Productivity depending risk minimization of production activities

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1 Productivity depedig risk miimizatio of productio activities GEORGETTE KANARACHOU, VRASIDAS LEOPOULOS Productio Egieerig Sectio Natioal Techical Uiversity of Athes, Polytechioupolis Zografou, Athes GREECE webaddress: Abstract: - Productio activities deped o a umber of importat parameters, oe beig the persoel s productivity. This parameter affects cosiderably the risk (techical ad/or fiacial risk) of the whole activity. For the computatio of the risk, a simple but comprehesive computer model has bee created allowig the time simulatio of the productio process for a umber of possible expected returs (scearios), as well as the computatio of the (well kow) fiacial idices ROI, IRR ad NPV. For the risk miimizatio the mi-max strategy for determiig optimal strategies has bee applied. The mi-max strategy" guaratees a worst-case risk/retur tradeoff i terms of the above fiacial idices, whichever of the specified scearios occurs. The above method is described i the followig ad correspodig results are preseted i this paper. Key words: - Risk modelig, Risk miimizatio, Mi-max strategy, Productivity 1 Itroductio For the plaig of productio activities (i.e. ivestmet activities, idustrial productio activities, etc.) computer models for computig the techical ad/or fiacial risk as well as riskmiimum strategies, are i use sice a log time. For the computatio of risk-miimum strategies umerous methods exist, like the mea variace approach, stochastic multi-period optimizatio, cotiuous time methods, factor models, etc. [1]- [10]. Newly, Rustem, Becker ad Marty [2] itroduced the mi-max strategy. The proposed method cosiders a fiite set of possible expected returs, called scearios, ad uses a strategy that guaratees a worst-case risk/retur tradeoff, whichever of the specified scearios occurs. I this paper we apply the mi-max strategy for computig the impact of persoel s productivity o the fiacial risk idices ROI, IRR ad NPV. This is importat for explorig the sesitivity of the fiacial risk from the qualificatio of the workig persoel ad from the maagemet s performace. I this ivestigatio we cosider the model already developed i [12], but with some modificatios that are described i sectio 2. The same holds for the optimizatio method ecessary for determiig the optimal strategy, which is already itroduced i [11]. Fially, the results of the study are preseted ad commeted i sectios 5 ad 6. 2 Model Descriptio We cosider (see [12]) the case of a productio activity for a time period of T years, subdivided i yearly quarters q (q=1,,q). The total ivestmet for the productio activity is equal to I Total. The ivestmet Iv is assumed to be realized withi N (N<Q) yearly quarters ad to vary liearly with q: Iv( = (2 ITotal / N) (1 q / N), q = 1,.., N (1) The maiteace costs MaiC for the productio equipmet (e.g. machies, etc.) i each yearly quarter q are equal to MaiC( = α I T, q = N + 1,... (2) with α deotig a fixed percetage of I Total. The expeses for the busiess operatio iclude the persoel costs PersC, the operatig costs

2 OperC ad the marketig costs MarkC. The persoel costs PersC deped o the umber of employees Empl, their iitial salary SalC, ad the salary raise SalR. Therefore: PersC( = Empl( * SalC( (3) SalC( = SalC( q 1) * (1 + SalR) The operatig costs OperC cosist of a ielastic part defied by the costat β 1 (e.g. ret, cleaig, security, etc.) ad of a variable part (e.g. telephoe, cosumables, travellig, etc) which depeds o the costat β 2 ad the umber of employees Empl. OperC = β + β Empl( ) (4) ( 1 2 q For the curret productio activity, the assumptio is made that the marketig costs MarkC are high at the begiig of the busiess (i order to stir the market) ad are decreasig durig the T years period, cotrolled by the costats γ 1 ad γ 2 : P = L( P ( (9) L 3 Scearios We cosider a fiite set of possible (icludig the worst case) expected returs or icomigs MaxIc(s), s=1,,s, called scearios. For these scearios the followig costrait holds: Ic MaxIc(s) (10) This meas that the icomigs defied through eq. (6) have MaxIc(s) as upper boud. For the case uder cosideratio, 10 MaxIc scearios are defied ad are show i Figure 1. Sceario 1 is cosidered to be the basic sceario. γ 2 q MarkC( = γ ( ) (5) 1 With respect to the icomigs Ic, we assume that each employee produces a costat icomig Ic E accordig to his productivity P. The the total icomig is give by the equatio (6). Ic( = IcE P Empl( (6) The productivity P depeds o two factors: At the begiig of the activity, the productivity is expected to follow the well kow learig curve L, which is assumed to be of expoetial type ad is defied by two costats λ 1 ad λ 2 : 1 2 λ2 q L = λ + (1 10 ) (1 λ ) (7) ( 1 1 The productivity P is also expected to drop with the icrease of the umber of the employees, sice maagemet ad coordiatio problems ecouter if the umber of employees is big. The productivity loss P L is assumed to be liear ad to is described give by the followig equatio: 3 4 PL 150 PL 30 PL ( = PL 30 + [ Empl( 30] 120 (8) 5 6 I equatio (8) P L30 is the estimated productivity loss for 30 employees ad P L150 the estimated productivity loss for 150 employees. Therefore: 7 8

