Problem solving with multiple interdependent
criteria ∗
Christer Carlsson
christer.carlsson@abo.fi
Robert Fullér
robert.fuller@abo.fi
Abstract
We consider multiple objective programming (MOP) problems with additive interdependences, i.e. when the states of some chosen objective are attained through
supportive or inhibitory feed-backs from several other objectives. We show how the
concept of interdependency can be applied to an n-party single-issue negotiation problem in which the negotiators cooperatively face a common problem or in which the
parties’s interests are not diametrically opposed (variable-sum bargaining situations).
1
MOP Problems with Interdependent Objectives
In their classical text Theory of Games and Economic Behavior John von Neumann and
Oskar Morgenstern (1947) described the problem with interdependence; in their outline of a
social exchange economy They discussed the case of two or more persons exchanging goods
with each others (page 11):
. . . then the results for each one will depend in general not merely upon his own actions
but on those of others as well. Thus each participant attempts to maximize a function
. . . of which he does not control all variables. This is certainly no maximum problem,
but a peculiar and disconcerting mixture of several conflicting maximum problems.
Every participant is guided by another principle and neither determines all variables
which affects his interest.
This kind of problem is nowhere dealt with in classical mathematics. We emphasize at
the risk of being pedantic that this is no conditional maximum problem, no problem of
the calculus of variations, of functional analysis, etc. It arises in full clarity, even in the
most ”elementary” situations, e.g., when all variables can assume only a finite number
of values.
The interdependence is part of the economic theory and all market economies, but
in most modelling approaches in multiple criteria decision making there seems to be an
implicit assumption that objectives should be independent. This appears to be the case, if
not earlier than at least at the moment when we have to select some optimal compromise
among the set of nondominated decision alternatives. Milan Zeleny [26] - and many others
- recognizes one part of the interdependence (page 1),
Multiple and conflicting objectives, for example, ”minimize cost” and ”maximize the
quality of service” are the real stuff of the decision maker’s or manager’s daily concerns.
∗
The final version of this paper appeared in: J. Kacprzyk, H.Nurmi and M.Fedrizzi eds., Consensus
under Fuzziness, The Kluwer International Series in Intelligent Technologies, Vol. 10, Kluwer Academic
Publishers, Boston, 1997 231-246.
1
Such problems are more complicated than the convenient assumptions of economics
indicate. Improving achievement with respect to one objective can be accomplished
only at the expense of another.
but not the other part: objectives could support each others.
Situations with multiple interdependent objectives are not only derived from some axiomatic framework as logical conclusions, or built as illustrations of complex decision problems in classical text books, there are real life situations which, if we ponder them systematically, reveal themselves to have interdependent objectives.
A well-known negotiation problem is the Buyer/Seller dilemma [24], in which it is unclear for both parties at which price they are going to settle when they start the negotiation
process (cf fig.1):
Reservation prices
Buyer
Seller
Fig. 1
Buyer/Seller negotiation problem.
Their objectives are clearly conflicting: the Buyer wants the price to be as low as
possible; the Seller tries to keep the price as high as possible. There are two points, the
reservation prices, beyond which the negotiations break down (cf fig.1). The Buyer will
not discuss a price higher than the (Buyer)-point; the Seller will find a price lower than
the (Seller)-point insulting. If both parties compromise they can eventually settle on a
price somewhere at the mid-point of the interval. The problem becomes interesting and
challenging when none of the parties can be sure of the other party’s reservation price, but
the setup is simple in the sense that the objectives are conflicting, and the problem can be
solved with standard methods.
Let us now assume that there is a third party, the Government, involved and that the
