_{1}

^{*}

In Multi-Criteria Decision Analysis, the well-known weighted sum method for aggregating normalised relative priorities ignores the unit of scale that may vary across the criteria and thus causes rank reversals. A new aggregation rule that explicitly includes the norms of priority vectors is derived and shown as a remedy for it. An algorithmic procedure is presented to demonstrate how it can as well be used in the Analytic Hierarchy Process when norms of priority vectors are not readily available. Also, recursion relations connecting two decision spaces with added or deleted alternatives give an opportunity to extend the idea of connectivity to a new concept of cognitive space. Expanded analytic modelling embracing multiple decision spaces or scenarios may assist in detecting deficiencies in analytic models and also grasping the big picture in decision making.

In Multi-Criteria Decision Analysis (MCDA), often a complex decision problem is decomposed into a hierarchy of criteria, sub-criteria and alternatives. For such decomposition, one assumes that the decision about alternati- ves in respect of one criterion (or sub-criterion) can be made independent of decisions made in respect of other criteria (or sub-criteria). It also becomes easier for a decision maker (DM) to understand and analyze the decision hierarchy in small segments at a time. Finally, results (i.e., criteria and sub-criteria weights and alternative priorities) from different segments are aggregated. Integrated values for the decision alternatives can be presented as aggregate scores, normalised relative priorities or ordinal ranks. In this paper, we consider a simple three level hierarchy without any loss of generality. The top level represents the overall goal of the decision. The second level represents the criteria, and the third level is for a list of alternatives. The criteria are characterised by weights that may sum to one. The priorities of the alternatives under each criterion may be expressed as un-normalised preference scores or relative priorities that are normalised such that they sum to one. The preference scores or the relative priorities are often derived from a Multi-Criteria Decision Model (MCDM). Finally, the criteria weights and alternative priorities are aggregated to generate an overall rating for alternatives.

A commonly used aggregation procedure, introduced in the Multi-Attribute Value/Utility Theory (MAVT/ MAUT), is the weighted sum method (see [

In any MCDM technique, since method b) may cause unwanted rank reversals, it is justified to question the validity of the aggregate values derived using this technique. This observation brings the Analytic Hierarchy Process (AHP) [

In addition, based on recursion relations that link two decision spaces having an added or deleted alternative, we introduce the concept of cognitive space spanning multiple decision spaces or scenarios. We present arguments for expanding the usual modelling experiences beyond a particular decision space into the cognitive space. Expanded modelling will help us detect shortcomings if any, of models developed on a particular decision space. In Section 2, we outline the new contributions of this paper. We provide a generalised derivation of a modified aggregation rule, further discuss recursion relations and overview the meaning of weights in this context. Section 3 and Section 4 demonstrate the use the new aggregation rule in the AHP with numerical examples. Section 5 delineates the concept of expanded analytical modelling in cognitive space and conclusions are made in Section 6.

In this section, starting from a generalised multi-criteria decision space, a modified aggregation rule is derived and contrasted with the familiar WSM. Calculations involve combining measurements from two different spaces (i.e., criteria space and alternative space), each having different units of scale and may have different dimensions, too. However, no mathematical derivation for the WSM which represents a complex measurement procedure is available in the MCDA literature. Perhaps, the formula was introduced as a simple extension of the expected value formula with the normalised weights replacing probabilities. The earlier work [

We consider a decision space with n alternatives judged with respect to m criteria as represented by the hierarchy diagram in _{i}, (_{j}, (

Let us assume that the criteria weights (implying the relative importance of the criterion) can be obtained from any available technique. We develop formulations of this section independent of any particular model (e.g.,

AHP). We represent the weights by a vector w and for the sake of convenience we assume that the weights are normalised such that they sum to one. The rest of the discussions in this section will not change even if the weights are not normalised (in that case weights will be a constant multiple of the normalised ones). So, the weight vector w is,

Similarly, with respect to i_{th} criterion, let the priority vector for n alternatives be

