leastSquaresCapaIdent

Provider:kappalab Version:1.0

Description

-Contact: Patrick Meyer (patrick.meyer@telecom-bretagne.eu)

-Reference: J-L. Marichal and M. Roubens (2000), Determination of weights of interacting criteria from a reference set, European Journal of Operational Research 124, pages 641-650.

Identifies a Mobius capacity by means of an approach grounded on least squares optimization. More precisely, given a set of overall values for each alternative, and possibly additional linear constraints expressing preferences, importance of criteria, etc., this algorithm determines, if it exists, a capacity minimising the sum of squared errors between overall scores as given by the data and the output of the Choquet integral for those data, and compatible with the additional linear constraints. The existence is ensured if no additional constraint is given. The problem is solved using quadratic programming.

Inputs

kAdditivity

Description:

Indicates the level of k-additivity of the Mobius capacity (the Mobius transform of subsets whose cardinal is superior to k vanishes).

XMCDA related:

-Tag: methodParameters

-Code:

<methodParameters>
    <parameter
         name="kAdditivity"> <!-- REQUIRED  -->
        <value>
            <integer>%1</integer>
        </value>
    </parameter>
</methodParameters>

GUI information:

-Status: define parameter value via XMCDA file or input by hand, by default via XMCDA file.

-Name: kAdditivity

-Constraint description: The value should be a positive integer, less than or equal to the number of criteria.

-Default value: 1

---

criteria

Description:

A list of criteria. Criteria can be activated or desactivated via the <active> tag (true or false). By default (no <active> tag), criteria are considered as active.

XMCDA related:

-Tag: criteria

-Code:

<criteria>
    <criterion>
        <active>[...]</active>
        [...]
    </criterion>
    [...]
</criteria>

---

alternatives

Description:

A list of alternatives. Alternatives can be activated or desactivated via the <active> tag (true or false). By default (no <active> tag), alternatives are considered as active.

XMCDA related:

-Tag: alternatives

-Code:

<alternatives>
    <alternative>
        <active>[...]</active>
        [...]
    </alternative>
    [...]
</alternatives>

---

performanceTable

Description:

A performance table. The evaluations should be numeric values, i.e. <real>, <integer> or <rational>.

XMCDA related:

-Tag: performanceTable

---

shapleyPreorder

-Optional: yes, enabled by default

Description:

A valued relation on criteria expressing importance constraints on the critera. A numeric <value> indicates a minimal preference threshold for each <pair>. One <pair> represents an affirmation of the type “the Shapley importance index of criterion g1 is greater than the Shapley importance index of criterion g2 with preference threshold delta”.

XMCDA related:

-Tag: criteriaComparisons

---

shapleyInterval

-Optional: yes, enabled by default

Description:

A list of <criterionValue> containing the constraints relative to the quantitative importance of the criteria. Each <criterionValue> contains an an <interval>. Each <criteriaValue> represents an affirmation of the type “the Shapley importance index of criterion g1 lies in the interval [a,b]”.

XMCDA related:

-Tag: criteriaValues

---

interactionPreorder

-Optional: yes, enabled by default

Description:

A valued relation on pairs of criteria expressing constraints on value of the the Shapley interaction index. A numeric <value> indicates a minimal preference threshold for each <pair> of the relation. One <pair> represents a constraint of the type “the Shapley interaction index of the pair (g1,g2) of criteria is greater than the Shapley interaction index of the pair (g3,g4) of criteria with preference threshold delta”.

XMCDA related:

-Tag: criteriaComparisons

---

interactionInterval

-Optional: yes, enabled by default

Description:

A list of <criterionValue> containing the constraints relative to the type and the magnitude of the Shapley interaction index for pairs of criteria. Each <criterionValue> contains an an <interval>. Each <criteriaValue> represents an affirmation of the type “the Shapley interaction index of the pair (g1,g2) of criteria lies in the interval [a,b]”.

XMCDA related:

-Tag: criteriaValues

---

overallValues

Description:

A list of <alternativeValue> containing the desired overall value of each alternative. The <value> should be a numeric value.

XMCDA related:

-Tag: alternativesValues

---

Outputs

messages

Description:

A list of messages generated by the algorithm.

XMCDA related:

-Tag: methodMessages

---

mobiusCapacity

Description:

The Mobius transform of a capacity.

XMCDA related:

-Tag: criteriaValues

-Code:

<criteriaValues mcdaConcept="mobiusCapacity">
    <criterionValue>
        <criteriaSet>
            <element>
                <criterionID>[...]</criterionID>
            </element>
            [...]
        </criteriaSet>
        <value>
            <real>[...]</real>
        </value>
    </criterionValue>
    [...]
</criteriaValues>

---

residuals

Description:

A list of <alternativeValue> representing the difference between the provided overall evaluations and those returned by the obtained model, for each alternative.

XMCDA related:

-Tag: alternativesValues

-Code:

<alternativesValues mcdaConcept="mobiusCapacity">
    <alternativeValue>
        [...]
    </alternativeValue>
    [...]
</alternativesValues>

---

For further technical details on the web service underlying this program, have a look at its documentation here.