THE APPLICATION OF UTA METHOD FOR SUPPORT EVALUATION NEGOTIATION OFFERS 1

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1 OPTIMUM. STUDIA EKONOMICZNE NR 2 (80) 2016 dr hab. Ewa ROSZKOWSKA, prof. UwB Faculty of Economcs and Management, Unversty of Balystok e-mal: erosz@o2.pl DOI: /ose THE APPLICATION OF UTA METHOD FOR SUPPORT EVALUATION NEGOTIATION OFFERS 1 Summary The MCDA technque has been extensvely and successfully appled for supportng decson makng n negotaton processes. The mostly used technques SAW, AHP or TOPSIS are based on drect preference nformaton whch requres from negotator a clear and precse defnton all the parameters of the preference model (e.g. ssue weghts, opton rates, aspraton and reservaton values etc.), so those technques can be successfully appled n well-structured negotaton problems. But, many real negotaton problems are llstructured, that means that the negotaton space s mprecsely defned, and the negotator s preferences the vagueness or mperfect. The man purpose of ths paper s to nvestgate the potentals and the applcablty the UTA method, one of the technques based on ndrect preference nformaton, n evaluaton of negotaton offers, especally n ll-structured negotaton problems. The UTA (Jacquet-Lagreze and Sskos, 1978, 1982, 2001) s a multcrtera decson makng method whch s based on the lnear programmng model for nferrng addtve utlty functons from a set of representatve decson data. The example s also presented to elaborate and demonstrate the holstc judgment and the usefulness UTA approach for evaluaton negotaton offers. Key words: UTA method, holstc judgments, support negotaton, preference analyss, scorng functon JEL: C02, C30 1. Introducton Varous MCDM technques have already been appled for solvng the evaluaton negotaton offers problems [see Salo, Hamalanen, 2010; Brzostowsk et al. 2012a, 2012b; Wachowcz et al. 2012; Wachowcz, Błaszczyk 2012; Roszkowska et al. 2012, 2014, 2015a, b; Górecka et al. 2014, 2016]. The scorng system can be determned by smply addtve weghtng method (SAW), [Kersten, Noronha, 1999; Schoop et al. 2003] whch s used, for nstance, n such negotaton support systems as Inspre or Negosst. Another technque s AHP [Saaty 1980] whch s appled n Web-HIPRE system [Mustajok, Hamalanen, 2000; Kersten, La, 2007; Brzostowsk, Roszkowska, 2012a]. In seres of papers [Roszkowska et al. 2012, 2014, 2015a, 2016; Wachowcz, Błaszczyk 1 Ths research was supported by the grant from Polsh Natonal Scence Centre (2015/17/B/HS4/00941).

2 The Applcaton of UTA method ] the technque for order preference by smlarty to deal soluton (TOPSIS) method was also dscussed n the perspectve of evaluaton negotaton offers. In general, the negotator can construct scorng system takng nto account drect or ndrect preference nformaton provded by hm [Greco et al. 2008; Fguera et al. 2009]. The drect preference elctaton requres from negotator a clear and precse defnton of all the parameters of the preference model (e.g. ssue weghts, opton rates, aspraton and reservaton values etc.), whereas the ndrect preference nformaton requres selected examples/negotaton offers whch are used to nducng the parameters of the preference model and all the remanng packages consdered wthn the decson makng problem may be evaluated and rank. The drect preference nformaton s used n the tradtonal aggregaton paradgm, accordng to whch the aggregaton model s frst constructed and then appled on set X to rank the alternatves, so for nstance n SAW, AHP or TOPSIS algorthm. Indrect preference nformaton s used n the dsaggregaton (or regresson) paradgm, accordng to whch the holstc preferences on a subset of alternatves R X are known frst, and then a consstent aggregaton model s nferred from ths nformaton to be appled on set X n order to rank the alternatves [Greco et al. 2008, Fguera et al. 2009, Sskos et al. 2001, Kadzńsk et al. 2012; Ishzaka and Nemery 2013]. Summng up, the mult-crtera technques based on drect preference nformaton can be appled manly n well-structured negotaton problems. However, most real negotaton decson problems take place n a complex envronment where conflctng systems of crtera, uncertan and mprecse knowledge, and possbly vague or mperfect preferences have to be consdered. To face such complexty and ll-structure of negotaton problems, the use of specfc tools, technques, and concepts whch allow the avalable nformaton to be represented s necessary. In ths paper we are partcularly nterested n the role of ndrect preference nformaton based on relatvely less cogntve negotator s effort and holstc judgment n mult-crtera technques whch can be apply n ll-structured negotaton problems for evaluaton negotaton offers. The holstc judgment approach n evaluaton negotaton offers were consdered prevously n some papers. In the paper [Górecka et al. 2014] a new algorthm for elctng the negotators preferences n mult-ssue negotatons that hybrdze the basc concepts of ZAPROS and MACBETH was proposed. Ths algorthm requres of the negotators to compare the reference alternatves that dffer from the deal one n the resoluton level of one ssue only (the ZAPROS-lke approach). Also, n the papers [Roszkowska, Wachowcz, 2015b], two technque MARS and GRIP based on holstc judgment, were dscuss n the context to support negotaton problems. In ths paper however, we propose an effectve applcaton of holstc judgment and dsaggregaton paradgm whch s apply n the UTA (UTltés Addtves) approach and UTA software to handle ndrect preference nformaton n ll-structured negotaton problems. More exactly, we proposed an applcaton of the UTA method to determnng scorng functons for the evaluaton of negotaton offers. The UTA method [Jacquet- Lagreze, Sskos, 1982] s based on preference dsaggregaton model and ams to assess decson models from a pror known decson or preference data n the form of ranked lsts of optons. At the frst step decson maker descrbe a set of crtera wth non-decreasng,

