IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

27 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 DECISION MAKING IN DYNAMIC ENVIRONMENTS: AN 

APPLICATION OF MACHINE LEARNING TO THE 

ANALYTICAL HIERARCHY PROCESS  

 

Caelum Kamps 

DRDC Ottawa Research Centre 

Canada 

caelum.kamps@drdc-rddc.gc.ca 

 

Rahim Jassemi-Zargani 

DRDC Ottawa Research Centre  

Canada 

rahim.jassemi@drdc-rddc.gc.ca 

 

ABSTRACT 

 

The purpose of this work is to propose a method of algorithmic decision making that 

builds on the Analytical Hierarchy Process by applying reinforcement learning. Decision 

making in dynamic environments requires adaptability as new information becomes 

available. The Analytical Hierarchy Process (AHP) provides a method for comparative 

decision making but is insufficient to handle information that becomes available over 

time. Using the opinions of one or many subject matter experts and the AHP, the relative 

importance of evidence can be quantified. However, the ability to explicitly measure the 

interdependencies is more challenging. The interdependency between the different 

evidence can be exploited to improve the model accuracy, particularly when information 

is missing or uncertain. To establish this ability within a decision-making tool, the AHP 

method can be optimized through a stochastic gradient descent algorithm. To illustrate 

the effectiveness of the proposed method, an experiment was conducted on air target 

threat classification in time series developing scenarios. 

 

Keywords: threat assessments; AHP; machine learning; decision making  

 

 

1. Introduction 

During complex decision making for classification, the ability of a subject matter expert 

(SME) to reach a conclusion depends on the availability of information. In dynamic 

situations, where information is developing over time, the decision maker must be 

flexible as information becomes available or is lacking. Air target threat assessment is 

such a situation. The available systems dictate the rate and amount of target information 

that accumulates. Additionally, with the increasing amount of air traffic, it is not feasible 

for an individual or even a team to be able to evaluate and classify each target that is 

detected. The need for machine-aided classification is evident and increasing. It is 

desirable that the method of algorithmic classification be as follows:  

  

mailto:rahim.jassemi@drdc-rddc.gc.ca


IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

28 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

1. Adaptive to updated information 

2. Predictive of unknown information 

3. Reflective of the classification of a SME 

 

The Analytic Hierarchy Process (AHP) is a general theory of measurement. The method 

facilitates deductive and inductive algorithmic decision making by comparing the relative 

importance of evidence (Saaty, 1987). Threat assessment (TA) is the fusion of many 

sources of information for the purpose of classifying and predicting the intent, capability, 

and proximity of detected entities. (Johansson & Falkman, 2008) This fusion of 

information is used to infer an associated target threat value (TV) for each entity at any 

given point in time. This work is particularly concerned with the application of 

algorithmic target threat assessment to airborne targets during states of uncertainty and 

deficient information. 

 

The objective of this paper is to present methods to improve the accuracy of the AHP 

network evaluations when not all the cues are known. The assumption is that under the 

complete information state (CIS), the state where all the cues are known, the optimal 

decision or reference decision is the complete AHP network.  

 

The problem that is being addressed is what to do when not all the cues are known. How 

can the network be adapted to best predict the reference decision? Given a state of some 

information, what are the optimal weights and best techniques to improve the accuracy of 

prediction for the correct threat classification?  

 

The proposed decision structure is a hierarchal decision tree. Through the AHP, the nodes 

are each assigned a weight corresponding to their importance. Machine learning 

techniques can then be used to optimize the node weights when not all the information is 

known.  This structure can then be further improved by introducing special learning 

nodes called sigmoid nodes. This will be shown in the following sections covering the 

AHP, the scenario description, the proposed methodology, and finally, the experiments 

and results.  

 

 

2. Analytical Hierarchy Process 

The Analytical Hierarchy Process (AHP) is a well-known algorithm for decision making 

support (Saaty, 1987).  It allows a SME to break the problem down into criteria groups to 

more easily compare evidences, called cues. The structure of the decision tree is shown in 

Figure 1. 

 

  



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

29 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 

 

 

A SME is given a list of cues to partition into criteria. Within each criteria, the cues are 

compared pairwise with one another. This information is stored in comparison matrices. 

Assuming the SME is relatively consistent in their comparisons, a principal eigenvector 

analysis on the comparison matrices can be used to obtain normalized weights for each of 

the cues. Once this is done, the same process is applied to the criteria to generate 

normalized criterion weightings. The structure then takes input from each of the targets or 

alternatives and makes a decision based on the previously defined weights. It is important 

to understand that since the weights are normalized, the sum of all the cue weights within 

a criterion is unity. Similarly, the sum of the weights of each of the criterion is unity 

(Saaty, 1987). 

