D4

D5

D6

D7 (b) C22 D1

D2

D6

D7 (c) C23 D1

D2

D3 D4

1/3 1/5 1/3 3 1/3 1/3 1/3 2 1/2 1/3 1/2 2

3 2 3

5 3 4

D1 1

1/7 1/5 1 3 1/5 1/5 2 3 1/4 1/5 1/2 2

3 4 3

5 4 2

D1

3 3 3

5 5 4

3 1 3 3 1/2 1

3 5 2

3 5 2

5 7 3

D2

7 5 4

1

5 2 2

7 3 3

7 5 4

7 5 3

7 6 2

D2

1/3 1/3 1 1/3

3 3 3

1/3 3 3 5 3 4

3 5 4

5 5 4

D3

5 5 5

1/5 1/2 1 1/2

5 3 3

7 5 2

7 5 2

7 6 3

D3

3 5 4

5 5 3

D4

1 1/7 1/5 1/2 1/3 1/3 1 2 1/3 1/3

3 3 3

3 3 3

5 4 4

D4

2 3 1/3 3 1 2

D5

1/3 1/7 1/7 1/3 1/3 1/5 1/5 1/3 1 1/2 1/4 1/2 1/3

3 2 3

5 2 3

D5

3 3 2

D6

1/3 1/7 1/7 1/3 1/3 1/4 1/5 1/5 1/3 1/2 1 1/3 1/3 1/2 1/3 1/3

3 2 2

D6

1/5 1/5 1/5 1/5 1/3 1/3 1/3 1/7 1/5 1/5 1/3 1/3 1 1/4 1/3 1/4 1/3 1/2 1/2

D7

1/5 1/7 1/7 1/5 1/5 1/3 1/4 1/6 1/6 1/4 1/2 1/2 1 1/2 1/2 1/3 1/4 1/3 1/2

D7

3 5 4

D5

1/3 1/3 1/3 1/3 1/2 1/5 1/5 1/5 1 1/2 1/2 1/4 1/4

D6

1/3 1/3 1/3 1/3 1/2 1/2 1/5 1/5 1/5 3 1 1/3 1/2 1/4 1/4 1

D7

D3

D4

D5

1

D5

D6

D7

(d) C24 D1

D2

1/5 3 1/3 2 1/2 3

5 2 1

5 5 3

D1

1/5 1/5 3 2 1/3 3 3 1/3 4

1/5 5 1/3 3 1/3 2

5 5 2

5 5 3

D2

1

1/5 1/3 1/5 1/3 1 1/4 1/3

1/7 1/3 1/3 1/2 D3 1/5 1/2 1/2 1 1/3 1/2 1 1

5 3 2

1

5 3 3

7 5 3

D3

D4

D5

D6

D7

(e) C25 D1

D2

D3

5 3 4

5 3 4

7 5 5

D1 1

1/7 5 1/5 5 1/5 4

7 5 5

1

5 7 7

D4

D5

D6

D7

1/9 5 1/7 5 1/7 2

5 7 3

5 7 3

1/5 5 1/5 7 1/5 4

5 7 4

5 7 4

5 1/2 1 1/3

1/3 5 1/3 2 1/3 3

5 5 4

5 3 4

7 5 4

D2

5 3 3

6 3 4

7 4 5

7 5 5

8 2 6

D3

1

1/5 1/5 1/5 1/7 1 1/4 1/7

1/9 1/3 2 1/5 1/3 3 1/6 1/3 2

3 3 2

1/3 1/5 1/6 1/3 1/2 1/3 1 1/4 1/3 1/4

3 3 4

3 3 4

5 5 5

D4

9 7 7

1

9 7 6

9 8 7

9 8 7

1/9 1/7 1 1/6

3 3 2

3 3 2

1/5 1/5 1/2 1/9 1/3 1/7 1/7 1/3 1/8 1/3 1 1/3 1/4 1/2 1/7 1/2

3 2 2

3 3 3

5 3 3

7 3 4

7 5 4

D4

1/3 1/5 3 1/2 1/3 2 1/3 1/2 2

1/5 1/3 1 1/3

3 1 2

5 3 2

D5

1/5 1/5 1/7 1/3 1/3 1/5 1/4 1/3 1 1/4 1/4 1/5 1/4

1/3 3 D5 1/2 1/3 1/2 2

1/5 1/5 3 1/2 1/5 2 1 1/2 1

1/7 1/3 1/3 1 1 1/4 1/2

3 3 2

D6

1/5 1/5 1/7 1/3 3 1/3 1/3 1/5 1/3 2 1/4 1/4 1/5 1/4 2

1

1/5 1/5 2 1/5 1/5 1 1/3 1/3 1

1/7 1/5 1/3 1/5 1/3 1/3 1 1/4 1/2 1/2

D7

3 D6 1/3 2

1/7 1/7 1/8 1/5 1/3 1/3 1/5 1/5 1/2 1/5 3 3 1 1/5 1/4 1/6 1/5 1/2 1/2

D7

5 5 5

9 5 6

1/5 1/5 3 1/5 1/7 3 1/2 1/4 3

1/5 1/5 1/3 1/9 1/3 1/3 1/7 1/7 1/3 1/8 1/3 1/2 1 1/3 1/4 1/2 1/7 1/2 1/2

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(a) C21 D1

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Y.-C. Tang / Neurocomputing 72 (2009) 3477–3489

1

1

C11

C1

C14 0.5

0 0.15 0.2 0.25 0.58 0.9 0.91 D7 D1 D2 D4 D6 D5

0 0.25 0.35 C11 C13

