Risk reducing actions: efficiency evaluation

Transcription

1 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva Ri reducing action: efficiency evaluation JOE PEREIR JR.,2,*, OUG BRBO,4, PTRICI BERNRE, TIGO BRUZEI 4,5, PETR EE,2, TIGO RIV,4 Graduate Program in Electrical Engineering Pontifical Catholic Univerity of Mina Gerai v. om Joe Gaar, 500, , Belo orizonte, MG BRZI * joelgj@gmail.com 2 dvanced ytem Otimization Technologie tda Rua om ucio ntune, 855/02, , Belo orizonte, MG BRZI Graduate Program in ditration Pontifical Catholic Univerity of Mina Gerai v. om Joe Gaar, 500, , Belo orizonte, MG BRZI 4 Mina Gerai tate Energy Comany v. Barbacena, 200, , Belo orizonte, MG BRZI 5 Graduate Program in ditration UMEC Univerity Rua Gonçalve ia,, , Belo orizonte, MG BRZI btract: - Thi aer reflect the reult of reearch related to evaluating oible conequence of action directed at the enterrie ri reduction. Thee reult are baed on alying the generalization of the claic aroach to deciion maing in condition of uncertainty to multicriteria roblem and ermit one to rovide a rational way to chooe robut action with the reence of multile cenario. The choice of the robut action are aociated with the ue of o-called choice criteria of ald, alace, avage, and urwicz. Conidering thee choice criteria a objective function ermit one to aly the Bellman-Zadeh aroach to deciion maing in a fuzzy environment to chooe a rational olution alternative. The aroach reented in thi aer wa alied in a cae tudy to imrove indicator of an enterrie liquidity ri. The reult of the aer are of a general character and can be ued to analyze variou tye of enterrie ri by conidering divere indicator. ey-ord: - Ri reduction, Robut olution, Multicriteria deciion maing, Information uncertainty, Multile cenario, Payoff matrice. Introduction The reent aer reent the reult of reearch related to evaluating oible conequence of realizing different action directed at the enterrie ri reduction with the goal to chooe the mot robut action. e ea about the neceity of evaluating of articular monocriteria conequence and aggregated multicriteria conequence. Thi neceity i aociated with the following conideration. it i indicated in [], to decribe or to meaure the ri or, in fact, to judge it a big or a mall, it i oible to ue everal metric. examle, the following characteritic can be indicated []:. The combination of robability and magnitude/everity of conequence. 2. The trilet t,, c, where t i the th cenario, i the robability of that cenario, and c i the conequence of the th cenario, =,...,.. The trilet C, Q,,, where C i ome ecified conequence, Q i a meaure of uncertainty aociated with C tyically robability and the bacground nowledge that uort C and Q which include a judgment of the trength of thi nowledge. E-IN: Volume, 206

2 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva Thu, the ri i characterized by more than one indicator criterion. Taing thi into account, the reent aer reflect reult aociated with evaluating the conequence of realizing different action directed at reducing the ri on the bai of generalizing the claical aroach to deciion maing in condition of uncertainty [2-4], taing into account the uggetion of [5]. The reult of [2-5] are aociated with the analyi of articular and aggregated ayoff matrice. 2 Claic aroach to dealing with the uncertainty of information The claic aroach [6-8] to dealing with the uncertainty of information i baed on the aumtion that the analyi i carried out for a given number of olution alternative, =,..., and a given number of rereentative combination of initial data the tate of nature or cenario Y, =,...,, which define the correonding ayoff matrix, reented in Table. The ayoff matrix reflect effect or conequence of one or other action, =,..., to the correonding tate of nature. The analyi of ayoff matrice and choice of the rational olution alternative are aociated with the ue of o-called choice criteria. In thi wor, we dicu the ue of the choice criteria of ald, alace, avage, and urwicz, whoe alication i jutified in [6-8]. Thee criteria are baed on alying the following characteritic etimate for the given olution alternative Table 2: - the imum objective function indicator, criterion, etc. level =, Y which i the mot otimitic etimate if the objective function i to be imized or the mot eimitic etimate if the objective function i to be imized; - the imum objective function indicator, criterion, etc. level =, Y 2 which i the mot otimitic etimate for the imized objective function or the mot eimitic etimate if the objective function i to be imized; - the average objective function indicator, criterion, etc. level =, Y = - the imum regret level: =, Y 4 where, Y i an over-exenditure which tae lace under combination of the tate of nature and the choice of the olution alternative intead of the olution alternative that i locally otimal for the given Y. To detere the regret, Y, one need to define the imum value of the objective function indicator, criterion, etc. if it i to be imized a in Table for each combination of the tate of nature Y for each column of the ayoff matrix: Y Y =, Y. 5 On the other hand, if the objective function indicator, criterion, etc. i to be imized, it i neceary to define it imum value for each combination of the tate of nature Y for each column of the ayoff matrix: Y =, Y. 6 The regret for any olution alternative any tate of nature Y can be evaluated a and, Y =, Y Y 7 if the objective function indicator, criterion, etc. i to be imized or, Y =, Y Y 8 if it i to be imized. Table. Payoff matrix... Y... Y 2, Y..., Y..., Y..., Y..., Y..., Y..., Y..., Y..., Y Y Y Y Y E-IN: Volume, 206