The priority vectors v^{i} can be obtained in various ways subject to a decision model. For example, it can represent a set of scores as obtained from a decision maker’s mental model or it can be obtained from the pairwise comparison matrix (P^{i}) as in the AHP. If one employs the eigenvector method in the AHP, then it can be just an eigenvector (not necessarily normalised) derived from the eigenvalue equation,

Other methods such as minimization of logarithmic least-square approach have been discussed in the AHP literature as well (see [

If u^{i} is the corresponding normalised priority vector, then

_{th} criterion. In the scoring method, ^{i}_{j} and can vary for different criteria. Therefore, the correct aggregation rule should incorporate

The combined decision space D is a direct product of the criteria space and the alternative space, i.e.,

Thus, the priority state y is derived from a direct product of w and v^{i} s so that it preserves the hierarchic structure of

Please note that the column vectors α_{i} s are defined in m dimensional criteria space and v^{i} s are defined in the n-dimensional alternative space.

It is worth pointing out that, in general, there are arbitrary constants multiplying each term in the expansion of Equation (6). For convenience, we absorbed them in v^{i} s and thus in the corresponding norms

Now expanding v^{i} s in terms of the basis vectors β_{j} given in Equation (3), we finally have,

We project out the intensity of priority corresponding to the p_{th} state of the criteria space and the q_{th} basis state of the alternative space by multiplying on the left by

In additive models, we sum over all criteria to obtain the aggregate priority R_{q} of alternative q,

Now, using the relation between the unnormalised local priority

The normalised aggregate priority r_{q} (i.e.,_{th} alternative is given by a Modified Aggregation Rule (MAR),

The above expression should be compared with the traditional weighted sum formula called Simple Aggregation Rule (SAR),

The MAR was proposed as the correct aggregation procedure for obtaining overall priorities in [_{p}), we would have

It is straightforward to check that the expression of the normalised aggregate priority r_{q} in Equation (11) would remain the same confirming that we did not sacrifice any generality by considering a normalised weight vector at the very beginning in Equation (6).

It should be further noted that the proposed aggregation rule (MAR) becomes identical with the familiar weighted sum rule (SAR) only when norms of all priority vectors are equal (i.e.,

The above result is independent of decision models that one would use to derive values for the alternative priority vectors. For example, Saaty [

Two recursion rules used in [

1) When a new alternative (i.e., the (n + 1)_{th} alternative) is added to the decision space (the primed items refer to the corresponding quantities in the new decision space with n + 1 alternatives), in a fully consistent case,

2) When an existing alternative (i.e., the n_{th} alternative) is deleted from the decision space (the primed items refer to the corresponding quantities in the new decision space with n − 1 alternatives), in a fully consistent case,

Using the MAR, it can be proved [

In MCDA research, it has been noted [_{i} of alternative A_{i} (

where, W^{ j} is the weight of the j^{th} criteria. Here, ^{ j} s can be norma- lised (i.e.,

the normalisation of weights does not affect the MAR that is being proposed here as the correct aggregation method as it prevents rank reversals. But, rank reversals may occur in the SAR where the _{i,k} and other related derived ones in [

Here, in this research, we may be tempted to redefine the criteria weights as follows absorbing the norms (i.e., additional information) of the priority vectors:

But, because of the recursion relations in Equation (14) and Equation (16), the new weights_{p} represent the importance of criteria as well and they get adjusted by the norms of priority vectors creating the adjusted weights_{p} is obtained from a DM’s inputs and

In the AHP, we can determine weights w_{p}_{ }and the local normalised alternative priorities

In the following formulations, we will assume that there are n alternatives, so we will suppress the index n and will restore it only in the final results. Estimating values for

In Step 3.1 of Text Box 1, m estimates for the preference scores for alternative 1 is required. In MAVT, such values are obtained from value functions. However, m estimates for the preference scores for alternative 1can be estimated assuming a Beta distribution as a possible choice if we want to deal with the underlying uncertainty stochastically.