 

Currently, the AHP assumes that all the inputs are known. It relies on the ability of a 

SME to rank each of the cues with respect to others within the category. One of the 

limitations of the AHP is that it does not consider the relative importance of cues with 

respect to multiple criteria. For example, if one cue is used when considering two criteria, 

then its relative importance might not be the same in both cases. The Analytical Network 

Process (ANP) offers the ability to do these multiple comparisons. In both methods, all of 

the cues must be known to make an accurate decision. This is not what is being addressed 

here. In the ANP and AHP community, dependency refers to the influence of the criteria 

under consideration on the importance of the cue (Mu, 2006). The type of dependency 

that is being addressed by this paper is different. Here we consider dependency in the 

calculation of the problem at hand. This means that cues might hold information about 

each other, either explicitly in their formulae for calculation or implicitly in correlation. 

The method proposed in this paper deals with accounting for the values of unknown cues 

that might have these dependency features with some of the known cues. 

 

 

3. Cues 

The calculation of some cues is dependent on others.  For example, to calculate the 

estimated closest point of approach of a target to a defended asset, the orientation, range, 

and track history of the target must be known.  The introduction of dependent cues 

creates potentially complex and unknown relationships between cues that are 

hypothetically accounted for during states of complete information but are not necessarily 

understood when not all the inputs are known. This is one of the challenges that will be 

addressed. Here, the difference between dependent and independent cues can be 

Figure 1 General AHP structure 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

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established. Cues that require that other information be known before a measurement can 

be made are dependent. Conversely, cues that can be measured on their own are 

independent. Table 1 shows the dependency and criteria for each of the cues that will be 

considered for the target threat assessment. 

 

Table 1 

Target threat assessment cues 

 

Cue Criteria Dependent/Independent Dependencies 

Speed (Sp) Capability Independent N/A 

Altitude (Al) Proximity Independent N/A 

Range (R) Proximity Independent N/A 

Orientation (O) Intent Independent N/A 

Closest Point of 

Approach (CPA) 

Capability Dependent Orientation & Range 

Angle of Attack 

(AOA) 

Intent Dependent AOA & Altitude 

Time Before CPA 

(TBCPA) 

Proximity Dependent CPA & Speed 

 

Using the cues from Table 1, a hierarchal decision tree is built and shown in Figure 2. 

  

 

Figure 2 Hierarchal decision tree 

 

To initialize the weights for each of the nodes in the decision tree using the AHP, four 

comparison matrices were generated and are shown in Table 2. The rankings provided by 

a study of cues and information order are used to create generalized pairwise 

comparisons for each cue (Liebhaber, Kobus & Feher, 2002). 

 

  



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

31 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

Table 2 

Comparison matrices for the hierarchy shown in Figure 2 

 

Threat Value (TV)  Intent 

 Proximity Capability Intent  AOA Orientation 

Proximity 1 2 2 AOA 1 2 

Capability  1 1 Orientation  1 

Intent   1    

  

Proximity Capability 

 Alt TBCPA Range  Speed CPA 

Alt 1 2 3 Speed 1 2 

TBCPA  1 2 CPA  1 

Range   1    

 

 

The weights generated through principal eigenvector analysis of the above comparison 

matrices are presented in Table 3. 

 

Table 3 

Weights of each criteria and cues 

 

Criteria Weight  Sub-Criteria (Cues) Weight 

Intent 0.25  Orientation 0.33 

 AOA 0.66 

 

Proximity 0.5  Range 0.54 

 Altitude 0.30 

 TBCPA 0.16 

 

Capability 0.25  Speed 0.66 

 CPA 0.33 

 

 

4. Information domain partitioning 

In real situations, information accumulates as a time series. Not all information is known 

right away because it takes time to run tests and acquire data. To represent each of the 

combinations of known information, the information domain is partitioned into an 

information state space (ISS). This is a list of every possible scenario that can occur with 

the information available. Each of the scenarios is referred to as an information state (IS). 

The number of information states is directly dependent on the number of independent 

cues. Each IS can be encoded as a binary string. If there are four independent cues, 

denoted 1, 2, 3, and 4, then the state in which cue 1 and 4 are known is 1001 or 

information state 9. Clearly, any combination of cues can be represented as a 4-bit string. 

Thus, there are 2
4
 information states. The ISS can now be explicitly defined. In Table 4, 

the dependent cues receive a designation of 1 (known) if their dependencies are satisfied. 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

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https://doi.org/10.13033/ijahp.v13i1.766 

Each IS can be referenced either by the string that is the first four binary digits in its row 

of the ISS table or by its state identification (ID). 