3 146 Ewa Roszkowska exhaustve and non-redundant utlty functons. Next, usng specal lnear programmng technques, one global and several partal addtve utlty functons are obtaned from a gven rankng of the reference set optons. At the end a stablty analyss check s ncluded, as a post-optmzaton step. The UTA method adopt the aggregaton-dsaggregaton approach whch consttutes a bass for the nteracton between the analyst and the DM, whch ncludes the consstency between the assessed preference model and the a pror preferences of the DM, and he overall evaluaton of potental actons [Sskos et al. 2005; Ishzaka, Nemery 2013]. The UTA-based methods have been appled n several real-world decson-makng problems n the felds of venture captal evaluaton, portfolo selecton and management, falure predcton, busness fnancng country rsk assessment marketng of new products, marketng of agrcultural products, consumer behavor customer satsfacton, sales strategy problems, job evaluaton, among many others [for more revew see Sskos et al. 2005]. Despte the UTA approach has already been consdered n the lterature wth many applcatons there s no research on applyng t to the support of the ll-structured negotaton problems. The man contrbuton of the paper s the dscusson about applcablty of holstc judgment and UTA technque as well preference dsaggregaton analyss n negotaton problems, and ts effcency. The paper s organzed as follows. Secton 2 outlnes the basc features, mathematcal formulaton and operaton of the UTA method. The negotaton scorng system based on UTA approach s presented n Secton 3. The example usng UTA software s dscussed n Secton 4. Fnally, Secton 5 concludes the artcle, summarzes the man fndngs of ths research and proposes some future research drectons. 2. The fundaments of UTA method The UTA (UTltés Addtves) method was proposed by Jacquet-Lagreze and Sskos [1978, 1982] to am at nferrng one or more addtve value functons from a gven rankng on reference set. Ths technque s based on MAUT, adopt the preference dsaggregaton prncple and uses lnear programmng technques n order to optmally nfer addtve value/utlty functons whch are as consstent as possble wth the global decsonmaker s preferences. The UTA algorthm have been mproved and extended for varous applcatons therefore resultng n a famly of UTA methods. Several varants of the UTA method ncorporate, for nstance, dfferent forms of optmalty crtera used n the LP formulaton, are developed. For an overvew of UTA famly methods [see Sskos et al. 2005; Sskos et al. 2001; Fguera et al. 2009; Greco et al. 2008; Kadzńsk et al. 2012]. The UTA s one of the mult-crtera decson makng technque whch s based on Mult-attrbute Utlty Theory (MAUT) and assess addtve value functons on the whole crtera. We have two man preferences approaches: tradtonal aggregaton paradgm as well dsaggregaton-aggregaton (or regresson) paradgm. In the aggregaton approach the crtera aggregaton model s frst constructed and then appled on set A to get

