decision problems with interdependent objectives

Decision problems with interdependent objectives∗

Christer Carlsson ([email protected]) and Robert Fuller ([email protected])

AbstractWe consider multiple objective programming

(MOP) problems with additive interdependences,i.e. when the states of some chosen objective areattained through supportive or inhibitory feed-backsfrom several other objectives. MOP problems withindependent objectives (i.e. when the cause-effectrelations between the decision variables and theobjectives are completely known) will be treatedas special cases of the MOP, in which we have in-terdependent objectives. We illustrate our ideasby a simple three-objective and a real-life seven-objective problem.

Keywords: Multiple objective programming, ad-ditive interdependences, compound interdependences

1 MOP Problems with Interdepen-dent Objectives

In their classical text Theory of Games and Eco-nomic Behavior John von Neumann and Oskar Mor-genstern (1947) described the problem with interde-pendence. In their outline of a social exchange econ-omy they discussed the case of two or more personsexchanging goods with each others:

. . . then the results for each one will depend ingeneral not merely upon his own actions but on thoseof others as well. Thus each participant attempts tomaximize a function . . . of which he does not controlall variables. This is certainly no maximum prob-lem, but a peculiar and disconcerting mixture of sev-eral conflicting maximum problems. Every partici-pant is guided by another principle and neither de-termines all variables which affects his interest. Thiskind of problem is nowhere dealt with in classicalmathematics. We emphasize at the risk of being pedan-tic that this is no conditional maximum problem, noproblem of the calculus of variations, of functional

∗The final version of this paper appeared in: C. Carlssonand R. Fuller, Decision problems with interdependent objec-tives, International Journal of Fuzzy Systems, 2(2000) 98-107.

analysis, etc. It arises in full clarity, even in the most’elementary’ situations, e.g., when all variables canassume only a finite number of values.

The interdependence is part of the economic the-ory and all market economies, but in most modellingapproaches in multiple criteria decision making thereseems to be an implicit assumption that objectivesshould be independent. This appears to be the case,if not earlier than at least at the moment when wehave to select some optimal compromise among theset of nondominated decision alternatives. MilanZeleny [14] - and many others - recognizes one partof the interdependence:

Multiple and conflicting objectives, for example,’minimize cost’ and ’maximize the quality of ser-vice’ are the real stuff of the decision maker’s ormanager’s daily concerns. Such problems are morecomplicated than the convenient assumptions of eco-nomics indicate. Improving achievement with re-spect to one objective can be accomplished only atthe expense of another.

but not the other part: objectives could supporteach others. We will in the following explore theconsequences of allowing objectives to be interde-pendent.

2 Additive Linear Interdependencesin MOP

Objective functions of a multiple objective program-ming problem are usually considered to be indepen-dent from each other, i.e. they depend only on thedecision variable x. A typical statement of an MOPwith independent objective functions is

maxx∈X

{f1(x), . . . , fk(x)

}(1)

where fi is the i-th objective function, x is the deci-sion variable, and X is a subset, usually defined byfunctional inequalities. Throughout this paper wewill assume that the objective functions are normal-ized, i.e. fi(x) ∈ [0, 1] for each x ∈ X .

1

However, as has been shown in some earlier workby Carlsson and Fuller [1, 2], and Felix [10], thereare management issues and negotiation problems,in which one often encounters the necessity to for-mulate MOP models with interdependent objectivefunctions, in such a way that the objective functionsare determined not only by the decision variablesbut also by another (or several other) objective func-tions.

Typically, in complex, real-life problems, thereare some unidentified factors which effect the valuesof the objective functions. We do not know them orcan not control them; i.e. they have an impact wecan not control. The only thing we can observe is thevalues of the objective functions at certain points.And from this information and from our knowledgeabout the problem we may be able to formulate theimpacts of unknown factors (through the observedvalues of the objectives).

Interdependences among the objectives exist when-ever the computed value of an objective function isnot equal to its observed value. In this paper weclaim that the real values of an objective functioncan be identified by the help of feed-backs from thevalues of other objective functions.

Suppose now that the objectives of (1) are inter-dependent, and the value of an objective function isdetermined by a linear combination of the values ofother objectives functions. That is

f ′i(x) = fi(x) +k∑

j=1, j 6=i

αijfj(x), 1 ≤ i ≤ k (2)

or, in matrix formatf ′1(x)f ′2(x)

...f ′k(x)

=

1 α12 . . . α1k

α21 1 . . . α2k...

......

...αk1 αk2 . . . 1

f1(x)f2(x)

...fk(x)

where αij is a real number denoting the grade ofinterdependency 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 ; ifαij = 0 then we say that fi is independent from fj

(or the states of fj are irrelevant to the states of fi).

