## Allocation of activities to spaces

In architectural design and urban planning, a dominant spatial dimension of resources is the position of resources in two- and three-dimensional space. This position is commonly expressed in floor plans, land use plans, and three dimensional models of buildings and their urban environments. In terms of allocation of resources, a floor plan is a proposal for allocation of architectural spaces to accommodate human activities such as living, shopping, eating, and office work. In terms of Open Design: Which spatial layout of the resources fits the activities to be accommodated best, in accordance with stakeholders' wishes, goals, and constraints, and with the architectural style chosen? Starting from the standard LP model:

subject to:

If we define the activities as demand (d) and the resources as supply (s) we can represent this problem (which is called in Operations Research literature the transportation problem, or the distribution problem) in an LP model as follows:

Minimise Z = ££ CijXij for i = 1,2,..., m, j = 1,2,..., n (8.2)

1

2

3

4 .

.. n

1

C11

c12

C13

c14 .

.. c1n

2

C21

C22

c23

c24 .

.. c2n

3

C31

c32

c33

c34 .

.. c3n

m

cm1

cm2

cm3

cm4 .

1

2

3

4 .

.. n

1

X11

x12

X13

x14 .

. . x1 n

>

d1

2

x21

x22

x23

x24 .

. . x2n

>

d2

3

X31

x32

x33

x34 .

. . x3n

>

d3

m

xm1

xm2

xm3

xm4 .

. . xmn

>

dm

<

<

<

<

<

S1

S2

S3

S4 .

. . Sn

In this model xij is the representation of an activity i in space j. cij is the representation of the cost (expressed in money, energy, appreciation, and the like) of the realisation of activity i in space j. This representation can be explained with two aspects of the relationship between activities and spaces as follows: Since in buildings and urban areas human activities are not fixed to one unique space - or in other words activities are spread out over more spaces, like rooms, auditoria, corridors, zones, areas - a design expresses, m among a lot of other things, a spatial pattern of different architectural and urban spaces to fit a set of different activities allocated to the designed spaces.

The second aspect concerns the fact that most of the architectural spaces are suited for more than one activity, but of course not all. This means that the designer can propose alternative arrangements of the activities required, for a given spatial arrangement of spaces. Also the other way around: for a given spatial arrangement of activities, alternative layouts of architectural spaces may be proposed. By changing the input values of cij, a representation of the design process on both aspects becomes available. With this mechanism, a designer can represent his pattern of possible activities in such a way that he can see how well this pattern fits the activities required.

While architectural spaces may be suited for more than one activity, they are not necessarily suited for all activities due to technical constraints such as daylight, noise hindrance, permitted location in the building, or conceptual constraints such as structure of spaces and patterns of connections.

The model for this design problem (the limited distribution problem) can be formulated as follows:

Minimise Z = EE cijxij for i = 1,2,..., m, j = 1,2,..., n (8.3)

subject to:

and xij > 0 for i = 1,2,..., m, j = 1,2,..., n aij = {0,1} for i = 1,2,..., m, j = 1,2,..., n

Due to the LP problem solving algorithm, xij will be zero (non Basic) if a j = 0, and xij will get a value if aij = 1. This means that if the designer decides that space sj is not suited or otherwise not appropriate for activity i, he sets aij = 0 and automatically xij becomes 0. In other words, using the zero and one value of aij, the designer uses the model to calculate the best allocation of activities to the designed pattern of spaces.

The function of the variable aij is explained with an example (Table 8.1): A floor plan F for building B consists of four spaces, si,s2, s3, s4. The floor plan should accommodate three different activities, d1, d2, d3. The designer of the

Table 8.1 Example allocation

Activity type 1 Activity type 2 Activity type 3

Table 8.1 Example allocation

Activity type 1 Activity type 2 Activity type 3

 X11 x12 *13 x14 x21 x22 x23 x24 X31 x32 x33 x34 1 0 1 1 > d1 1 1 1 0 > d2 0 1 1 1 > d3 1 1 0 < S1 0 1 1 < S2 1 1 1 < S3 1 0 1 < S4

floor plan decides that si is suited for d1 and d2, s2 is suited for d2 and d3, s3 is suited for d1, d2, and d3, and s4 is suited for d1 and d3. The optimal allocation then follows from equation (8.3).

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