Proceedings of the 2005 Systems and Information Engineering Design Symposium Ellen J. Bass, ed.

Transcription

1 Proceedngs of the 5 Systems and Informaon Engneerng Desgn Symposum Ellen J. Bass, ed. TOWARDS A MODEL FOR SELECTION AND SCHEDULING OF RISKY PROJECTS ABSTRACT The plannng process of publc and prvate companes reles on opmal project selecon and schedulng and the effcent allocaon of scarce resources. Ths process s complcated due n part to the fact that project nvestment must consder mulple crtera, project cash flows are uncertan, and there are several operaonal busness and techncal constrants. The proposed mxed-nteger programmng model asssts the plannng manager/analyst by choosng from a bank of projects n whch projects to nvest and when to nvest. The model maxmzes the sum of net present values of the chosen projects whle mnmzng ther varance. The model sasfes smultaneously a set of precedence relaons among projects; early and tardy project starng dates; exogenous budget lmts; and endogenous project cash flow generaon. Fnally, by quanfyng the opportunty cost, the model shows how arbtrary project selecon and sequencng can reflect non-desrable soluons for the company and the socety. 1 INTRODUCTION Jorge A. Sefar Industral Engneerng Department School of Economcs Unversdad de los Andes Bogotá, Colomba The plannng process for companes s complex due to the great amount of nvestment projects, the nterrelaon between them (Vonortas and Hertzfeld, 1998; Chlds, Ott, and Trans, 1998), the mulple crtera that can be relevant when evaluang each alternave (Benjamn, 1985; Ehe et. al 199), and the constrants nherent to the corporate operaon (resources, me, regulaon, among others). The mathemacal models proposed n the project selecon lterature facltate the decson-makng process avodng the use of subjecve crtera that may generate certan states suscepble of mprovement or sub-opmal soluons wth harmful consequences for the company or the socety. Snce the poneer work of Lore & Savage (1955), the project selecon problem has attracted several researchers. Many technques have been appled to the project selecon problem: lnear programmng (Benhard, 1969; Freeland and Rosenblatt 1978; Myers, 197), mulobjecve lnear programmng (Rnguest and Graves 1989; Rnguest and Graves, 199), nteger programmng (Beged-Dov, 1965), goal programmng (Benjamn 1985; Mukherjee and Bera, Andrés L. Medagla Industral Engneerng Department Unversdad de los Andes Bogotá, Colomba 1995), and evoluonary algorthms (Medagla, 3) among others. Benl and Yavuz () have addressed mng and sequencng n project selecon problems usng zero-one programmng. Ther model provdes starng dates for the projects, whle maxmzng the net present value (NPV). Gupta, Kyparss and Ip (199) and Kyparss, Gupta and Ip (1995) use the same crtera n order to select and sequence projects. Mulple crtera have been used to quanfy the projects performance both n selecon and sequencng problems. The most frequently used crtera are NPV (Gupta, Kyparss and Ip 199; Kyparss, Gupta and Ip, 1995; Wengartner, 1967) and rsk (Kangar and Boyer, 1981; Orman and Duggan, 1999; Stone 1973). Several researchers have handled rsk usng the tradonal Markovtz methodology (Markowtz, 195). To the best of our knowledge, the proposed model s a novel addon to the exsng project selecon lterature. It combnes the project selecon and sequencng decsons, whle consderng rsk and proftablty as opmzaon crtera. The uncertanty present n the forecasts of the projects cash flows s the source of the NPV s varance. The model provdes an ntertemporal rsk dversfcaon by ncludng NPV covarance terms for all projects and all starng dates. The proposed model s an extenson of our experence n one of Colomba s largest water and sewage companes, where a determnsc model was successfully desgned and mplemented (Medagla et. al, 5). Ths arcle s dvded nto four secons. Secon defnes the portfolo s expected return and varance when usng forecasts. Secon 3 contans the formulaon of the proposed mxed-nteger programmng model. In Secon 4, we provde computaonal experments usng a set of sample projects. In ths Secon, we also perform a sensvty analyss and show how to construct an effcent froner. We conclude n Secon 5. PROFITABILITY AND RISK UNDER THE FORECAST APPROACH The portfolo s proftablty s measured by the net present value (NPV). Let P be the set of nvestment projects to be consdered. Let T be the plannng horzon (no nvestments