4 The Applcaton of UTA method 147 nformaton about the comprehensve preference. Contrary to the dsaggregaton-aggregaton model where the comprehensve preference on a subset s known a pror (but the global preference s unknown) and a consstent crtera aggregaton model s nferred from ths nformaton [Greco et al. 2007; Fguera et al. 2009; Ishzaka, Nemery, 2013] In UTA technque the preference dsaggregaton approach refers to the analyss (dsaggregaton) of the global preferences (judgment polcy) of the DM n order to dentfy the crtera aggregaton model that underles the preference result. Let us assume that we have mult-crtera decson-makng problem, where the decson maker consders a set of alternatves, called, whch s valued by a famly of crtera F={ f 1, f 2,..., f j,..., f n }. Let us use the followng notaton: A= {a 1, a 2,..., a,..., a m } s fnte set of alternatves, R- reference set F={ f 1, f 2,..., f j,..., f n } s fnte set of crtera, f j (a ) s the evaluaton of the alternatve a on crteron f j, {1,2,..n}, n X j - doman of crteron f j, X X the evaluaton space, * f *, 1 * * f the nterval of crteron evaluaton scale f f, {1,2,..n}, u j are non-decreasng real margnal value functon, j {1,2,..n}, U(x) the global utlty functon. The am of the UTA method [Jacquet-Lagreze and Sskos 1982] s to nfer the margnal utlty functons of U through the ordnal rankng gven by the decson maker on the reference set RA. The preference nformaton s gven n the form of a complete preorder on a subset of reference alternatves R A, called reference preorder. A decson maker (DM) s wllng to rank the alternatves n R from the best to the worst, accordng to hs/her preferences. The goal s to construct such an addtve utlty functon U whch reflect to decson maker s preferences. The UTA methods used an unweghted form of addtve value functon, where for each x X, U x n u n f x u x, j j j 1 j 1 where u j are non-decreasng real margnal value functons, u : F R for -1,,n satsfyng normalzaton constrans: j j n j 1 u j f * 1 j

5 148 Ewa Roszkowska and u j f * 0 for =1,2,,n * where: f *, f nterval values for crteron f. The UTA margnal value functons are assumed to be pecewse lnear. Ths requres the analyst or decson maker to defne the number of endponts for each crteron. When a crteron accepts only dscrete scores, ths parameter can also correspond to the number of possble values for ths crteron. Ths endponts allows to determned unknown margnal utlty scores by the lnear program. For each crteron, the nterval * *, j f f s dvded nto α 1 equal sub-ntervals, so that end ponts f are gven by the formula: j j 1 * f f* f f* 1 for j = 1,2,,α. If the margnal utlty scores of these endponts are known, the utlty score of any j j 1 f x f f s calculated n the followng way: pont ; j j f x f j 1 j u f x u f u f u f f j1 f for j = 1,2,,α The decson maker needs to provde a complete rankng on the reference (learnng) set R whch avods any ncomparablty amongst alternatves. For all the alternatves of the learnng set R, the approxmate margnal utlty functon can be calculated and next the margnal value functon are fnally estmated by means of the lnear program [for detals see: Sskos et al 2005]. The stablty analyss of the results provded by LP s consdered as a post-optmalty analyss problem. Thus, we can obtan utlty functon by solvng lnear problem task mnmzng sum of estmaton errors, under constrants that provdes enforced propertes of the addtve utlty functon. The addtve global utlty functon U has followng propertes: normalzed values of utlty, for worst evaluaton equals 0, whle the total utlty for the best evaluatons equals 1; concordance of preferences defned by decson maker s on the R set, preferences monotoncty for each defned crteron, mnmal utlty estmaton error (defned as sum of gathered estmaton errors). Each alternatve of set X s evaluated on the bass of functon U and receves a utlty score U(a) whch allows the rankng of all alternatves from best to worst. The preference and ndfference relatons amongst the alternatves of X are thus defned as follows: for every a, b X a P b U(a) > U(b) : a s preferred to b, for every a, b X : a I b U(a) = U(b) : a and b are ndfferent. Each alternatve of set X s evaluated on the bass of functon U and receves a utlty score U(a) whch allows the rankng of all alternatves from best to worst. j

