**4. CS‐AHP model for optimal selection problem**

In this section, we formally construct the quality model under the presen**c**e of variability and uncertainty among demands in decision‐making process. The CS‐AHP algorithm is adopted for specification of stakeholders' preferences and for further measurements during the optimal selection. Furthermore, we analyze different kinds of preferences and define final configura‐ tion goals induced by the constructed quality model.

#### **4.1. The two‐layered selection criteria structure**

In a given model, let each demand will be available with a finite set of options characterized with respect to selection criteria. Having in mind that stakeholder defines his/her own preferences about overall values (i.e., person A specified overall budget and expectations regarding the level of personal satisfaction and conformity), we will create the structure of concerns and qualifier tags with respect to aggregated values of each selection‐making criterion:

**Step 1.** Generate *lower and upper bound values of each selection criteria dimension* (*Qagg LB*, *Qagg UB*) =(*Q*<sup>1</sup> *LB*, ..., *Qk LB*, *<sup>Q</sup>*<sup>1</sup> *UB*, …, *Qk UB*)∈*R <sup>k</sup> xR <sup>k</sup>*

**Step 2.** Stakeholders should define own preferences about aggregated values presenting own attitudes and personal fillings about each selection criteria dimension, in the form of covering subintervals *QT* =*QT*<sup>1</sup> 1 , .., *QTk*<sup>1</sup> 1 , …, *QT*<sup>1</sup> *k* , .., *QTkn <sup>k</sup>* . *Each subinterval QTi <sup>j</sup> is open, semi‐open or*

combination of options. The formal foundations of measurements are presented later in Section

Additionally, as usual for the optimal selection tasks, *hard* constraints might be defined for special demands for limiting the corresponding selection‐making criteria. That is, those preferences are not only defined as appropriate relative importance in the selection criteria model. For example, person A defined that total costs of stay should not be over 5000. It means that any combination of options which has the best characteristics with respect to the stake‐ holders' other preferences and which violates this specified value of price should be eliminated

{ 1 , .. ,*l*}, is a constant limitation value. In our illustrative example, both total budget limitations

The *optimal selection problem* can be defined as a problem of unique options derivation, such that stakeholders' preferences are satisfied. In our approach, once the preferences are obtained, genetic algorithm (GA) is used for the selection of the most desirable demands and the relevant options that collectively maximize the stakeholders' satisfaction. Furthermore, constraints defined in GA will guarantee that every combination of options will be valid with respect to the interdependencies and other relations between demands, as proved in reference [17].

In this section, we formally construct the quality model under the presen**c**e of variability and uncertainty among demands in decision‐making process. The CS‐AHP algorithm is adopted for specification of stakeholders' preferences and for further measurements during the optimal selection. Furthermore, we analyze different kinds of preferences and define final configura‐

In a given model, let each demand will be available with a finite set of options characterized with respect to selection criteria. Having in mind that stakeholder defines his/her own preferences about overall values (i.e., person A specified overall budget and expectations regarding the level of personal satisfaction and conformity), we will create the structure of concerns and qualifier tags with respect to aggregated values of each selection‐making

**Step 1.** Generate *lower and upper bound values of each selection criteria dimension*

*UB*)∈*R <sup>k</sup> xR <sup>k</sup>*

( *o*1, … , *o*n, ≤ *u*<sup>i</sup>

, *i* ∈

Formally, *hard constraints* can be defined as a set of constraints: *cl*<sup>i</sup>

200 Applications and Theory of Analytic Hierarchy Process - Decision Making for Strategic Decisions

(defined by person A) should be considered as hard constraints.

**4. CS‐AHP model for optimal selection problem**

tion goals induced by the constructed quality model.

**4.1. The two‐layered selection criteria structure**

criterion:

(*Qagg LB*, *Qagg*

*UB*) =(*Q*<sup>1</sup>

*LB*, ..., *Qk*

*LB*, *<sup>Q</sup>*<sup>1</sup>

*UB*, …, *Qk*

4.

and should not be evaluated any further.

**3.3. Problem of optimal selection**

*close interval* (*ai j* , *bi j* ), (*ai j* , *bi <sup>j</sup>* , *ai j* , *bi j* ), *ai j* , *bi <sup>j</sup> which satisfies the conditionsU i*=1 *kj QTi j* = *Qj LB*, *Qj UB* : *and QTi j* ∩ *QTl j* = ∅, 1≤*i* <*l* ≤*kj* , *for each fixed j* ∈{1, …, *n*}.