3 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva Table 2. Matrix with characteritic etimate The choice criteria, which are baed on the ue of the characteritic etimate are rereented below under the aumtion that the objective function i to be imized. The choice criterion of ald ue the etimate and ermit one to chooe the olution alternative imum:, for which thi etimate i Y =, Y. 9 The choice criterion of alace ue the etimate and i oriented to chooe the olution alternative, for which thi etimate i imum:, Y = 0 The choice criterion of avage i aociated with the ue of etimate and allow one to chooe the olution alternative etimate i imum:, for which thi =, Y. inally, the choice criterion of urwicz utilize a convex combination of and and ermit one to chooe the olution alternative, for thi combination i imum: [ α [ α + α, Y + α ] =, Y 2 where α [0,] i the index "eimim-otimim" whoe magnitude i defined by the deciion maer. The advantage and diadvantage of the choice criteria conidered above are dicued in [4,5,8]. Thee criteria have found the broad ractical alication for intance, [8] for monocriteria deciion maing under uncertainty. Multicriteria deciion maing a alied to generalizing the claic aroach to dealing with the uncertainty of information In the equel, we reent the information on the ue of the Bellman-Zadeh aroach to deciion maing in a fuzzy environment [9,0] for generalizing the claic aroach to deal with information uncertainty. hen uing the Bellman-Zadeh aroach for analyzing multiobjective roblem [4,], objective function, =,..., q are relaced by fuzzy et, =,..., q, where µ i the memberhi function of {, }, [4,0]. fuzzy olution i defined a with the memberhi function µ = µ,. q = µ = The ue of allow one to get the olution µ =. µ q I q = 4 Therefore, the roblem of multicriteria deciion maing i reduced to earch for 0 = arg µ. 5 q To obtain 5, one need to build the memberhi function µ, =,..., q, which reflect a degree of achieving own otima by,, =,..., q. Thi condition i atified if one chooe [4,]: µ = 6 for imized objective function or µ = 7 for imized one. In 6 and 7, λ, =,..., q are the imortance factor for the correonding objective function. λ λ E-IN: Volume, 206

4 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva 4 Choice criteria a objective function in multicriteria deciion maing under information uncertainty The reult of ection have erved [2-4] for generalizing the claical aroach to dealing with the uncertainty of information, decribed in ection 2, to analyze multicriteria roblem. owever, the reult of [5,2] indicate ome limitation of [2-4]. One way to overcome thee limitation i baed on conidering the choice criteria 9-2 for a given objective function in an environment with everal Y, =,..., [5,2]. tate of nature Therefore, conidering the choice criteria of ald, alace, avage, and urwicz, reectively, a objective function, one can obtain the following correlation: and = =, Y, 8 = =, Y, 9 = = =, Y, 20 = α + α =. 2 α, Y + α, Y Thi conideration of the choice criteria of the claic aroach ermit one to contruct q roblem, generally, including four or le objective function if not all choice criteria are ued in the analyi a follow: r, extr, r =,..., t 4, =,..., q, 22 where the objective function are, =, =, =, 2, 4, and =., lying 6 to contruct the memberhi function for r,, r =,..., t, =,..., q, one can olve the roblem 4 for the olution alternative, =,...,. The analyi, realized in thi way, guarantee the choice of the rational olution alternative in accordance with the rincile of the Pareto otimality [5] and allow one to overcome the limitation of the generalization of the claic aroach to deal with information uncertainty to multicriteria deciion maing, a reviouly dicued. Conidering thi, the matrix with the characteritic etimate Table 2 i reented a the matrix with the choice criteria etimate for =,..., q, in Table. Therefore, uing q matrice for the choice criteria etimate, we can contruct q modified matrice of the choice criteria etimate when alying 6, a hown in Table 4. inally, with thoe q modified matrice of the choice criteria etimate, after alying 4, we can contruct the aggregated matrix of the choice criteria etimate, a it i given in Table 5. Thi matrix include the etimate calculated on the bai of 4 and ued to chooe the olution alternative. Table. Matrix of choice criteria etimate for the th objective function Table 4. Modified matrix of choice criteria etimate for the th objective function µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ Table 5. ggregated matrix of the choice criteria etimate µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ E-IN: Volume, 206