Let a decision maker estimates the norm (n = 1), following a Beta distribution, as follows:

b_{p} = high end estimate for_{p} = low end estimate for_{p} = most likely estimate for ^{ }

Then, then

Average score for

Zahir [

In the preceding sections, we observed that as we move from one decision space to another (obtained by adding or deleting an alternative), the decision model can generate meaningful results only if the aggregation rule is the correct one. This “correctness” is based on our reasoning, i.e., ranks should not reverse when they are not expected to. We illustrated with numerical examples how the MAR supported our correct expectations. Deficiencies of the SAR were detected only when logical reasoning could not be satisfied by calculated results as an alternative was added to or deleted from a decision model. This observation emphasises the need for testing the validity of analytic modelling over an expanded model space. In the AHP (or MAVT using the SAR), if the early developers considered results of two decision spaces (obtained by adding and subtracting alternatives), they would have identified the deficiency of the SAR. An improved aggregation rule that incorporated the unit of scale would have been used in the normalisation of alternatives such as the MAR being discussed in this paper. Such an expanded “big picture” framework is outlined below.

In the context of an MCDM, decision making can be discussed in terms of the decision space that is cha- racterised by the number of alternatives and criteria. Alternatives and criteria have numerical values or subjective values (derived from a decision maker’s cognitive process and its interaction with the external world) in a quantitative or a qualitative decision model respectively. However, in this paper, since we are dealing with quantitative decision modelling we assume that we can assign numerical values to our judgements. In general, there can be a large number of choices for the number of alternatives and criteria making a decision problem complex. Often the problem is represented by a model that is nothing but an abstraction of the reality. However, the analysing capability of humans as rational beings, even with technological assistance, is limited and thus we are inclined to “satisfice” by setting a boundary to the decision spaces. The boundary can change as we acquire more knowledge about the problem or are subject to different external influences. This effect can be understood in light of Simon’s bounded rationality [

As more alternatives (or criteria) are added or some are deleted, the decision space expands or contracts creating a new decision space for the underlying decision model. The set of related decision spaces collectively is the cognitive space of the decision problem, and we try to link them through reasoning and understanding. Each decision space is also a cognitive sub-space where model formulations are defined in terms of logic, reasoning and facts. Often, the links among decision spaces are done in human minds and are left out of the decision models and thus each model is treated independently in its own decision space. But, together they are discussed by researchers in their cognitive domain via reasoning processes. Our memory is a repository of information and it plays a significant role in linking the analyses of the decision spaces via reasoning. Thus, it helps us connect scenarios in decision spaces logically to the satisfaction of our rational minds.

We can think of a Gedanken experiment in reference to the example in Section 4. We keep the inputs same as in Section 4. Suppose a DM computes the relative priorities using the SAR for three alternatives. And then she takes a nap and wakes up when she has no memory of the priorities derived before the nap. Now, she computes the relative priorities using the SAR again for four alternatives and obtains new results. Since she has no memory of the results she obtained before the nap, does it matter even if the ranks now are reversed? Most probably it does not. However, in reality we have memory, and our cognitive process finds reasoning to link findings and results from related decision spaces. The links of reasoning connecting the results from one decision space to another as they reside in our memory are not incorporated in the decision models or, may be, models are deficient in part. We reason in our minds. Sometimes, we also fail to satisfy our reasoning and then we raise the question about the validity of a decision model as it stands. It may be due to a flaw in parts of the model and therefore it needs to be corrected. Another approach that has not been widely pursued in practice may be to extend the decision model across the cognitive space that would also model the reasoning links among the decision spaces. That would also help us find a solution to an apparent imperfection in a decision model.

As it has been discussed earlier, when a new alternative is added or an existing alternative is deleted (i.e., moving from one decision space to another in the cognitive space of the decision problem (see _{ij} representing the reasoning that links DS-i and DS-j in the cognitive space. CL_{ij} represents the recursion rules that asserts that norms (setting the scale of normalisation) of priority vectors change as an alternative is added or deleted, and the MAR incorporates this assertion preserving ranks when it should. The SAR is deficient as it does not incorporate the scale of normalisation and hence produces unwanted rank reversals.

