Asia
Pacific Management
Review Review
(2006) 11(3),
Iraj Mahdavi
et al./Asia
Pacific Management
(2006)155-162
11(3), 155-162
Conceptual Framework for Quantitative Modeling of Semi-structured MADM
Iraj Mahdavia∗, Babak Shirazia, Namjae Chob and Nezam Mahdavi-Amiric
a
Department of Industrial Engineering, College of Technology, Mazandaran University of Science & Technology, PO Box734, Babol, Iran
b
School of Business, Hanyang University, 17 Haegdang-dong, Seongdong-gu, Seoul, Korea
c
Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
Accepted in June 2006
Available online
Abstract
In some situations, MADM matrix is not distinguished completely at the first stage of decision making, because of the complexity of
environment. These complexities lead to incomplete cognition and non-optimal decision making. In such “semi-structured” environment,
due to its high degree of complexity, the whole environment is not identifiable for Decision Maker (DM). We design an autonomous agent
for semi-structured MADM that solves problems when alternatives have incomplete structure and DM is not able to recognize the whole
alternatives of the environment for optimal decision making. The proposed model is a systematic approach for semi-structured MADM
with multi-layer mathematical model. The Agent’s Stepwise Response Generator (ASRG) moves in semi-structured environment over
decision surface step by step to generate hidden alternatives. The new alternatives are designed to go through Feasibility Analyzer and
Dynamic Filter Module. The procedure is continued with a closed loop feedback which results in the construction of the Meta-Decision
phase.
Keywords: Autonomous agent; Decision surface; Meta-decision; Semi-structured MADM
represent different dimensions of an alternative, they may
conflict with each other. For instance, cost may conflict
with profit, etc. Most of the MADM methods require that
attributes be associated with weights of importance. Usually, these weights are normalized to add up to one. Several methods have been proposed for solving
multi-attribute decision making problems. A major criticism to MADM is that different techniques yield different
results when applied to the same problem. A simulation
comparison of selected methods was performed by
Zanakis et al. (1998).
1. Introduction
Multi Attribute Decision Making (MADM) is the most
well known branch of decision making. It is a branch of a
general class of Operations Research (OR) models which
deal with decision problems under the presence of a number of decision criteria (Triantaphyllou et al., 1998). This
class of models is very often called Multi Criteria Decision
Making (MCDM). According to many authors
(Zimmermann, 1996), MCDM is divided into Multi Objective Decision Making (MODM) and Multi Attribute
Decision Making (MADM). MODM studies problems in
which the decision space is continuous. A typical example
is mathematical programming problems with multiple objective functions. On the other hand, MADM concentrates
on problems with discrete decision spaces. In these problems the set of decision alternatives tends to be predetermined. Although MADM methods vary widely, many of
them have certain aspects in common (Chen and Hwang,
1992). Each MADM problem is associated with multiple
attributes. Attributes are also referred to as “goals” or “decision criteria”. Attributes represent different dimensions
from which the alternatives can be viewed and measured.
In cases where the number of attributes is large, attributes
can be arranged in a hierarchical manner. That is, some
attributes are defined as major attributes. Each major attribute is associated with several sub-attributes. Similarly,
each sub-attribute may be further associated with several
sub-sub-attributes and so on. Since different attributes
∗
Email:
2. Multi Attribute Decision Making
During the recent decades, the classical decision
making of optimization with one criterion or one objective
function has evolved into Multiple Criteria Decision
Making (MCDM) models for complex decision making
problems. These models can be linear, nonlinear, or hybrid.
Two categories of decision making with multiple criteria
are identified. Multiple Objective Decision Making
(MODM) model is used to support planning and MADM
is designed to select the best alternative (Hwang and Yoon,
1971; Triantaphyllou et al., 1998).
Full-structured
MADM model is formulated in the form of decision making matrix as shown in Table 1.