5 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva 5 Cae tudy Thi ection reent a cae tudy aociated with the evaluation of action to reduce the liquidity ri of an enterrie to illutrate the ue of the aroach reented in thi aer. The tudy i related to an enterrie lanning for a 2-month eriod from 206 to 207. Thi mean that the action defined in 206 are to be evaluated for oible future cenario for 2 month later in 207. In the firt tage, the ri management ecialit defined four action that will be evaluated to attemt to reduce the iquidity Ri: Roll the debt; Perform the caital contribution; ecreae ditribution of dividend. Caitalization through dividend from ubidiarie. The ri management ecialit identified the uncertainty variable related to the iquidity Ri analyi. In articular, the following uncertain variable were elected for the tudy: v Inflation rate in % by year; v 2 interet rate in % by year; v exchange rate BR/U; v 4 tate debt, ince the comany i controlled by the tate; v rofit rate evolution of ubidiarie; 5 v 6 enterrie' rating, by credit rating agencie. In the econd tage, the ri management ecialit have been aed to create three cenario rereenting ditinct view of the future baed on the ix variable elected for the tudy: Y Otimitic; Y 2 Balanced; Y Peimitic. Each cenario rereent a oible future baed on the temoral behavior of the uncertainty variable. The variable v 4, v5 and v 6 are defined by two or more comonent, comlicating the quantification of the variable tate in the future. Therefore, thee variable are rereented by the following qualitative etimate, indicating the variable variation in the lanning eriod: IC their value will increae; M their value remain the ame level; C their value will decreae. The three cenario created by the ecialit are reented in Table 6: Table 6. cenario definition v v 2 v v 4 v 5 v 6 Y C IC IC Y M M M Y IC C C In the third tage, the ecialit evaluated the oible conequence of the action in each cenario. The conequence of the action will be evaluated alying two criteria indicator, widely ued to analyze ri: Probability and Imact. The Table 7 how the ayoff matrix for the criterion of Probability: Table 7. Payoff matrix for the criterion of Probability, % Y Y Y Baed on the exreion 8-2, with α = 0.75 for the choice criterion of urwicz, it i oible to contruct Table 8 with the choice criteria value for the indicator of Probability. Conidering that thi indicator i to be imized, uing 6, it i oible to define the normalized value of the choice criteria for each action, reented in Table 9. Table 8. Value of the choice criteria for the criterion of Probability % Table 9. Normalized value of the choice criteria for the criterion of Probability µ µ µ µ E-IN: Volume, 206

6 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva et u analyze the olution alternative for the criterion of Probability. Table 9 how that the ue of the criterion of ald lead to = }, ince µ 4 = { 4 for it column. The criterion of alace roduce the election of = }, ince { µ =. It i not difficult to undertand that the alication of the criterion of avage lead to =, }. inally, the criterion of urwicz { generate = { 4 }. Thu, the olution 2 i clearly the wort olution from the oint of view of the criterion of Probability, ince it wa not elected by alying all choice criteria. et u conider the ayoff matrix for the criterion of Imact, reented in Table 0. Thu, baed on the exreion 8-2, with α = 0. 75, we can contruct the Table with the choice criteria value for the criterion of Imact. Table 0. Payoff matrix for the criterion of Imact 6 BR 0 Y Y 2 Y Table. Value of the choice criteria for the 6 criterion of Imact BR inally, a the criterion of Imact i to imized, uing 6, it i oible to define the normalized value of the choice criteria for each action, reented in Table 2. Table 2. Normalized value of the choice criteria for the criterion of Imact µ µ µ µ The analyi of Table 2 how that the alication of the criterion of ald ermit one to chooe the olution alternative = { 4 }. The ue of the criterion of alace roduce the election of = } a well. The utilization of { 4 the criterion of avage generate = { }. inally, the ue of the criterion of urwicz alo generate = { 4 }. Thu, the analyi of the criterion of Imact indicate that the olution and 4 are to be conidered a referable olution alternative. et u conider the multicriteria roblem, which i aociated with the imultaneou imization of the indicator of Probability and Imact of the iquidity Ri. Uing we can aggregate the normalized value of the choice criteria given in Table 9 and Table 2 to obtain Table. It rereent the erformance of the action for the criteria of Probability and Imact imultaneouly. Table. ggregated choice criteria value for the multicriteria analyi µ µ µ µ nalyzing Table 5, it i oible to obtain the following reult: = }, = } { 4 { = { }, and = { 4 }. Thee reult indicate that the action 2 and can be excluded from the future conideration. t the ame time, the action and 4 hould be ubjected to the additional analyi, baed on alying other E-IN: Volume, 206

