Abstract:
Decision-making processes appear intuitive but can easily become complex when
abstracted to theory. Therefore in this paper we will propose a simple hybrid
methodology that can efficiently resolve multi-attribute multivariate (MM) problems. We
do this by linking classical Shortest Path optimization with the practical business decision
methods for Multi-Attribute trade-off analysis. A simple example will demonstrate and
validate this approach for the reader.
Why this Approach?
To optimize large (multi-node) mono-attribute trades, an analyst can use a number of
approaches, classic linear programming approaches such as Shortest Path Multiple
Objectives (SPMO) method. However classical SPMOs based optimization has two
shortcomings; 1) the algorithms require that the user have specialized knowledge of
optimization theory and at a minimum some programming expertise not readily available
in most organizations also 2) in practical application (time and budget limited) SPMO
problems become inefficient as the meta data or attributes associated with the arcs
become more complex. Table 1 shows how the optimization times increase as the
network complexity grows:
Nodes
100
10000
100
10000
100
10000
100
10000
100
10000
Arcs
Attributes
1
1
2
2
3
3
5
5
10
10
Time
Notes:
Table1; Network complexity vs. time to Optimize. (this might be better shown as a
graph)
Methods for applied business decision making should to be simple to apply and intuitive.
However as Table 1 demonstrates as the number of attributes increase the networks
complexity no longer lends itself to simple optimization methods.
Here discuss a summary of what section 3 demonstrates.
What is the Approach?
Using a holistic approach as embodied in the Systems Engineering philosophy, this paper
is proposing a hybrid methodology referred to as System Heuristic Approach to Shortest
Path (SHASP). SHASP links a linear program (LP) for optimization with proven
decision methods for trade-off analysis.
The goal so this paper is to demonstrate the value of the SHASP approach which is:
(1) Intuitive; decision analyses trade methods make sense.
(2) Theoretically sound; each of the proposed pieces of SHASP are accepted in
Theory and practical application
(3) Comprehensive; the scaled arc values can integrate qualitative as well as intuitive
attributes. (This can be discussed here but is really the basis of a another paper)
Here discuss the benefits of SHASP from section 3.
Here include a process Diagram; Step 1, Step 2 to the SHASP process.
.
How does the Approach work? (Here we should decompose the
approach into its components)
This paper we will utilize Decision Analysis trade techniques to conjoin the defined
attributes referenced in the original network arc (arc prime) into a single scaled value for
that arc (arc double prime). These scaled arc values (arc double prime) will then be
substituted into the shortest path optimization equation to develop our final solution.
Decision Analysis Trades
Since the 1950s decision theory has been a very attractive area of research in fields such
as economics, statistics, psychology, and engineering. As a result, decision theory has
become a useful tool to many professionals including engineers. (i) In simple terms
decision theory provides a logical framework for solving real-life problems. (ii)
Smith[iii]summarized the benefits of decision theory into the following:
(1) The decision maker’s best course of action is derived based on his/her objectives
and knowledge of the problem.
(2) The decision maker communicates to the other specialists the best course of
action and justifies why the chosen action is optimal.
(3) A framework is provided within which the decision maker’s ideas can be
critically evaluated and modified, especially if new information is to be
incorporated or other than common decisions have to be made.
In this paper we will incorporate the basics of decision theory by setting up a descriptive
framework for the decision process. In this framework we will apply the concepts of
multi-attribute utility theory (MAUT) but the concepts described in this paper are not
limit to this single approach.
MAUT is closely related to multi-criteria decision making and allows quantification and
aggregation of multiple objectives even when these objectives are composed of
conflicting attributes, or when they are subjective. More specifically, the multi-attribute
utility theory is related to the case where our consequence arc(r) is a vector variable, i.e.
there is more than one attribute for r = (ci,j, ti,j, di,j,)
Multi-attribute utility theory strikes a trade-off so that the optimization procedure obtains
the best combination of the consequences with the choice of an appropriate action. In all
cases however the decision maker must also consider the independence of the arc
attributes that will be used to develop the final value. For example, in the case of our
Figure 1.1 below, r might be associated with three conflicting attributes, the time ti,j
associated with the, and the cost ci,to implement the option and finally the distance
traveled di,j
Origin
(1, 9, 7)
(2,
1
3
5
(0, 8, 1)
(3, 5, 1)
(ci,j, ti,j, di,j,)
,3
)
2
(0, 4, 3)
4,
(0,
5)
5
Dest.
4
Figure 1.1 Network with Variables Cost ci, Time ti, and Distance di
In the case that ci,j, ti,j and di,j are single-attribute utility functions then, ci,j , ti,j and di,j are
additive independent utility functions and therefore the scaling constants k for each
attribute kc + kt + kd = 1. If, however, ci,j , ti,j and di,j are not mutually independent, then
a k4 scaling constant will be required to normalize the arc vector (r) kc + kt + kd + k4 =
1.
Explanation of Muli-Attibution Function vs Muli-Objective Functions
f(x)  Min f(x)
Consider value(f(x))=v(f(x))=v(y), where
x =Design variable (objective)
y =Performance variable (attribute)
v(y) =value function
With respect to multi-attribute problems, we have:
yi, with i=1,2,...,n;
SAU: Single Attribute Utility/Value function
MAU: Multiple Attribute Utility/Value function, with
MAU =v(y1,y2,...,yn) =fn[v(y1),v(y2),...,v(yn)]
There are two forms often used to calculate MAU:
1.
