IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 28 Vol. 3 Issue 1 2011 ISSN 1936-6744 DECISION-MAKING WITH MODELING OF PROBLEM SITU- ATIONS USING THE ANALYTIC NETWORK HIERARCHY PROCESS Tatiana Kravchenko State University – Higher School of Economics Moscow, Russia E-mail: tkravchenko@hse.ru Natalia Seredenko State University – Higher School of Economics Moscow, Russia E-mail: alia_nata@mail.ru ABSTRACT The Modeling of problem situations is a very important issue in decision-making the- ory. Actually, there are no decision support systems which include decision making methods under risk and uncertainty. The main advantage of a proposed approach is the ability to process dependences and feedbacks which may exist between condi- tions, sub-conditions and their realizations. Keywords: modeling of economic decision-making problem situations, analytic hier- archy process (AHP), analytic network process (ANP), relevant significance esti- mates of problem situations. 1. Introduction The decision maker (DM) has to consider interrelation of his particular case with oth- er areas of concern during the decision process. Thereto DM needs a comprehensive model of an external environment, which is called a scenario. In practice the DM constructs a more limited model focused on his specific problem – the local scenario (Figure 1). Figure 1 Model of the local scenario Rob Typewritten Text http://dx.doi.org/10.13033/ijahp.v3i1.81 IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 29 Vol. 3 Issue 1 2011 ISSN 1936-6744 When DM doesn’t model even a local scenario, then, in fact, the decision is accom- plished in “ideal conditions” without considering an external environment. Quite often the local scenario is represented in the form of variant problem situations, which DM should take into account. In this case the choosing of an efficient alterna- tive becomes more complicated because experts should evaluate alternatives consid- ering all possible problem situations and DM should compare all the estimates ob- tained. Nevertheless, only the modeling of alternative problem situations can increase the efficiency of decision-making. There are two approaches which are mainly used for the modeling of problem situa- tions: simulation modeling and expert forecasting. It is necessary to assign a set of conditions and dependences between them to deal with a simulation model. Then we may model a problem situation by combining dif- ferent results of implementation. If it is difficult to specify all dependences in the formal way, the modeling of problem situations are often based on expert forecast. There are questions which have to be solved for successful forecast: • Organizational support of forecasting; • Stating demands for forecasting; • Organizing an analytical work group; • Setting up an expert committee; and • Preparing all needed background materials, software and data. Precise expert forecast can be stated only if well prepared and competent experts en- gaged. Information source must be reliable and evaluations must be correctly col- lected and processed. Only experienced specialists are invited to join the expert committee. The main advantage of problem situations modeling approach is ability to process dependences and feedbacks which may exist between conditions, sub-conditions and their realizations (most probable results of checking sub-conditions). The methodological basis of proposed approach to the modeling of problem situations and evaluating judgments are the Analytical hierarchy process (AHP) and Analytical network process (ANP) (Saaty, 1993; Saaty, 2003; Saaty, 2008). 2. Theoretical background of the decision-making conditions model- ing We use AHP and ANP for building hierarchy of conditions, which are decomposed into sub-conditions and realizations (probable results of checking sub-conditions). All these items are considered in the problem to be solved. There is a main decision goal “to find priorities of conditions, sub-conditions and their realizations” on the top of the hierarchy structure. The number of levels of con- ditions depends on the statement of the problem. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 30 Vol. 3 Issue 1 2011 ISSN 1936-6744 We assume that the main goal and levels of conditions, which are located under the main goal, are composed in the Control Hierarchy of Conditions. Figure 2 Examples of the sub-conditions networks We use Analytical hierarchy process for modeling sub-conditions and their realiza- tions. Under each condition of the control hierarchy the sub-conditions network should be modeled. Type of networks under the control hierarchy depends on the structural complexity of the problem. Figure 2 shows possible variants of the sub- conditions networks. To process the control hierarchy of conditions we may use the Analytical hierarchy process (Saaty, 1993). To get results (to calculate the relevant significance estimates of sub-conditions and their realizations) we may use the Analytical network process (Saaty, 2008). 