# One speaks decision

under security of decisions under security in the context of the decision theory if the decision maker knows the occurring environmental condition with security ([itex] w_j=1< /math>) and it thus all consequences from an action to forecast can. Decisions with several objectives (multikriterielle decision problems) play thereby the most important role.

## general

the acceptance that in advance admits all consequences of an action are, appears intuitively completely unrealistically. The importance of decision rules under security is nevertheless very great. So for example certain linear decision problems can be transferred with decisions under uncertainty into a decision situation under security with several objectives.

Examples: (Laux, 2003, S. 65ff.):

• The goal actually aimed at cannot be measured directly, besides the influence of certain parameters of action on this goal is not with security well-known.
• The complexity with a decision on the basis a global goal (maximization of profit) is not controllable any longer, so that individual Unterziele is defined, whose influence on the global goal admits at least tendentious is.

Thus for example an enterprise will make a decision over a new location as a rule under the goal of the long-term maximization of profit, the respective influence of the possible locations on the profit is however not not directly assignable. However the location characteristics are like for example infrastructure, labour costs, taxes, granted subsidies, construction costses etc. with security admits, equally knowledge of the connection between these factors and the global goal exists. From the linear decision under uncertainty so a multidimensional (multikriterielle) decision becomes with security.

## linear decision problems

a little relevant problem is present, if only one goal is pursued and goal development with the choice of different alternatives admits is. Two cases can be differentiated:

• Unlimited objective: A maximization or a minimization of goal development is aimed at. Example: Maximization of profit, minimizing the risk.
• Limited objective: At least/at the most here a goal is to be achieved either exactly (adjustment) or (Satisfizierung). Example: In order to reach a flight, one must be at the latest 1 hour before at the airport, it is however no matter whether one is in former times there (Satisfizierung). Starting from a certain quantity of sweets one with one more becomes rather bad at sweets, it gives an optimal quantity, with which deviations are bad upward and down (adjustment).

## Multikriterielle of decision problems

a decision (the selection of an alternative course of action from several alternatives the available is called) has usually consequences for several goals, so that a multikriterielles decision problem is present. Since for each alternative course of action the available all consequences admits are (goal sizes) and admits is, which development of the individual goals is desired (preference relation) is present a target system. The goals of this target system can stand to each other for target system the goal sizes

in a target system

### to place

, which consequences of its action alternatives a Entscheider meaning in different relations [to work on] measures out and form the yardstick for the evaluation of the alternative by the evaluation of the action alternatives regarding in each case a consequence. The target system depends on individual Entscheider.

Example:

For the way to the work are available the travel with the car or the ÖPNV. The target system of the Entscheiders looks then for example in such a way:

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math>
[itex] a_1< /math> 40 5 8
[itex] a_2< /math> 45 4 4

with [itex] z_1< /math> to [itex] z_3< /math> the different goals (here: [itex] z_1< /math> = travel duration in minutes, [itex] z_2< /math> = costs for each travel into euro, [itex] z_3< /math> = comfort on a scale from 1 to 10) and [itex] a_1< /math> = travel with the car and [itex] a_2< /math> = travel with the ÖPNV.

Another Entscheider at another place of the city can have another target system, for example:

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math>
[itex] a_1< /math> 30 4 -5
[itex] a_2< /math> 60 6 -2

this Entscheider thereby the comfort could be completely all the same, so that for it [itex] z_3< /math> an expression for the ecological consequences of the respective means of transport is.

Regarding desired goal developments said the above applies: Are possible maximization, minimization, adjustment or Satisfizierung.

### goal relations

of goals can stand in different relations to each other (Laux, 2003, S. 67ff.):

• Zielindifferenz and/or. Goal neutrality: Will not affected the reaching of the goal by the other goal, the decision problem can into linear in each case sub-problems be divided.
• Goal complement airty: The reaching of the goal facilitates the reaching of the other goal. Example: English knowledge and vacation in England. If a goal is 1, as well as possible English to be able and a goal 2 is to spend as much as possible vacation in England then a high reaching of the goal 2 (much vacation in England) improves automatically the Zielerreichung with a goal 1.
• Conflicting aims and/or. Goal competition: Those actually problematic situation develops, if goals are conflict acre to each other, thus the Zielerreichung of a goal 1 on the goal 2 affects itself negatively. Example: Money earn and spare time: To have, earns the less can one wants more spare time one work, the fewer moneys one.

## decision rules for multikriteriellen decision problems

### the dominance principle

for the simplification of the decision problem should not be regarded those alternatives, which are dominated by other alternatives. An alternative is dominated if there is at least a further alternative, which fares in all goals at least just as well and is better in at least one goal.