Figure 1: Linear feed-back from fj to fi (αij > 0).

The matrix (αij) is then called the interdependencymatrix of MOP (1).

In such cases, i.e. when the feed-backs from theobjectives are directly proportional to their indepen-dent values, then we say that the objectives are lin-early interdependent. It is clear that if αij = 0, ∀i 6=j, then we have an MOP problem with independentobjective functions. For the time being we will as-sume that αij is found through preference elicita-tions from decision makers working with the ob-jective functions. The preference elicitation can bedone by pairwise comparisons over a large enoughintervals of a set of objective functions used for reallife decisions.

Taking into consideration the linear interdepen-dences among the objective functions (2), the inter-dependent problem (1) turns into the following prob-lem (which is treated as an independent MOP)

maxx∈X

{f ′1(x), . . . , f

′k(x)

}(3)

It is clear that the solution-sets of (1) and (3) areusually not identical.

A typical case of interdependence is the follow-ing (almost) real world situation. We want to buy ahouse for which we have defined the following threeobjectives

• f1: the house should be non-expensive

• f2: as we do not have the necessary skills, thehouse should not require much maintenanceor repair work

• f3: the house should be more than 10 year oldso that the garden is fully grown and we neednot look at struggling bushes and flowers

We have the following interdependences:

• f1 is supported by both f2 and f3 as in certainregions it is possible to find 10 year old houses

which (for the moment) do not require muchrepair and maintenance work, and which arenon-expensive.

• f2 can be conflicting with f3 for some housesas the need for maintenance and repair workincreases with the age of the house; thus f3 isalso conflicting with f2.

• f3 is supporting f1 for some houses; if the gar-den is well planned it could increase the price,in which case f3 would be in partial conflictwith f1; if the neighbourhood is completedand no newbuilding takes place, prices couldrise and f3 be in conflict with f1.

To explain the issue more exactly, consider athree-objective problem with linearly interdependentobjective functions

maxx∈X

{f1(x), f2(x), f3(x)

}(4)

Taking into consideration that the objectives are lin-early interdependent, the interdependent values ofthe objectives can be expressed by

f ′1(x) = f1(x) + α12f2(x) + α13f3(x),f ′2(x) = f2(x) + α21f1(x) + α23f3(x),f ′3(x) = f3(x) + α31f1(x) + α32f2(x).

That is f ′1(x)f ′2(x)f ′3(x)

=

1 α12 α13

α21 1 α23

α31 α32 1

f1(x)f2(x)f3(x)

In this case, depending on the value of alpha1,

we can have the following simple additive linear in-terdependences between the objectives of (4):

• if αij = 0 then we say that fi is independentfrom fj ;

• if αij > 0 then we say that fj unilaterally sup-ports fi;

• if αij < 0 then we say that fj hinders fi;

• if αij > 0 and αji > 0 then we say that fi andfj mutually support each others;

1Here the αij can be found (in an imprecise manner)through the pairwise evaluations of the objectives

Figure 2: A three-objective interdependent problemwith linear feed-backs.

• if αij < 0 and αji < 0 then we say that fi andfj are conflicting;

• if αij + αji = 0 then we say that fi are fj arein a trade-off relation;

It is clear, for example, that if f2 unilaterallysupports f1 then the larger the improvement of f2

(supporting objective function) the more significantis its contribution to f1 (supported objective func-tion).

To illustrate our ideas consider the following sim-ple decision problem.

max{x, 1− x}; subject to x ∈ [0, 1]. (5)

Choosing the minimum-norm to aggregate thevalues of objective functions this problem has a uniquesolution x∗ = 1/2 and the optimal values of the ob-jective functions are (0.500, 0.500).

Figure 3: Independent problem.

Suppose that for example f1 is unilaterally sup-ported by f2 on the whole decision space [0, 1] andthe degree of support is given by2

f ′1(x) = f1(x) + 1/2f2(x) = 1/2 + x/2.

2In this case we have simply assumed that α = 1/2.

Then (5) turns into the following problem

max{1/2 + x/2, 1− x}; subject to x ∈ [0, 1].

Choosing the minimum-norm to aggregate thevalues of objective functions this problem has a uniquesolution x∗ = 1/3 and the optimal values of the ob-jective functions are (0.667, 0.667).

Figure 4: Unilateral support.

Suppose now that f1 and f2 support each othermutually, i.e. the better the value of f1 the moresignificant is its support to f2 and vica versa. Thedegrees of supports are given by

f ′1(x) = f1(x) + 1/2f2(x)= x+ 1/2(1− x) = 1/2(1 + x),

f ′2(x) = f2(x) + 1/2f1(x)= (1− x) + 1/2x = 1− x/2.