3. Calculating the relevant significance estimates of decision-making conditions, sub-conditions and their realizations Control Hierarchy of Conditions GOAL(to find priorities of conditions, sub-conditions and their realizations) Conditions of decision-making Examples of the sub-conditions networks under the control hierarchy of conditions : - Sub-conditions composed of their realizations IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 31 Vol. 3 Issue 1 2011 ISSN 1936-6744 3.1 Calculating the relevant significance estimates of decision-making conditions Notation • MmUUUU Mm ...1),...,,...,,( 1 == — decision-making conditions; • Pmn — elements of the pairwise comparison matrix for conditions, formed by DM, ;...1, Mnm = • )...,,( .1.. MeigeigMeig ppp = — principal eigenvector of the pairwise com- parison matrix for conditions; • Mmaxλ — principal eigenvalue of the pairwise comparison matrix for condi- tions; and • )...,,( 1 MM ppP = — vector of the relevant significance estimates (weights) of conditions. Input data • mU — decision-making conditions, Mm ...1= ; • mnP — elements of the pairwise comparison matrix for conditions, ....1 Mm = All ratios should be estimates as numbers using the Fundamental Scale of the AHP, consisting of: {1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, 1/2, 1, 2, 3, 4, 5, 6, 7, 8, 9}. There is a correlation between evaluations mnP and nmP : .1 mnnm PP = It means that if ,7=nmP then the relevant significance estimates of m- condition is very strong do- minate n- condition. Note that in this case nmP = 1/7, so it means that the relevant significance estimates of n- condition is very strongly dominated by m- condition. Solution algorithm 1. DM forms input data. 2. DM fills up the pairwise comparison matrix with elements ,mnP where ,...1, Mnm = to evaluate the relevant significance estimates of m- and n- condi- tions. 3. Then the principal eigenvector of the pairwise comparison matrix for conditions Meigp . has to be obtained (Saaty, 2003). There is a general equation for obtaining the principal eigenvector (in compliance with the principal eigenvector definition): (1) MeigMMeigmn ppP .max. ×=× λ 4. The elements of obtained eigenvector have to be normalized: (2) ∑ = m meig meig m p p p . . 5. The vector )...,,( 1 MM ppP = is the required vector relevant significance estimates of conditions, which will be used further for calculating the relevant significance es- timates of sub-conditions and their realizations. The example of forming and processing the control hierarchy of conditions is de- scribed in chapter 5.2. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 32 Vol. 3 Issue 1 2011 ISSN 1936-6744 Table 1 The Fundamental Scale of absolute numbers Rating Interpretation of the evaluations 1 relevant significance estimate is equal importance 2 relevant significance estimate is weak 3 relevant significance estimate is moderate importance 4 relevant significance estimate is moderate plus 5 relevant significance estimate is strong importance 6 relevant significance estimate is strong plus 7 relevant significance estimate is demonstrated importance 8 relevant significance estimate is very, very strong 9 relevant significance estimate is extreme importance 3.2 Calculating relevant significance estimates of decision-making sub-conditions and their realizations After processing the control hierarchy of conditions, DM should form and make cal- culations over all sub-networks, presenting the sub-conditions structure for each con- dition. As stated above, the structure of sub-networks under the control hierarchy of condi- tions depends on the decision-making problem. One of the possible views of decision sub-conditions and their realizations sub-networks is shown in Figure 3. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 33 Vol. 3 Issue 1 2011 ISSN 1936-6744 Sub-goal To find priorities of sub- conditions and their realizations Sub-condition Um1 Realization Um11 Realization Um12 Realization Um1k Realization Um1Ky . . . . . . . . . . . . . . . . . . Sub-condition Um2 Realization Um21 Realization Um22 Realization Um2k Realization Um2Ky . . . . . . . . . . . . . . . . . . Sub-condition UmYm Realization UmYm1 Realization UmYm2 Realization UmYmk Realization UmYmKy . . . . . . . . . . . . . . . . . . ... Figure 3 Example of the sub-networks under the control hierarchy of conditions Such sub-network structures can have the following dependences between the sub- goal, sub-conditions and their realizations: • Connections between the sub-goal and sub-conditions, influencing the sub- goal. This type of outer dependence is shown by the solid arrows outgoing the sub-goal in Figure 3 (dependences of type I). • Connections between the sub-conditions and their realizations: • Connections between two different sub-conditions