2 are made after perod T). Let y t be a varable that takes the value of 1 f the project ( P) begns n the year t (t {,1,,T}); t takes the value of, otherwse. Let NPV t be the net present value of the project gven that t starts n the year t of the plannng horzon. Therefore, t s possble to express the portfolo s NPV as follows: NPV % T = P NPV y When dealng wth nvestment projects wthout hstorcal nformaon (e.g., nfrastructure projects), the NPV for each project must be modeled accordng to the followng steps: Step 1 Varable Idenfcaon: Varables that can be related to the project s cash flows must be denfed. Step. Varable Forecast: The denfed varables are forecasted. The varance assocated wth the forecast must be captured for each perod. Step 3 Varable-Cash Flow lnkage: Each project s cash flows must be bult n accordance to the forecasts found n Step. Step 4 Smulaon: Usng each forecast and ts varance we must smulate varous realzaons for each varable. Ths step provdes mulple realzaons for the NPV, one for each starng date and project. Wth these realzaons, t s possble to calculate the NPV s mean, varance, and covarance. Lets assume that s the starng date of project P. Even though s a decson to be determned, we assume that t s known to llustrate a set of mportant defnons used from ths pont on. Let β be the dscount factor. Let v be the length or lfespan of the project P, meanng the number of perods ncluded between the frst nvestment and the last posve cash flow (project s return). Let F t (?) be the cash flow for the project n perod t {,1,,T}, whch s affected by the random component? present n the forecast. The NPV for the project starng n perod, follows the formula: NPV v t = β Ft t t ( ω) We measure rsk by compung the NPV s volalty. The nature of ths varablty arses from the cash flows forecast. Let σ t be the NPV varance for the project starng n year t. Let cov(npv,npv jt(j) ) be the covarance between the NPV for the project starng n and the NPV for the project j starng n t(j). The varance for the portfolo s return can be expressed as follows: σ% = E ( NPV % E NPV % ) { } T T = E NPV y E NPV y t t t t P P Sefar Medagla 159 T T = y yjt( j) cov( NPV, NPVjt( j) ) (1) P j P = t( j) = Note that the terms n the form of cov(npv,npv jt(j) ) for =j and t(j) are equal to zero due to the fact that a project s started only once durng the plannng horzon. The varance for the NPV of project gven that t starts n perod t(j) can be expressed as follows: v t σ = Var β Ft ( ω) An approxmaon of ths expresson (and the covarance) can be tackled va Monte Carlo smulaon usng forecasts. Let H be the forecast horzon for the varables affecng the cash flows. Let f t be the forecast value for perod t {,1,,H} for a gven varable and σ (f t ) ts varance. The proposed model does not strctly requres an specfc dstrbuon for the forecasts, but for reasons explaned n Secon 4.1, we could assume that the forecast value s normally dstrbuted wth parameters (f t,σ ft). In general, a Monte smulaon experment can be conducted to esmate the average NPV t, ts varance (σ t) and covarance (cov(npv,npv jt(j) )). 