δ

2.72 C12

4

2

4

2

4

δ

1 C21

C12

C2

1

1.05 C15

D3

0.5

0.5

0.5

D2

0

0 0.6 0.78 0.8 C24 C25 C22

0.030.28 0.45 0.65 0.7 1.43 D1 D4 D3 D5 D6 D7

4

1.76 C23 δ

δ

1

Fig. 5. Set of weight values from judgements on two criteria over 0rdr4.

4.2. Results from the ANN An ANN does not have any knowledge at the beginning. Learning process starts on entering data into the input layer of the network. The error back-propagation algorithm is a popular algorithm to adjust the interconnection weights during training. Based on the case study as described in Section 3, the results from ANN analysis are illustrated in the following steps. Step 1: Present input and desired outputs: The input data set is obtained from Tables 3–5. It has 540 input variables and three samples. The desired outputs based on the three DMs’ opinion are found by using AHP as presented in Table 7. Step 2: Create BPN model: One hidden layer is set up after the input and desired output are arranged. The activation functions

0.5

0 0.2 D2

0.75 1.2 1.35 1.81.82 D1 D5 D6 D7 D3

4

δ 1

C14

D2 0.5

0 0.15 0.570.75 0.76 D4 D1 D3 D7

1.65 D5

2.05 D6 δ

4

1 D7 C15

C12, C13, and C15. ‘‘Strong preference’’ is give by DM1 to C12 and C13. ‘‘Equal preference’’ is given by DM1 to C11 and by DM3 to C13 and C15. The least preference shown in Fig. 5a is ‘‘the relations between government and company strategy’’ C12. This also can be verified from Table 3a. In row C12 presented in Table 3a, there is no preference made by DMs among all the sub-criteria. Similarly, Fig. 5b shows that C21 has the most preference made by DMs. It also can be found in Table 3b. In row C21 of Table 3b, five ‘‘strong preferences’’ made by DMs when compared with some of sub-criteria. For example, in the C21 row, DM1 and DM2 have ‘‘strong preferences’’ when comparing C21 with C22. ‘‘Strong preferences’’ is given by DM1 and DM3 to C21 and C23, and by DM1 to C21 and C24. The comparisons between C21 and other subcriteria have positive preferences, apart from the decision made by DM3 to C21 to C24. The least preference shown in Fig. 5b is C23. This also can be examined in Table 3b. In the row C23 presented in Table 3b, there is no preference made by DMs among all the sub-criteria, apart from 3 ‘‘equal preferences’’ when compared with some of subcriteria. For example, in row C23, DM1 has ‘‘equal preference’’ when comparing C23 with C24 and C25. ‘‘Equal preference’’ is given by DM2 to C23 and C22. This method is compared to the well known method, BPN as shown below.