 

Table 4 

ISS table 

 

 

Independent Dependent 

State ID Speed Altitude Orientation Range AOA CPA TBCPA 

1 0 0 0 1 0 0 0 

2 0 0 1 0 0 0 0 

3 0 0 1 1 1 0 0 

4 0 1 0 0 0 0 0 

5 0 1 0 1 0 0 0 

6 0 1 1 0 0 0 0 

7 0 1 1 1 1 1 0 

8 1 0 0 0 0 0 0 

9 1 0 0 1 0 0 0 

10 1 0 1 0 0 0 0 

11 1 0 1 1 1 0 0 

12 1 1 0 0 0 0 0 

13 1 1 0 1 0 0 0 

14 1 1 1 0 0 0 0 

15 1 1 1 1 1 1 1 

 

 

5. Methodology 
5.1 Initialization and base cases 

To prove the validity of the method proposed in later sections, a base case must be 

generated in order to define some reasonable minimum accuracy of classification. A 

decision structure for each IS used to evaluate relative improvement is created using the 

AHP method. This ensures that the base case still reflects the relative weights of each of 

the known cues and is the best estimate available for the optimal classification given the 

information provided by the SME. Information state 11 or 1011 can be considered as an 

example to illustrate how a base case decision structure for that state can be generated. In 

this example, the CIS decision structure is recreated, however, all the cues that are not 

available and each criterion that becomes empty as a result of this process are removed. 

For IS 11, the decision structure is shown in Figure 3. 

  



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

33 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 

 

Figure 3 Decision structure 

 

To produce weights for this decision structure, the AHP method must be applied again, 

and this time each comparison matrix only contains elements that appear in the decision 

structure. Since all the comparisons are pairwise, they still reflect the relative weights of 

each cue and criteria within their respective category. Thus, the AHP method is still a 

valid procedure for calculating the decision of this structure. This process is then applied 

to every other IS to produce a different base case for each. This base case initialization 

process is outlined in the following information flow chart shown in Figure 4. 

 

 

 
 

Figure 4 Initialization process 

 
5.2 Mathematical representation of decision trees 

When considering the state of complete information, the decision structure needs to be 

explicitly defined to be able to apply machine learning for optimization purposes. Taking 

the inputs from a specific target, a method to easily describe the propagation of the inputs 

to the outputs is described below.  

 Calculate the threat value of target x1 

𝑇(𝑥)   =   ∑ 𝜃𝑖 ∙ ( ∑ 𝜔𝑖𝑗 𝜇𝑖𝑗 (𝑥) 

𝑗 𝜖 𝐾

) 

𝑖 ∈ 𝑁

 

 

Where: 

 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

34 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 𝜃 = [𝜃1, … , 𝜃𝑁 ] is the vector of criteria weights 
 

 N is the number of criteria  
 

 𝜔 = [

𝜔11 ⋯ 𝜔1𝐾
⋮ ⋱ ⋮

𝜔𝑁1 ⋯ 𝜔𝑁𝐾
] is a matrix of cue weights 

 

 The rows of 𝜔 represent the cues within each criteria. 
 K is the size of the criteria with the most cues 
 Smaller criteria have trailing zero entries 

 

 𝜇(𝑥) =  [
𝜇11(𝑥) ⋯ 𝜇1𝐾 (𝑥)

⋮ ⋱ ⋮
𝜇𝑁1(𝑥) ⋯ 𝜇𝑁𝐾 (𝑥)

] is the measurement of each of 

the cues from target x and has the same shape as omega with the 

same zero entries 

 

It is important to note that because of the properties of the AHP method, the sum of θ or 

any row in ω is always unity. Additionally, it is desirable for the threat value to be 

normalized, which also normalizes the measurements 𝜇𝑖𝑗 (𝑥). With these properties, the 

threat value T, is always in the interval [0, 1]. 

 

For every other information state, different theta and omega values are generated using 

the AHP method. To create scenarios in which only some of the values of 𝜇(𝑥) are 
known, each information state is associated with a variable denoted as an input shape 

parameter. These are binary matrices with entries of either 0 or 1. They have the exact 

shape of μ, but if the cue exists in that information state then the entry in the position of 

the cue is 1, otherwise it is 0. This allows an easy partitioning of μ to create scenarios into 

a lesser IS for training purposes. 

 
5.3 Weight optimization 

When considering an incomplete IS, any IS that is not the CIS, the weights are calculated 

using the AHP method and are therefore intelligent in the sense that they take into 

consideration the relative importance of one another. In this situation, the problem is the 

uncertainty of the information that is lacking. Using weight redistribution through mini 

batch stochastic gradient descent (Ruder, 2017), it is possible to improve the accuracy of 

a correct threat classification of any information state that considers a minimum of two 

cues. To accomplish this, a cost function needs to be implemented to measure the error 

between the predicted threat value of an information state and the reference threat value.  

  



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

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 Quadratic Cost: 

𝐶1 =
1

2𝐵
 ∑ ( 𝑌(𝑥) − 𝑇(𝑥) )2

𝑥 ∈ 𝐵

 

 

 

 Entropic cost function: 

𝐶 =  −
1

𝐵
∑ [𝑌(𝑥) ln(𝑇(𝑥)) + ( 1 − 𝑌(𝑥) ) ln( 1 − 𝑇(𝑥) )]

𝑥 ∈ 𝐵

 

Where: 

 

 C is the cost to be minimized 

 B is the batch of training targets 

 X is a target 

 Y(x) is the reference threat for target x 

 T(x) is the threat calculated in the information state being trained 
 

Both cost functions have their advantages; however, the entropic cost has a much steeper 

derivative and provides improved optimization when learning slowdown occurs. The 

method of stochastic gradient descent takes the gradient of the cost with respect to the 

weights θ and ω for that information state and shifts them in the opposite direction of the 

gradient with respect to some learning rate, α. 

