## Assessment of the order of preference through tradeoffs and weight factors

Once the preferences for various alternatives according to conflicting criteria have correctly been established, the stakeholder's order of preference for the available alternatives can be calculated by means of weight factors, determined by trade-offs defining how much the stakeholder is prepared to sacrifice on one criterion to achieve a certain gain in another one. A software package for this purpose is available. This procedure can be summarised as follows (for details see Barzilai (1997)).

Given a number of m Alternatives Ai and a number of n Criteria Cj, the preferences pi,j of each alternative Ai under criterion Cj can been specified by the stakeholder. We want to calculate the total preference P for alternative Ai *:

We have to determine the weight factor Wj of criterion Cj, for j = 1,2,3,..., n. The weight factors can be derived using 'trade-offs'. This means the ratio of two weight factors j = Wj must be calculated. The criteria will be compared pairwise, asking the stakeholder concerned how much 'gain' in one criterion must compensate for a 'loss' in the other one or vice versa.

Example: Let P\,j ,P1;c ,P2,j and P2,k be the preferences of alternatives A! en A2 according to the criteria Cj and Ck, then:

P( A1) = WjP1,j + WkP1,k P( A2) = WjP2,j + WkP2,k (5.2) These are equally preferred alternatives:

WjP1,j + WkP1,k = WjP 2,j + WkP2,k (5.3) which gives the ratio between the weight factors Wj and Wk:

When a stakeholder has specified the values of ratios rj,l and rl,k the value of ratio rj,k can simply be derived by multiplying the former ones:

If the stakeholder is consistent when specifying the ratios, then:

A ratio-matrix R satisfying this rule is called consistent, if not inconsistent. In case of an inconsistent R the final ratios can differ from the specified ones due to least squares averaging.

Note that the weight ratios rj,k are calculated without any manipulation of the stakeholder's input. If we use these weight ratios to reduce the multi-criteria problem to a single-criterion one by optimising the weighted sum, we

*V means: for all

Table 5.1 Part of the enquiry in use on the criterion Reliability

### Skill of office workers 1 2 3 4 5 6 7 8 9 10

are still respecting the Open Design principle that no preference of the modeller should be allowed. In this way, the preference method of using the weight factors arbitrarily chosen by the modeller is replaced by a nonpreference method which exclusively reflects the preferences of the stakeholder concerned. For a more detailed discussion on preference and non-preference methods, see Chapter 6.

If the problem is 'over-defined' in the sense that a stakeholder specifies more equivalent alternatives than necessary to compute the weight ratios, that means the number of trade-offs is larger than (n — 1), least squares fitting is in order as long as the preferences are related to one stakeholder only. Averaging preferences belonging to different stakeholders using least squares curve fitting would be against the basic philosophy of Open Design.

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