Furthermore, each combination of options *o*1, .., *on* is characterized by aggregated values of selection‐making criteria, *Qagg* =*q*<sup>1</sup> *agg*, …, *qk agg* ∈*R <sup>k</sup>* [18], which belongs to exactly one combina‐ tion of covering subintervals *QTi* 1 1 *x* … *xQTi k k* .

Accordingly, aforementioned combination of options in our running example, *o*13 and *o*<sup>22</sup> (with total comfort of 10 and travel costs of 850) belongs to combination of covering subintervals [8, 10] *x* (700, ‐) defined by stakeholder A. Since stakeholder A mostly prefers values from combination of subintervals [8, 10] *x* (‐, 500) (see Section 3.2), other combinations of available options should be analyzed in order to check if any belongs to the most preferable combination of subintervals. Furthermore, combination of options belonging to the most reachable combination of subintervals should be considered as the most appropriate combination of options. Finally, if we have several combinations of options belonging to the same combinations of covering subintervals, the difference of their selection criteria values should be quantitatively measured and compared.

Thus, based on the output ranks of the CS‐AHP method, we define measures of selection‐ making criteria fitting degrees, as follows:

**Definition 1 (CS‐AHP selection making degree measurements).***For a given model (C, Qagg LB*, *Qagg UB*, *QT, P), where C is a set of concerns, QT is a set of qualifier tags over aggregated intervals Qagg LB*, *Qagg UB for k selection‐making criteria, and P represents the set of specified preferences, the standard CS‐AHP algorithm defines the measurements obtained on the bases of output ranks over the set of selection‐making criteria (written asr*<sup>1</sup> *<sup>C</sup>*, …, *rk <sup>C</sup>*) *and a collection of covering subintervals (written asr*<sup>1</sup> 1 , …, *ri* 1 1 , .., *r*<sup>1</sup> *k* , …, *ri k k* ), *as follows:*

*(1') 1‐dimension selection criteria fitting degree of covering intervalQTj i :<sup>r</sup> QT* (*QTij i* ) =*ri <sup>C</sup>∙rj i*

*(1) Selection criteria fitting degree of combination of covering subintervalsQTi* 1 1 *x* … *xQTi k k :*

$$r^{QT}\left(\mathbf{Q}T^1\_{i\_1}\mathbf{x}\dots\mathbf{x}\mathbf{Q}T^k\_{i\_k}\right) = \frac{1}{k}\sum\_{j=1}^k r^{QT}\left(\mathbf{Q}T^j\_{i\_j}\right), i\_j \in \{1, \dots, k\_1\}, l \in \{1, \dots, k\}, i$$

*(2') Fitting degree of combination of options for the ith selection criteria dimension:*

$$\boldsymbol{r}\_{i}^{S}\left(\boldsymbol{o}\_{1},\ldots,\boldsymbol{o}\_{n}\right) = \begin{cases} \boldsymbol{r}\_{i}^{c} \cdot \left[\boldsymbol{r}\_{j}^{i} + \frac{\boldsymbol{m}\_{i}^{j} - \boldsymbol{q}\_{i}^{\text{agg}}}{\boldsymbol{m}\_{i}^{j} - \boldsymbol{m}\_{i}^{f^{i-1}}} \left(\boldsymbol{r}\_{j}^{i} - \boldsymbol{r}\_{j-1}^{i}\right)\right], & \boldsymbol{q}\_{i}^{\text{agg}} \le \boldsymbol{m}\_{i}^{f}, \\\ \boldsymbol{r}\_{i}^{c} \cdot \left[\boldsymbol{r}\_{j}^{i} + \frac{\boldsymbol{q}\_{i}^{\text{agg}} - \boldsymbol{m}\_{i}^{f}}{\boldsymbol{m}\_{i}^{f^{i+1}} - \boldsymbol{m}\_{i}^{f}} \left(\boldsymbol{r}\_{j+1}^{i} - \boldsymbol{r}\_{j}^{i}\right)\right], & \boldsymbol{q}\_{i}^{\text{agg}} \ge \boldsymbol{m}\_{i}^{f}, \end{cases}$$