[email protected]
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Iraj Mahdavi et al./Asia Pacific Management Review (2006) 11(3), 155-162
Table 1. Full-structured Static MADM Matrix
then we could use the decision making techniques as SAW,
ISAW, TOPSIS, LINMAP, ELECTRE, etc. (Hwang and
Yoon, 1971; Triantaphyllou et al., 1998). Classical MADM
techniques using deterministic and specific mathematical
models are applicable when all system variables are determined and there are no uncontrollable variables. In contrast,
the uncertainty and complexity in problems and alternatives cause incomplete understanding and thus, incomplete
decision making (Gu and Zhu, 2006; Kaebling et al., 1998;
Leung et al., 2006). These complexities can be the cause
for a decision bottleneck of a system. Complexities arise
from the relationship between DM and Environment
(Kendall, 2000). Specifically the complexity is related to
the following aspects:
1. DM intelligence: Inability in recognizing problems, and
in defining problems, prioritization, and organizing information.
A1 , A2 , …, Am in matrix D are predefined alternatives
and x1 , x2 , …, xn represent utility attributes applicable to
the alternatives. Each component aij stands for the coefficient value of j-th attribute for i-th alternative. If a problem
does not have a clear structure due to environmental complexity, we can not construct matrix D completely. The
problem domain can also take static or stochastic dynamic
nature (Howard, 1971; Kaebling et al., 1998; Leong, 1994;
Monahan et al., 2000). So we can apply MADM to different types of problems as shown in Figure 1.
2. Problem planning: Inability in generating alternatives,
and in explaining and evaluating alternatives.
3. Selecting: Inability in selecting a right solution method
and selecting alternatives.
4. Counting:
Large number of alternatives.
5. Environment: Environment with uncontrollable variables, environmental disturbance.
Here we focus on the static structure. In static MADM
system, decision making does not depend on time and the
matrix has non-dynamic structure. But it can be incomplete when the problem is semi-structured MADM. If the
decision making system were full-structured static
MADM,
These factors lead to an “environment with high degree of complexity”. In these cases a part of MADM matrix is invisible. This structure is called “semi-structured”
decision model as shown in Table 2. Alternatives Am+j , for
j ≥ 1, are invisible.
Table 2. Semi-structured MADM Matrix
Figure 1. MADM Classification
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Figure 2. High-level Descriptive Functions System
the system) that may naturally possess nonlinear elements
(Boutilier et al., 1999). We can use regression methods to
fit a function in an n-dimensional Euclidean space. Some
high level functions which describe the system behavior
(i.e. sub-goals in our model) are stability, productivity,
performance and flexibility:
Because of the invisibility in this incomplete structure,
the analyst’s task is to help DM to make better decision
through incremental exploration of alternatives (Leung et
al., 2006). This proposed model should be in a way that
help decrease “decision bottleneck”. For this purpose we
suggest an autonomous decision maker agent. This agent
recognizes the incomplete structure of design environment, and begins to interpret the environment step by step
and generate unknown alternatives and provides feedback
to redesign the alternatives to complete the matrix structure. The agent, thus, determines the unspecified portion of
the decision for a DM and helps him to proceed in decision making. We design and suggest a multi-layer architecture of agent and the basic rules of each layer. Each
layer core uses OR rules for designing sub-modules.
Subgoal1 = Stability =F1 (x1 , x2 , …, xn);
Subgoal2 = Productivity= F2 (x1 , x2 , …, xn) ;
Subgoal3 = Performance = F3 (x1 , x2 , …, xn) ;
Subgoal4 = Flexibility= F4 (x1 , x2 , …, xn).
3.
Autonomous Decision Maker Agent
Decision making in semi-structured environments
needs a support for searching unknown portion of the decision environment. This support can be either an automated decision making process or a mechanism for generating unknown alternatives (Alami et al., 1998; Baroni et
al., 1995; Ferber, 1999; Leger, 1999). Here we focus on
the autonomous decision maker agents. An agent acts as
an interface to reduce complexity of the semi-structured
decision environment. Agent helps DM in making decision
so that the relationship between DM and Agent is depicted
as shown in Figure 3.
Under a semi-structured environment without a clear
structure we need to analyze the system step by step following the hierarchical structure. The hierarchical diagram
describes the behavior of the system as shown in Figure 2
(Alpert, 1971; Armacost and Hosseini, 1994; Youngpil et
al., 2003). This structure has 3 layers of Goal, Sub-Goals
(F1 , F2 , …, Fk) and attributes (x1 , x2 , …, xn).