6 The Applcaton of UTA method The evaluaton negotaton offers usng UTA procedure In the prenegotaton phase the negotaton template s evaluated and the negotaton offer scorng system s bult to support negotators n ther decsons [Raffa, 1982]. Negotaton template descrbes the structure of the negotaton problem n the form of defned lst of ssues and feasble optons of these ssues [Raffa et al. 2002]. In the case of bg negotaton problems or contnuous optons the wde ranges of optons are reduced to salent optons only to dscretze the negotaton problem and make t easer to analyze. The evaluated template s used to support the negotators n evaluatng the offers, analyzng the negotaton progress, scale of concessons among many others [Roszkowska, Wachowcz, 2015b]. To formalze the model of negotaton template and buldng scorng functon we start wth the followng defntons: a negotaton package (opton) s an offer, whch negotator may send to or receve from ther opponent, an ssue s a crteron negotator use to evaluate the offers. Let us denote by: F f, f 2,..., f, the set of negotaton ssues, 1 n j X x j 1,..., X the sets of feasble resoluton levels (optons) for each ssue f, n X X the set of all feasble negotaton offers (packages) 1 x x 1, x 2,..., x n, where x X s a vector of optons whch represents negotaton package. The negotaton offer scorng systems, s determned by a set of non-decreasng margnal j value functons u : X R, for = 1,..., n that allow to assgn scores u x to each opton defned n the negotaton template and evaluate each negotaton offer x X by means of addtve value functon U n x u x. 1 We assume here, that accordng to the general assumptons of addtve preference aggregaton the DM s preferences over the resoluton levels are mutually preference ndependent. The UTA procedure of ndrect elctng preferences and buldng the negotaton offer scorng systems conssts of the steps (see Schema 1).

7 150 Ewa Roszkowska SCHEMA 1. The process of buldng negotaton offer scorng system based on UTA approach Determnng set of crtera F, set of negotaton packages (alternatves) and the evaluaton scale X for each ssue (decson crteron). Defnng the set R of the reference negotaton packages (alternatves). Provdng the preference nformaton on the reference set R. Checkng for exstence of at least one value functon compatble wth the preference nformaton provded by the DM and next buldng a set of compatble addtve value functons (a lnear program s bult and solved). Extendng the scorng system for all feasble offers from X and rankng alternatves from the set usng the most representatve value functon. Source: own elaboraton. We descrbe her more exactly all steps: Step 1. Determnng set of crtera F, set of negotaton packages (alternatves) and the evaluaton scale for each ssue (decson crteron) X. The negotaton template s a smplfed defnton of the negotaton space and specfes the set of negotaton ssues and the sets of feasble resoluton levels (optons) for each ssue. Three dfferent types of crtera n UTA software can be used: cost (less s better), gan (more s better) and ordnal (non numercal crteron n vew of classfed negotaton offers).two frst types of crtera can be used especally n the case of quanttes ssues, and the ordnal scale n case qualtatve ssues. It s word notng that the ordnal scale can be also represented easy by lngustc labels. Step 2. Defnng the set R of the reference negotaton packages (alternatves) from the set X. From the total set of packages X to be evaluated, a small number s selected as a reference set R. The set of the reference alternatves conssts of those packages for whch negotator s judgment s certan and preference nformaton s easy to obtan. Alternatvely, a set of vrtual offers whose values on the crtera are assgned delberately can be constructed by the negotator to ad the decson makng hs global evaluaton (rankng). The offers

