USING A MULTICRITERIA INTERACTIVE APPROACH IN SCHEDULING NON-CRITICAL ACTIVITIES

Transcription

1 OPERATIONS RESEARCH AND DECISIONS No DOI: /ord Mace NOWAK 1 Krzysztof S. TARGIEL 1 USING A MULTICRITERIA INTERACTIVE APPROACH IN SCHEDULING NON-CRITICAL ACTIVITIES A typcal proect conssts of many actvtes. Logcal dependences cause some of them to be crtcal and some non-crtcal. Whle crtcal actvtes have a strct start tme, n some proects the problem of selectng the start tme of a non-crtcal actvty may arse. Usually, t s possble to use the as soon as possble or as late as possble rules. Sometmes, however, the result of such a decson depends on external factors, e.g., an exchange rate. In ths paper, we consder the mult-crtera problem of determnng the start tme of a non-crtcal actvty. We assume that the earlest start and the latest start tmes of the actvty have been dentfed usng the crtcal path method, but the proect manager s free to select the tme when the actvty wll actually be started. Ths decson, however, cannot be changed later, as t s assocated wth the allocaton of key resources. The crtera that are usually consdered n such a stuaton are cost and rsk. We assume that the cost depends on an exchange rate. We also consder the rsks of proect delay and a decrease n qualty. Ths paper formulates the selecton of the start tme for a non-crtcal actvty as a dscrete dynamc multcrtera problem. We solve t usng an nteractve procedure based on the analyss of trade-offs. Keywords: proect schedulng, trade-offs, nteractve approach, CRR bnomal method 1. Introducton One of the most mportant processes n proect management, executed durng the plannng phase, s schedulng. The crtcal path method (CPM), proposed n the late 1950s by Walker and Kelley [7], s one of the oldest tools for schedulng but s stll wdely used. Knowng actvtes duratons and the logcal dependences between them, we can calculate the earlest start and latest fnsh tme for each actvty, whch consttutes the schedule. 1 Department of Operatons Research, Unversty of Economcs n Katowce, ul. 1 Maa 50, Katowce, Poland, e-mal address: mace.nowak@ue.katowce.pl

2 44 M. NOWAK, K. S. TARGIEL The CPM defnes two types of actvtes: crtcal and non-crtcal. Actvtes whch have a strct start and fnsh tme are crtcal. An actvty s crtcal n the sense that any delay n ts mplementaton results n a delay of the whole proect. The start tme for a non-crtcal actvty can be selected from a specfc perod. In the lterature, lttle attenton s pad to the selecton of the start tme of a non-crtcal actvty. Two classcal approaches are usually appled: as soon as possble (ASAP), as late as possble (ALAP). Fgure 1 presents three versons of a schedule for a proect consstng of four actvtes: A, B, C and D. Whle A, B and D are crtcal actvtes, C s a non-crtcal one. The graph on the left represents a schedule prepared accordng to the ASAP rule actvty C starts at the earlest possble start tme. The graph presented on the rght llustrates the schedule prepared n accordance wth the ALAP rule actvty C s fnshed at the latest fnsh tme. Fnally, the graph presented n the mddle llustrates a schedule n whch actvty C starts somewhere between the earlest and latest start tmes. Fg. 1. Approaches to the choce of when to begn a non-crtcal actvty (C) ASAP s a more approprate approach when t s mportant to complete a proect wthn a stpulated tme lmt. Selectng ths approach mnmzes the rsk of exceedng the deadlne. ALAP, the more rsky approach, may be chosen because of the avalablty of resources. There s also a thrd opton: to start a non-crtcal actvty at some tme between these extremes. The am of ths paper s to propose a method for selectng ths moment. In busness practce, sometmes we are free to select the start tme for noncrtcal actvtes anywhere between the earlest start tme (ASAP) and the latest start tme (ALAP). However, we need to be aware that we ncrease the rsk of proect delays by delayng the start of a non-crtcal actvty. In some cases, the result of an actvty may depend on ts completon tme. An example s gven by constructon proects, where total costs depend on the prces of materals whch vary seasonally. In such a stuaton, the problem of selectng the approprate tme to start a non-crtcal actvty arses. Ths s an nterestng research problem whch rases the queston of whether t s possble to determne the optmal startng tme

