## Multiple objectives

So far, we have assumed that the wishes of stakeholders are incorporated in the mathematical model through the constraints, while choosing financial return as the objective to be optimised. The reason for doing so is that in Open Design all stakeholders should be treated equally. Concessions to the wishes of stakeholders should only be asked for if that is necessary to achieve a solution at all, and not because some wishes are, by nature, more important than others. In principle, all stakeholders are equal. The reality is usually quite opposite. Financial return is a boundary condition - it should be sufficient to allow the project to be financed - while a number of stakeholders' objectives have to be optimised. Those stakeholders' objectives are usually conflicting. The situation is like a zero-sum game: the more one stakeholder gives in, the less concessions are required from the others.

The sensitivity analysis described in the preceding section has given us a limited number of crucial stakeholders who will have to make concessions to allow a solution at all. How to deal with the conflicting requirements from these crucial stakeholders will be described in this section on Multiple Objectives, also called Multi-Criteria Optimisation.

In optimisation models for multiple objective problems, we can distinguish non-preference and preference methods (Radford and Gero, 1988). With the non-preference approach, we limit the model to the production of information on non-dominated (Pareto) performances. A non-dominated (Pareto optimal) solution is one for which no other solution exists that is capable of providing a better performance in one criterion and no worse performance in all other criteria. Given criteria that completely express the goals of a design problem and a complete Pareto set of solutions for those criteria, the best solution for the stakeholders concerned must lie within the Pareto set. Which member of the set this is, is still open to question.

In the preference approach, the designer's trade-off preferences are incorporated in the model. For instance, he can reduce the multi criteria problem to a single-criterion problem by assigning weight factors to the criteria and optimise the weighted sum. The choice of the weight factors remains rather arbitrary however. Even if there were a rationale for a certain choice - quod non - it would be extremely difficult for the open designer to explain why the interests of some crucial stakeholders get less weight than those of others. We therefore recommend that the preference approach - in whatever form -is used only to explore possibilities in the Open Design laboratory, but never in the dialogue between the open designer and the crucial stakeholders.

The general (non-preference) Pareto optimisation problem with n decision variables, m constraints, and p objectives is:

subject to n

Consider the criteria space for a problem with two criteria, Zi and Z2 (Fig. 7.1). As a corollary of the definition of Pareto optimality, the set of Pareto optimal performances lies along the northeast boundary of the criteria space (indicated with a thick line in Figure 7.1).

The generalisation to the p-dimensional criteria space is similar to generalising a linear programming problem with two decision variables to one with n decision variables.

In Open Design, it is not necessary to generate the complete Pareto set. We are only interested in the range where concessions are made related to all objectives (no stakeholder is given a preference position). We can, therefore, iterate to the final solution using the so-called Constraint method in a straight forward manner starting with the outcome of the sensitivity analysis as described in Section 7.1.

Figure 7.1 Pareto optimal set for a problem with two objectives

The Constraint method retains one objective as primary while treating the remaining objectives as constraints. By doing this in turns for the various objectives, the relevant part of the Pareto set is found.

Let us consider the case of a two-objective problem, for instance:

Z1 = number of low cost houses to be built on the location;

Z2 = financial return of the project.

The associated stakeholders are:

• The housing co-operation wishing as many low-cost houses as possible (Z1);

• The financial institution wishing the highest possible financial return (Z2).

The two objectives are conflicting in the sense that the more low cost houses are built, the lower the financial return of the project will be (Figure 7.2).

The calculation with the ideal values of the two crucial stakeholders, say Zi = 2 000 low-cost houses, and Z2 = 0.18 (IRR) - i.e. the point (Z1, Z2)0 -gives no solution. In a first negotiation with the financier the financial return is reduced to IRR= 0.15 and put into the model as a constraint. The number of low-cost houses is then optimised with result Z1 = 1400. This is not accepted by the housing cooperation, which is only prepared to go down (from their

 |_{Z1,Z2)0, c CD (Z1,Z2)1 ideal values c ® fz^//^///////^ (Z1,Z2)3 % m^mmrn^ i I(Z1,Z )2 ¡ /> y | c itt g ® —number of low cost houses Zi

Figure 7.2 Iterative procedure for (two) crucial stakeholders negotiation ideal of 2 000) to 1700 houses. This is put into the model as a constraint, while optimising the financial return with result Z2 = 0.09. This is not acceptable to the financier, but he will accept a return of 0.12. This is put into the model as a constraint, while optimising the number of low-cost houses. The result, 1500 low-cost houses, is finally accepted by the housing cooperation.

To summarise:

(Zi, Z2)0 = (2000, 0.18) (Zi, Z2)1 = (1400, 0.15) (Z1, Z2)2 = (1700, 0.09)

ideal values (no solution);

result after first negotiation (with financier);

result after second negotiation (with housing co-operation);

result after third negotiation (with financier): solution acceptable to both parties.

It should be noted that the shadow prices related to the objectives that were treated as constraints are the basis for the negotiations (recall: shadow price = change in objective function per unit change of a constraint).

The procedure for more than two objectives is quite similar. In turn, all objectives except one are incorporated into the model as constraints. Shadow prices are used in the negotiations on new values for the objectives which have to be accepted by the stakeholders concerned.

We close by noticing two important observations:

1. The sensitivity analysis takes place in the open designer's laboratory; it can be done as homework. The multi-criteria optimisation has to take place in the meeting of Crucial Stakeholders as it is closely intertwined with the negotiations among them;

2. It is completely unpredictable how much individual crucial stakeholders will be prepared to compromise, because their willingness to do so is heavily influenced by the outcomes of the pilot runs, i.e. by the effect they see that the compromises requested from them will have on resolving the problem of achieving a solution at all.

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