3 MODEL Let P be the set of projects to be selected and sequenced. Let A be the set of precedence relaons between projects, that s, f project P precedes project j P, then (,j) A. Let u be the length or lfespan of negave cash flows measured n perods of me (months, years), that s, the number of perods between the frst and last nvestment costs. Let t be the earlest starng date for project, meanng the earlest perod n whch the project can be started. Let t be the latest starng date for project, meanng the tardest perod n whch the project can be started. Therefore, must be guaranteed. Let N l and N u be the mnmum and maxmum number of projects to be carred out, respecvely. Let g j be the gap allowed between precedence relaons, meanng that f project precedes project j, g j ndcates the number of perods of separaon or overlap between them. For example, f the overlap of nvestment perods s allowed between projects and j, then g j =-1. In case, a strct precedence s requred (no overlap or separaon), then g j =. Let r o t be the amount of avalable resources for nvestment (.e., budget) for year t, t {,1,,T}. Let c k be the nvestment cost (negave cash flow) for project n perod k, k {,1,,u -. On the other hand, let b t be the expected fnancal ncome (posve cash flow) generated by project n perod t. Let us recall that y t s the bnary decson varable that takes the value of 1 f project starts on year t ( t,,mn{ t, T-u ); t takes the value of, otherwse. Let x kt be the bnary decson varable that takes the

3 value of 1 f for project, the perod k (k=, v -1) s assgned to year t n the plannng horzon; ( t,,mn{ t v -1, T}); t takes the value of, otherwse. Let r t be the amount of nvestment resources not used at the end of perod t, and carred over as budget for the next perod t1. Fnally, let z jt(j) be a bnary varable wth value of 1 f the projects and j begn n perods and t(j), respecvely; t takes value of, otherwse. The two objecve funcons are shown n equaons and 3. max mn{, T u NPVtyt () P mn{ t, mn{ tj, T uj T u mn z cov( NPV, NPV ) P j P = t( j) = tj j( t j) jt( j) (3) Equaon () s the crteron that maxmzes the portfolo s NPV or sum of net present values of the chosen projects, whereas equaon (3) mnmzes the portfolo s NPV varablty. The frst set of constrants tells the model that every project may or may not be selected. It ndcates that only some of the projects wll be part of the opmal portfolo. Ths s expressed n Equaon 4. mn{, T u yt 1, P (4) The number of projects n the portfolo can be controlled mposng lower and upper bounds K l and K u, respecvely (see Equaon 5). mn{, T u (5) N y N l t u P As shown n Equaon 6, t s necessary to acvate the correspondng perods of nvestment and ncome generaon, once a project starts on a gven perod. Equaon 7 guarantees that every perod of nvestment s assgned (to a gven year n the plannng horzon) at most once for each project. y x ; k =,..., v 1, t = t,...,mn{ t, T u (6) t kt k mn{ v1, T} xkt 1 ; P, k =,..., v 1 (7) Equaon 8 shows how the precedence relaons are modeled. Equaon 9 explctly forbds starng projects when t s not possble for them to be carred out accordng to the precedence relaons. From a computaonal pont of vew, ths varable fxng could be acheved effcently usng preprocessng and not explct constrants as shown n Equaon 9. tugj t t' (, ), j ; = j,..., j t' = y y j A t u g t t t (8) y = ; (, j) A, tu g < ; t = t,..., t (9) t j j j Sefar Medagla Equaon 1 shows the budget constrant. Ths constrant ncludes the resources comng from the prevous perod and the fnancal ncome generated by the projects prevously carred out. u1 v1 t 1 t t k kt t kt Pk= Pk= u (1) r = r r c x bx ; t =,.., T Fnally, the famly of constrants n Equaon 11 shows the exsng relaon between varables y s and z s, allowng a lnearzaon of the non-lnear rsk measure shown n Equaon 1. y yjt( j) zj( t j) (11) y y z 1 jt( j) j( t j), j P, th ( ) = th,...,mn{ th, T uh for h =, j The bobjecve mxed-nteger problem descrbed by Equaons through 11, can be solved n two phases. In the frst phase, one of the crtera s maxmzed or mnmzed (dependng f t s Equaon or 3), subject to all constrants (Equaons 4 through 11). In the second phase, the second crteron s opmzed ncludng an addonal restrcon that avods the deteroraon of the frst objecve, * guaranteeng Pareto opmalty (Steuer, 1986). Let NPV % * and σ% be the opmal values for each crteron obtaned n the frst phase. Equaons 1 and 13 show the constrants used n the second phase, dependng on the objecve selected n the frst phase. Note that just one and only one wll be used n the second phase. mn{, T u * NPVtyt NPV % (1) P mn{ t, mn{, T u tj T uj * zj( t