In this case our interdependent problem turns into

max{1/2(1 + x), 1− x/2}; subject to x ∈ [0, 1].

Choosing the minimum-norm to aggregate the val-ues of objective functions this problem has a uniquesolution x∗ = 1/2 and the optimal values of the ob-jective functions are (0.750, 0.750).

Suppose now that f2 hinders f1, i.e. the betterthe value of f2 the more significant is its negativefeed-back to f1. The degree of hindering is

f ′(x) = f1(x)− 1/2(1− x) = 3/2x− 1/2.

Figure 5: Mutual support.

So our interdependent problem turns into

max{3/2x− 1/2, 1− x}; subject to x ∈ [0, 1].


Figure 6: Hindering.

Suppose now that f2 hinders f1, but f1 supportsf2

f ′1(x) = f1(x)− 1/2f2(x)= x− 1/2(1− x) = 3/2x− 1/2,

f ′2(x) = f2(x) + 1/2f1(x)= (1− x) + 1/2x = 1− x/2

So our interdependent problem turns into

max{3/2x− 1/2, 1− x/2}; subject to x ∈ [0, 1].


These findings are summarized in Table 1.

3 Additive Nonlinear Interdependencesin MOP

Suppose now that the objectives of (1) are interde-pendent (in the way we introduced in Section 2),and that the value of an objective function is deter-mined by an additive combination of the feed-backsof other objectives functions

f ′i(x) = fi(x) +k∑

j=1, j 6=i

αij [fj(x)], (6)

for 1 ≤ i ≤ k, where αij : [0, 1] → [0, 1] is a - usu-ally nonlinear - function defining the value of feed-back from fj to fi.

case compromise solution optimal values

independent objectives 0.5 (0.500, 0.500)

f1 is supported by f2 0.333 (0.667, 0.667)

mutual support 0.5 (0.750, 0.750)

f2 hinders f1 0.6 (0.400, 0.400)

f2 hinders f1 and f1 supports f2 0.75 (0.625, 0.625)

Table 1: Cases and solutions.

If αij(z) > 0,∀z we say that fi is supported byfj ; if αij(z) < 0,∀t then we say that fi is hinderedby fj ; if αij(z) = 0,∀z then we say that fi is inde-pendent from fj . If αij(z1) > 0 and αij(z2) < 0for some z1 and z2, then fi is supported by fj if thevalue of fj is equal to z1 and fi is hindered by fj ifthe value of fj is equal to z2.

Figure 7: Nonlinear unilateral support.

Consider again the three-objective problem (4)with nonlinear interdependences. Taking into con-sideration that the objectives are interdependent, theinterdependent values of the the objectives can beexpressed by

f ′1(x) = f1(x) + α12[f2(x)] + α13[f3(x)],f ′2(x) = f2(x) + α21[f1(x)] + α23[f3(x)],f ′3(x) = f3(x) + α31[f1(x)] + α32[f2(x)].

For example, depending on the values of the cor-relation functions α12 and α21 we can have the fol-lowing simple interdependences between the objec-tive functions f1 and f2 of (6)

• if α12(z) = 0,∀z then we say that f1 is inde-pendent from f2;

• if α12(z) > 0, ∀z then we say that f2 unilat-erally supports f1;

Figure 8: A three-objective interdependent problemwith nonlinear feed-backs.

• if α12(z) < 0,∀z then we say that f2 hindersf1;

• if α12(z) > 0 and α21(z), for all z > 0, thenwe say that f1 and f2 mutually support eachothers;

• if α12(z) < 0 and α21(z) < 0 for each z, thenwe say that f1 and f2 are conflicting;

• if α12(z) + α21(z) = 0 for each z, then wesay that f1 are f2 are in a trade-off relation;

However, differently from the linear case, wecan here have more complex relationships betweentwo objective functions, e.g.

• if for some β ∈ [0, 1]

α12(z) ={

positive if 0 ≤ z ≤ βnegative if β ≤ z ≤ 1

then f2 unilaterally supports f1 if f2(x) ≤ βand f2 hinders f1 if f2(x) ≥ β.

Figure 9: fj supports fi if fj(x) ≤ β and fj hindersfi if fj(x) ≥ β.

• if for some β, γ ∈ [0, 1]

α12(z) =

positive if 0 ≤ z ≤ β0 if β ≤ z ≤ γnegative if γ ≤ z ≤ 1

then f2 unilaterally supports f1 if f2(x) ≤ β,f2 does not affect f1 if β ≤ f2(x) ≤ γ andthen f2 hinders f1 if f2(x) ≥ γ3.