3 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes 45 takng nto account the hstory of changes n the factors that determne the result of the actvtes. The fnancal lterature proposes varous solutons to a smlar problem, called the tmng problem, based on the valuaton of fnancal optons. A well-known soluton s gven by the Cox Ross Rubnsten (CRR) method [2], based on bnomal trees. The next secton presents the modellng of future changes n parameters by a bnomal tree. In ths paper, the selecton of the start tme of a non-crtcal actvty s defned as a dynamc mult-crtera decson makng problem subect to rsk. We consder three crtera: the expected cost of the actvty, probablty of delay and probablty of a declne n qualty. When multple crtera are consdered, t s usually mpossble to dentfy a soluton whch s optmal n relaton to all of the crtera. Instead, we can try to dentfy non-domnated solutons ones for whch t s not possble to mprove the value accordng to one crteron wthout decreasng the value accordng to any of the others. Usually, the number of non-domnated solutons s so large that t s not easy to decde whch one should be selected. Thus, solvng a multcrtera problem requres nformaton about a decson-maker s preferences. Two man approaches can be used n multple crtera decson makng [10]. The frst assumes that the decson-maker artculates hs/her preferences on an a pror bass. In such a case, the procedure s dvded nto two dstnct phases: (1) acquston of nformaton on preferences, (2) computatons. Ths approach s often crtczed. Frst, the decson-maker has to consder all knds of choces and trade-offs whch mght be relevant, and as ths nformaton s acqured before knowng whether the alternatves are nfluenced by these preferences, t may be redundant. Moreover, the decson-maker may fnd the choces he/she faces to be purely hypothetcal, whch results n a reduced level of concentraton, thereby reducng the qualty of the nformaton obtaned. An nteractve approach s an alternatve to methods based on an a pror bass. Usng such an approach, nformaton on preferences s acqured step by step. At each teraton, the dalog and computaton phases are repeated. The decson-maker s more closely nvolved n the process of solvng the decson problem and, as a result, mproves hs/her knowledge about the structure of the problem. Two man paradgms are used for ganng nformaton on preferences: drect and ndrect [6]. The former assumes that the decson-maker expresses hs/her preferences n relaton to the crtera themselves. Such an approach s used, e.g., by Benayoun et al. [1]. Indrect collecton of nformaton on preferences means that the decson maker has to determne whch trade-offs between attrbutes are acceptable at each teraton, gven the current canddate soluton. The method proposed by Geoffron et al. [3] s an example of such an approach. Technques combnng both approaches have also been proposed, for nstance n [5]. As was shown n [8] and [9], trade-offs can also be used to solve a dscrete stochastc multcrtera decson makng problem. Ths study s an extenson of the work presented n papers [14] and [15], where a bcrtera problem was consdered. Here, we propose a technque that can be used when

4 46 M. NOWAK, K. S. TARGIEL more than two crtera are analysed. In a b-crtera problem, the stuaton s clear: at a non-domnated soluton mprovng the value of f 1 requres worsenng the value of f 2 and vce versa. If both crtera requre maxmsaton, t s qute sensble to dentfy a soluton for whch the ncrease n f 1 per unt decrease n f 2 s maxmal. However, when more than two crtera are analysed, the problem becomes more complcated. Frst, comparng trade-offs for varous pars of crtera requres evaluatons to be standardzed. Second, t can be possble to mprove a soluton accordng to more than one crteron at the same tme. In ths study, we propose a new nteractve technque based on trade-offs that can be used when at least three crtera are consdered. We use ths technque for selectng the start tme of a non-crtcal actvty. The paper s structured as follows. The problem s formulated n Secton 2. In Secton 3, we present the methodology. A numercal example llustratng the applcablty of the procedure s presented n Secton 4. The last secton contans conclusons. 2. Formulaton of the problem Let us assume that the cost of an actvty s expressed n a foregn currency (e.g., EUR). The cost n the domestc currency (e.g., PLN) depends on the exchange rate, whch s constantly fluctuatng. Snce we assume that ths actvty s non-crtcal, the problem of selectng ts start tme arses. If the probablty that the exchange rate wll fall s greater than the probablty of ts ncrease, t s qute clear that (based purely on the crteron of expected cost) the actvty should be started as late as possble. On the other hand, the later the actvty s started, the hgher the rsk that t wll not be completed on tme. In ths paper we consder a mult-crtera problem of schedulng a non-crtcal actvty. Our assumptons are as follows: The cost of an actvty s expressed n foregn currency, and does not depend on the actual completon tme. The mnmal completon tme (t mn ) and the latest fnsh tme (LF) for the actvty consdered have been estmated. For organzatonal reasons, the actvty can only be started at the begnnng of one of the followng perods: k = 1, 2,..., LF t mn. The actual cost of the actvty n domestc currency depends on the exchange rate at the end of the perod n whch the actvty s started. For each perod, expert estmates of the probablty that the actvty s fnshed on tme, assumng that t s started at the begnnng of perod n, are avalable. For each perod, expert estmates of the probablty that a declne n qualty occurs, are avalable. The problem conssts of decdng when to start the actvty takng nto account three crtera based on: f 1 the cost of the actvty, f 2 the probablty that the actvty