Government reacts to the negotiation process by working out consequences of both the
reservation prices and the offers made during the negotiations. Let us also assume that the
Government wants to intervent in the process in order to promote its own objectives.
In this way the negotiation problem becomes much more complex as there are a number
of new objectives involved over which the primary parties have no control, cf fig.2.
Seller
(2)
(3)
Buyer
(1)
Reservation prices
G1
Obj 1
G2
Obj 2
Obj 3
Fig. 2
A modified Buyer/seller negotiation problem.
The Buyer and the Seller influence two different objectives of the Government respectively (Obj 1 and Obj 2):
(i) a low Seller reservation price will support Obj 1, but a high Seller reservation price
will be in conflict with Obj 1 after some point G1 ;
2
(ii) a high Buyer reservation price will support Obj 2, but a low Buyer reservation price
will be in conflict with Obj 2 after some point G2 ,
(iii) Obj 1 and Obj 2 are conflicting (as often is the case with political objectives).
The negotiation process ((1), (2), (3)) influences Obj 3 of the Government:
(iv) short steps and a small interval both support the attainment of Obj 3, but
(v) large, oscillating steps and a large interval are hinders for the attainment.
There are a number of cases of labor market negotiations in which these types of interdependences are present. Because Obj 1-3 are influenced by the Buyer-Seller negotiations
the Government can not remain passive, but will influence the objectives of the primary
negotiators. Then we get a set of decision problems for all parties involved, in which we
have multiple interdependent objectives; these problems are not easy to cope with and
to resolve. In some labor market negotiations in the spring 1996 the Finnish government
managed to both define such levels of its own objectives and to push the reservation prices
of the primary negotiators to levels which were unattainable. When the government finally
agreed to face-saving compromises the country was 12 hours from a general strike.
We will in the following explore the consequences of allowing objectives to be interdependent.
2
Additive linear interdependences in MOP
Objective functions of a multiple objective programming problem are usually considered to
be independent from each other, i.e. they depend only on the decision variable x. A typical
statement of an MOP with independent objective functions is
max{f1 (x), . . . , fk (x)}
x∈X
(1)
where fi is the i-th objective function, x is the decision variable, and X is a subset, usually
defined by functional inequalities. Throughout this paper we will assume that the objective
functions are normalized, i.e. fi (x) ∈ [0, 1] for each x ∈ X.
However, as has been shown in some earlier work by by Carlsson and Fullér [4, 5, 6],
and Felix [14], there are management issues and negotiation problems, in which one often
encounters the necessity to formulate MOP models with interdependent objective functions,
in such a way that the objective functions are determined not only by the decision variables
but also by one or more other objective functions.
Typically, in complex, real-life problems, there are some unidentified factors which effect
the values of the objective functions. We do not know them or can not control them; i.e.
they have an impact we can not control. The only thing we can observe is the values of the
objective functions at certain points. And from this information and from our knowledge
about the problem we may be able to formulate the impacts of unknown factors (through
the observed values of the objectives).
First we state the multiobjective decision problem with independent objectives and then
adjust our model to reality by introducing interdependences among the objectives.
Interdependences among the objectives exist whenever the computed value of an objective function is not equal to its observed value. In this paper we claim that the real values
3
of an objective function can be identified by the help of feed-backs from the values other
objective functions.
Suppose now that the objectives of (1) are interdependent, and the value of an objective
function is determined by a linear combination of the values of other objectives functions.
That is
k
fi (x) = fi (x) +
αij fj (x), 1 ≤ i ≤ k
(2)
j=1, j=i
or, in matrix format