8 The Applcaton of UTA method 151 composng the reference set have to be: at frst known to the evaluators, n order to facltate the negotator n expressng her/hs overall evaluaton (rankng) as well representatve of the whole set of offers, n order to take nto account the dfferent levels of the decson space, coverng ts whole range. From ths pont of vew n ths work we can recommend the followng reference sets, based on ZAPROS approach [Larchev, Moshkovch, 1995, 1997; Moshkovch et al. 2005]: R= Y nirs { P I }, where Y nirs s the set of all hypothetcal negotaton packages wth the best evaluaton for all the crtera but one, so t s a set of packages beng near to the Ideal Reference Stuaton, P I - the deal package, that s means package wth the best evaluaton for all the crtera R= YnAIRS PAI, where Y nairs s the set of all hypothetcal negotaton packages wth the worst evaluaton for all the crtera but one, so t s a set of alternatves beng near to the Ant-Ideal Reference Stuaton, PAI - the ant-deal package, that s means package wth the worst evaluaton for all the crtera. It s worth nothng that set R s used n MARS procedure [Górecka et al. 2014, 2016]. The advantages of UTA approach s fact that the preference nformaton s collected n a very easy way and concerns a small subset of negotatons offers and playng the role of a tranng sample. Elctaton of holstc parwse comparsons the negotaton packages from Reference Set requre from the negotator relatvely small cogntve effort. From the perspectve of negotator the set represent the offers whch are near the Ideal Reference Stuaton, so those whch can be easy accept by hm. On the other way the set wth offers closed the Ant-Ideal Reference Stuaton represent offers whch are defntely reject by negotator. Step 3. Provdng the preference nformaton on the reference set R The preference nformaton descrbes a preference relaton on a set of reference alternatves from R, e.g. alternatve a s at least as good as b. In the UTA software the weak order, defned on the subset of alternatves have to be ntroduced by negotator. The UTA technque does not requre to evaluate the weghts of ssues separately, but derves them from package-to-package comparsons. It s especally comfortable n negotatons stuatons, where t s very dffcult for hm/her to elct exactly crtera weghts. The process nvolved to assess an overall preference usng a decomposed analytcal procedure may be not natural to hm. The negotator can say only that one crteron s more mportant that the other, but most aggregaton methods requre a more precse nformaton, so negotator mght reject any model based on weghts whch are less meanngful to hm. On the other hand, especally when some offers are known he can more easly compare these all packages n a holstc manner. Offer a s better than b. Takng ths nto account, UTA technque allow negotator to evaluate trade-offs by comparng complete packages, so seems more natural for negotators snce t s close to the actual decson makng analyss conducted n a real-lfe negotaton [Roszkowska, Wachowcz, 2015].

9 152 Ewa Roszkowska Step 4. Checkng for exstence of at least one value functon compatble wth the preference nformaton provded by the DM and next buldng a set of compatble addtve value functons (a lnear program s bult and solved). In ths steep the non-decreasng margnal value functons for each ssues are constructed whch allow to assgn scores to each opton defned n the negotaton template. Next all packages can be evaluate by means of addtve value functon. The number of lnear peces of margnal utlty functon used to model ts margnal utlty value should be ntroduced n UTA software by negotator. As result we obtan the margnal utlty functon dagrams for every crteron and the rank of alternatves evaluated by these utlty functons. The fnal rankng descrbng ordered alternatves wth calculated comprehensve utltes for every alternatve s also gven. What s mportant the UTA software allow modfy value of the margnal utlty functon wthn the range of for a gven crteron n characterstc ponts. The scope of possble modfcatons, assumng the unchanged value of Kendall coeffcent calculated for the ntal soluton. Modfyng n such scope the values of margnal utlty functon n characterstc ponts we can estmate the robustness of obtaned rank n relaton to the values of margnal utlty for a gven crteron. Of course, negotator can modfy more deeply the margnal utlty functon values n characterstc pont what can result n changes of ntally obtaned rank of alternatves 2 Step 5. Extendng the scorng system for all feasble offers from X and rankng alternatves from the set usng the most representatve value functon. 4. Illustratve example and dscusson In ths secton, we present the numercal example to llustrate applcaton holstc judgment n evaluaton negotaton packages. We testfy practcally all steps of negotator preference analyss to show usefulness UTA approach. Let as assume that Seller and Buyer negotate the condtons of the potental busness contract. The followng negotaton ssues are dscussed: f 1 untary prce (EUR), f 2 tme of delvery (days), f 3 tme of payment (days). The negotaton template s defned by the numercal value for all crtera Prce (EUR): 20, 40 for both partes; Tme of delvery (days): 3, 30 for both partes, Tme of payment (days): 3, 21 for both partes. To buld the ntal set of packages n the pre-negotaton phase Seller defnes the salent optons for each ssue n the followng way: 2 See descrpton Vsual UTA method n Vsual UTA software avalable at: ste/vsualuta.htm (date of access: ).