5 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes 47 s delayed and f 3 the probablty that a declne n qualty occurs. Our goal s to mnmze the values of all three functons Modellng the future usng of a bnomal tree We assume that the future value of our parameter (X) can be modelled usng stochastc dfferental equatons. For ths purpose, we choose geometrc Brownan moton (GBM) based on the equaton: dxt () Xtdt () XtdWt () () (1) where: W(t) s the Wener process, X(t) s the value of the parameter X, at tme t, s the drft parameter, s the volatlty parameter whch determnes the varablty of the process. Implementaton of ths process s shown n Fg. 2 up to the pont t = 0. The same fgure shows smulatons of three paths of the process after the pont t = 0. Ths contnuous process can be approxmated by a dscrete structure, namely a bnomal tree. In Fg. 2, we present such a tree usng arrows whch cover future changes n the process, startng from the pont t = 0. Fg. 2. Bnomal tree coverng the stochastc process One problem that arses s to select an approprate model of a stochastc process and then to calbrate a bnomal tree. Ths ssue was dscussed n [13] and prevously n [12]. The nodes of such a graph can be calculated from the formula:

6 48 M. NOWAK, K. S. TARGIEL x e k2 ˆ Δtp k, x0,0 (2) where: x,k s the value of the parameter x after k perods and declnes, Δt p s the amount of tme n years represented by one perod n the tree, ˆ s the estmated volatlty parameter for GBM. We can estmate such parameters on the bass of hstorcal data. The estmated volatlty ˆ of the process s calculated on the bass of hstorcal data regardng ther varablty: d ˆ (3) Δ where: Δt d s the amount of tme n years between observatons, t d d s the standard devaton n hstorcal data. Knowng ths, we can calculate the typcal growth factor (u) (together wth the fall factor 1/u): ˆ Δt p u e (4) The probablty of an ncrease can be calculated usng the followng formula: 1 Δt q m (5) 2 2 ˆ We can also calculate the probablty of reachng node (, k), = 0, 1,, k, after k perods [4]: k! k Px at (, k) = q (1 q)!( k )! (6) Ths leads us drectly to the expected value of the parameter X at stage k: k k! k E X ( k) = q (1 q) xk, (7)!( k )! 0 Usng formula (7), we can calculate the expected cost of the actvty when t starts at a partcular moment k, whch gves us the obectve functon for the frst crteron: f ( ) ( ) 1 ak K X k (8)

7 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes 49 where K denotes the fxed cost n EUR, and the parameter X s the EUR/PLN exchange rate Modellng the rsk of a delay The second crteron s rsk of delay, measured as the probablty of a delay. A non- -crtcal actvty must be fnshed before the latest completon tme. A longer delay causes a delay n the entre proect. Ths probablty can be derved from the expected value and standard devaton of the duraton, estmated usng the PERT method (program evaluaton and revew technque) [11] but t s better to nform ths calculaton usng expert knowledge and ntuton. We ask an expert to defne the probablty of delay for each alternatve start tme. In the example presented below, t s assumed that a non-crtcal actvty can be started between January and October (Table 1). When the actvty s started n January (alternatve a 1 ), there s a 1% chance of a delay past the end of the year, but when t starts n October (alternatve a 10 ), there s a 20% chance that not only the analysed actvty, but also the entre proect, wll be delayed Modellng the rsk of bad qualty The thrd crteron s the rsk of poor qualty, measured as the probablty of such qualty. In some stuatons, the value of ths probablty s nfluenced by the start tme of an actvty. For example, n a constructon proect, the rsk of poor qualty depends on the weather, whch changes durng the year. Smlarly to the rsk of delay, we assume that the rsk of poor qualty s estmated by an expert. In the example presented below, t s assumed that the probablty of poor qualty s the lowest f the actvty s completed n the summer months. 3. Multcrtera procedure for schedulng non-crtcal actvtes To solve ths problem, we use the nteractve approach wdely dscussed n [9]. Let A { a1, a2,..., a m } be the set of effcent (non-domnated) alternatves representng the perod n whch the actvty s started and F { f1, f2,..., f n } be the set of obectve functons for each crteron. By f ( a ) we denote the evaluaton of alternatve a wth respect to crteron f.