and their realizations. This type of outer dependence is shown in Figure 3 by the dotted arrows, drawn from one sub-condition to another sub-condition whose elements influence it (dependences of type II.a); • Interconnections between two different sub-conditions and their realiza- tions. This type of outer dependence is shown in Figure 3 by the two dot- ted multidirectional arrows between two different sub-conditions (de- pendences of type II.b). • Connections between realizations within some sub-condition. This type of in- ner dependence is shown in Figure 3 by the dotted arrows, drawn from one sub-condition to the same sub-condition (dependences of type III). As stated above, it is applicable to use ANP for processing such types of sub- networks. In this paper we give the interpretation of ANP for modeling sub- conditions and their realizations. Notation IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 34 Vol. 3 Issue 1 2011 ISSN 1936-6744 Dependences of type I • myU — decision-making sub-conditions of m- condition, where ,...1 mYy = for all Mm ...1= ; • mykU — k- realizations, formed within y- sub-condition, within m- condition, where ,...1 yKk = ,...1 mYy = ....1 Mm = • I tymP ),( — elements of the pairwise comparison matrix for y- and t- sub- conditions influencing m- condition. These matrixes reflect the dependences of type I. The number of such type of ma- trixes equals to the number of conditions, which are considered in the decision- making problem. The general view of the pairwise comparison matrix of type I is il- lustrated in Table 2. Table 2 General view of the pairwise comparison matrix for the sub-conditions of m- condi- tion 1mU … mtU … mmYU 1mU … … … myU … … I tymP ),( … mmY U • I meigp . — principal eigenvector of the pairwise comparison matrix for Umy sub-conditions influencing m- condition; ....1 Mm = • I mmaxλ — principal eigenvalue of the pairwise comparison matrix for sub- conditions influencing m- condition; ....1 Mm = • I myP — vector of the relevant significance estimates (weights) of sub- conditions influencing m- condition; ,...1 mYy = ....1 Mm = Dependences of type IIa • t — sub-conditions of m-condition, influencing on the other sub-conditions; • y — sub-conditions of m-condition, being influenced by the other sub- conditions; • mykU — k- realizations of y- sub-condition (of m-condition), being influ- enced by the other sub-conditions, yKk ...1= ; • mthU — h- realization of t- sub-condition (of m-condition), influencing the realizations of other sub-conditions, tKh ...1= ; and IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 35 Vol. 3 Issue 1 2011 ISSN 1936-6744 • IIa UUU mtlmthmyk P ),( — elements of the pairwise comparison matrix for h- and l- realizations of t- sub-condition of m- condition, influencing k- realizations of y- sub-condition. Note that in compliance with the notation conventions, as shown in Figure 3, y- sub- conditions are the sub-conditions from which the dotted arrows come and t- sub- conditions are the sub-conditions in which the dotted arrows are. Each generated dependence (dotted arrow) would be defined by such number of pairwise comparison matrixes for h- and l- realizations of t- sub-conditions, what number of k- realizations is contained in y- sub-condition, being influenced by t- sub- condition. The general view of the pairwise comparison matrixes of type IIa is illus- trated in Table 3. Table 3 General view of the pairwise comparison matrix for the realizations of t- sub- condition, influencing the realizations of y- sub-condition 1mtU … mtlU … tmtKU 1mtU … … … mthU … … IIa UUU mthmyk P ,( … tmtK U • pIIаeig.m(y,t) — principal eigenvectors of the pairwise comparison matrixes in which the realizations of t- sub-condition, influencing the realizations of y- sub-conditions, are estimated; • λIIаmax.m(y,t) — principal eigenvalues of the pairwise comparison matrixes; and • PIIаm(y,t) — vectors of the relevant significance estimates (weights) of deci- sion-making realizations. Dependences of type IIb In case of interdependence between two different sub-conditions and their realiza- tions (dependences type IIb, as shown in Figure 3 by the two dotted multidirectional arrows), besides matrixes of type IIa, additional matrixes should be completed. IIb UUU mylmykmth P ),( — elements of the pairwise comparison matrix for k- and l- re- alizations of y- sub-condition of m- condition, being influenced by the h- realizations of t- sub-condition. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 36 Vol. 3 Issue 1 2011 ISSN 1936-6744 Note that in compliance with the notation conventions, as shown in Figure 3, t- sub- conditions are the sub-conditions, from which the dotted arrows come and y- sub- conditions are the sub-conditions in which the dotted arrows are. Each generated dependence (dotted arrow) would be defined by such number of pairwise comparison matrixes for k- and l- realizations of y- sub-conditions, what number of h- realizations is contained in t- sub-condition, being influenced by y- sub- condition. The general view of the pairwise comparison matrixes of type IIb is illus- trated in Table 4. Table 4 General view of the pairwise comparison matrix for the realizations of t- sub- condition, being influenced by the realizations of y- sub-condition 1myU … mylU … ymyKU 1myU … … … mykU … … ( mUmthU IIbP … ymyK U Thus, Tables 3 and 4 show the type II dependences. • pIIbeig.m(t,y) — principal eigenvectors of the pairwise comparison matrixes, in which the realizations of t- sub-condition, being influenced by the realizations of y- sub-conditions of m- condition, are estimated; • λIIbmax.m(t,y) — principal eigenvalues of the pairwise comparison matrixes; and • PIIbm(t,y) — vector of the relevant significance estimates (weights) of decision- making realizations. Dependences of type III • t — sub-conditions of m-condition with internal dependences between reali- zations; • Umtk — k- realizations of t- sub-condition of m- condition, being influenced by the other realizations of the same sub-condition, where tKk ...1= ; • Umth , Umtl – h- and l- realizations of t- sub-condition of m- condition, influ- encing the other realizations of the same sub-condition, where tKk ...1= ; and • ),( mtlmthmtk UUU IIIP – elements of the pairwise comparison matrix for h- and l- realizations of t- sub-condition of m- condition, influencing k- realizations of the same t- sub-condition. Note that each generated dependence (shown as dotted arrow drawn from one sub- condition to the same sub-condition) would be defined by such number of pairwise comparison matrixes for k- and l- realizations, what number of realizations is con- tained in t- sub-condition. The general view of the pairwise comparison matrixes of type III is illustrated in Table 5. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 37 Vol. 3 Issue 1 2011 ISSN 1936-6744 Table 5 General view of the pairwise comparison matrix for the realizations of sub-condition with internal dependences and feedback III mtU 1 … III mtlU … III mtK t U III mtU 1 … … … III mthU … … III UU mthmtk P ( … III mtK t U • III Ueig mt P . — principal eigenvectors of the pairwise comparison matrixes for the realizations of each t- sub-condition of m- condition; • III U mtmax λ — principal eigenvalues of the pairwise comparison matrixes for the realizations of each t- sub-condition of m- condition; • III U mt P — vector of the relevant significance estimates (weights) of decision- making realizations of sub-conditions with internal dependences of m- condi- tion; • PmSuperMatr – weighted supermatrix composed of the eigenvectors derived from the pairwise comparison matrixes for the realizations of each sub-condition of m- condition with elements: IIa UUU mtlmthmyk P ),( , IIb UUU mylmykmth P ),( , ),( mtlmthmtk UUU IIIP ; • limmSuperMatrP — weighted supermatrix being raised to powers (the limit superma- trix); and • mykP — required vectors of the global relevant significance estimates (weights) of decision-making realizations. Input data • myU — decision-making sub-conditions of m- condition, where ,...1 Mm = mYy ...1= ; • mykU — k- realizations of t- sub-condition of m- condition; and • PIm(y,t) , IIa UUU mtlmthmyk P ),( , IIb UUU mylmykmth P ),( , ),( mtlmthmtk UUU IIIP — elements of the pairwise comparison matrixes formed by DM. Solution algorithm 1. DM forms input data. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 38 Vol. 3 Issue 1 2011 ISSN 1936-6744 2. DM fills up the pairwise comparison matrixes with elements I tymP ),( to evaluate the relevant significance estimates of sub-conditions. All ratios should be estimates as numbers using the Fundamental Scale of the AHP. Interpretation of the evaluations is illustrated in Table 1. 3. Then the principal eigenvector of the pairwise comparison matrix for sub- conditions I meigp . has to be obtained (Saaty, 2003). There is a general equation for obtaining the principal eigenvector (in compliance with the principal eigenvector de- finition): (3) ..max.),( I meig I m I meig I tym ppP ×=× λ 4. The elements of obtained eigenvector have to be normalized: (4) . 1 . . ∑ = = mY y I myeig I myeig my I p p p 5. DM forms the pairwise comparison matrixes with elements ),( mtlUmthUmykU IIaP , ),( mylUmykUmthU IIbP and ),( mtlmthmtk UUU IIIP . Filling the matrixes, the DM has to answer the basic question in all pairwise comparisons: «How many times more dominant is one element than the other with respect to a certain element (sub-condition or condition)? ». All ratios should be estimates as numbers using the Fundamental Scale of the AHP. This allows to take into account the mutual influences and inner dependences of sub- conditions. Interpretation of the evaluations is illustrated in Table 1. 