C13

D4

0.5

0 0.86 1.351.51.65 2 D2 D3 D1 D4

2.7 D5

3.65 4 D6

δ Fig. 6. (a–e) Graphs of weight values between the decision alternatives on operating budgets’ sub-criteria.

utilized here are sigmoid, hyperbolic tangent and linear activation functions. The learning rule is using gradient descent methods. The epochs are 10 000 times. Step 3: Training samples: Training of the ANNs is done using the commercial software NeuroSolutions 5 [56]. BPN was the basis of training for this supervised neural network. The active performances and the MSEs are shown in Figs. 8–10.

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Table 6 The sets of weight values for all fuzzy comparison matrices and the final results obtained where d ¼ 3.65 based on the DM’s opinions.

C21

1

0.5

Weight values Weight for criteria values for subcriteria

D3

0 0.08 0.1 0.48 0.95 1 D2 D4 D1 D6 D5

1.7 2 D7 δ

4

1

Weight values for decision alternatives

D1

D2

D3

D4

D5

D6

D7

C1 0.4

C11 C12 C13 C14 C15

0.2496 0.0531 0.2409 0.2680 0.1883

[0.1554, [0.1709, [0.1618, [0.1617, [0.1571,

0.1527, 0.1721, 0.1904, 0.1883, 0.2038,

0.1661, 0.1500, 0.0906, 0.1531, 0.1670,

0.1364, 0.1581, 0.2003, 0.1823, 0.1426,

0.1143, 0.1365, 0.1374, 0.0991, 0.0684,

0.1161, 0.1317, 0.1276, 0.0661, 0.0004,

0.1591] 0.0808] 0.0919] 0.1492] 0.2577]

C2 0.6

C21 C22 C23 C24 C25

0.2539 0.2006 0.1266 0.2164 0.2025

[0.1536, [0.1586, [0.1774, [0.1828, [0.1944,

0.1730, 0.2011, 0.1703, 0.1826, 0.2296,

0.1756, 0.1933, 0.0872, 0.1993, 0.1056,

0.1715, 0.1624, 0.1923, 0.1637, 0.2644,

0.1245, 0.1323, 0.1464, 0.0844, 0.1409,

0.1256, 0.0996, 0.1316, 0.1161, 0.0619,

0.0763] 0.0526] 0.0949] 0.0711] 0.0032]

D2 C22

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0.5

Final results [0.1672, 0.1879, 0.1531, 0.1802, 0.1174, 0.0975, 0.0967] Final ranking [D2, D4, D1, D3, D5, D6, D7]

0 0.17 0.68 0.75 1.15 D3 D4 D1 D5

1.6 D6

2

δ

4

2.35 D7

Table 7 The desire output from the analysis of AHP.

1

C23

D4

DM1 DM2 DM3

0.5

D1

D2

D3

D4

D5

D6

D7

0.116 0.152 0.161

0.264 0.274 0.255

0.167 0.138 0.135

0.224 0.263 0.256

0.053 0.058 0.063

0.043 0.043 0.051

0.133 0.074 0.080

0 0.3 0.43 0.851.05 1.6 1.605 2 D1 D2 D5 D6 D7 D3 δ

4

Table 8 The desired and actual output based on different activation functions. Activation functions

1 Sigmoid Tangent Linear

C24

D3 0.5

0 0.35 0.37 0.72 D1 D2 D4

4

1.5 1.85 1.98 D6 D5 D7 δ

1

C25

D4 0.5

0 0.57 D2

1.1 D1

1.8 2.25 D5 D3

2.75 D6

3.65 D7

4

δ Fig. 7. (a–e) Graphs of weight values between the decision alternatives on capital budgets’ sub-criteria.

