## Sensitivity analysis

How can the open designer find out which constraints are crucial to arrive at a positive solution space and invite their associated stakeholders to the negotiation table? In other words: how can he or she establish the sensitivity of the solution space to variations in the constraints?

If the model is still relatively simple, he or she can vary one constraint -by say 10% - while keeping all other constraints unchanged, and see what happens with the solution space. When the model includes a large number of constraints, however, this primitive method of analysing the sensitivity of the solution space to the various constraints soon becomes very cumbersome if not prohibitive. Fortunately, the Simplex method used to resolve the inequalities of the LP model almost automatically provides the information needed to cope with this problem, as we will describe in this section.

As described in Chapter 4, stakeholders should preferably specify three values for their constraints:

• an acceptable value;

During the initial phases of the Open Design process, these values may coincide for certain constraints - that means only one value is used - while for other constraints ranges are given.

Only in exceptional cases will calculating exclusively with ideal values yield a positive solution space. If so, everybody will be happy and some money will still be left. If not, the question arises to what degree constraints will have to be alleviated within the range between the ideal value and the walk-out value.

When the model is relatively simple, in the sense that the number of constraints is limited, a trial-and-error approach may suffice, but in cases involving many constraints this would become extremely cumbersome. In such cases we need to know which constraints have a great impact and which ones are relatively unimportant for achieving a positive solution space. Sensitivity analysis provides the answer to this question.

Sensitivity analysis in linear programming is closely linked to the concept of duality which will be described below.

Recall that our LP model can be formulated as:

j=i subject to:

We call this the primal problem. The dual problem (connected to this primal problem) is obtained by interchanging the C vector and the b vector and minimising instead of maximising (the a-matrix remains unchanged):

subject to:

From the symmetry apparent from the table, we can derive the Dual Theorem (the asterisk refers to optimality):

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