Here, we may attempt to relate decision spaces to a simplified form of scenarios as a parallel concept. The idea of scenario has been discussed for over 40 years (see [

concept of a scenario is still an evolving one but it is clear that each scenario has a context. Tversky and Kahnemann [

We have stressed the need for linking decision spaces so as to test the validity of mathematical models. The analysis presented in the preceding sections lead us to the correct aggregation rule called the MAR. The simplified version of the scenario that we adopt is defined by a set of alternatives and criteria and preferences assigned to them by a DM, who sets the context of the scenario. The context emerges as the DM assigns to alternatives and criteria subjective and objective judgements that are influenced by his/her value system and sense of uncertainty. Mathematically speaking, in a modelling environment the context is created by a set of variables, operators and constraints formulated as mathematical expressions. Thus, as we navigate from one decision space to another, some of the things that can happen are,

Parameter values change.

New parameters enter.

Old parameters may drop out.

Expressions are re-evaluated (e.g., the recursion relations in preceding sections).

New constraints may emerge.

And so on.

If we have a valid model, its predictions would satisfactorily track our cognitive expectations not only in a particular decision space but also as navigate to others. Gomes and Costa [

While we discussed the multi-criteria decision problem in light decision spaces, we can extend the concept to scenarios applicable to other areas of decision sciences. In management science and operational research, we make models for inventory/supply chain, resource allocation, financial market analysis, portfolio management, etc. based on a particular scenario (analogous to a decision space). Then, we change models as scenarios change (often probabilities are assigned to each scenario in scenario analysis). We try to understand the varying results in our reasoning as we connect scenarios utilising our memory and cognitive strength. It may be possible to include intra-scenario reasoning by having cognitive links in the models spanning more than one scenario. If the models cannot explain results from two scenarios, that can be attributed to a deficient component or weakness of the model itself and thus will call for improvement in the model. We can also try to understand the significance of such integrated multi-scenario decision modelling in light of system thinking [

We identified the deficiency of the traditional weighted sum aggregation rule used in the AHP and possibly in the MAVT with the prior normalisation as the cause for rank reversals and proposed a modified aggregation rule as a remedy. Utilising recursion rules, we analyzed the additive aggregation process in light of the new aggregation rule. We also suggested ways how it could be used with the AHP or MAVT with prior local normalisation. The ideas are illustrated with numerical examples. Also, we discussed possibilities for extending usual modelling techniques developed in the context of a particular decision space or scenario to an expanded form in the cognitive space. Concepts of satisficing and bounded rationality were invoked for managing such expansions. Testing the validity of mathematical models in decision sciences, operations management and possibly economics may need the support of a “big picture” beyond a single scenario.

Sajjad Zahir, (2016) Aggregation of Priorities in Multi-Criteria Decision Analysis (MCDA): Connecting Decision Spaces in the Cognitive Space. American Journal of Operations Research,06,317-333. doi: 10.4236/ajor.2016.64030

The aggregation rule that we derived in the text (also see [

1) An alternative is added

Let

From Equation (14), we have

However, in a fully consistent case, we have from Equation (5) and Equation (14),

So, finally we get from Equation (A2),

2) An existing alternative is deleted

Let

From Equation (15) and Equation (A5), we have

However, in a fully consistent case, we have from eqs (5) and Equation (15),

So, finally we get from Equation (A6),

A Numerical Example for Using the MAR

For the sake of illustration, in this section we demonstrate the use of the algorithm given in Section 3 for computing aggregate relative priorities in the AHP using a numerical example. It is a two-level hierarchy having three criteria and three alternatives. It is the same example as the one employed in [

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Alternative B | |||

Alternative C | |||

Norms |

With absolute scores known, we can construct pairwise comparison matrices under each criterion and compute normalised local relative priorities as in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Alternative B | |||