6. Then the principal eigenvectors of the pairwise comparison matrixes for realiza- tions IIa tymp ),(eig. , IIb ytmp ),(eig. and III mtpeig. have to be obtained (Saaty, 2003). There are general equations for obtaining the principal eigenvectors: (5.1) ),(.),(max),(.),( ** tymeigIIatymIIatymeigIIamtlUmthUmykU IIa ppP λ= ; (5.2) ),(.),(max),(.),( ** ytmeigIIbytmIIbytmeigIIbmylUmykUmthU IIb ppP λ= ; (6) mteig III mt III mteig III UUU III ppP mtlmthmtk .max.),( ** λ= . 7. The elements of obtained eigenvectors have to be normalized: (7.1) ∑ −= = tKk ktymeig IIa ktymeig IIa ktym IIa p p p 1 ),(. ),(. ),( ; (7.2) ∑ −= = tKk kytmeig IIb kytmeig IIb kytm IIb p p p 1 ),(. ),(. ),( ; (8) ∑ −= = tKk mtkeig III mtkeig III mtk III p p p 1 . . . 8. All derived vectors of the relevant significance estimates (weights) of decision- making realizations and sub-conditions (pt. 4 and pt. 7) should be placed in the col- IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 39 Vol. 3 Issue 1 2011 ISSN 1936-6744 umns of the supermatrix PmSuperMatr (Table 6). The blocks of the supermatrix are to be filled according to the following rules: • The vectors PIm presenting dependences of type I are used to weigh the ele- ments of the corresponding blocks of the supermatrix. • The vectors PIIam(y,t), PIIbm(t,y) presenting dependences of type IIa and type IIb are placed within the outside of the main diagonal blocks according to the fol- lowing rules: • In the case of outer dependence between two different sub-conditions and their realizations (dependence of type IIa), only the one block of the supermatrix has to be filled. The symmetry with respect to the main diagonal block is all zero. This case is illustrated in Table 6 by blocks 2.1 and 2.2; and • In the case of interdependence between two different sub-conditions and their realizations (dependence of type IIb), both blocks symmetry with respect to the main diagonal of the supermatrix have to be filled. This case is illustrated in Table 6 by blocks 4.1 and 4.2. • The vectors IIImtP presenting dependences of type III should be placed on the main diagonal blocks. This case is illustrated in Table 6 by block 3. In case there are no inner dependences within the sub-condition then the correspond- ing block is all zero. This case is illustrated in Table 6 by block 1. 9. Then, to obtain the limit supermatrix, it is necessary to raise the supermatrix to powers until it is orderly cyclic (Saaty, 2008): (9) . 1 lim 1 lim ∑ = ∞→ = N k k SuperMatrkSuperMatr P N P 10. In compliance with ANP the required vectors of the global relevant signifi- cance estimates (weights) of decision-making realizations Pmyk may be obtained from the limit supermatrix limmSuperMatrP (Saaty, 2008). Note that all inner and outer depend- ences between sub-conditions and their realizations are taken into account. The simi- lar calculations should be implemented for each m- condition, where m=1...M. Table 6 General view of the Weighted Supermatrix Sub-conditions Realizations … Realizations … Realizations R ea liz a- tio ns (1) Zero entries (2.2) Zero entries R ea liz a- tio ns (2.1) Zero entries (3) Nonzero entries (4.2) Nonzero entries Su b- co nd iti on s R ea liz a- tio ns (4.1) Nonzero entries The example of processing the sub-conditions and their realizations network structure is described in chapter 5.3. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 40 Vol. 3 Issue 1 2011 ISSN 1936-6744 4. Modeling of decision-making problem situations In fact, the problem situation is a set of different realizations of all considered sub- conditions of all considered decision-making conditions, between which there is a logical conjunction “AND”. Besides, one problem situation should contain only one realization of each sub-condition. Because of the logical connective “and” it is rea- sonable to add the relevant significance estimates (weights) of decision-making reali- zations which make a problem situation in aggregate. Thus, the indicator of the relevant significance estimate (weight) of decision-making problem situation may be calculated as a sum of weights of all realizations which are made in aggregate this problem situation. If there is a considerable quantity of different sub-conditions and their realizations, then the quantity of decision-making problem situations will be very great. In addi- tion, after forming a set of problem situations every expert should provide all corre- sponding judgments (evaluating alternatives with respect to the criteria etc.) consider- ing all problem situations. So the labour-intensiveness of decision-making can be un- reasonably high. Therefore only six-to-eight problem situations with the highest weights are usually considered. The example of forming a set of problem situations is described in chapter 6.4. 