Example:

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math>
[itex] a_1< /math> 10 5 8
[itex] a_2< /math> 10 6 8
[itex] a_3< /math> 7 7 7

alternative 1 by alternative 2 and no more is dominated here should not be regarded. Alternative 2 is better in a goal 1 and a goal 3 than alternative 3, however not in a goal 2, so that alternative 3 is not dominated.

### lexicographical order

with this procedure is provided a ranking of the goals. First only the most important goal is regarded and evaluated, therefore the procedure is called also goal suppression. If one does not come thereby to a result, because more than one alternative is equivalent regarding the most important goal, then the next-most important goal looked at and so on. This can lead to implausible results.

Example: (A goal 1 is most importantly before a goal 2 before a goal 3)

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math>
[itex] a_1< /math> 101 0 0
[itex] a_2< /math> 100 ,100 ,100

although to alternative 2 in the goal 1 only scarcely more badly, in the two other goals however clearly better, according to the lexicographical order alternative 1 cuts off selected.

### goal weighting

during the goal weighting is provided also a ranking of the goals, however a weighting factor must be intended for each goal. With the decision the different goals are multiplied and summed with each alternative by the respective weighting factor. The alternative, which obtains here the highest value, is selected. Contrary to the lexicographical order with each alternative however all goal developments are considered, i.e. a particularly high development of the secondarymost important goal can compensate a low development of the most important goal.

### Körth rule

it is aimed at the maximization of the minimum (relative) goal reaching degree. In addition the maximum development of a goal in all alternatives is looked for in each case and all values of goal development in the column by this value is divided. In the use matrix the values are now nomiert on the interval [0..1], it no more goal development are thus indicated, but the relative Zielerreichung in the comparison to the possible maximum. Each alternative (line) is evaluated after the minimum relative goal reaching degree (line by line the minimum searched). The alternative, which exhibits here the highest value, is left

< selected> math \ max_i \ (\ min_p \ (\ frac {u_ {IP}} {\ max_h u_ {hp}} \ right) \ right) /math< left>

whereby [itex] u_ {jk}< /math> the use of the alternative j regarding a goal k is.

Example:

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math>
[itex] a_1< /math> 16 20 5
[itex] a_2< /math> 4 10 10
[itex] a_3< /math> 8 8 8
maximum 16 20 10

this matrix now one transforms:

[itex] z_1< /math> [itex] z_2< /math> [itex] z_3< /math> Minimum
[itex] a_1< /math> 1 1 0.5 ,0.5
[itex] a_2< /math> 0,25 0.5 1 0.25
[itex] a_3< /math> a preference order results

0,5,0.4,0.8,0.4 thereby: Alternative 1 (0,5) better than alternative 3 (0,4) better than alternative 2 (0,25).

### efficiency analysis

the efficiency analysis is called also point evaluation or Scoringmodell. Here goal criteria a point value from 1 to 5 is assigned. These are then weighted and added.

### goal programming

the method of goal programming is called also Goal Programing. Here one tries to minimize the sum of the weighted deviations. There with for a goal a reaching value is specified. The deviations of the alternatives are then weighted (also different weights upward and down). These values can be strengthened and become then still in the end summed up. The smallest sum wins.

[itex] z_i=z_i^*-z_i (x)< /math> with [itex] z_i^*< /math> as target and [itex] z_i (x)< /math> as goal function value of the alternative i

< math> \ min {G= \ left [\ sum (\ underline w_i \ underline z_i^p+ \ overline w_i \ overline z_i^p) \ right] ^ {\ frac {1} {p}}}< /math>

whereby

1. [itex] \ underline z_i< /math> the deviation of the goal i downward,
2. [itex] \ underline w_i< /math> the weighting of the deviation that of a goal i downward,
3. [itex] \ overline z_i< /math> the deviation upward,
4. [itex] \ overline w_i< /math> the weighting of the deviation that of a goal i upward and
5. [itex] p< /math> the discrepancy factor (usually = 1) is.

Mostly is [itex] \ overline w_i= \ underline w_i=1< /math>, with which not between the deviation upward anddown one differentiates.

### analytics Hierarchy Process

of analytics Hierarchy Process (AHP) offer support for a hierarchical target system and are mathematically more fastidious, in addition, more precise.

## literature

• v. Zwehl, W.: Decision rules, in: Hand dictionary of the marketing and management, volume 1, 5. Aufl., Schäffer Poeschel, 1993
• Laux, H.: Decision theory, 5. Edition, Berlin and others, Springer, 2003
• Bamberg, G. and Coenenberg, A. G.: Economical decision teachings, 12. Edition, Munich: To Vahlen, 2004