3 a opt is the ami J [ ami ] =smallest icrease m=m+1 for which 9` 10 Figure 1. The te idividual s=1-10 scearios MaxIc (i ) ad all i cocert. For a give distributio of the umber of employees over the total time period T: Empl( (q=1,,q), the followig fiacial risk idices - NPV (Net Preset Value), - IRR (Iteral Rate of Retur) ad - ROI (Retur of ivestmet) ca be computed. It is clear that these fiacial risk idices mirror through the modelig of eq. (1)-(9) also the orgaizatioal ad productivity aspects. 4 Risk Miimizatio I the preset case the risk miimizatio problem cosists i maximizig the miimum NPV(s) accordig to equatio (11): Typical ζ ad ξ values are ζ=0.05 ad ξ=1.15. For further iformatio see [11]. 5 A First Optimizatio Result If P=1 ad the basic sceario s=1 is cosidered, the the optimal strategy u s=1 ca be easily computed (see Figure 2a): The NPV(s=1) value results i NPV s=1=1.8303e+006 (14) 2a mi[ NPV ( s), s = 1,.., S] = max (11) The above mi-max problem has Q idepedet variables, amely u(=empl(, q=1,,q. For the solutio of this problem a ovel hybrid evolutioary strategy is proposed [11]. It cosists of the Evolutio Strategy (ES) with oe paret ad oe offsprig combied with the determiistic Nelder- Mead algorithm. Accordig to this method, for each vector a ev (created by the ES) 2b a [ u(1), u(2),..., u( Q)] (12) ev = the Nelder-Mead algorithm takes over ad yields the earest local miimum a mi. The, if two successive radom vectors lead to the same local miimum, the stadard deviatio σ (ad thus the search area) is icreased. The proposed hybrid optimizatio method reads: set iitial vector a 0 m =1 (cycle) set σ 0 for =1:N create radom vectors a ev (a 0, σ 0 ); fid the earest local miimum a mi ; 1 1 if J[ a mi ] J[ a mi ] ζ J[ a mi ] the σ = ξ σ +1 (13) 2c Figure 2. Computed optimum strategy u s=1 (2a), productivity P (2b) ad NPV, IRR ad ROI values for s=1-10 (2c). If the strategy u s=1 is applied to all scearios s=1-10, the the NPV(s), IRR(s) ad ROI(s) values show i Figure 2c are obtaied. The miimum NPV-value is obtaied for sceario 2 ad is equal to NPV s=2= e+006 (15)

4 Applyig ow eq. (11), a ew strategy u s=1-10 (Figure 3a) is computed. u s=1-10 is as expecteddifferet from u s=1, as it serves ot oly oe, but all 10 scearios. The maximum miimum NPV-value is ow reduced to: max(mi(npv s=1-10))= e+005 (16) from the value of eq. (15). I order to facilitate the compariso of the basic sceario eq. (15) with other results, the u s=1 strategy ad the NPV s=1 value for P=1 are displayed i the relevat Figures, characterized by curves with black squares. 4a 4b 3a 3b 4c Figure 4. Case λ 1 =0.8, λ 2 =0.2 ad P L150 =0. Strategy u s=1-10 (4a), productivity P (4b) ad NPV, IRR ad ROI values for s=1-10 (4c). max(mi(npv s=1-10))= e+005 (17) 3c Figure 3. Case λ 1 =1.0 ad P L150 =0. Strategy u s=1-10 (3a), productivity P (3b) ad NPV, IRR ad ROI values for s=1-10 (3c). 5a 6 Variatio of the Productivity P I the followig Figures, the λ 1, λ 2 ad P L150 values are chaged (P L30 =0). The results are displayed i the ext Figures. 5b