𝜃𝑖 =  −𝛼𝜃
𝜕𝐶

𝜕𝜃𝑖
 

 

𝜔𝑖𝑗 =  −𝛼𝜔
𝜕𝐶

𝜕𝜔𝑖𝑗
 

 

Where:  

 C is entropic cost 

 𝛼𝜃   is the learning rate for the criteria weights  
 𝛼𝜔  is the learning rate for the cue weights 

 
𝜕𝐶

𝜕𝜃𝑖
=  −

1

|𝑁|
∑ (

𝑌(𝑥)−𝑇(𝑥)

𝑇(𝑥)(1−𝑇(𝑥))
)

𝜕𝑇(𝑥)

𝜕𝜃𝑖
𝑥 ∈ |𝑵|  

 

 
𝜕𝐶

𝜕𝜔𝑖𝑗
=  −

1

|𝑁|
∑ (𝑌(𝑥) − 𝑇(𝑥))𝑥 ∈ |𝑵|

𝜕𝑇(𝑥)

𝜕𝜔𝑖𝑗
 

 

 
𝜕𝑇

𝜕𝜃𝑖
= ∑ 𝜔𝑖𝑗 𝜇𝑖𝑗𝑗 ∈ 𝐾 (𝑥) 

 

 
𝜕𝑇

𝜕𝜔𝑖𝑗
= 𝜃𝑖 𝜇𝑖𝑗 (𝑥)  

 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

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Analytic Hierarchy Process 

36 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

This process is repeated until some end condition is met to produce optimized 

weights for that information state. These weights are then saved and are the new 

weights used to calculate threat for that information state. The reason the cost 

function is averaged over a batch of training examples is to reduce the noise of 

any single scenario which helps avoid overfitting. It smooths the learning process 

and under some conditions results in better weight optimization. The weights of 

the CIS do not change. They are the weights which are used to calculate the 

reference threat value. The weights that do change are those of each information 

state. 

 
5.4 Adapted decision method  

One of the largest limitations of the above process is the inability to improve the 

classification accuracy unless many cues exist in the information state that can be 

reweighted. Next, a new structure of the decision tree is proposed that allows for much 

greater improvement in accuracy after optimization.  

 
5.4.1 Sigma node 

The sigma node is a function that is frequently used in machine learning. Specifically, its 

properties are used for training neural networks. The sigmoid node is a special case of the 

logistic function. It is a simple, non-linear function that provides a balance between linear 

and non-linear behavior (Menon, 1996). Sigma functions are of the following form. 

 

 Sigma function 

𝑆(𝜌, 𝛽, 𝑥) =
1

1 + 𝑒 −𝜌(𝜇− 𝛽)
 

Where: 

 ρ is a shape parameter 

 μ is the input 

 β is a shift parameter 

For these purposes, it is useful to consider the standard values for the shape and shift 

parameter as: 

 

 ρ=4.5   

 β= 0.5 

The sigma function with these parameters is denoted the standard sigma function (SSF). 

This is because of the similarity of this function to the identity function on the interval 

[0,1], as seen in the Figure 5. This is desirable because it means that the use of SSF nodes 

within the decision structure will result in a largely unchanged threat value from the 

original method given the same cue measurements. 

  



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

37 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 

Figure 5 Sigma function 

 
5.4.2 Updated decision structure 

To improve the ability of a learning function to optimize the threat value prediction in 

any IS, an update to the decision structure needs to be made. When considering the CIS, 

the decision shape takes the same form as in the previous case but with an added step. 

Before input to any of the weight nodes, the input passes through an SSF node. This 

structure is called a sigmoid decision tree as seen in Figure 6. 

 

When sigmoid operations are being considered (i.e., in the adapted decision method), the 

reference threat value becomes the output of the decision tree as seen in Figure 6. To 

create the sigmoid decision trees for each IS, the process described in section 5.1 is 

applied. The difference is the addition of SSF nodes before input to any weight node. 

These decision structures become the new base cases for each of the ISs used as 

comparisons for how much the learning algorithm that is described in the following 

section can improve the prediction. 

 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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38 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

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Figure 6 Sigmoid decision tree 

 

 
5.5 Sigma node justification 

The sigmoid decision structure offers a method to quantify sensitivity to dependency 

within the cues. Input from cues with more influence can be adjusted to reflect their 

impact on future calculations. Using the same learning process described in section 5.3, 

the value of the sigma parameters can be optimized to produce further improved threat 

value predictions. Using the shape and shift parameters, a broad spectrum of I/O relations 

can be achieved through sigma optimization. To illustrate the flexibility of the sigma 

node to adapt to countless circumstances, Figure 7 offers some examples of different I/O 

relationships. 