4 Compound Interdependences

Let us now (more) consider the case with compoundinterdependences in multiple objective programming,which is - so far - the most general case. Assumeagain that the objectives of (1) are interdependent,and the value of an objective function is determinedby an additive combination of the feed-backs fromother objectives functions

f ′i(x) =k∑

j=1

αij [f1(x), . . . , fk(x)], (7)

for 1 ≤ i ≤ k, where αij : [0, 1]k → [0, 1] is a- usually nonlinear - function defining the value offeed-back from fj to fi. We note that αij dependsnot only on the value of fj , but on the values of otherobjectives as well (this is why we call it compoundinterdependence [5]).

Let us again consider the three-objective prob-lem (4) with nonlinear interdependences With theassumptions of (7) the interdependent values of the

3Here β is found in the same manner as αij , i.e. by elici-tating the preferences of the decision makers. This change ofpreferences was also discussed in [13].

i-th objective can be expressed by

f ′i(x) = αi1[f1(x), f2(x), f3(x)]+ αi2[f1(x), f2(x), f3(x)]+ αi3[f1(x), f2(x), f3(x)],

for i = 1, 2, 3.

Figure 10: A three-objective interdependent prob-lem with compound interdependences.

Here we can have more complicated interrela-tions between f1 and f2, because the feedback fromf2 to f1 can depend not only on the value of f2, butalso on the values of f1 (self feed-back) and f3. Un-fortunately, in real life cases we usually have com-pound interdependences (cf [3]).

5 Biobjective Interdependent Deci-sion Problems

Let us now illustrate how the solution of a biob-jective decision problem changes under interdepen-dent objectives. A biobjective independent decisionproblem in a (normalized) criterion space can be de-fined as follows

max{f1, f2}; subject to 0 ≤ f1, f2 ≤ 1. (8)

As f1 and f2 take their values independently of eachothers, we can first maximize f1 subject to f1 ∈[0, 1] then f2 subject to f2 ∈ [0, 1] and take the (fea-sible) ideal point, (1, 1), as the unique solution toproblem (8). A typical two-dimensional indepen-dent problem is

max{x1, x2}; subject to 0 ≤ x1, x2 ≤ 1,

that is, f1(x1, x2) = x1 and f2(x1, x2) = x2.

More often than not, the ideal point is not feasi-ble, that is, we can not improve one objective with-out giving away something of the other one. If wehappen to know the exact values of the objectives inall conceivable cases then a good compromise so-lution can be defined as a Pareto-optimal solutionsatisfying some additional requirements specified bythe decision maker [9, 11].

Suppose that the interdependent values f ′1 andf ′2 of f1 and f2, respectively, are observed to satisfythe following equations

f ′1 = f1 + α12f2, f ′2 = f2 + α21f1

The following two theorems characterize the behav-ior of max-min compromise solutions of biobjectivelinearly interdependent decision problems with con-flicting and supporting objective functions.

Theorem 5.1. If f ′1 = f1 − αf2 and f ′2 = f2 −αf1, i.e., f1 hinders f2 globally and f2 hinders f1

globally with the same degree of hindering 0 ≤ α ≤1 then

max min{f ′1, f ′2} ≤ (1− α) maxmin{f1, f2};

subject to 0 ≤ f1, f2 ≤ 1.

Proof. Suppose f1 ≤ f2. Then

f ′1 = f1−αf2 ≤ f ′2 = f2−αf1 ⇒ min{f ′1, f ′2} = f ′1

and

(1− α) maxmin{f1, f2} = (1− α)f1

≥ f1 − αf2 = f ′1,

That is,

(1− α) min{f1, f2} ≥ min{f1 − αf2, f2 − αf1}= min{f ′1, f ′2}.

Similar reasoning holds for the case f1 ≥ f2.

In other words, Theorem (5.1) shows an upperbound for the max-min compromise solution, whichcan be interpreted as ’the bigger the conflict betweenthe objective functions the less are the criteria satis-factions’. For example, in the extremal case, α = 1we end up with totally opposite objective functions

f ′1 = f1 − f2, f ′2 = f2 − f1,

and, therefore, we get

max min{f ′1, f ′2} = maxmin{f1 − f2, f2 − f1}≤ (1− α) maxmin{f1, f2} = 0.

On the other hand, if α = 0 then the objective func-tions remain independent, that is,

max min{f ′1, f ′2} ≤ (1− α) maxmin{f1, f2}= maxmin{f1, f2}.

Theorem 5.2. If f1 supports f2 globally with thedegree α ≥ 1 then

max min{f ′1, f ′2} = max f1;

subject to 0 ≤ f1, 0 ≤ f2.

Proof. From α ≥ 1 we get

f1 ≤ f2 + αf1

and therefore,

max min{f1, f2 + αf1} = max f1,

which ends the proof.