5. The example of forming a set of problem situations and their re- levant significance estimates To illustrate the stated theoretical basis consider IT outsourcing decision making. This decision-making problem has been described by the scientific group of Youxu Tjader, Jennifer Shang, Luis Vargas and Jerry May. These researchers made a report at the international symposium ISAHP2009. In that paper (Tjader, Shang, Vargas and May, 2009) the ANP and the Balanced Scorecard (BSC) are used for IT outsourcing decision making. According to the current article, it is offered to use the BSC for revealing a set of de- cision-making conditions and their sub-conditions. Thus the concept of “perspective” is interpreted as a decision-making condition and concept of “key indicator” is inter- preted as a decision-making sub-condition. The AHP/ANP are used for the modeling of decision-making problem situations and calculating their relevant significance es- timates. 5.1 Target setting IT outsourcing decision making is a really topical problem for any large-scale enter- prise. It is necessary to analyze all circumstances around the company which may occur in the future to make an effective decision. So, as shown in Figure 4, we con- sider perspectives (financial, customer, company learning and growth, internal opera- tions) as decision-making conditions and their key indicators as decision-making sub- conditions (Tjader, Shang, Vargas and May, 2009). IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 41 Vol. 3 Issue 1 2011 ISSN 1936-6744 Financial Cash Flow Industry Leader Profitability Cost Savings Customer Availability Prod/Serv Customer Satisfaction Price stability Customer database Employee Competency Company Learning and Growth Employee Satisfaction Technology RD Management expertise Internal Operations Core Focus Quality Internal Control Agility Certificates Figure 4 Framework of perspectives and key indicators 5.2 Forming the Control Hierarchy of Conditions Figure 5 gives the Control Hierarchy of Conditions for the IT outsourcing decision making. The Control Hierarchy GOAL (to calculate the relevant significance estimates of conditions, sub-conditions and their realizations) Decision-making conditions: 1. Financial 2. Customer 4. Company Learning and Growth 3. Internal Operations Figure 5 The Control Hierarchy for the IT outsourcing decision making IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 42 Vol. 3 Issue 1 2011 ISSN 1936-6744 After forming the set of conditions it is necessary to model network sub-conditions structure for each of the conditions. 5.3 The example of the sub-conditions network structure for the Financial condition Figure 6 gives the sub-conditions network structure for the Financial condition for the IT outsourcing decision making. SUB-GOAL To calculate the relevant significance estimates of the IT outsourcing decision making sub- conditions and their realizations for the condition 1. Financial 1.1. Cash Flow ---------------------------------------- Investing in IT infrastructure will be reduced Investing in IT infrastructure will be remained unchanged Investing in IT infrastructure will be increased 1.2. Industry Leader ---------------------------------------------------- The company will become an industry leader The company will not become an industry leader 1.4. Cost Savings -------------------------------------------- Cost saving will be improved Cost saving will remain unchanged Cost saving will increase 1.3. Profitability ---------------------------------------------------- Profitability will increase Profitability will remain unchanged Profitability will increase Figure 6 The sub-conditions network structure for the Financial condition As shown in Figure 6, the Financial condition consists of the following sub- conditions: • 1.1. Cash flow. • 1.2. Industry Leader. • 1.3. Profitability. • 1.4. Cost Savings. In turn, each of the sub-conditions consists of the appropriate set of realizations. For example, the sub-condition “Cash flow” may subdivide into the following realiza- tions: • Investing in IT infrastructure will be reduced. • Investing in IT infrastructure will remain unchanged. • Investing in IT infrastructure will be increased. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 43 Vol. 3 Issue 1 2011 ISSN 1936-6744 5.4 The example of modeling the problem situations According to the rules stated above, we may form the required set of problem situa- tions for the IT outsourcing decision making. The elements in this set are as follows: • S1: {Investing in IT infrastructure will be reduced; the company will become an industry leader; profitability will increase; …}. • S2: { Investing in IT infrastructure will remain unchanged; the company will become an industry leader; profitability will increase; …}. • S3: { Investing in IT infrastructure will be increased; the company will be- come an industry leader; profitability will increase; …}. • S4: { Investing in IT infrastructure will be reduced; the company will not be- come an industry leader; profitability will increase; …}. • … By this way all possible combinations of