## Integrating LP Multi Criteria optimisation with Preference Modelling using fixed weight factors

The LP model that was used for the project developer's problem of Section 1.2 maximised the profit of the project developer. The municipality, however, is interested in maximising the number of parking places. This means that we want to maximise both the profit and the number of parking places. An objective function should not contain variables with different units. To overcome this problem weight factors can be used to express the relative importance of both decision criteria 'profit' and 'number of parking places'.

Suppose we assume that both stakeholders agree that the criteria 'number of parking places' has a weight factor of 1 and the criteria 'profit' has a weight factor of 2. The original objective function was:

We can then change the objective function to reflect these weight factors:

The modified model structure can be seen in Figure 6.1. Figure 6.2 shows a screenshot of the actual solved model. [O e_pm-1.xls]

As can be seen, developing 20 houses of type A and 50 houses of type B will yield the highest preference.

So far, the assumption was made that the scaling of preferences is not affected by any variations in the decision variables. This assumption has to be removed whenever a preference is dependent on the value of a decision variable.

A difficulty is, however, that more often than not stakeholder's preferences depend on other decision variables. How to allow for this, is explained in the next example.

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Arial - 10 ' I B I U I m @ = Ü I \$ % * « ¿S I it it I EE ' & - A -1 WBMAX T £ =SUMPRODUCT(BS2:C\$2.B4:C4)_

Fie Edit View Insert Format Tools Data Window WB! Help Type a question for help - _ iS X

J ¡3 A Là ¿J I d! iA I y £11 * ^ a - * I *J - -1 & x - il il I S - © 1

Arial - 10 ' I B I U I m @ = Ü I \$ % * « ¿S I it it I EE ' & - A -1 WBMAX T £ =SUMPRODUCT(BS2:C\$2.B4:C4)_

A

B

c

D

E

F

G

H

1

J

K 3

1

Endogenous variables

N A

N B

2

Outcome

GO

45

3

4

Objective function

30000

50000

4050000

5

required

available

6

Max. type A

1

60

=<= 60

7

Max. type B

1

45

<=

50

8

Max. parking-places

1

2

150

-<-

150

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

?4

► ir

il A ► n|\ WB! Status \Sheetl/Sheet2 /Sheet3 /

Figure 6.2 Screenshot solved model (project developer's problem using stakeholder's weight factors)

 Alternative ¿1 ¿2 ¿3 Profit 2.6 3.1 3.2 Number of affordable houses 60 20 40

Table 6.1 Numerical output of each alternative

Table 6.1 Numerical output of each alternative

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