Theorem 5.2 demonstrates that if one objectivesupports the other one globally with a degree of sup-port bigger than one, then the independent maxi-mum of the supporting objective gives the max-mincompromise solution to the interdependent problem.In other words, in the case of unilateral support, it isenough to maximize the supporting objective func-tion to reach a max-min compromise solution.

It should be noted that Theorem 5.2 remains validfor non-negative objective functions (that is, the con-straints f1 ≤ 1 and f2 ≤ 1 can be omitted). If, how-ever f1 does satisfy the constraint f1 ≤ 1 then theideal point (1, 1) will be the unique max-min solu-tion.

6 Fuzzy Reasoning for Interdepen-dent Biobjective Decision Problems

Suppose we do not know exactly the values of theobjectives, but we are able to identify some (linguis-tic) interdependences between them, such as ’f1 hin-ders f2 if the value of f1 is big’ or ’f2 supports f1

if the value of f2 is medium’, where the linguisticterms are represented by fuzzy numbers.

Then we use fuzzy rules for knowledge repre-sentation and the interdependent values of the ob-jectives will be calculated by simplified fuzzy rea-soning schemes [12].

For example, if we have two linguistic valuessmall and big for f1 and f2 then the interdepen-dences between the objective functions can be de-scribed by the following fuzzy rule base4

<1: if f1 is small then f ′2 = f2 + α21f1

<2: if f1 is big then f ′2 = f2 − β21f1

<3: if f2 is small then f ′1 = f1 − β12f2

<4: if f2 is big then f ′1 = f1 + α12f2

where the rules are interpreted as

<1: f1 supports f2 if f1 is small, where α21 > 0 isthe (linear) feedback from f1 to f2

<2: f1 hinders f2 if f1 is big, where −β21 < 0 isthe (linear) feedback from f1 to f2

<3: f2 hinders f1 if f2 is small, where −β12 < 0is the (linear) feedback from f2 to f1

<4: f2 supports f1 if f2 is big, where α12 > 0 isthe (linear) feedback from f2 to f1

For example, if small(t) = 1 − t and big(t) =t then the interdependent values f ′1 and f ′2 are ob-tained from the rule base < = {<1, . . . ,<4} as

f ′1 = (1− f2)(f1 − β12f2) + f2(f1 + α12f2).

and

f ′2 = (1− f1)(f2 + α21f1) + f1(f2 − β21f1),

In this way the original interdependent problem turnsinto the following optimization problem

max min{f1 − β12f2 + (α12 + β12)f22 ,

f2 + α21f1 − (α21 + β21)f21 }

subject to 0 ≤ f1, f2 ≤ 1.

For more results on fuzzy reasoning for optimizationsee [7, 8].

4Here we use αij and βij to denote the fact that the supportand the hindering may be different if the objective functionshave ’small’ or ’big’ values.

7 Nordic Paper Inc.: A Case Study

Nordic Paper Inc. (NPI) is one of the more suc-cessful paper producers in Europe5 and has gained areputation among its competitors as a leader in qual-ity, timely delivery to its customers, innovations inproduction technology and customer relationships oflong duration. Still it does not have a dominating po-sition in any of its customer segments, which is noteven advisable in the European Common market, asthere are always 2-5 competitors with sizeable mar-ket shares. NPI would, nevertheless, like to have aposition which would be dominant against any cho-sen competitor when defined for all the markets inwhich NPI operates.

We will consider strategic decisions for the plan-ning period 2000-2003.

Decisions will be made on how many tons of 6-9 different paper qualities should be produced for3-4 customer segments in Germany, France, UK,Benelux, Italy and Spain. NPI is operating 9 pa-per mills which together cover all the qualities tobe produced. Price/ton of paper qualities in differ-ent market segments are known and forecasts for theplanning period are available. Capacities of the pa-per mills for different qualities are known and pro-duction costs/ton are also known and can be fore-casted for the planning period. The operating re-sult includes distribution costs from paper mills tothe markets, and the distribution costs/ton are alsoknown and can be forecasted for the planning pe-riod.

Decisions will also have to be made on how muchmore added capacity should be created through in-vestments, when to carry out these investments andhow to finance them. Investment decisions shouldconsider target levels on productivity and competi-tive advantages to be gained through improvementsin technology, as well as improvements in prices/tonand product qualities.