*whereqi agg* <sup>∈</sup>*QTj i is the aggregated value of the ith selection criteria dimension,mi j* = *ai j* <sup>2</sup> <sup>+</sup> *bi j* <sup>2</sup> *‐middle of the jth covering subintervalQTj i* , *r*<sup>0</sup> *i* =*r*<sup>1</sup> *<sup>i</sup>* <sup>+</sup> *<sup>r</sup>*<sup>2</sup> *<sup>i</sup>* <sup>−</sup> *<sup>r</sup>*<sup>1</sup> *i m*2 *<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*<sup>1</sup> *<sup>i</sup>* (*a*<sup>1</sup> *i* −*m*<sup>1</sup> *i* )*and rki* +1 *<sup>i</sup>* <sup>=</sup>*rk <sup>i</sup>* <sup>+</sup> *rk* <sup>−</sup><sup>1</sup> *<sup>i</sup>* <sup>−</sup> *rl i mk* <sup>−</sup><sup>1</sup> *<sup>i</sup>* <sup>−</sup> *mk <sup>i</sup>* (*bk i* −*mk i* ).

*(2) Fitting degree of combination of options: if the overall selection criteria values Qagg* =*q*<sup>1</sup> *agg*, …, *qk agg* ∈*R <sup>k</sup> of the combination of optionso*1, …, *onbelongs to the combination of covering subintervalsQTi* 1 1 *x* … *xQTi k k , then its selection criteria fitting degree is measured by: <sup>r</sup> <sup>S</sup>* (*o*1, …, *on*) <sup>=</sup> <sup>1</sup> *<sup>k</sup>* ∑ *i*=1 *k ri <sup>S</sup>* (*o*1, …, *on*).

For better clarity, measurements are at first defined for one selection criteria dimension (*1'* and *2'*), and then generalized for *k* selection criteria dimensions (*1* and *2*). It is succeeded by considering all interested selection criteria dimensions in the weighted sum in the func‐ tions *rQT()* and *rS()* under the hypothesis that the aggregated values for quality dimensions can be evaluated as the average of the corresponding quality dimensions of component options [18, 19].

The main aim of these two measurements are to (i) measure **stakeholders' interests over selection criteria** (as quantified with measure *rQT* measure), and (ii) measure **how the selected combination of options fulfill defined preferences** (as quantified with measure *rS* measure).

#### **4.2. Characteristics of the two‐layered model for selection‐making criteria**

Created two‐layered structure with both measurements represents an integral **selection‐ making model,** with the following characteristics induced by the basic characteristics of its integrated components:


not required to be monotonically increasing or decreasing, and, thus, observed ranks do not correspond to any monotonic function;

**iv.** The ranks obtained by CS‐AHP algorithm are also assigned the middle values of covering subintervals with uniform distributions to ranks of the first neighbors. Thus, higher values of selection criteria fitting degree measurement for combination of options *rS()* correspond to the **combination of options that are better suited for the stakeholders' preferences with respect to selection criteria** [20];

( )

*S i i <sup>i</sup> agg j*

202 Applications and Theory of Analytic Hierarchy Process - Decision Making for Strategic Decisions

*m*2 *<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*<sup>1</sup> *<sup>i</sup>* (*a*<sup>1</sup> *i* −*m*<sup>1</sup> *i* )*and rki* +1 *<sup>i</sup>* <sup>=</sup>*rk*

*<sup>n</sup> o o*

, ,

*i* , *r*<sup>0</sup> *i* =*r*<sup>1</sup> *<sup>i</sup>* <sup>+</sup> *<sup>r</sup>*<sup>2</sup> *<sup>i</sup>* <sup>−</sup> *<sup>r</sup>*<sup>1</sup> *i*

*<sup>S</sup>* (*o*1, …, *on*).

1

*whereqi*

*Qagg* =*q*<sup>1</sup>

*subintervalsQTi*

*<sup>r</sup> <sup>S</sup>* (*o*1, …, *on*) <sup>=</sup> <sup>1</sup>

options [18, 19].

integrated components:

*agg* <sup>∈</sup>*QTj i*

*jth covering subintervalQTj*

*agg*, …, *qk*

1 1

*<sup>k</sup>* ∑ *i*=1

*k ri*

*agg* ∈*R <sup>k</sup>*

*x* … *xQTi k k* ( )

,

,

*<sup>i</sup>* <sup>+</sup> *rk* <sup>−</sup><sup>1</sup> *<sup>i</sup>* <sup>−</sup> *rl i*

*mk* <sup>−</sup><sup>1</sup> *<sup>i</sup>* <sup>−</sup> *mk <sup>i</sup>* (*bk i* −*mk i* ).