  

Threat Value (TV) 

Capability Proximity Intent 

TBCPA Al R CPA Sp AOA O 

SSF SSF SSF 

SSF SSF SSF SSF SSF SSF SSF 

Target 



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                                                       (a)                                                                  (b) 

 

Figure 7 (a) Varying shape parameters, (b) Varying shift parameter 

 

An added improvement of including sigma nodes in the threat calculation lies in the 

ability of the sigma function to easily adjust the threat value output even when few cues 

are available. This means the structure can adapt to reflect a more accurate prediction 

even when only one cue is known. 

 
5.6 Sigma calculations and optimization 

Given the new decision structure in Figure 6, the threat value calculation must be 

established. 

 

 Expanded Sigma threat calculation 

𝑇(𝑥) =  ∑ [𝜃𝑖 ∗ 𝑆 (𝜑𝑖 , 𝛿𝑖 ∑ (𝜔𝑖𝑗 𝑆(𝜌𝑖𝑗 , 𝛽𝑖𝑗 , 𝜇𝑖𝑗 (𝑥)))

𝑗 ∈ 𝐾

)]

𝑖 ∈ 𝑁

 

 

 Substitute 

𝛾𝑖 (𝑥) = ∑ (𝜔𝑖𝑗 𝑆(𝜌𝑖𝑗 , 𝛽𝑖𝑗 , 𝜇𝑖𝑗 (𝑥)))

𝑗 ∈ 𝐾

 

 

 

 Simplified version 

 

𝑇(𝑥) =  ∑ 𝜃𝑖 ∗ 𝑆(𝜑𝑖 , 𝛿𝑖 , 𝛾𝑖 (𝑥))

𝑖 ∈ 𝑁

 

Where: 

 

 θ, ω, and μ are as defined in 2.2 

 S is a sigma function 
 



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 𝛽 =  [
𝛽11 ⋯ 𝛽1𝐾

⋮ ⋱ ⋮
𝛽𝑁1 ⋯ 𝛽𝑁𝐾

] and 𝜌 =  [

𝜌11 ⋯ 𝜌1𝐾
⋮ ⋱ ⋮

𝜌𝑁1 ⋯ 𝜌𝑁𝐾
] are the 

matrices of cue shift and shape parameters, respectively. 

 

 𝛿 = [𝛿1, … , 𝛿𝑁 ] and 𝜑 = [𝜑1, … , 𝜑𝑁 ] are the vectors of criteria 
shift and shape parameters, respectively.   

The relationship between the sigma threat structure and the original structure is highly 

correlated. The following plot (Figure 8) outlines the correlation between threat value 

calculations of one versus the other. 

 

 

 
Figure 8 Correlation of sigmoid and standard decision structures 

 

The correlation shows that in the intermediate threat value section the output is 

functionally the same. At the extremes, there is a slight deviation from the perfect 

correlation. The Pearson correlation coefficient for the two outputs is 0.971. This is a 

strong indication that the sigma threat calculation very strongly supports the results from 

the original decision structure method. Therefore, the argument that the validity of the 

SSF supplemented decision tree is representative of the original AHP method is very 

strong. 

 
5.7 Sigmoid (SGD) optimization learning 

Through SGD, the ability of any IS to predict the correct classification can be greatly 

improved. The entropic cost function is used for its ability to improve the speed at which 

the algorithm converges to optimization. Consider the following progression of 



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application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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calculations that will lead to the gradient of the cost function with respect to the sigma 

parameters.  

 Partials of sigma function with respect to generic placeholder 
variables, x, y, and z. 

 






x

zyxS ),,(
[𝑆(𝑥, 𝑦, 𝑧)]2(𝑧 − 𝑦)𝑒 −𝑥(𝑧−𝑦) 

 






z

zyxS ),,(
[𝑆(𝑥, 𝑦, 𝑧)]2𝑥𝑒 −𝑥(𝑧−𝑦) 

 






y

zyxS ),,(

z

zyxS






),,(
 

 

1. Partial of sigma threat with respect to a criteria shape parameter 
 

 

i

T




=  𝜃𝑖

 

i

iii
S







 ,,
 

 

Where: 

 γ_i is defined above in section 2.6 

 α_i is the criteria shape parameter  

 φ_i is the criteria shift parameter 

 

2. Partial of sigma threat with respect to cue shape parameter 
 






































ij

i

iij

TT






 

 

Where: 

 

i

iii

i

i

ST






 






 ),,(
     and     

ij

ijiij

ij

ij

i
S

















 ),,(
 

 

3. Partial of sigma threat with respect to a criteria shift parameter 
 

𝜕𝑇

𝜕𝛿𝑖
= 𝜃𝑖

𝜕𝑆(𝜑𝑖 , 𝛿𝑖 , 𝛾𝑖 )

𝜕𝛿𝑖
  

 

4. Partial of sigma threat with respect to a cue shift parameter 
 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

International Journal of the 

Analytic Hierarchy Process 

42 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

https://doi.org/10.13033/ijahp.v13i1.766 

 






































ij

i

iij

TT






 

Where: 

 

i

iii
i

i

ST






 






 ),,(
      and       

ij

i








=  𝜔𝑖𝑗  

ij

ijijij
S







 ),,(
 

 
5.8 Input generation  

All the independent cues are randomly generated based on their own probability density 

functions so that there are an infinite number of scenarios.  