Each sub-goal is a function of attributes describing
system behavior. These functions reflect past experiences
of the system (DM’s and analyst’s beliefs, past activities of
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Iraj Mahdavi et al./Asia Pacific Management Review (2006) 11(3), 155-162
Figure 3. «DM – Agent » Relationship
Autonomous agent generates new alternatives that
DM could not recognize independently from other agents
in the multi-agent system and as a partial substitute for
DM. Due to the semi-structured nature of problems,
autonomous agent decision making is sequential (stepwise) to reach the final goal of a system (Littman et al.,
1996). As an agent receives additional information from
the environment, he helps or even replaces DM in an evolutionary manner (Cardon et al., 2000). Leger (1999)
shows comparison between human DM and autonomous
DM agent. As to be seen in the next section, a DM agent
has a multilayered structured environment that searches
through a vague space to generate alternatives (Littman et
al., 1996). The agent search locally on the decision surface
and varies the search incrementally using different paths
as shown in Figure 4 (Barraquand et al., 1992; Menczer et
al., 2001).
Figure 5. Autonomous Decision Maker Agent Architecture
The concept of the modules in the architecture can be
summarized as follows:
Module 1- Start from an initial point: Provides initial
starting point on the decision surface for beginning to
move.
Module 2- Autonomous Stepwise (Sequential) Response
Generator (ASRG): Autonomously generates new responses step by step away from the initial starting point.
Module 3- Feasibility Analyzer (FA): Tests the feasibility
of the generated alternatives in the solution space of a
given problem.
A0 is one of m predefined alternatives (A1 , A2 , …, Am)
and A11, A12 are new generated alternatives.
An autonomous agent for decision making in a
semi-structured environment with high degree of complexity should be able to deal with the vague nature of the
structure. This agent should be structured as given in Figure 5.
Module 4- Dynamic Filtering System (DFS): Presents the
new alternative to DM, eliminates irrelevant alternatives
based on the interaction with DM and suggests the selection of an alternative that satisfies a specific condition.
Module 5- Meta-Decision Synthesizer: Extracts effective
alternatives and adds them to the initial matrix.
Re-Decision: Revises MADM matrix based on newly
generated alternatives and applies classic MADM techniques to the incrementally expanded matrix.
4. Decision Surface and Autonomous Stepwise (Sequential) Response Generator
The method we use for generating the alternatives in a
vague semi-structured environment follows the sequential
movement logic. Since the process of alternative generation is a step by step movement on the decision surface, a
stepwise response generator is needed. The agent uses a
movement method on the decision surfaces as shown in
Figure 6.
Figure 4. Type of Agent Search Via Path ω1 or ω
The environmental complexity is interrelated to the
movement on the surface. As for the modeling of the
movement on this surface, we can use the regression
method to fit the function in an n-dimensional Euclidean
space.
1
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Ω(A)
ω1
A1
A11
A2
Ω (A) = Ω ( x1 , x2 , …, xn)
x2
x1
Figure 6. Agent Movement and Generation
of New Alternatives
Figure 7. Agent Movements Over Decision Surface Ω
To solve a semi-structured problem, we begin with the
decision surface which contains the initial point of the
system with m predefined alternatives, and we can assign
more complex functions to this surface to deal with a more
complex environment.
Agent begins movement on the decision surface Ω
from specified points A1 , A2 , …, Am through the path ω
constructed by algorithms φh-1 (to be described in the
next section). Figure 7 shows how the agent starts its
movement from A1 and generates A11 through path ω1 over
decision surface Ω (A).
4.1 Definition: Decision Surface
4.2 Sequential (Stepwise) Algorithms for Moving on a Decision Surface
F is a general form of the descriptive function and
the set of points A= {x1 , x2 , …, xn } is considered to satisfy
the followings:
Rc = {A | A ∈ R n, Fi (A) ∈ R}
Suppose that Ω(A) is the decision surface. An agent
starts from a selected and applicable point Ak on the decision surface. We define ξ=ξk as the amount of step size. If
it is too small, the speed of agent’s movement will be too
slow. On the other hand, if it is too large the system will
face overshooting. For this reason, the selection of ξk is of
special importance. Algorithm φh -1 is a suggestion for
generating alternatives as given below:
and Ω is the decision surface,
4.2.1 Algorithm φh -1
F: R n → R
( x1 , x2 , …, xn ) ∈ R n | xi ∈ R Î
A ∈ Rn
Î Fi(A) ∈ R , i=1, 2 , …, k ( k descriptive functions )
1- Begin
decision surface = Ω (A) = {A | A satisfies the above conditions}
Ω(A) = Ω( x1 , x2 , …, xn ) is one of the descriptive
functions Fi (A); A = ( x1 , x2 , …, xn ).