10 The Applcaton of UTA method 153 Prce: 20, 30, 40. Tme of delvery: 3 (days), 7 (days), 14 (days), 30 (days), Tme of payment: 3 (days), 14 (days), 21 (days). The scorng system for the feasble negotaton offers can be created out of varous combnatons of the salent optons (3 4 3 = 36 dfferent packages). Snce we assume that the Buyer consders f 1, f 2 to be the cost ssue, and f 3, to be the beneft one. The Ideal Reference Vector has the form W(20,3,21), and the Ant-Ideal Reference Vector s the followng: W(40,30,3). Havng such defned negotaton template Seller mplemented VISUAL UTA software to elct the preferences and generate the scorng system of the negotaton offers. We wll testfy three dfferent Reference sets. Reference Set 1. Let us assume that the buyer would lke to compare offers wth the Ideal Reference Vector W(20,3,21) and s able to evaluate the optons by declarng the occurrence of preferences (one opton s better than another). The reference set RS 1 conssts of all hypothetcal negotaton packages wth the best evaluaton for all the crtera but one so the set of packages beng near to the Ideal Reference Stuaton and the Ideal Reference Vector. The reference set has the form: RS 1 ={W(20,3,21), W(20,3,14), W(20,7,21), W(20,14,21), W(30,3,21), W(20,30,21), W(20,3,3), W(40,3,21)}. Reference Set 2: Let us assume that the buyer would lke to compare offers wth the Anty-Ideal Reference Vector W(40,30,3) and s able to evaluate the optons by declarng the occurrence of preferences (one opton s better than another). The reference set RS 2 conssts of the set of all hypothetcal negotaton packages wth the best evaluaton for all the crtera but one, so the set of packages beng near to the Ideal Reference Stuaton and the Anty-Ideal Reference Vector The reference set has the form: RS 2 ={W(20,30,3), W(30,30,3), W(40,3,3), W(40,7,3), W(40,14,3), W(40,30,21), W(40,30,14), W(40,30,3)}. Reference Set 3: conssts of those packages for whch negotator s judgment preference nformaton was easy to obtan. The reference set has the form: RS 3 ={W(20,30,3), W(40,30,21), W(40,30,14), W(40,3,3), W(40,7,3), W(30,30,3)}. The rankorderng packages from consdered reference sets are presented n Table 1. The rankorderng packages for Reference Sets Range Reference Set 1 (RS 1 ) Reference Set 2 (RS 2 ) TABLE 1. Reference Set 3 (RS 3 ) 1 W(20,3,21) W(20,30,3) W(20,30,3) 2 W(20,3,14) W(30,30,3) W(40,30,21) 3 W(20,7,21) W(40,3,3) W(40,30,14) 4 W(20,14,21) W(40,7,3) W(40,3,3) 5 W(30,3,21) W(40,14,3) W(40,7,3) 6 W(20,30,21) W(40,30,21) W(30,30,3) 7 W(20,3,3) W(40,30,14) 8 W(40,3,21) W(40,30,3) Source: own elaboraton.

11 154 Ewa Roszkowska As result UTA 3 software we obtan the margnal utlty functon for every crteron and reference set as well the rank of alternatves evaluated by these utlty functons. The results llustratng the pecewse lnear margnal utlty functon for the crteron Prce, Tme of delvery and Payment are gven n the tables 2-4 and Schemas 2-4 Margnal utlty functon for crteron Prce TABLE 2. Value Reference Set 1 (RS 1 ) Reference Set 2 (RS 2 ) Reference Set 3 (RS 3 ) 20 0, ,5 0, , , , Source: own elaboraton usng VsualUTA software. TABLE 3. Margnal utlty functon for crteron Tme of delvery Value Reference Set 1 (RS 1 ) Reference Set 2 (RS 2 ) Reference Set 3 (RS 3 ) 3 0, , , , , , , , , Source: own elaboraton usng VsualUTA software. Margnal utlty functon for crteron Payment TABLE 4. Value Reference Set 1 (RS 1 ) Reference Set 2 (RS 2 ) Reference Set 3 (RS 3 ) , , , , , , Source: own elaboraton usng VsualUTA software. The fnal rankng descrbng ordered negotaton packages wth calculated scorng ponts (comprehensve utltes) for every negotaton packages s gven n Table 5. 3 Software VISUAL UTA avalable at: (date of access: ).

12 The Applcaton of UTA method 155 TABLE 5. The global utlty functons and rankorderng negotaton offers for all reference sets Packages Crteron Reference Set 1 Reference Set 2 Reference Set 3 (RS 1 ) (RS 2 ) (RS 3 ) f 1 f 2 f 3 Value Value Value Range Range U(x) U(x) U(x) Range W W , , ,929 2 W , , , W , , ,929 3 W , , ,857 4 W , , , W , , ,839 5 W , , ,768 7 W , , , W , , ,786 6 W , , ,714 8 W , , , W , , ,643 9 W , , , W , , , W , , , W , , ,5 16 W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W , , , W Kendall Coeffcent Source: own elaboraton usng VsualUTA software.