7 E TRNCTION on BUINE and ECONOMIC Joel Pereira Jr., ougla Barboa, Patricia Bernarde, Thiago Bruzadelli, Petr Eel, Tiago araiva deciion maing technique, for examle, dicued in [4,2].. 6 Concluion In thi aer, the reult related to analyzing the conequence of realizing action to enterrie ri reduction have been reented. The rooed aroach i baed on the contruction of ayoff matrice, which ermit one the rational conideration of the uncertainty factor. The analyi of ayoff matrice, baed on the ue of ecific choice criteria, allow one to find the robut action, conidering divere oible future cenario. The ue of the choice criteria a objective function ermit the correct evaluation of the conequence of the action alternative. n imortant benefit of thi aroach i the oibility to evaluate the action alternative from the oint of view of a articular criterion indicator a well a from the oint of view of multile criteria indicator on the bae of generalizing the clai aroach to conidering the information uncertainty with uing rocedure of multicriteria deciion maing in a fuzzy environment. The aer reult have been illutrated by the cae tudy related to imroving indicator of the enterrie liquidity ri. The reult of the aer demontrate their alicability to analyze variou tye of enterrie ri, conidering divere indicator which characterized thee ri. Reference: [] T. ven, Ri aement and ri management: Review of recent advance on their foundation, Euroean Journal of Oerational Reearch, in re. [2] P.Ya. Eel, J..C. Martini, and R.M. Palhare, Multicriteria analyi in deciion maing under information uncertainty, lied Mathematic and Comutation, 2002, 2008, [] P. Eel, I. ohenev, R. Palhare, R. Parreira, and. chuffner Neto, Multicriteria analyi baed on contructing ayoff matrice and alying method of deciion maing in fuzzy environment, Otimization and Engineering, 2-2, 20, [4]. Pedrycz, P. Eel, and R. Parreira, uzzy Multicriteria eciion-maing: Model, Method, and lication. New Yor/Chicheter/Bribane: John iley & on, 20. [5] J.G. Pereira Jr., P.Ya. Eel, R.M. Palhare, and R.O. Parreira, On multicriteria deciion maing under condition of uncertainty, Information cience, 24, 205, [6] R.. uce and. Raiffa, Game and eciion. New Yor/Chicheter/Bribane: John iley & on, 957. [7]. Raiffa, eciion nalyi. Reading/Menlo Par/yudney: ddion-eley, 968. [8].. Belyaev, Practical roach to Chooing lternative olution to Comlex Otimization Problem under Uncertainty. axenburg: II, uchta, Choice of the bet alternative in cae of a continuou et of tate of nature alication of fuzzy number, uzzy Otimization and eciion Maing, 6, 2007, [9] R.E. Bellman and.. Zadeh, eciionmaing in a fuzzy environment, Management cience, 7 4, 970, [0]. Pedrycz and. Gomide, n Introduction to uzzy et: nalyi and eign. Cambridge: MIT Pre, 998. [] P.Ya. Eel, uzzy et and model of deciion maing, Comuter and Mathematic with lication, 447, 2002, [2]P. Eel, I. ohenev, R. Parreira,. Pedrycz, and J. Pereira Jr., Multiobjective and multiattribute deciion maing in a fuzzy environment and their ower engineering alication, Information cience, 6-62, 206, E-IN: Volume, 206