j) cov( NPV, NPVjt( j) ) σ% P j P = t( j) = tj 4 COMPUTATIONAL EXPERIMENTS 4.1 A Ten-project Example (13) Consder a company wth 1 project nvestment opons. The company wshes to carry out ts nvestment plan for the next 13 years starng n 4. Fgure 1 shows the forecast for the ncome (cash) flows and ther assocated confdence nterval (dotted lnes). For nstance, f the me seres methodology s used for forecasng, t s possble to model the ncreasng uncertanty of dstant perods. We assume that the forecasts are normally dstrbuted to use the NPV varance as a measure of rsk. For a thorough dscusson about measures of rsk the reader s referred to Szëgo, 4. 16

4 Sefar Medagla Table1: Investment Costs Project Perods of Investment 1 p1 46 p 53 p3 18 p p p6 1 p p p p COP Mllons Fgure : Investment Budget Fgure 1: Forecasts (wth a 95% confdence nterval) of the ncome (cash) flows for each project Let us assume, that the company has certanty of the nvestment costs for each project. Ths negave cash flows are shown n Table 1. Wth these nvestment costs and forecasted (posve) cash flows t s possble to obtan, va Monte Carlo smulaon, the average NPV t, ts varance (σ t) and the covarance matrx (cov(npv,npv jt(j) )). The dscount rate used n ths example s 14%. Fgure shows the avalable nvestment budget durng the plannng horzon. For the frst few years, the company shows an hgh budget that tends to stablze by 6. A shock that reduces the budget s foreseeable at the year 13. Table shows the early ( t ) and tardy ( t ) starng dates, nvestment lengths (u ), lfespans (v ), and precedence gaps (g j ) for the set of 1 projects. These exogenous restrcons orgnate from prevous plannng exercses, market knowledge, techncal needs or polcal rules. The company requres to undertake all of ts projects (N l =N u =1). Table : Tme wndows, nvestment length, project lfespan and precedences Project u v t t g j p p p p g 8,4 = p p p p p g 9,1 = p Results Fgures 3a and 3b show the results obtaned for the opmal sequencng. We show n gray only the perods wth nvestments. Note the complance to the precedence relaons. Fgure 3a shows the results for the case when the maxmzaon of the portfolo s NPV s used as the crteron for the frst phase. Fgure 3b shows the results when the mnmzaon of varance s used n the frst phase. The portfolo s NPV as well as ts varance are shown for each case (see lower rght of each fgure). 161

5 Sefar Medagla 4.3 Sensvty Analyss Fgure 3: Opmal sequencng (a) Maxmzng NPV n the frst phase (b) Mnmzng varance n the frst phase These results suggest that the company should choose the project sequencng that balances the exsng tradeoff between rsk and proftablty. Fgure 4 shows the set of opmal portfolos generated through an terave procedure. Ths procedure s dvded nto two parts. Frst, the portfolo s varance s mnmzed subject to a fxed NPV * o value ( NPV % = NPV n equaon 1). Second, the portfolo s NPV s maxmzed subject to the varance obtaned n the prevous step. Ths two-step procedure s repeated for each value of the dscrezed NPV s doman. Ths terave process, over non-decreasng values of the project portfolo % *, ulmately produces the effcent froner shown n NPV Fgure 4. Note that the second step of ths process assures that the ponts obtaned are ndeed Pareto opmal. To assure that a good sample of the effcent froner s obtaned, the same process s carred out starng wth the NPV maxmzaon subject to fxed varance values. By dscrezng the varance doman and terang over t, t s possble to unvel some new ponts of the effcent froner. Abrupt changes may be due to the nature of the problem tself; small changes n the NPV s target may generate a radcal change n the project sequencng. When ncorporang the company s ndfference curve, we obtan that the best portfolo s tangent to the hghest ndfference curve (pont A). Fgure 4 shows the effcent froner formed by a dscrete set of black damonds and dots (extreme effcent ponts). Furthermore, the connuous lne shows the hypothecal ndfference curve for the company. Suppose that the tardy starng dates for all the projects ( t =T, P) are relaxed. Fgures 