13 156 Ewa Roszkowska SCHEMA 2. Rankorderng negotaton packages from the set X and margnal utlty functon for the Reference Set 1 W(20,3,21) W(20,3,14) W(20,7,21) W(20,7,14) W(20,14,21) W(30,3,21) W(20,14,14) W(30,3,14) W(30,7,21) W(30,7,14) W(30,14,21) W(30,14,14) W(20,30,21) W(20,30,14) W(20,3,3) W(40,3,21) W(20,7,3) W(40,3,14) W(40,7,21) W(40,7,14) W(20,14,3) W(30,30,21) W(30,30,14) W(40,14,21) W(30,3,3) W(40,14,14) W(30,7,3) W(30,14,3) W(20,30,3) W(40,30,21) W(40,30,14) W(40,3,3) W(40,7,3) W(30,30,3) W(40,14,3) W(40,30,3) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Utlty Utlty Utlty Margnal utlty functon: Prce 0,4 0,3 0,2 0, Prce Margnal utlty functon: Tme of delvery 0,4 0,3 0,2 0, Tme of delvery Margnal utlty functon: Payment 0,4 0,3 0,2 0, Payment Source: own elaboraton.

14 The Applcaton of UTA method 157 SCHEMA 3. Rankorderng negotaton packages from the set X and margnal utlty functon for the Reference Set 2 W(20,3,21) W(20,3,14) W(30,3,21) W(20,7,21) W(30,3,14) W(20,3,3) W(30,7,21) W(20,7,14) W(30,3,3) W(30,7,14) W(20,7,3) W(20,14,21) W(30,7,3) W(20,14,14) W(30,14,21) W(20,14,3) W(30,14,14) W(20,30,21) W(30,14,3) W(30,30,21) W(20,30,14) W(20,30,3) W(30,30,14) W(40,3,21) W(30,30,3) W(40,3,14) W(40,7,21) W(40,3,3) W(40,7,14) W(40,7,3) W(40,14,21) W(40,14,14) W(40,14,3) W(40,30,21) W(40,30,14) W(40,30,3) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Utlty Utlty Utlty 0,5 0,4 0,3 0,2 0,1 Margnal utlty functon: Prce 0,6 0,5 0,4 0,3 0,2 0, Prce Margnal utlty functon: Tme of delvery 0,12 0,1 0,08 0,06 0,04 0, Tme of delvery Margnal utlty functon: Payment Payment Source: own elaboraton.

15 158 Ewa Roszkowska SCHEMA 4. Rankorderng negotaton packages from the set X and margnal utlty functon for the Reference Set 3 W(20,3,21) W(20,3,14) W(20,7,21) W(20,7,14) W(20,14,21) W(20,30,21) W(20,14,14) W(20,30,14) W(30,3,21) W(20,3,3) W(20,7,3) W(30,3,14) W(40,3,21) W(30,7,21) W(40,3,14) W(30,7,14) W(40,7,21) W(20,14,3) W(30,14,21) W(30,30,21) W(20,30,3) W(40,7,14) W(30,14,14) W(40,14,21) W(30,30,14) W(40,30,21) W(40,14,14) W(40,30,14) W(30,3,3) W(30,7,3) W(40,3,3) W(40,7,3) W(30,14,3) W(30,30,3) W(40,14,3) W(40,30,3) 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Utlty Utlty Utlty 0,5 0,4 0,3 0,2 0,1 0 0,25 0,2 0,15 0,1 0,05 Margnal utlty functon: Prce Prce Margnal utlty functon: Tme of delvery 0,4 0,3 0,2 0, Tme of delvery Margnal utlty functon: Payment Payment Source: own elaboraton.