 

1. Orientation: The angle at which the target is oriented with respect to some 
defended asset.  

2. Range: The horizontal distance of the target from the defended asset 
3. Speed: The magnitude of the velocity of the target with respect to the ground 
4. Altitude: The vertical distance of the target from the ground 

 

Each cue is generated using a realistic interval for random value selection. Depending on 

the cue, the probability of the value is either based on a uniform or normal distribution. 

Speed and altitude are based on a normal distribution since most targets will exhibit 

standard speed and altitude values. Range and orientation are generated with a uniform 

distribution since there is no bias towards a particular position of the target.  

 

Using this generation method for the independent cues, CPA, AOA and TBCPA can be 

calculated when the other required cues are available. Figure 9 visually represents the 

geometric cues. 

 

 

 

Figure 9 Graphical representation of cues 

 

 

 

 

CPA 

Target 

Defended Asset 

 

Heading 

Distance to CPA 

Range at CPA 

AOA Range 

Orientation 



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43 Vol. 13 Issue 1 2021 

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The following formulae are used to calculate the dependent cues. 

 

1. 𝐴𝑂𝐴 = |𝐻𝑒𝑎𝑑𝑖𝑛𝑔 − 𝑂𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛| 

2. 𝐶𝑃𝐴 = (||𝑥𝑐𝑝𝑎, 𝑦𝑐𝑝𝑎 ||
2

+ 𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒 2)

1

2

  

Where: 

(𝑥𝑐𝑝𝑎 , 𝑦𝑐𝑝𝑎 ) = min
t > 0

 ||𝑅 ∗ cos(𝑂) + 𝑡 ∗ cos(𝐻) , 𝑅 ∗ sin(𝑂) + 𝑡 ∗ sin(𝑅)|| 

 

 R = Range 

 O = Orientation 

 H = Heading 

3. 𝑇𝐵𝐶𝑃𝐴 =
𝐶𝑃𝐴

𝑆𝑝𝑒𝑒𝑑
 

 

Heading is not used in threat value structure, but is used in the intermediate calculation of 

cues. It is also generated randomly and is independent of every other cue.  

 

Once the cues have been generated or calculated, they need to be normalized with respect 

to the severity of their threat value. Recall that all the inputs need to be normalized to a 

[0, 1] scale so that the decision structure can work properly. The general process is to 

assume some average case and relate the severity of the cue to how improbable the value. 

For this example, the severity normalizations are very simple. For example, when 

considering altitude, there is some PDF for the random generation of altitude. The 

severity of the altitude is directly related to the deviation from the average value.  If a 

target is much higher or much lower than generally expected, then it is considered 

suspicious. 

 

𝑃(𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒 = 𝑥) =  𝑁𝑜𝑟𝑚𝑎𝑙. 𝑃𝐷𝐹𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 (𝑚𝑒𝑎𝑛 = 𝑎, 𝑆𝑇𝐷 = 𝑏, 𝑥) 
 

𝑆𝑒𝑣𝑒𝑟𝑒𝑖𝑡𝑦 𝑜𝑓 𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒 = 𝐶𝐷𝐹𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 (𝑚𝑒𝑎𝑛 = 𝑎, 𝑆𝑇𝐷 = 𝑏, (𝑏 + |𝑏 − 𝑥|)) 
 

Where:  

 Severity of altitude is the normalized threat value for altitude 

 𝑁𝑜𝑟𝑚𝑎𝑙. 𝑃𝐷𝐹𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 is a normal probability distribution 
function for the generation of altitude values 

 𝐶𝐷𝐹𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒  is the cumulative density function of altitude 
 a is the mean of the PDF 

 b is the standard deviation of the PDF 

In a similar way, each of the cues can be assigned a threat value given their randomized 

measurements. As stated above, this is a very simple method which is not to say that it is 

a good method to evaluate a target, but more to illustrate the ability of the method to 

predict the outcome using cue dependencies and conditional probabilities. 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Analytic Hierarchy Process 

44 Vol. 13 Issue 1 2021 

ISSN 1936-6744 

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5.9 Optimization algorithm 

SGD is applied to both the sigma parameters and the weight nodes so that an optimal 

structure for each IS can be generated. First, the cues are segmented into their base case 

information states as described in section 5.1. The structures are built and the SSF nodes 

are inserted to prepare each IS for training. These base cases are saved in a database as 

the reference decision structures for later comparison with the optimized versions. The 

second step is the optimization process. This process occurs in the following three 

phases: training data pre preparation, training, and evaluation.  