f1 (x)
f2 (x)
..
.
fk (x)


 
 
=
 
 
1 α12 . . . α1k
α21 1 . . . α2k
..
..
..
..
.
.
.
.
αk1 αk2 . . . 1






f1 (x)
f2 (x)
..
.






fk (x)
where αij is a real numbers denoting the grade of interdependency between fi and fj .
If αij > 0 then we say that fi is supported by fj ; if αij < 0 then we say that fi is
hindered by fj ; if αij = 0 then we say that fi is independent from fj (or the states of fj
are irrelevant to the states of fi ).
αijfj(x)
1
1 fj(x)
Fig. 3
fj(x)
α ijfj(x)
Linear feed-back with αij > 0 and αij < 0
In such cases, i.e. when the feed-backs from the objectives are directly proportional to
their independent values, then we say that the objectives are linearly interdependent.
Taking into consideration the linear interdependences among the objective functions
(2), (1) turns into the following problem (which is treated as an independent MOP)
max{f1 (x), . . . , fk (x)}
(3)
x∈X
It is clear that the solution-sets of (1) and (3) are usually not identical.
To explain the issue more exactly, consider a three-objective problem with linearly
interdependent objective functions
max{f1 (x), f2 (x), f3 (x)}
(4)
x∈X
Taking into consideration that the objectives are linearly interdependent, the interdependent values of the objectives can be expressed by





f1 (x)
1 α12 α13
f1 (x)
  


1 α23   f2 (x) 
 f2 (x)  =  α21
α31 α32 1
f3 (x)
f3 (x)
For example, depending on the values of αij we can have the following simple linear interdependences among the objectives of (4)
4
• if α12 = 0 then we say that f1 is independent from f2 ;
• if α12 > 0 then we say that f2 unilaterally supports f1 ;
• if if α12 < 0 then we say that f2 hinders f1 ;
• if α12 > 0 and α21 > 0 then we say that f1 and f2 mutually support each others;
• if α12 < 0 and α21 < 0 then we say that f1 and f2 are conflicting;
• if α12 + α21 = 0 then we say that f1 are f2 are in a trade-off relation;
It is clear, for example, that if f2 unilaterally supports f1 then the bigger the value of
f2 (supporting objective function) the bigger its contribution to f1 (supported objective
function).
f2
α2 1
α1 2
f1
Fig. 4
3
α2 3
α3 2
α1 3
α3 1
f3
A three-objective interdependent problem with linear feed-backs.
Additive nonlinear interdependences in MOP
Suppose now that the objectives of (1) are interdependent, and the value of an objective
function is determined by an additive combination of the feed-backs of other objectives
functions
fi (x) = fi (x) +
k
αij [fj (x)], 1 ≤ i ≤ k
(5)
j=1, j=i
where αij : [0, 1] → [0, 1] is a - usually nonlinear - function defining the value of feed-back
from fj to fi .
If αij (z) > 0, ∀z we say that fi is supported by fj ; if αij (z) < 0, ∀t then we say that fi is
hindered by fj ; if αij (z) = 0, ∀z then we say that fi is independent from fj . If αij (z1 ) > 0
and αij (z2 ) < 0 for some z1 and z2 , then fi is supported by fj if the value of fj is equal to
z1 and fi is hindered by fj if the value of fj is equal to z2 .
αijfj(x)
1
1 fj(x)
Fig. 5
α ijfj(x)
Nonlinear unilateral support and hindering.
5
fj(x)
β
1
fj(x)
αijfj(x)
Fig. 6 fj supports fi if fj (x) ≤ β and fj hinders fi if fj (x) ≥ β.
Consider again a three-objective problem
max{f1 (x), f2 (x), f3 (x)}
x∈X
with nonlinear interdependences. Taking into consideration that the objectives are interdependent, the interdependent values of the objectives can be expressed by
f1 (x) = f1 (x) + α12 [f2 (x)] + α13 [f3 (x)]
f2 (x) = f2 (x) + α21 [f1 (x)] + α23 [f3 (x)]
f3 (x) = f3 (x) + α31 [f1 (x)] + α32 [f2 (x)]
For example, depending on the values of the correlation functions α12 and α21 we can have
the following simple interdependences among the objectives of (4)
• if α12 (z) = 0, ∀z then we say that f1 is independent from f2 ;
• if α12 (z) > 0, ∀z then we say that f2 unilaterally supports f1 ;
• if α12 (z) < 0, ∀z then we say that f2 hinders f1 ;
• if α12 (z) > 0 and α21 (z), ∀z > 0 then we say that f1 and f2 mutually support each
others;
• if α12 (z) < 0 and α21 (z) < 0 for each z then we say that f1 and f2 are conflicting;
• if α12 (z) + α21 (z) = 0 for each z then we say that f1 are f2 are in a trade-off relation;
However, despite of the linear case, we can have here more complex relationships between
two objective functions, e.g.
• if for some β ∈ [0, 1]
α12 (z) =
positive if 0 ≤ z ≤ β
negative if β ≤ z ≤ 1
then f2 unilaterally supports f1 if f2 (x) ≤ β and f2 hinders f1 if f2 (x) ≥ β.
• if for some β, γ ∈ [0, 1]