8 50 M. NOWAK, K. S. TARGIEL In the procedure descrbed below, we wll also use standardzed evaluatons of the realzatons wth respect to each crteron g ( a ) whch are determned from the followng formula: max{ f ( a)} f ( a) 1, m g ( a) max{ f ( a )} mn{ f ( a )} 1, m 1, m (9) Let A be the set of alternatves consdered n teraton l. In each teraton, a canddate alternatve a and a potency matrx M s presented to the decson maker (DM). The potency matrx conssts of two rows: the frst contans the best values accordng to the crtera attaned wthn the set A, and the second one, the worst ones: M f f f f n n (10) Snce n ths study we assume that all the crtera nvolve mnmsaton, the followng formulas are used for determnng the best and worst, respectvely, values accordng to each crteron: f mn f ( a ), 1, n (11) n ( l ) aa f max f ( a ), 1, n (12) n ( l ) aa Our procedure conssts of the followng steps: Prelmnary phase 1. Usng formula (9), for each alternatve a calculate the standardzed values of the evaluatons wth respect to each crteron. 2. Determne the frst canddate, alternatve a (1), usng the mn-max crteron: For each alternatve, determne the mnmum of the standardzed evaluatons wth respect to the crtera: mn g a g a 1, n ( ) mn{ ( )} (13) Set the alternatve a that maxmzes the value g mn ( a ) to be the frst canddate a (1).

9 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes Set l = 1 and (1) A A and start the frst teraton. Iteraton l 1. Determne the potency matrx M. 2. Present the values of the obectve functons obtaned for alternatve a and the potency matrx M to the DM. If the DM s satsfed wth the proposal, end the procedure. 3. Ask the DM to assgn each crteron to one of the followng three sets: F 1 the set of crtera accordng to whch mprovement s requred n comparson wth alternatve a. F 2 the set of crtera accordng to whch there should not be any deteroraton n comparson wth a. F 3 the set of crtera accordng to whch deteroraton s acceptable n comparson wth alternatve a. 4. Determne the set A consstng of all the alternatves from the set A whch satsfy the followng condtons: f a f a (14) f F1 ( ) ( ) f a f a (15) f F2 ( ) ( ) 5. If A, nform the decson maker that no alternatve exsts satsfyng the requrements specfed n step 4. Return to step If A only conssts of one alternatve, take ths alternatve as the next proposal a. Proceed to step For each alternatve A and each par of crtera ( f, f ) such that a f F1, fk F3 and fk( a) fk( a ), calculate the value of the trade-off tk( a ) usng the followng formula: k t g ( a) g ( a ) k ( a ) gk( a ) gk( a) (16) 8. For each par of crtera ( f, f k ) such that f F1, fk F3, check whether there ( l exsts at least one alternatve a 1) A, for whch the value of tk( a ) was calculated n step 7. If so, then for each alternatve A such that f ( a ) f ( a ), assume am k m k

