## Open Designs structuring of a design problem

The structuring commonly used in design problems for the purpose of mathematical optimisation originates from the domain of operations research (OR). In that domain, reference is made to 'design' problems and 'decision(-making)' problems. By doing so, OR experts do not recognise the obvious relationship between a design problem and a decision problem. After all, one cannot make a decision about something without having 'conceived' (designed) it first. Design problems and decision problems are connected, so it would be more accurate to refer to 'design-decision problems'.

Operations research bases the structuring of a problem on the collection of all the alternatives available to the decision-makers. This collection (A) contains the possibilities from which a choice must be made. It is assumed that this collection is finite and discrete. It is formally represented as:

Table 1 Alternative factors

Type of property |
Price (nlg) |
Site area (ha) |
Number of units |

5 x 106 |
0.5 |
25 | |

32 |
3 x 106 |
0.2 |
40 |

a3 |
4 x 106 |
0.3 |
30 |

Table 2 Preference score

Type of property |
Price |
Site area |
Number of units |

31 |
6 |
8 |
7 |

32 |
9 |
7 |
6 |

33 |
9 |
6 |
8 |

Every alternative ai is an element of the collection A: ai e A

For example, an architect who wants to decide what type of residential block is best could take as a basis the following collection of alternatives:

ai = a block of semi-detached residential units a2 = an apartment block a3 = a block of terraced housing

In order to make a choice based on the collection, the architect must arrange the alternatives in order of preference, thus allowing him to select the one that is most preferable.

Of course this example is very simple. In practice, each alternative will have many attributes, for instance price, site area, and number of units. These are shown in Table 1.

In mathematical terms the price, site area, and number of units are the variables, and a1, a2, and a3 are the vectors. In this example each vector has three components (the permitted value for each variable).

To render this problem - what is the best type of housing? - open to a decision and therefore to (formally) solve it, the components will have to be assigned a preference score (Table 2). If the order of dominance is price, site area, and number of units, type a2 will receive highest preference.

In practice, the number of alternatives and of components might be so large that the issue of choice becomes too complex. This can be resolved by accepting Simon's 'satisficing' principle, that a decision-maker will be happy to consider only a limited number of alternatives - the alternatives with which

Figure 1 The realisation area (shaded)

he will be satisfied. It is then possible to make the best choice from this limited number. In mathematical terms this is represented in the form of constraints applied to the vectors, which divides the decision-making area into a 'permitted' (or realisation) area and a 'forbidden' area. Figure 1 shows an example in which the cost of the residential property must be between 3 million and 4 million guilders, and the site area between 0.2 and 0.4 hectares.

In OR, the following formula is used for a general (symbolic) notation of the structure of a decision-making problem (Ackoff and Sasieni, 1968, page 9):

where:

U = the utility or value of the system's performance Xi = the variables that can be controlled

Yj = the variables (and constraints) that are not controlled but do affect U

f = the relationship between U and Xi and Yj

This indicates that a decision-making problem consists of two types of elements (variables): the elements Xi that can be determined by the decision-maker, and the elements Yj that the decision-maker cannot determine. The elements Yj are given, they come from 'outside' and are immutable as far as the decision-maker is concerned. Furthermore, it is indicated that these two groups of elements are 'ordered' in such a way (in the function f) that the whole has a value, or utility U. The decision-maker has the task of selecting the 'free' elements in such a way that the whole, together with the 'fixed'

elements, produces the best outcome (design). Expressed mathematically, the decision-maker has the task of finding the values of the variables Xi that, with the given function f and the given values of the variables Yj, produces the desired, best value of the variable U.

In Open Design, the variables Xi are split into two groups: the decision variables Di, which are the variables whose value can be influenced outside the model; and the result variables Rk, whose values are determined by the model. Expressed as a formula (Yj has been replaced by fixed variable Fj):

where:

Di = decision variables; input variables which can be influenced outside the model Rk = result variables; output variables resulting from the model Fj = fixed input variables

We label this function the modified Ackoff-Sasieni utility function. By considering the values of the decision variables Di to be negotiable, a feasible solution space can be found.

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