 positive
if 0 ≤ z ≤ β
0
if β ≤ z ≤ γ
α12 (z) =

 negative if γ ≤ z ≤ 1
then f2 unilaterally supports f1 if f2 (x) ≤ β, f2 does not affect f1 if β ≤ f2 (x) ≤ γ
and then f2 hinders f1 if f2 (x) ≥ γ.
6
It should be noted that so far we have considered only pairwise interdependences between
the objectives, i.e. when the feedback from f2 to f1 depends only on the value of f2 , but
in real life cases we usually have compound interdependences (cf. [6, 8]), i.e. when the
feedback from f2 to f1 can depend not only on the value of f2 , but also on the values of f1
(self feed-back) and/or f3 .
4
Consensus reaching with interdependent utilities
Negotiations arise from a variety of different types of disputes. For simplicity, we focuse on
an n-party, single-issue negotiation in which individuals are in conflict because they want
different things, but must settle for the same thing. The resolution of negotiations requires
parties to reach a join decision about a settlement. Potential settlements consist of different
combinations of values for the issue explicitly or implicitly under negotiation.
In a simple n-party, single-issue negotiation problem, each negotiator judges the utility
of potential settlements. Classically, judgments of utility are assumed to be a function of
the values of the issue (independent utilities). Let ui (x) represent the judgment by the i-th
negotiator of the utility of the potential settlement x ∈ [0, 1]. The reference points, zero
and one, indicate the two possible extreme settlements of the issue.
1
1
u1(x)
1
Fig. 7
u2(x)
x
1 x
Linear utility functions.
To have a uniform (monoton increasing) presentation for utility functions ui (x), we
introduce application functions
hi : IR → [0, 1]
such that hi (ui (x)) measures the degree of satisfaction of the i-th party with the utility
value ui (x) for a settlement x ∈ [0, 1], i = 1, . . . , n.
Usually, the application functions are of the form (the more the better)