Step 4: Testing model: After training the neurons, this step uses two samples and one sample for training and testing, respectively. The desired and actual outputs are presented in Table 8. In step 3, the data are trained themselves by using different activation functions. In Fig. 8, the MSE curve is not approaching

Output D1

D2

D3

D4

D5

D6

D7

Desired Actual Actual Actual

0.255 0.270 0.270 0.271

0.135 0.153 0.138 0.143

0.256 0.263 0.263 0.274

0.063 0.058 0.058 0.053

0.051 0.043 0.044 0.038

0.080 0.104 0.567 0.079 0.791 0.065 1.163

0.161 0.152 0.152 0.157

Testing MSE

zero. It means the network is not learning the problem. Three possible reasons for an unsuccessful training: (1) the network is capable of learning the problem but has not been trained long enough; (2) the network is capable of learning the problem but is stuck in a local minima and (3) the network is not powerful enough to learn the problem. However, the MSE curves in Figs. 9 and 10 approach zero. It seems the hyperbolic tangent and linear activation functions are learning the problem better in this step. It also shows that different activation functions will produce different results. In step 4, the desired and actual output based on different activation functions are listed in Table 8. In Table 8, each of different activation functions’ actual output is approaching the desired output. The testing MSE presented in Table 8 reveals that the sigmoid function has the least MSE. The least MSE has the least difference between the actual and the desired. For this step, the sigmoid activation function was found to be the most suitable activation function and was used for the modeling.

4.3. Comparing the FAHP and ANN methods The results based on FAHP presented in Table 6 and based on ANN presented in Table 8 have similar ranking to each other. The

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first two preferences in Tables 6 and 8 are D2 and D4, respectively. The FAHP method provided in this paper has successfully helped DMs to make a proper judgment. In ANN method, training is required to operate and learn the problem. Therefore, the more data to be trained, the more operations to be learnt. Then the results will be found after the testing. However, if the quantity of data is not enough then the ANN method might not learn the problem well. In the three samples in this case study, the actual outputs are approaching the desired output as presented in Table 8. It is found the ANN also learns the problem when the sample is greater than three. 5. Conclusions The purpose of this study is to investigate the application of the fuzzy analytic hierarchy process (FAHP) method of multicriteria decision-making within a budget allocation problem. The application problem in question is to allocate the company’s budgets to the subordinate sections. Due to uncertainty and the fuzzy nature of the complex problem to the decision makers, the FAHP is used to allow for imprecision in the judgements made. The issue of imprecision is reformulated in this paper which further allows for a sensitivity analysis on the preferences’ weights on changes in the levels of imprecision. It is found the DMs of an aerospace company successfully made the necessary judgements utilizing the FAHP. This included an allowance to not make specific pairwise comparisons between all pairs of decision alternatives—incompleteness being another aspect of the possible inherent uncertainty in the decision process. The redefinition of the degree of fuzziness—d associated with the preference judgements allowed for the change of imprecision (fuzziness) to be succinctly reported. It is suggested that the suitable degree of fuzziness, to be utilized within the fuzzy scale values to obtain the sets of weights, is the maximum of the minimum workable values of d. Moreover, where there are different maximums of the minimum workable values of d for different scales or different models of aggregation, as in the comparisons in this paper, it is suggested that the highest of the maximums of the minimum workable values of d should be chosen. The proposed method is also compared with one of the AI techniques, ANN. The results from ANN are similar to FAHP. In summary the proposed FAHP can be used to evaluate various management strategies and thus resources can be effectively deployed to strengthen these aspects of project management.

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Yu-Cheng Tang received her Ph.D. in accounting from the University of Cardiff (Wales, UK). Currently she is an assistant professor at National Changhua University of Education (Taiwan). Her research interests are in the general area of financial management, in particular in the capital investment, human perceptions on decision-making, enterprise resource planning (ERP), ethic position and budgetary system, etc. Specific methodology investigations include fuzzy set theory, analytical hierarchy process and balanced scorecards. Her study is at the theoretical development and application based level, including business and other topics.