5 Figure 6. Case λ 1 =0.8, λ 2 =0.2 ad P L150 =0.1. Strategy u s=1-10 (6a), productivity P (6b) ad NPV, IRR ad ROI values for s=1-10 (6c). max(mi(npv s=1-10))= e+005 (19) 5c Figure 5. Case λ 1 =1, λ 2 =0.2 ad P L150 =0.1. Strategy u s=1-10 (5a), productivity P (5b) ad NPV, IRR ad ROI values for s=1-10 (5c). max(mi(npv s=1-10))= e+005 (18) From Figures 4-5 but also from Figure 6 we observe how the P-parameters ifluece the results. The max(mi(npv s=1-10 )) value for P=1 decreases from e+005 to e+005 accordig to Figure 6 ad to e+005 accordig to Figure 7. 7a 7b 6a 6c 6b 7c Figure 7. Case λ 1 =0.0, λ 2 =0.4 ad P L150 =0.1. Strategy u s=1-10 (7a), productivity P (7b) ad NPV, IRR ad ROI values for s=1-10 (7c). max(mi(npv s=1-10))= e+005 (20) I 6 Coclusios I this paper we applied the mi-max strategy to a productio problem usig a determiistic modelig of the possible expected returs (future scearios) ad a ovel hybrid parameter optimizatio method for the computatio of the optimal risk-miimum strategy. The umerical results preseted show that the learig effect ad possible productivity losses greatly ifluece the risk of the plaed productio operatio.

6 Fially, the method performs well ad possesses a iterestig potetial for modellig more complex risk miimizatio problems. Computig to Computatioal Egieerig, Athes, 8-10 September, 2004 Literature: [1] Markowitz H. (1959), Portfolio Selectio: Efficiet Diversificatio of Ivestmets. Joh Wiley, New York, NY. [2] Rustem B., Becker R. ad Marty W. (2000), Robust mi-max portfolio strategies for rival forecast ad risk scearios, Joural of Ecoomic Dyamics & Cotrol 24, pp [3] Chag T.-J., Meade N.,Beasley J.E. ad Sharaiha Y.M (2000), Heuristics for cardiality costraied portfolio optimizatio. Computers & Operatios Research 27, pp [4] Korczak J., Lipiski P. ad Rogger P. (2002), Evolutio Strategy i Portfolio Optimizatio. R. Collet et al. (Ads.): EA 2001, pp [5] Mulvey J.M., Rosebaum D.P. ad Shetty B. (1997), Strategic fiacial risk maagemet ad operatios Research, Europea Joural of Operatioal Research 97, pp [6] Voeker P., Werers B. (2000), A decisio support system for busiess process activities, Europea Joural of Operatioal Research 125, pp [7] Fowler A. (2003), Systems modelig, simulatio, ad the dyamics of strategy, Joural of Busiess Research 56, pp [8] Gruee L., Semmler W. (2004), Usig dyamic programmig with adaptive grid scheme for optimal cotrol problems i ecoomics, Joural of Ecoomic Dyamics & Cotrol ( to be published). [9] McQuitty S. (2004), Statistical power ad structural equatio models i busiess research, Joural of Busiess Research 57, pp [10] Mulvey J.M. ad Shetty B. (2004), Fiacial activities via multi-stage stochastic optimizatio, Computers & Operatios Research 31, pp [11] Kaarachou G. ad Leopoulos V. (2003), Mi-max optimizatio of ivestmet processes usig a hybrid evolutio strategy, WSEAS Trasactios o Circuits ad Systems, Special Issue: Optimizatio, Simulatio, Modelig ad Cotrol i Systems Applicatios, pp [12] Kaarachou G. ad Leopoulos V. (2004), Optimizig fiacial idices i the case of a ew ecoomy ivestmet, Proceedigs of the 1st Iteratioal Coferece From Scietific

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