Alternative C | |||

1 | 1 | 1 |

Aggregate relative priorities using the SAR and the MAR (Equation (12) and Equation (11) respectively) are given in

Using the SAR | Using the MAR | |
---|---|---|

r_{1 } | 0.4512 | 0.4500 |

r_{2} | 0.4697 | 0.4750 |

r_{3} | 0.0791 | 0.0750 |

In

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Norms |

We add another alternative and the PCMs are given in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | ||||||
---|---|---|---|---|---|---|---|---|

- | A | B | - | A | B | - | A | B |

A | 1 | 1/9 | A | 1 | 9 | A | 1 | 8/9 |

B | - | 1 | B | - | 1 | B | - | 1 |

The normalised relative priorities obtained from the PCMs using the eigenvalue method are calculated in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Alternative B | |||

1 | 1 | 1 |

Using the norms in the bottom row of

Next we add a third alternative and the PCMs are given in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | A | B | C | A | B | C | |||

A | 1 | 1/9 | 1 | A | 1 | 9 | 9 | A | 1 | 8/9 | 8 |

B | 1 | 9 | B | 1 | 1 | B | 1 | 9 | |||

C | 1 | C | 1 | C | 1 |

The normalised local relative priorities derived from the PCMs using the eigenvalue method are presented below in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Alternative B | |||

Alternative C | |||

1 | 1 | 1 |

Using the numbers in

Finally, the aggregate relative priorities computed using the MAR are (compare with

Using MAR | |
---|---|

r_{1 }(Alternative A) | 0.4500 |

r_{2} (Alternative B) | 0.4750 |

r_{3} (Alternative C) | 0.0750 |

The above numerical exercise demonstrates how the MAR can be used in the AHP as explained in the algorithm of Text Box 1. We performed extensive numerical analysis and confirmed that the final results are independent of the relative order of the alternatives. Rank reversals are avoided using the MAR as long as the comparisons are consistent. In the presence of inconsistency, rank reversal can happen even with the MAR; but that is acceptable as imperfections in judgments. We performed simulation of the algorithm with random presence of inconsistencies and observed rank reversals varying with amount of inconsistencies (see [

have a simple hierarchy with one level of m criteria and n alternatives, we need m additional inputs for

However, the procedure involves extra computational steps but generates correct aggregate relative priorities for the alternatives. To substantiate this claim, let us add another alternative (D) to the example above. The PCMs are given in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A | B | C | D | A | B | C | D | A | B | C | D | |||

A | 1 | 1/9 | 1 | 1/9 | A | 1 | 9 | 9 | 9 | A | 1 | 8/9 | 8 | 8/9 |

B | 1 | 9 | 1 | B | 1 | 1 | 1 | B | 1 | 9 | 1 | |||

C | 1 | 1/9 | C | 1 | 1 | C | 1 | 1/9 | ||||||

D | 1 | D | 1 | D | 1 |

The normalised relative priorities obtained by eigenvalue method are calculated in

Criteria 1 (C_{1}) | Criteria 2 (C_{2}) | Criteria 3 (C_{3}) | |
---|---|---|---|

w_{1} = 1/3_{ } | w_{2} = 1/3 | w_{3} = 1/3 | |

Alternative A | |||

Alternative B | |||

Alternative C | |||

Alternative D | |||

1 | 1 | 1 |

Using the numbers in

The aggregate relative priorities are (using both the MAR and the SAR) presented in

Using SAR | Using MAR | |
---|---|---|

Alternative A r_{1 } | 0.3654 | 0.3051 |

Alternative B r_{2} | 0.2889 | 0.3220 |

Alternative C r_{3} | 0.0568 | 0.0508 |

Alternative D r_{4} | 0.2889 | 0.3220 |

The important point to note is that the ranks of A and B are reversed by the SAR but are preserved by the MAR! Therefore, if we use the MAR that was derived in Section 3, we can avoid the rank reversal controversy not only of the AHP but also other additive MCDM s.

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