realizations must be searched. Then all the relevant significance estimates (weights) of decision-making realizations of sub- conditions for each condition have to be calculated. As an example the relevant sig- nificance estimates of decision-making realizations of sub-conditions for the Finan- cial condition are resulted in Table 7. All calculations are made by the decision sup- port system SuperDecisions (Saaty, 2002). Table 7 The relevant significance estimates of decision-making realizations of sub-conditions for the Financial condition Realizations Estimates of sub- conditions weighted by the es- timates of conditions Local es- timates of realiza- tions (within sub- conditions) Global estimates of realiza- tions 1.1. Cash flow: 0,04 Investing in IT infrastructure will be reduced 0,6 0,024 Investing in IT infrastructure will re- main unchanged 0,2 0,008 Investing in IT infrastructure will be increased 0,2 0,008 1.2. Industry leader: 0,09 The company will become an industry leader 0,3 0,027 The company will not become an indus- try leader 0,7 0,063 1.3. Profitability: 0,19 Profitability will increase 0,5 0,095 Profitability will remain unchanged 0,4 0,076 Profitability will increase 0,1 0,019 … … … … IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 44 Vol. 3 Issue 1 2011 ISSN 1936-6744 After calculating such values for every condition, the relevant significant estimates of problem situations may be obtained. As stated above, these coefficients may be calcu- lated as a sum of the relevant significant estimates of realizations of which the prob- lem situations are composed. We may cite as an example the problem situation 1S : investing in IT infrastructure will be reduced; the company will become an industry leader; profitability will in- crease; cost saving will be improved; availability of product will decrease; customer satisfaction will decrease; price stability will remain unchanged; customer database will be reduced by 10 percent; employee satisfaction will remain unchanged; technol- ogy RD will be developed; employee competency will remain unchanged; manage- ment knowhow will increase; certifications will remain unchanged; core focus will remain unchanged; quality will increase; internal control will be strengthened; agility will reduce. The relevant significant estimate of 1S P may be calculated this way: (10) 0,742 0,0140,016 0,008 0,022 0,087 0,014 0,012 0,009 0,035 0,016 0,055 0,033 0,077 0,054 0,095 0,027 0,024 P 1S =+++++++++ ++++++++= In this sum (10) the first four terms represent the relevant significant estimates of re- alizations of sub-conditions for the Financial condition, the next four are the esti- mates of realizations of sub-conditions for the condition Customer, the next five are the estimates of realizations of sub-conditions for the condition Internal Operations and the last four are the estimates of realizations of sub-conditions for the condition Company Learning and Growth. By the same way the relevant significant estimates of all the other problem situations should be calculated. Then the problem situations have to be sorted by the coeffi- cients and the most significant of them are to be included in the task of the IT out- sourcing decision making. 6. Conclusion This paper provides an approach to the modeling of economic decision-making prob- lem situations using the Analytical hierarchy process and the Analytical network process. The calculation algorithm of the relevant significant estimates of decision-making conditions, sub-conditions and their realizations is described. The decision support system SuperDecisions has been used for the auxiliary calculations. The development of an algorithm and software for modeling of problem situations and finding among them the most significant are the directions for future researches. REFERENCES Saaty, T.L. (1993). Принятие решений — Метод Анализа Иерархий. М.: Радио и Связь. [in Russian]. IJAHP ARTICLE: Kravchenko, Seredenko/Decision-Making With Modeling of Prob- lem Situations Using the Analytic Network Hierarchy Process International Journal of the Analytic Hierarchy Process 45 Vol. 3 Issue 1 2011 ISSN 1936-6744 Saaty, T.L. (2008). Принятие решений при зависимостях и обратных связях. Аналитические сети. М.: Изд-во ЛКИ. [in Russian]. Saaty, T.L. (2003). Decision-making with the AHP: Why is the principal eigenvector necessary. European Journal of Operational Research. 145 (1), 85-91. Youxu Tjader, Jennifer Shang, Luis Vargas, Jerry May. (2009). Integrating the Ana- lytic network process and the balanced scorecard for strategic IT outsourcing decision making. The Joseph M. Katz Graduate School of Business University of Pittsburgh. Saaty, R.W. (2002). Decision Making in Complex Environments: The Analytic Net- work Process (ANP) for Dependence and Feedback; A Manual for the ANP Software SuperDecisions. Creative Decisions Foundation, 4922 Ellsworth Avenue, Pittsburgh, PA 15213.