 
5.9.1 Training data preparation 

SGD attempts to optimize a cost function with respect to some training parameters. If 

there is some underlying tendency of the function under evaluation to have a specific 

distribution, then SGD will recognize this and optimize with respect to that distribution. 

Given a large sample of target input cases, the target threat value calculation converges to 

the probability distribution shown in Figure 10. 

 

 
Figure 10 Probability distributions of target threat value 

 

The threat value output approximately takes the form of a normal distribution with a 

mean of 0.43 and standard deviation of 0.14. If SGD were applied to the information 

states with a training batch that was randomly sampled, the optimization might learn to 

predict the distribution as opposed to the dependencies and relationships within the cues 

themselves since the probability that the correct classification will be within the standard 

deviation is so high. In other words, any underlying interdependencies of the threat 

calculation might get overpowered by the predisposition of the output to reflect a normal 

distribution.  

 

To address this problem, a partitioning algorithm was implemented to create a uniform 

set of training targets from which to sample from during training. A random sample of 

1000 reference targets were selected to mimic a uniform distribution of threat value. This 

is known as a training set. Using this set during training will not reveal the underlying 



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probability of the outcome to the decision structure and consequently requires that the 

prediction be based on cue interdependency and cue conditional probabilities.  

 
5.9.2 Training method 

The training process is when the optimization occurs. A pseudo code for learning 

optimization is outlined below. The training set is described in section 5.9.1 and all the 

Greek variables have been defined in the sections about weight and sigma parameter 

optimization. The length of the batch is the number of cost gradients that are averaged per 

iteration. As mentioned before, this averaging promotes smoother learning due to reduced 

noise. 

 

WHILE (NOT stop condition):  

FOR EACH (information state IN information state space): 

   Average Cost Gradient(𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽) = [0, 0, 0, 0, 0, 0] 
FOR (length (batch)): 

FROM uniform training set SELECT 𝜇(𝑥) 

Reference Threat = CIS Threat Value ( 𝜇(𝑥) ) 

PARTITION  𝜇(𝑥) into IS shape 

    FROM database SELECT 𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽 for IS 

Calculate IS Threat Value:  𝑇(𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽, 𝜇(𝑥) ) 

    Calculate ∇𝐶𝑜𝑠𝑡(𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽) 

Average Cost Gradient = Average Cost Gradient + 
∇𝐶𝑜𝑠𝑡(𝜃,𝜔,𝜑,𝛿,𝜌,𝛽)

𝐿𝑒𝑛𝑔𝑡ℎ 𝐵𝑎𝑡𝑐ℎ
 

  [𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽] =  −[𝛼𝜃 , 𝛼𝜔, 𝛼𝜑 , 𝛼𝛿 , 𝛼𝜌, 𝛼𝛽 ] −  ∇(𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽) 

 

   UPDATE database with new 𝜃, 𝜔, 𝜑, 𝛿, 𝜌, 𝛽 for current IS 
 

A skeleton outline of the pseudo code is depicted in the information flow graphic in 

Figure 11. Below, the optimization of the weights is shown; however, the process for the 

sigma parameters is much the same. 

 

 



IJAHP Article: Kamps, Jassemi-Zargani/Decision making in dynamic environments: an 

application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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46 Vol. 13 Issue 1 2021 

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Figure 11 Graphic representation of information flow 

 

For the example scenario in the following sections, the following hyper parameters were 

used. These might not be the optimal values; however, they are enough to illustrate the 

results in the following sections. There is a set of selected hyper parameters which were 

found using a grid search and a set of random parameters that were found using random 

generation shown in Table 5. Both sets produce similar results. 

 

Table 5 

Hyper parameters values 

 

Learning Rate Best Selected Values Best Random Values 

𝜃 0.005 0.00661 

𝜔 0.01 0.000795 

𝛿 0.05 0.11 

𝛽 0.05 0.00661 

𝜑 0.5 1.645 

𝜌 1.5 9.03 
Batch 3 3 

Iterations 800 1000 

 

 

6. Experiment and results 
6.1 Measures of performance 

Three measures of performances were used to compare the base case to the trained 

structures. The first is range classification. A target threat prediction is considered correct 

if the difference between the reference threat value and the predicated threat value are 

within 0.1 of each other. 

= |𝑅𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 − 𝑇ℎ𝑟𝑒𝑎𝑡| < 0.1 



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The second measure of performance is the bin classification method. The target threat 

values are partitioned into five bins of length 0.2.   

 

𝐹𝑟𝑖𝑒𝑛𝑑𝑙𝑦 → 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 → 𝑈𝑛𝑘𝑛𝑜𝑤𝑛 → 𝑆𝑢𝑠𝑝𝑖𝑐𝑖𝑜𝑢𝑠 → 𝑇ℎ𝑟𝑒𝑎𝑡𝑒𝑛𝑖𝑛𝑔 
 

If the reference threat value and the predicted threat value fall within the same bin, the 

prediction is considered correct. Finally, the mean error of prediction and the standard 

deviation of that error is calculated. For each of these measures of performance, the result 

will be considered as an average over all of the information states and for each individual 

state. 