There are about 6 significant competitors in eachmarket segment, with about the same (or poorer)production technology as the one operated by NPI.Competition is mainly on paper qualities, just-in-time deliveries, long-term customer relationships andproduction technology; all the competitors try to avoidcompeting with prices. Competition is therefore com-

5NPI is a fictional corporation, but the case is realistic andquite close to actual decisions made in the forest products in-dustry.

plex: if NPI manages to gain some customers forsome specific paper quality in Germany by takingthese customers away from some competitor, the com-petitive game will not be fought in Germany, butthe competitor will try to retaliate in (for instance)France by offering some superior paper quality atbetter prices to NPI customers; this offer will per-haps not happen immediately but over time, so thatthe game is played out over the strategic planninginterval. NPI is looking for a long-term strategy togain an overall dominance over its competitors in theEuropean arena.

Decisions will have to be made on how to attainthe best possible operating results over the planningperiod, how to avoid both surplus and negative cashflow, how to keep productivity at a high and stablelevel, and how to keep up with market share objec-tives introduced by shareholders, who believe thatattaining dominating positions will increase shareprices over time.

There are several objectives which can be de-fined for the 2000-2003 strategic planning period.

• Operating result [f1]

should either be as high as possible for the pe-riod or as close as possible to some acceptablelevel.

• Productivity [f2],

defined as output (in ton) / input factors, shouldeither be as high as possible or as close as pos-sible to yearly defined target levels.

• Available capacity [f3]

defined for all the available paper mills, shouldbe used as much as possible, preferably at, orclose to their operational limits.

• Market share [f4]

objectives for the various market segments shouldbe attained as closely as possible.

• Competitive position [f5]

assessed as a relative strength to competitorsin selected market segments, should be builtup and consolidated over the planning period.

• Return on investments [f6]

should be as high as possible when new pro-duction technology is allocated to market seg-ments with high and stable prices and growingdemand.

• Financing [f7]

target levels should be attained as closely aspossible when investment programs are decidedand implemented; both surplus financial as-sets and needs for loans should be avoided.

There seems to be the following forms of inter-dependence among these objectives:

• f1 and f4 are in conflict, as increased marketshare is gained at the expense of operating re-sult; if f5 reaches a dominant level in a chosenmarket segment, then f4 will support f1; if f5

reaches dominant levels in a sufficient num-ber of market segments, then f4 will supportf1 overall.

• f4 supports f5, as a high market share willform the basis for a strong competitive posi-tion; f5 supports f4 as a strong competitiveposition will form the basis for increasing mar-ket shares; there is a time lag between theseobjectives.

• f3 supports f2, as using most of the availablecapacity will increase productivity.

• f2 supports f1 as increasing productivity willimprove operating results.

• f3 is in conflict, partially, with f1, as using allcapacity will reduce prices and have a nega-tive effect on operating result.

• f6 is supporting f1, f4 and f5, as increasingreturn on investment will improve operatingresult, market share and competitive position;f4 and f5 support f6 as both objectives willimprove return on investment; f6 is in conflictwith f3 as increasing return on investment willincrease capacity.

• f7 supports f1, as a good financial stabilitywill improve the operating result.

• f5 supports f2, as a strong competitive posi-tion will improve productivity, because priceswill be higher and demand will increase, whichis using more of the production capacity.

• f4 and f6 are in conflict, as increasing marketshare is counterproductive to improving returnon investment, which should focus on gainingpositions only in market segments with highprices and stable growths.

7.1 Outline of a macro algorithm

Let X be a set of possible strategic activities of rele-vance for the context in the sense that they are instru-mental for attaining the objectives f1− f7. Strategicactivities are decisions and action programs identi-fied as appropriate and undertaken in order to estab-lish positions of sustainable competitive advantagesover the strategic planning period. As the objectivesare interdependent the strategic activities need to bechosen or designed in such a way that the interde-pendences can be exploited, i.e. we can make theattainment of the various objectives more and moreeffective. In the following we describe a macro al-gorithm for solving the NPI problem. There is a nu-merical case available on request (too extensive tobe used here), which proved to have several alter-nate solutions.

Let X be composed of several context-specificstrategic activities:

X ⊂ {XMP , XCP , XPROD, XINV , XFIN , XPROF },

where the context-specific activities are defined asfollows:

• XMP , market-oriented activities for demand,selling prices and market shares

• XCP , activites used for building competitivepositions

• XPROD, production technology and productivity-improving activities

• XINV , investment decisions

• XFIN , financing of investments and opera-tions

• XPROF , activities aimed at enhancing and con-solidating profitability

It is clear that these activities have some tempo-ral interdependences; it is, for instance, normally thecase that a market position will influence the corre-sponding competitive position with some delay - in

some markets this can be 2-3 months, in other mar-kets 6-12 months. In the interest of simplicity wewill disregard these interdependences.