 1
hi (t) =
if t ≥ Mi
vi (t) if mi ≤ t ≤ Mi


0
if t ≤ mi
where mi denotes the reservation level (which represents the minimal requirement about the
issue), Mi stands for the desired level on the issue and vi is a monoton increasing function.
1
1
h1(x)
m1
M1
Fig. 8
h2(x)
m2
1 u1(x)
M2
1 u2(x)
Linear application functions.
Negotiations consist of a ”dance” involving a sequence of proposals and counterproposals, offers and counteroffers. The uncertainty and cognitive complexity entailed in most negotiations prevent the parties from leaping to a joint agreement on their first move. Instead,
7
negotiations tend to proceed incrementally and cautiously with the parties, attempting to
”feel their way along” to a settlement, unsure of when the level of concessions they offer
meets the other’s minimum reservation level, and hoping not to be taken advantage of.
The resolution of negotiations ordinarily requires settlement of differences by mutual
concessions. For disputes involving multiple issues, two fundamental strategies of concession
can be identified. The first consists of compromise - agreeing to a value intermediate between
each negotiator’s initial bargaining positions for each issue under dispute. The second can
be described as horsetrading - the parties agree to trade-offs such that each obtains what
he or she bargains for on certain issues, in exchange for granting the other what he or she
wants on other issues.
There are two main approaches to the classification of negotiations ([20, 22]):
Distributive bargaining (DB) is commonly identified as involving the division of resources; it refers to situations in which there is a fixed supply of some resource, and one’s
gain is the other’s loss. DB appears typically in the form of a single-issue negotiation, such
as bargaining for a used car. Any reduction in the price of the car removes money from
the pocket of the salesman, while any increase in the price paid removes money from the
pocket of the buyer. This is also known as a win-lose situation. DB can also be applied
to multi-issue negotiations wherein each issue is handled singularly. In DB the issues are
assumed to be areas in which the parties are in conflict. As a technique, the function of
DB is to resolve pure conflicts of interest in a fixed-sum negotiation.
Integrative bargaining (IB) refers to situations in which the negotiators cooperatively
face a common problem or in which the parties’ interests are not diametrically opposed.
These constitute variable-sum bargaining situations. IB is the system of activities which
is instrumental to the attainment of objectives which are not in fundamental conflict with
those of the other party and which therefore can be integrated to some degree (win-win
situation). IB may occur in a multi-issue negotiation; however, negotiators in multi-issue
negotiations often bargain in a distributive manner. The IB approach attempts to steer the
parties toward problem solving and away from the traditional zero-sum horse trading.
Suppose now that we have a 3-party single-issue integrative bargaining situations in
which we want to find a good compromise settlement.
Suppose that the first party is being supported by the others until it reaches the level
β12 and β13 , respectively.
h1(x)
h1(x)
1
m1 β12 M1
Fig. 9
1
m1 β13 M1
1 u1(x)
1 u1(x)
Feed-backs from the second and third parties.
In general, the benchmark βij denotes the level until the i-th party is being supported
by the j-th party for i, j = 1, 2, 3. Any negotiated settlement (α1 , α2 , α3 ) will satisfy the
following condition (see fig. 10)
m1 ≤ β12 ∧ β13 ≤ α1 ≤ β12 ∨ β13 ≤ M1 ,
m2 ≤ β21 ∧ β23 ≤ α2 ≤ β21 ∨ β23 ≤ M2 ,
m3 ≤ β32 ∧ β31 ≤ α3 ≤ β31 ∨ β32 ≤ M3 ,
8
Let us define the overall satisfaction of the group, α, with a negotiated settlement (α1 , α2 , α3 )
by the minimum operator, i.e.
α = α1 ∧ α2 ∧ α3
It is clear that α satisfies the following inequalities
min{βij } ≤ α ≤ max{βij }
i,j
i,j
because from none of the parties are able to get more than max{βij }, but each party can
reach at least the level min{βij }.
If for certain i and j the benchmark level βij is given in such a way that βij < mi then
it means that even the minimal wish of the i-th party is absolutely unacceptable for the
j-th party.
M3
β32
β31
m3
β12
m1
β13
M1
m2
β21
β23
M2
Fig. 10
The efficient frontier for the negotiation problem.
If for certain i all the benchmark levels satisfy the condition
βij < mi , j = i
(6)
then we say that the other two parties form a coalition againts the i-th party. However,
this coalition (which is usually temporal) can result in overall conflict if (6) holds for all
parties. In this case the negotiation dance will result in a settlement which is absolutely
not acceptable by any parties.
The bencmarks βij and βji are usually determined in a preliminaries between the i-th
and j-th parties before the group meeting starts.
5
Summary and Conclusions
We have introduced a class of negotiation problems in which it can be shown that we have
multiple interdependent objectives, and in which negotiaitons become fatally deadlocked
unless the interdependences are understood and resolved as part of the process.
In order to deal with the interdependences we have introduced methods for both additive
linear and additive nonlinear interdependences, and shown how the complexities growing
from the interdependences can be dealt with rather effectively. We have also shown how
consensus can be found in negotiation problems when working with interdependent utilities
9
as described in the original von Neumann-Morgenstern discussion. Unfortunately, we do
not yet have enough facts available to show how the Finnish labor market conflict could
(or should) have been resolved, but we will return to this issue in a forthcoming paper.
6
Acknowledgment
The second author, who is presently a Donner Visiting Professor at Institute for Advanced
Management Systems Research, Åbo Akademi University, has been partially supported by
OTKA T 14144 and T 019455.
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