2- Initialize A, σ
3- While || ∂Ω (A) || > σ do
4- Determine ξ
5- AÅ A – ξ ∂Ω (A)
Decision surface is an n-dimensional subspace of the n+1
dimensional space which is divided into Rc and R n-Rc.
Depending on the level of complexity we can associate
several forms of functions with the decision surface:
Ω(A) = b0 + ∑ bi xi
6- Return A
7- End.
(A: new generated alternative)
1≤ i≤ n
To design a method for setting ξ , suppose that the
function can be approximated by a second-order expansion
around the value A = Ak :
( linear decision surface for low complexity)
1≤ i≤ n
, 1≤ j≤ n
Ω(A) = b0 + ∑ bi xi +∑ ∑ bij xi xj
( quadratic decision surface for medium complexity)
Ω(A) ≈ Ω(Ak)+ ∂ Ω t(A-Ak) + ½ ( A-Ak) t H (A-Ak)
Ω(A) = b0 + ∑ bi xi +∑ ∑ bij xi xj +∑ ∑ ∑ bijk xi xj xk
1≤ i ≤ n , 1≤ j ≤ n , 1≤ k ≤ n (highly complex environment).
where H is the Hessian matrix (second partial derivatives
∂2 Ω / ∂Ai∂Aj ) evaluated at Ak .
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Using the above approximation, we will have:
Ω(Ak+1) ≈
Ω(Ak) - ξ k || ∂Ω || 2 + ½ ξ k 2 ∂Ω t H ∂Ω .
Feasibility Analysis of A11 :
1) x1L ≤ x111≤ x1R
2) x2L ≤ x211≤ x2R
Minimizing with respect to ξ k we will obtain:
ξ = ξ k = || ∂Ω || 2 / ∂Ω t H ∂Ω.
Of course, one may use a line search strategy to set ξ
in each iteration to guarantee reduction in every step (see
Nocedal and Wright, 1999).
Ω(A)
ω
11
A
As a second-order approach for minimization, we can
minimize the quadratic approximation in each iteration
obtaining the following algorithm (the so-called modified
Newton’s method):
A0
x111
4.2.2 Algorithm φh -2
x211
1- Begin
x2
2- Initialize A, σ
x1
3- Repeat
4- p= -H -1∂Ω(A) ( p solution of Hp= - ∂Ω(A))
Figure 8. Extracting New Alternative Attributes
5- A Å A+ α p (α obtained from a line search)
6- Until || p || ≤ σ
-
F(A ) = min F(A), F(A+ ) = max F(A) .
7-Return A
We can design three general forms of filters as shown
in Figure 9:
8-End.
In the above algorithm, the step size in each iteration
can be obtained using a line search strategy. Alternatively,
one may use robust minimization techniques such as secant (quasi-Newton) methods or, to avoid line search, trust
region methods (Nocedal and Wright, 1999).
5.
a) Low–pass filter: Eliminates all alternatives which satisfy F(A) ≥ F(A+) or passes all alternatives which satisfy
F(A) <F(A+ ).
b) Medium–pass filter: Eliminates all alternatives which
do not satisfy F(A ) ≤ F(A)≤ F(A+) or passes all alternatives which satisfy F(A ) < F(A) < F(A+).
Feasibility Analyzer
For some newly generated alternatives, some attributes xi may not satisfy xiL ≤ xi ≤ xiR . For this reason we use
the Feasibility Analyzer to find feasible solutions among
the newly generated alternatives. A new alternative A is
feasible if all of its attributes are in the feasible region
(xiL ≤ xi ≤ xiR , for all i ,1 ≤ i ≤ n ). Over the decision surface Ω, the new alternative A11 is generated following path
ω. To test the feasibility of A11 as appeared in Figure 8 we
consider the following constraints:
c) High–pass filter: Eliminates all alternatives which satisfy F(A) ≤ F(A ) or passes all alternatives which satisfy
F (A) > F (A ).