1.1 check through the database on markets, cus-tomers for an intuitive view on potential changesin demand, prices, sales;

1.2 work out XMP and list expected consequenceson demand, selling prices and market shares;

1.3 work out consequences for f4 and check if theobjective will be attained during the planningperiod; if not got to 1.1, otherwise proceed;

1.4.1 work out the impact of f4 on f1; if f1 is un-tenable, go to 1.2, otherwise proceed;

1.4.2 work out the impact of f4 on f5, and the im-pact of f5 on f4; if f5 is tenable, proceed, oth-erwise go to 1.2;

1.4.3 work out the impact of f4 on f6; if f6 is ten-able, proceed, otherwise go to 1.2;


1.4.5 if 1.4.1-1.4.4 have iterated n times, then stop;

2.1 check through the database on markets, cus-tomers for intuitive view on the positions ofkey competitors;

2.2 work out XCP and list expected consequenceson overall status on critical success fac-torsand competitive positions;


2.4.1 work out the impact of f5 on f4 and f1; if f1,f4 are untenable, go to 2.2, otherwise proceed;





3.1 check through the database on markets, cus-tomers for an intuitive view on potential changesin product demand, quality constraints, require-ments on technology;

3.2 work outXPROD and list expected consequenceson the production program, required sellingprices and market shares;






4.1 check through XMP , XCP , XPROD;

4.2 work outXINV and list expected consequenceson productivity, competitive position and mar-ket position;


4.4.1 work out the impact of f6 on f1, f4 and f5; ifall of them are tenable, proceed; otherwise goto 4.2;

4.4.2 work out the impact of f4 and f5 on f6; if f6

is tenable, proceed, otherwise go to 4.2;



5.1 check through XMP , XCP , XPROD, XINV ;

5.2 work outXFIN and list expected consequenceson profitability and cash flow;



5.4.2 if 5.4.1 has iterated n times, then stop;

6.1 check through XMP , XCP , XPROD, XINV ;

6.2 work outXPROF and list expected consequenceson profitability, capital structure, cash flow andkey ratios;

6.3 work out consequences for f1 and check if theobjective will be attained during the planningperiod; if not got to 6.1 (or possibly 1.1), oth-erwise proceed;








There are second and third degree interdepen-dences between the objectives, and there are degreesto the interdependences; all with an impact on thedesign of the set of strategic activities:

X ⊂ {XMP , XCP , XPROD, XINV , XFIN , XPROF }.

These will not be worked out here, as this illustrationis sufficient to show the inherent complexity. Thereare 4-5 different numerical solutions to the NPI case,which can be obtained from the authors.

Remark 7.1. In this paper we have considered onlyadditive interdependences and time independent feed-backs. It should be noted, however, that in negotia-tion processes the feed-backs from other objectivesare always time-dependent. Time-dependent addi-tive linear interdependences in MOP (1) can be de-fined as follows

f ′i(x) = fi(x) +k∑

j=1, j 6=i

αij(t)fj(x), 1 ≤ i ≤ k

where αij(t) denotes the dynamical grade of inter-dependency between fi and fj at time t.

8 Conclusions

Interdependence among criteria used in decision mak-ing is part of the classical economic theory even ifmost of the modelling efforts in the theory for mul-tiple criteria decision making has been aimed at (thesimplified effort of) finding optimal solutions for caseswhere the criteria are multiple but independent.

Decision making with interdependent objectivesis not an easy task. However, with the methods pro-posed in this paper we are able to at least start deal-ing with interdependence. If the exact values of theobjective functions can be measured (at least par-tially, or in some points), then from this informa-tion and some (incomplete or preliminary) model wemay be able to approximate the effects of other ob-jective functions, and of the set of decision variableswe have found to be appropriate for the problem. Inthis way we will be able to deal with more complexdecision problems in a more appropriate way.

In this paper we have tried to tackle interdepen-dence head-on, i.e. we have deliberately formulateddecision problems with interdependent criteria andfound ways to deal with the ’anomalies’ thus cre-ated. We have demonstrated with a fairly exten-sive case, called Nordic Paper Inc, that the situationswe first described as just principles do have justifi-cations in real world decision problems. It turnedout that the introduction of interdependences createscomplications for solving the decision problem, andthere are no handy tools available for dealing withmore complex patterns of interdependence. We canhave the case, in fact, that problem solving strate-gies, which decide the attainment of some subset ofobjectives will effectively cancel out all possibilitiesof attaining some other subset of objectives.

Allowing for additive, interdependent criteria ap-pears to open up a new category of decision prob-lems.

References

[1] C.Carlsson and R.Fuller, Interdependence infuzzy multiple objective programming, FuzzySets and Systems 65(1994) 19-29.

[2] C.Carlsson and R.Fuller, Multiple CriteriaDecision Making: The Case for Interdepen-dence, Computers & Operations Research22(1995) 251-260.