10 52 M. NOWAK, K. S. TARGIEL t ( a ) to be twce as great as the maxmal value of the trade-off calculated for the par k ( f, f ) n step 7. If there does not exst an alternatve k a A such that the value of ( l tk( a ) was calculated n step 7, assume that tk ( am) 1 for all a 1) A. ( l 9. For each a 1) A, determne the average of the trade-offs calculated n steps 7 ( l and 8. Set the alternatve a maxmzng ths average to be the next proposal A 1). 10. Set l = l + 1 and proceed to the next teraton. The frst canddate s determned usng the mn-max crteron. In each teraton, the evaluatons of the obectve functons for the proposed alternatve and the potency matrx are presented to the DM. The DM can ether accept the proposed alternatve as the soluton of the problem, or else determne the drecton of mprovement by ndcatng the followng: Accordng to whch crtera are mprovements requred n comparson to the canddate? Accordng to whch crtera should there be at least no deteroraton n comparson to the canddate? Accordng to whch crtera can there be deteroraton n comparson to the canddate? Snce we are operatng wthn the set of effcent alternatves, the decson maker must ndcate at least one crteron accordng to whch deteroraton s permssble. Ths procedure contnues untl the decson maker s satsfed wth the proposed alternatve (step 2). If, as a result of ths analyss, the set of optons consdered s reduced to one, the decson-maker may accept t, or consder the alternatves proposed at an earler step once agan and decde to select one of them. a 4. A numercal example We consder a non-crtcal actvty that should be completed by no later than December 31st. The cost of the actvty s 50 mllon and does not depend on the completon tme. The ntal PLN/ rate s As there s not enough space to show how real data can be used to estmate u usng formula (4), we assume that the probablty of an ncrease q s equal 0.4. The nomnal completon tme s 3 months. Obvously, the sooner the actvty s started, the lower the rsk that the actvty wll be delayed. In the ntal phase, a bnomal tree s used to generate the probablty dstrbutons of the PLN/EUR rate accordng to the amount of tme that passes. Next, these dstrbutons are used to dentfy the dstrbutons of the actvty s cost. Table 1 presents the expected costs for varous startng tmes and the values of the two other obectve functons for each alternatve.

11 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes 53 The expected cost, calculated accordng to the procedure descrbed n 2.1, decreases over tme. The probablty of delay grows wth tme, but the exact values must be declared by an expert. The probablty of low qualty s lowest durng the summer months, as we are consderng a constructon proect. It s qute clear that all of the alternatves are non-domnated, snce the later the actvty starts, the lower s the expected cost and the hgher the rsk of delay. Identfcaton of the fnal soluton proceeds accordng to the followng scenaro presented below. Alternatve Startng month Table 1. The set of alternatves Expected cost (10 6 PLN) Probablty of delay Probablty of poor qualty a1 January a2 February a3 March a4 Aprl a5 May a6 June a7 July a8 August a9 September a10 October Source: authors calculatons. Prelmnary phase. Usng formula (9), we calculate the standardzed values of the evaluatons of the effcent alternatves wth respect to each of the crtera g ( a ), as presented n Table 2. Alternatve Table 2. The standardzed values g ( a ) Startng month g ( ) 1 a g ( ) 2 a g ( ) 3 a mn{ g ( a )} 1, n a1 January a2 February a3 March a4 Aprl a5 May a6 June a7 July a8 August a9 September a10 October Source: authors calculatons.

12 54 M. NOWAK, K. S. TARGIEL 2. Alternatve a 5 s assumed to be the frst proposal, 3. We set l = 1 and A (1) A. Iteraton 1 1. We calculate the potency matrx M (1). 2. The potency matrx (Table 3) and the canddate soluton a (1) = a 5 are presented to the decson-maker. a (1). Table 3. Potency matrx n teraton 1 Expected cost f1 Probablty of delay f2 Probablty of low qualty Present proposal, a Optmstc value Pessmstc value Source: authors calculatons. The decson-maker s not satsfed wth the proposal. 3. The decson-maker s wllng to accept an ncrease n the value of f 3, wants to mprove f 2 and to retan the value of f 1. So we have: F1 { f2}, F2 { f1}, F3 { f3}. ( l 4. We determne A 1). 5. A, so we go back to step The potency matrx (Table 3) and the canddate soluton a (1) = a 5 are agan presented to the decson-maker. The decson-maker agan s not satsfed wth the proposal. 7. The decson-maker decdes to mprove the expected cost f 1, and retan the value of f 3. So we have F1 { f1}, F2 { f3}, F3 { f2}. (2) 8. A { a6, a7}. (2) 9. The trade-offs for the par of crtera (f 1, f 3 ) are calculated for a A (Table 4). Alternatve a 7 s dentfed as the new canddate soluton. f3 Table 4. Trade-offs n teraton 1 Alternatve a6 a7 Trade-off Source: authors calculatons. 10. l := 2 and the procedure goes to the next teraton.