Filtering newly generated alternatives is a dynamic
activity because the alternatives are generated autonomously by the agent without the DM’s control. Filters may
have different structures when compared with the above
general conditions, but they should eliminate some unnecessary alternatives. So we can consider different types of
filters according to DM’s perception and judgment.
Constraints x iL ≤ x i ≤ x iR in the n+1 dimensional
space (due to n attributes) form a subspace intersecting
the decision surface.
6.
7. Meta-Decision Synthesizer and Re-Decision
Dynamic Filtering System
Definition: Meta-decision
Designing a filter is a multi-stage, repetitive activity
that requires an interaction with DM. A filter has a dynamic nature due to the changes in priorities and attitude
of the DM. The filter eliminates some alternatives so that
the remaining alternatives satisfy DM’s needs. Let A and
+
A be arguments of minimum and maximum limits of the
descriptive function describing the attitude of a DM:
The movements on the decision surface Ω from certain points A1 , A2 , …, Am through the path ω with the algorithm φh-1 generate new alternatives which DM did not
have at the beginning. These additional alternatives form a
newly updated matrix that can lead to a new decision.
Besides the responses generated by traditional approaches,
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Step 4. Use sequential algorithms to move on the decision
surface from the best existing alternative as
described in Section 4.2 and obtain new alternatives.
Step 5. Apply feasibility analyzer and dynamic filtering to
the new alternatives as illustrated in sections 5 and
6 respectively.
Step 6. Obtain an acceptable new alternative and create a
new MADM matrix. If there is no new alternative,
then stop or jump to the next movement with subsequent permutation.
Figure 9. Low-Pass-Filter, Medium-Pass-Filter
and High-Pass-Filter (left to right)
Step 7. Let I=I+1. If I is less than NA then go to Step 1,
otherwise Stop.
new alternatives are added by the agent to the decision
matrix as shown in Figure 10. We call this process to improve the accuracy of a decision through a repeated
re-creation of the decision matrix as the “meta-decision”.
8. Conclusions
This study suggests the use of a Decision Maker
Agent, which autonomously generates unspecified alternatives independent of DM. The agent adds newly generated
alternatives to the initial matrix. MADM matrix extends to
contain more alternatives for the decision maker. For generating alternatives not recognized by the DM from the
beginning, the autonomous agent moves step by step on
the decision surface and uses the path that has been calculated by the corresponding algorithm. All newly generated
alternatives are not added to the initial matrix, because the
agent generates them autonomously without the control of
DM. For this reason, the agent has a layer to filter out alternatives which do not satisfy the perceptual requirements
of the decision maker (Feasibility Analyzer and Dynamic
Filter).
Newly generated alternatives created by the stepwise
generator that pass through Feasibility Analyzer and Dynamic Filtering System can be added to the initial decision
system. For this newly generated matrix, we design a new
decision surface on the basis of previously obtained stage
alternatives. Sequential algorithms are then applied for
movement on the decision surface to obtain next new alternatives. MADM technique is used here to make a decision again (Re-Decision). Improved outcomes are produced with the repetition of the Meta-Decision process.
The solution can lead to an effective and flexible selection
through an active interaction with the DM. Finally the
steps of the proposed conceptual framework for quantitative modeling of semi-structured MADM follow (NA is
the number of new alternative to be created).
The autonomous Decision Maker Agent interacts with
DM to increment the decision matrix to be reconsidered on
the basis of the classical MADM to achieve the decision
goal. Future research can focus on the development of an
extended model to incorporate the stochastic dynamic
evolution of the logic by taking into account the possibility
of the changes in attribute values and the existence of
missing attribute values of known alternatives.
Step 1. Let I=0.
Step 2. Identify decision surface on the basis of existing
alternatives as defined in Section 4.1.
Step 3. Obtain the best existing alternative using classic
MADM technique.
Figure 10. Meta-Decision
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