Additive form
2.
Multiplicative form:
with   i=1, where  i is weight coefficient for i-th attribute
Multi-objective decision analysis in general prescribes theories for quantitatively
analyzing decisions involving multiple, interdependent objectives. The essence of multiobjective decision analysis is to break decisions down into small pieces that a decisionmaker can deal with individually and then recombine them logically. For this paper, cost,
time, and distance (ci,j, ti,j, di,j,) are the selected attributes for which we will determine
attribute relationships and trade-offs.
Throughout this paper we will focus on multi-objective decisions that have independent
utility functions and are evaluated under certainty which Bichler iv states is widely used
in business and government decision-making. Here, the basic assumptions about
preferences and utility are relaxed and therefore less controversial.
Our concept is to simulate the decision makers overall utility as the combination of the
decision makers objectives and attributes. This “bundling” of objectives and attributes
result in a scaled arc value based on the decision makers system of interest a standard
paradigm of the systems engineering approach.
Decision analysis techniques such as the Multi-attribute Utility Theory MAUT vvi, the
Analytic Hierarchy Process AHPvii and conjoint analysis viii are used in a broad range of
software packages for decision-makers and can be used to determine the utility function
of a decision-maker see ix,x.These ordinal utility functions are only required to rank-order
certain outcomes in a way that is consistent with the decision-maker’s preferences for
those outcomes. The model used in MAUT, AHP and conjoint analysis is basically an
additive utility function where some kind of subjective judgment forms the basis of the
weights, and yet the interpretation of the weights is not always clear. Therefore, extreme
care must be exercised in making the judgments on which the additive utility function is
based.xi
For our example, we acknowledge that assumptions affect the interdependence of the
attributes but will not include them in our decision model. It is fair to say that
preferential independence holds for many situations and in the following we will
concentrate on cases with preferential independence. Keeney and Raiffa provide a more
detailed discussion of multi-attribute utility modeling.xii
Our objective “bundling” of the arc value will be handled through the development of an
additive utility function. Bichler notes that an additive utility function is both easy to
implement and intuitive to the user; however, the additivity assumption implies that
attributes are preferentially independent and there are no interaction effects.
V(y1)>V(y2) ==> Preferred to
V(y1) V(y2) ==> Equivalent to
Decision analysis can not replace decision making it can serve as a tool to making better
decisions and to avoiding “analysis paralysis”. In our example we will therefore assume
that the attributes (cost, time and distance) are preferentially independent and that there
are no international effects. Our additive utility function is composed of scores on
individual attribute scales also called (individual utility functions or IUF) and weights for
these attributes.
The decision maker evaluates the relevant attribute value ci,j through an IUF (kc * ci,j) and
indicates its relative value by a weight w(i). All weights are positive and sum to 1.
Optimization for Decision Analysis
The purpose of decision theory is to obtain an optimum solution to a given problem.
Once the basic ingredients have been established, the fundamental question “how to
choose the action so that the consequence will provide maximum benefits to the decision
maker?” arises. The optimal solution in decision theory can be obtained by applying a
number of procedures according to the characteristics of the problem. A choice for multi
criteria optimization includes the use of Utility Theory, Weighted Sum, Min-Max and
Goal Programing. For the optimization process of this paper we are using SHASP as
defined in the next section.
i
Smith, J.Q., Decision Analysis – A Bayesian Approach, Chapman & Hall, London, 1988.
ii
A.T. de Almeida and G.A. Bohoris - Decision Theory in Maintenance Decision Making, Journal of
Quality in Maintenance Engineering. Vol. 1 No. 1, 1995 pp 39-45 MCB University Press.
iii
Smith, J.Q., Decision Analysis – A Bayesian Approach, Chapman & Hall, London, 1988.
M. Bichler – An experimental analysis of multi-attribute auctions, Decision Support Systems 29
R.T. Clement, Making Hard Decisions: An introduction to Decision Analysis, Wadsworth Publishing,
Belmont, CA, 1996.
vi
R.L. Kenny, H. Raiffa, Decision Making with Multiple Objectives: Preferences and Value Tradeoffs,
Cambridge University Press, Cambridge, UK, 1993
iv
v
vii
T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill,New York, USA, 1980.
viii
K. Backhause, B. Erichson, W. Plinke, R. Weiber, Multivariate Analysemethoden, 8 th edn., Springer,
Berlin, Germany, 1996.
ix
M. Bichler, Decision analysis — a critical enabler for multi-attribute auctions, Proceedings of 12th
Electronic Commerce Conference, Bled, Slovenia, 1998 .
x
G.E. Kersten, S.J. Noronha, Negotiation and the Internet: Users’ Expectations and Acceptance vol. 1998
Carleton Unversity, Canada 1998.
M. Bichler – An experimental analysis of multi-attribute auctions, Decision Support Systems 29 page 256
R.L. Keeny, H. Raiffa, Decision Making with Multiple Objectives: Preferences and Value Tradeoffs,
Cambridge Univ.Press, Cambridge, UK, 1993.
xi
xii