13 Usng a multcrtera nteractve approach n schedulng non-crtcal actvtes 55 Iteraton 2 (2) 1. The potency matrx M s calculated. 2. The potency matrx (Table 5) and the canddate soluton a (2) = a 7 are presented to the decson-maker: Table 5. Potency matrx n teraton 2 Expected cost f1 Probablty of delay f2 Probablty of low qualty Actual proposal a Optmstc value Pessmstc value Source: authors calculatons. The decson-maker s satsfed wth ths proposal. As the DM s satsfed wth the proposed alternatve, we end the procedure. Accordng to the DM s preferences, the best opton s to start the non-crtcal actvty n July. The expected cost s equal to mllon PLN, the probablty of low qualty s mnmsed and the probablty of delay s at an acceptable level (0.16). 5. Concluson The problem of specfyng the start tme of a non-crtcal actvty has been defned as a multple crtera dynamc decson makng problem under rsk. The man and orgnal contrbuton of our work s a new nteractve procedure that can be used for solvng such problems. It uses trade-offs to dentfy proposals for the decson maker. Consderng more than two crtera creates addtonal problems n the analyss of trade-offs. These problems are solved usng the method presented n the paper. We have consdered a three-crtera problem, assumng that the decson maker s nterested n mnmzng the cost of the actvty, the rsk of delay and the rsk of low qualty. Ths procedure uses a bnomal tree to model the stochastc process descrbng the change n the actvty s cost. On the other hand, we assume that experts estmate the rsk of delay and the rsk of low qualty. The latter assumpton may be consdered as a weakness of the proposed approach. In future research, we plan to consder more sophstcated methods for rsk evaluaton, takng nto account prevous experence and the evaluatons of multple experts. f3 References [1] BENAYOUN R., DE MONTGOLFIER J., TERGNY J., LARICHEV C., Lnear programmng wth multple obectve functons. Step Method (STEM), Math. Program., 1971, 8,

14 56 M. NOWAK, K. S. TARGIEL [2] COX J.C., ROSS S.A., RUBINSTEIN M., Opton Prcng. A Smplfed Approach, J. Fn. Econ., 1979, 7, [3] GEOFFRION A.M., DYER J.S., FEINBERG A., An nteractve approach for mult-crteron optmzaton wth an applcaton to the operaton of an academc department, Manage. Sc., 1972, 19, [4] GUTHRIE G., Real Optons n Theory and Practce, Oxford Unversty Press, Oxford [5] KALISZEWSKI I., MICHALOWSKI W., Searchng for psychologcally stable solutons of multple crtera decson problems, Eur. J. Oper. Res., 1999, 118, [6] KALISZEWSKI I., MIROFORIDIS J., PODKOPAEV D., Interactve multple crtera decson makng based on preference drven evolutonary multobectve optmzaton wth controllable accuracy, Eur. J. Oper. Res., 2012, 216 (1), [7] KELLEY J., WALKER M., Crtcal-path plannng and schedulng, Proc. Eastern Jont Computer Conference, Boston, MA, [8] NOWAK M., Trade-off analyss n dscrete decson makng problems under rsk, [In:] D. Jones, M. Tamz, J. Res (Eds.), Lecture Notes n Economcs and Mathematcal Systems, Vol. 638, New Developments n Multple Obectve and Goal Programmng, Sprnger, Berln 2010, [9] NOWAK M., Interactve multcrtera decson adng under rsk. Methods and applcatons, J. Bus. Econ. Manage., 2011, 12 (1), [10] ROY B., Problems and methods wth multple obectve functons, Math. Program., 1971, 1 (1), [11] STAUBER B.R., DOUTY H. M., FAZAR W., JORDAN R. H., WEINFELD W., MANVEL A.D., Federal statstcal actvtes, Am. Stat., 1959, 13 (2), [12] TARGIEL K.S., Multple crtera decson makng n the valuaton of real optons, Mult. Crt. Dec. Mak., 2013, 8, [13] TARGIEL K.S., Real optons n the tmng problem of non-crtcal actvtes, Proect Management Development. Practce and Perspectves, 2015, [14] TARGIEL K.S., NOWAK M., TRZASKALIK T., Choosng the start tme of a proect usng an nteractve multcrtera approach, Opt. Stud. Ekon., 2017, 87, (n Polsh). [15] TARGIEL K.S., NOWAK M., TRZASKALIK T., Schedulng non-crtcal actvtes usng multcrtera approach, Centr. Eur. J. Oper. Res., do.org/ /s y. Receved 16 November 2017 Accepted 10 Aprl 2018