16 The Applcaton of UTA method 159 The scores from Table 5 provde the negotator wth all relevant cardnal data suffcent to decde whch of any two packages (offers) s better and by how much. They may also be used to perform the symmetrc analyss to determne the far soluton for both partes durng the medaton or arbtraton process or to vsualze the negotaton progress and the concesson paths. However the negotator have to take nto account that the fnal rankng depends on reference set, so he could accept t or modfy f necessary reference set. Let us consder, for nstance the packages W(20,3,3) and W (20,30,14) to show that offers can be qute dfferent evaluated wth respect to reference sets. In the scorng system 1 (based on Reference Set 1) those packages are fully comparable wth 0,667 pont. That mean the concesson from 30 days to 3 days n tme of delvery s equvalent to the concesson from 3 days to 14 days n tme of payment. In the scorng system 2 (based on Reference Set 2) offers W(20,3,3) s scored for 0,989 pont and s more preferable than offer W (20,30,14) wth 0,56 pont. Fnally, n the scorng system 3 (based on the Reference 3) offer W(20,3,3) wth 0,643 pont s less preferable than offer W (20,30,14) wth 0,714 pont. The fnal rankng of the offers as well as ther scores for consdered reference sets are presented n Schema 3-6. Let us assume now that durng negotaton process negotator can search and construct new packages and evaluate t usng scorng functon. The optmal scorng functon should produce consstent rankng after new packages are added (or removed) and should not provde to rank reversal. Ths means that n the case of addng or removng new package the negotator does not need to re-evaluate the prevously evaluated packages, as well the score ponts of all packages are stable. It s worth nothng that the UTA technque allows ntroduced any package from negotaton space to set of packages new ones wthout rescorng other packages. Summng up, a negotator s evaluaton of packages has been conducted by usng holstc judgment on reference set of packages. The negotaton support by UTA algorthm takes nto consderaton the knowledge and preferental data provded by a negotator. The whole mplementaton process s supported by UTA software due to ts vsualzed envronment that enables the nteracton wth the DM (manly durng the selecton and orderng reference set, so constructon of the addtve value functons) and ts practcablty at solvng the UTA model and graphcally presentng the results. 5. Conclusons In the paper a framework for the evaluaton of negotaton offers based on holstc judgment and UTA methodology and UTA software have been proposed. We used the UTA method for rankng a fnte set of negotaton offers, evaluated by the fnte set of crtera The UTA method s based on ndrect preference nformaton and requred from the negotator a rankng of small subset of reference negotaton packages from the least preferable to the most preferable. The man goal of the UTA algorthm s the constructon of utlty functon, accordng to ordnal regresson approach, as compatble as possble wth ths reference rankng. The utlty functon compatble wth the reference

17 160 Ewa Roszkowska rankng s then used to the whole set of negotaton packages gvng to each package a value that ranks them The man key advantages of the UTA approach are the followng: 1) The UTA approach can support the DM to mprove her (hs) knowledge about the decson makng problem and her (hs) way of preferrng n order to allow a consstent decson to be acheved n evaluaton negotaton offers. 2) The UTA technque allow for elctaton attractveness of packages wth procedure that transform ordnal nformaton for reference set for cardnal nformaton by a non-numercal parwse comparson questonng. It s very useful approach especally were problem s poorly defned, n the context of qualtatve ssues whch often appear n negotaton template, as well n the case of mprecse nformaton. 3) The decson maker needs to rank the best to the worst of the alternatves of R, by gvng each alternatve a rank. So, havng a weak-order preference structure on a set of actons, the problem s to adjust addtve value or utlty functons based on multple crtera, n such a way that the resultng structure would be as consstent as possble wth the ntal structure. 4) The holstc ndrect evaluaton and applcaton UTA approach allow for: buldng the scorng system based on the preferental nformaton defned for a predefned set of reference packages P X, extendng the scorng system for all feasble offers from X, takng nto account ether complete or ncomplete preferental nformaton defned for P, defnng aspraton and reservaton levels as the addtonal preferental nformaton and elmnatng the necessty of explct defnton of the ssue weghts. 5) The computaton processes of determnng the scorng functon take nto account the negotaton space of each ssue as well the concepts of reservaton and aspraton levels. 6) The UTA procedure makes possble to expand the negotaton template by ntroducng new package after the preference elctaton has been conducted (wthn the actual negotaton space) wthout modfyng rankng prelmnary estmated packages. That means that proposed scorng functon produces consstent rankng after new packages are added (or removed) and does not provde to rank reversal. The man advantages of VISUAL UTA software are that the procedure s straghtforward and frendly for decson maker and allow for useful vsualzatons results. The dsadvantages of the UTA technque s fact that the UTA procedure depend on the reference set, so t may produced not satsfactory scorng functons. Moreover, n UTA, the DM s overall preferences are expressed on a set R of real or fcttous reference packages, n the form of a rankng or parwse comparson of these packages, so can used only partally ndrect preference nformaton. One of the soluton n such case s usng the other technques from the UTA famly, for nstance GRIP procedure whch adopts all features of UTA approach and takes addtonal nformaton nto account, n the form of comparsons of ntenstes of preference between actons. These comparsons

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