[3] C. Carlsson and R. Fuller, Additive inter-dependences in MOP, in: M.Brannbackand M.Kuula eds., Proceedings of theFirst Finnish Noon-to-noon seminar onDecision Analysis, Abo, December 11-12,1995, Meddelanden Fran Ekonomisk-Statsvetenskapliga Fakulteten vid AboAkademi, Ser: A:459, Abo Akademistryckeri, Abo, 1996 77-92.

[4] C.Carlsson and R.Fuller, Fuzzy multiple cri-teria decision making: Recent developments,Fuzzy Sets and Systems, 78(1996) 139-153.

[5] C. Carlsson and R. Fuller, Compound in-terdependences in MOP, in: Proceedings ofthe Fourth European Congress on IntelligentTechniques and Soft Computing (EUFIT’96),September 2-5, 1996, Aachen, Germany, Ver-lag Mainz, Aachen, 1996 1317-1322.

[6] C. Carlsson and R. Fuller, On interdepen-dent biobjective decision problems, in: Pro-ceedings of the Seventh European Congresson Intelligent Techniques and Soft Comput-ing (EUFIT’99), Aachen, September 13-16,1999, Verlag Mainz, Aachen, 1999, (Proceed-ings on CD-Rom).

[7] C. Carlsson and R. Fuller, Optimization underfuzzy if-then rules, Fuzzy Sets and Systems,119(2001) 111-120.

[8] C. Carlsson and R. Fuller, Multiobjective lin-guistic optimization, Fuzzy Sets and Systems,115(2000) 5-10.

[9] M. Delgado, J.L.Verdegay and M.A.Vila,Solving the biobjective linear programmingproblem: A fuzzy approach, in: M.M.Guptaet al. eds., Approximate Reasoning in ExpertSystems, North-Holland, Amsterdam, 1985317-322.

[10] R.Felix, Relationships between goals in mul-tiple attribute decision making, Fuzzy setsand Systems, 67(1994) 47-52.

[11] P. Korhonen and J. Karaivanova, An Algo-rithm for Projecting a Reference Directiononto the Nondominated Set of Given Points,International Institute for Applied SystemsAnalysis (IIASA), Interim Report, IR-98-011/March.

[12] M. Mizumoto, Method of fuzzy inferencesuitable for fuzzy control, J. Soc. InstrumentControl Engineering, 58(1989) 959-963.

[13] J.von Neumann and O.Morgenstern, Theoryof Games and Economic Behavior, PrincetonUniversity Press, Princeton 1947.

[14] M.Zeleny, Multiple Criteria Decision Mak-ing, McGraw-Hill, New-York, 1982.

Robert Fuller is an associateprofessor at Department of OREotvos Lorand University,Budapest. He received his Ph.D.from Moscow State Universityin 1988. He was a postdoc

fellow at Department of OR,RWTH Aachen, Germany

(1990-91), a visiting researcher at Institute for Ad-vanced Management Systems Research at Abo Aka-demi University, Finland (1992-93), a visiting pro-fessor at Department of Computer and ManagementScience, University of Trento, Italy (1994), a Don-ner Visiting Professor at Abo Akademi University(1995-96). His current research interests include neu-ral fuzzy systems, fuzzy multiple criteria decisionmaking and approximate reasoning for optimizationand control.

Christer Carlsson is a professorof management science at AboAkademi University and directorof the Institute of Advanced Mana-gement Systems Research. He is amember of the Steering Committeeof ERUDIT, an ESPRIT Networkof Excellence, and chairman of the

BISC-SIG on Soft Decision Analysis. He got hisD.Sc(BA) from Abo Akademi University in 1977,and has lectured extensively at various universitiesin Europe, in the U.S., in Asia and in Australia. Hehas organised and managed several research programsin industry in his specific research areas: knowl-edge based systems, decision support systems and

expert systems, and has carried out theoretical re-search work also in multiple criteria optimisationand decision making, fuzzy sets and fuzzy logic, andcybernetics and systems research. Some recent re-search programs, which include extensive industrialcooperation, include Smarter (reducing fragmenta-tion of working time with modern information tech-nology), EM-S Bullwhip (eliminating demand fluc-tuations in the supply chain with fuzzy logic), Waeno(improving the productivity of capital in giga-invest-ments using hyperknowledge) and Imagine21 (fore-sight of new telecom services using agent technol-ogy). He is on the editorial board of several journalsincluding the EJOR, Fuzzy Sets and Systems, ITOR,Cybernetics and Systems, and Intelligent Systems inAccounting, Finance and Business. He is the authorof 3 books, and an editor or co-editor of 5 special is-sues of international journals and 12 books, and haspublished more than 200 papers.

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