*j* = *ai j* <sup>2</sup> <sup>+</sup> *bi j*

<sup>2</sup> *‐middle of the*

1 1


*C i i i i i agg j i j jj i i j j*

*C i i i i i agg j i j jj i i j j*

*q m rr r r q m m m*

( )

*of the combination of optionso*1, …, *onbelongs to the combination of covering*

*, then its selection criteria fitting degree is measured by:*

1 1

+ +

<sup>ì</sup> é ù - <sup>ï</sup> ×+ - £ ê ú ï ë - <sup>û</sup> ¼ = <sup>í</sup>

*m q r r rr q m m m <sup>r</sup>*

*is the aggregated value of the ith selection criteria dimension,mi*

*(2) Fitting degree of combination of options: if the overall selection criteria values*

For better clarity, measurements are at first defined for one selection criteria dimension (*1'* and *2'*), and then generalized for *k* selection criteria dimensions (*1* and *2*). It is succeeded by considering all interested selection criteria dimensions in the weighted sum in the func‐ tions *rQT()* and *rS()* under the hypothesis that the aggregated values for quality dimensions can be evaluated as the average of the corresponding quality dimensions of component

The main aim of these two measurements are to (i) measure **stakeholders' interests over selection criteria** (as quantified with measure *rQT* measure), and (ii) measure **how the selected combination of options fulfill defined preferences** (as quantified with measure *rS* measure).

Created two‐layered structure with both measurements represents an integral **selection‐ making model,** with the following characteristics induced by the basic characteristics of its

**i.** Fitting degree measurements *rQT()* and *rS()* are unit measurements, i.e.,

**ii.** The ranks obtained by CS‐AHP algorithm are at first assigned to the covering

**iii.** Quality measure *rQT()*, in general case, does not correspond to monotonic tendency

0≤*r QT* (), *r <sup>S</sup>* ()≤1, according to the basic characteristics of the CS‐AHP algorithm [2];

subintervals for creation of selection criteria fitting degree measurement *rQT()*, and thus its higher values correspond to the **more preferable combination of covering**

of selection criteria dimensions (i.e., increasing and decreasing) introduced in reference [18]. The CS‐AHP framework takes as input the set of stakeholders' preferences about the relative importance between covering subintervals which are

**4.2. Characteristics of the two‐layered model for selection‐making criteria**

**subintervals according to stakeholders' preferences**;

*j agg*

ï é ù ï ×+ - ³ ê ú ë û - î

*i i*


The characteristics of introduced measurements reflect uncertainty and conditionality in user preferences (which, to the best of our knowledge, mostly correspond to realistic scenarios [2]), and thus the *OptSelectionAHP* approach is developed in a manner which uses identified char‐ acteristics for faster convergence in the heuristic approach for optimal selection problem.

**Figure 3.** Schematic representation of *rQT* and *rs* measures for one selection criteria dimension.

#### **4.3. Optimal selection goals**

Finally, in the proposed selection criteria model, the final optimal selection goal is defined as the determination of the most preferable combination of available options based on both measurements representing stakeholders' requirements and preferences. Thus:

**Definition 2** (**optimal selection goal**). *Given a set of user demands*{*di* }*i*=1,…,*<sup>n</sup> with interdependencies defined with q logical statements, and associated with available set of options* {*oij* }*i*=1…*n*, *<sup>j</sup>*=1…*mi , where mi is the number of available options for ith demand, and CS‐AHP selection criteria model (C, Qagg LB*, *Qagg UB*, *QT, P), where C is a set of concerns, QT is a set of qualifier tags over aggregated intervals Qagg LB*, *Qagg UB for k selection‐making criteria, and P represents the set of specified preferences; the* **optimal selection goal** *is to find a valid combination of options which maximizes the overall selection criteria fitting degree <sup>r</sup> <sup>S</sup>* (*o*1, …, *on*), *subject to its affiliation to the most preferable combination of selection criteria and the hard constraints satisfaction.*

It is necessary to notice that in comparison to standard optimization goals widely used in the literature [21–23], defined as *maximization of the overall aggregated values*, definition 2 addition‐ ally requires affiliation to the most preferable combination of selection criteria.

In the following, we propose a meta‐heuristic search approach that overcomes the aforemen‐ tioned complexities.