 
6.2 Results 

Using the hyper parameters described in section 5.9, an experiment was run in which 

targets from the training set were randomly sampled to apply SGD optimization over the 

learning parameters to the information states. The progression of the average cost at 

every 10 cycles of the 800 iteration learning progression is depicted in Figure 12. 

Learning occurs very quickly at the start, followed by a gradual reduction in cost as 

learning continues. The cost used to create this plot is the average quadratic cost. The 

yellow line represents a moving average. It can also be seen that as learning progresses, 

the cost becomes smoother with fewer large upward spikes, indicating a smaller standard 

deviation on error. 

 

 

 
Figure 12 Learning progression in 800 iterations with 10 per cycle 

 

The final optimized structures were saved in a database. Based on a trial of 1000 targets 

sampled randomly to form a validation set, the following results were generated for each 

measure of performance as shown in Table 6. 

 

  



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Table 6 

Performance results 

 

Metric Untrained Trained  

Information State 1 Mean 6 1 Mean 6 

Mean Absolute 

Error  

0.285 0.152 0.073 0.121 0.096 0.066 

Error Standard 

Deviation  

0.161 0.120 0.053 0.082 0.069 0.044 

Range 

Classification 

Accuracy 

12 % 37 % 62 % 46 % 55 % 79 % 

Bin Classification 

Accuracy 

13 % 33 % 72 % 45 % 48% 69 % 

 

Using range classification, there is an average improvement of 18%. Bin classification 

showed an improvement of 15%. The range classifications are higher which is expected 

since when using bin classification, there are cases when even if the prediction is very 

accurate, the deviation causes the prediction to fall within the wrong bin. 

 

The mean error and standard deviation also give merit to the performance of the 

algorithm. The mean error is reduced by 37% from 0.152 to 0.096. Additionally, the 

average standard deviation is almost halved. This is an indication that the learned 

structure is not only more accurate, it also attests to the fact that when it is wrong, it is 

still more precise. The error is much more highly concentrated meaning that even an 

incorrect classification is likely not far from a correct one. The distribution of target 

threat values that are output during the validation set have the following distributions: 

green is the base case, red is the trained case and blue is the reference threat output. 

 

 
 

Figure 13 Target threat value output distributions 

 



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application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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Figure 13 graphically shows the reduction of not only the error of classification but also 

the standard deviation. It is important to remember that the trained case was trained on a 

training set that had a uniform distribution. It was not given any information about what 

the probability distribution might look like. 

 
6.3 Analysis of the information states 

One of the other goals of this study was to prove the ability to predict dependent cues. As 

shown in Figure 14, the base case accuracy for each information state is represented in 

red and the optimized value is in green. Organizing the information states by the number 

of known independent cues produces the results shown in Figure 14. 

 

 

 
 

Figure 14 Independent cues accuracy 

 

 

7. Conclusions  

The AHP is a useful tool for synthesizing a decision based on the opinions of a SME. 

Supplementation of SSF nodes into the standard decision structure preserves the 

decisions that would have been made before the supplementation. The goal is to be able 

to predict the decision of a complete SSF supplemented decision structure with weights 

generated through the AHP when not all the information is available. This paper has 

shown that by applying the AHP to derive the weights and using machine learning to 

adjust the parameters of the sigmoid nodes in the deficient decision structures, the 

predictive capabilities can be vastly improved. This method is useful for all states of 

information availability and shows promise for use in practice. 

 



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application of machine learning to the Analytical Hierarchy Process 

 

 

 

 

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REFERENCES 

 

Johansson, F. & Falkman,G. (2008). A Bayesian network approach to threat evaluation 

with application to an air defense scenario. 11th International Conference on Information 

Fusion, 1-7. Cologne. 

 

Liebhaber, M.J., Kobus, D.A., & Feher, B.A. (2002). Studies of U.S. navy cues, 

information order, and impact of conflicting data. Technical Report 1888. San Diego: 

SPAWAR Systems Center. Doi: https://doi.org/10.21236/ada406334 

 

Menon, A., Mehrotra, K., Mohan, C.K., & Ranka, S. (1996). Characterization of a class 

of sigmoid functions with applications to neural networks. Neural Networks, 9(5), 819-

835. Doi: https://doi.org/10.1016/0893-6080(95)00107-7 

 

Mu, E. (2006). A unified framework for site selection and business forecasting using 

ANP. Journal of Systems Science and Systems Engineering, 15(2), 178–188. Doi: 

https://doi.org/10.1007/s11518-006-5006-6 

 

Ruder, S. (2017). An overview of gradient descent optimization algorithms. Computer 

Research Repository (CoRR), 1609.04747. 

 

Saaty, T.L. (1987). The Analytical Hierarchy Process – What it is and how it is used. 

Pittsburgh, PA: Pergamon Journals Ltd.