5a and 5b show the results for ths scenaro. Consderng the NPV as a crteron for the frst phase, we observe that project p7 s pushed away from ts nal starng date, causng an ncrement of % n the portfolo s NPV and a reducon of 58% n the portfolo s varance. Takng nto account the mnmzaon of the varance n the frst phase, we observe that 4 projects are postponed, ncreasng the portfolo s NPV by 61% and decreasng the rsk by 33%. Ths case shows that f the restrcons mposed on the tardy starng dates are not real or respond to subjecve crtera (perhaps polcal), t s not possble to effcently allocate the nvestment resources, resulng n states suscepble of mprovement and prevenng the company or the socety as a whole from accessng better returns or lower uncertanty. Fgure 5: Late starng date relaxaon scenaro (a) NPV n the frst phase (b) Varance n the frst phase. Now assume that the company wshes to program a varable number of projects, allowng the selecon of the number of projects between N l = and N u = 1. Also assume that we allow the overlap of nvestment perods between projects p9 and p1, that s, g 9,1 = -1. The remanng parameters are not altered (ncludng the orgnal restrcons for the starng dates). Fgure 6 shows the results for ths new scenaro. We observe that project p7 s removed from the opmal bank of projects, resulng on almost doublng the portfolo s NPV and an even more sgnfcant rsk reducon. The allowed overlap between projects p9 and p1 also contrbutes to the portfolo s mprovement. Ths shows that by arbtrarly fxng a project, we may cause the resource allocaon to be neffcent by undertakng rsky projects that do not generate added value. Fgure 4: Opmal portfolos and the company s ndfference curve Fgure 6: Opmal sequencng after relaxng the lmts on the number of projects 16

6 The model uses the concept of endogenous resources represented by the projects posve cash flows and by carryng over nto the next perod the resources that were not used n the prevous perod. Note that the opmal sequencng shown n Fgures 3a and 3b s feasble due n part by consderng the nal budget of the company (r o t ) along wth the endogenous resources. Fgure 7 emphaszes the fact that the proposed sequencng s not feasble wth just the nal allocated budget Investment nal Budget Inal Budget Fnancal Income Inal Budget Fnancal Income Cumulave Fgure 7. Budget and nvestment Sefar Medagla nclude a great number of projects, busness constrants, and nterrelaons that would have otherwse been dffcult to nclude. The model deals wth projects that have not been prevously carred out. We have outlned a methodology that uses forecasts of underlyng varables lnked to the projects cash flows. To generate the cash flows, we propose the use of Monte Carlo smulaon, makng the calculaon of the portfolo s varance and NPV possble. The model carres out an ntertemporal rsk dversfcaon by ncludng the covarance between the projects returns gven ther respecve starng years. Thus, the model does not only accounts for a negave correlaon between returns referrng to the same me perod, but also through the varaon of the project s starng dates. A negave covarance reduces the portfolo s volalty, schedulng projects n a me perod where the covarance s as negave as possble. Ths model also shows that arbtrary decsons concernng projects may provde non-desrable soluons. For companes n the publc sector, the selecon and sequencng of nvestment projects s subject to external pressures that can nfluence the selecon of sub-opmal portfolos that do not guarantee the best allocaon of resources. Usng the proposed model, a company can quanfy the sacrfced profts and the addonal rsk caused by arbtrary (perhaps polcal) decsons. Fnally, ths model contrbutes to the promoon of a plannng culture nsde the companes, especally those n the publc sector, by usng techncal crtera that wll allow them to reach the best possble allocaons for the company and the socety as a whole. 5 CONCLUDING REMARKS The proposed model can become part of a larger decson support tool for companes nterested n selecng and schedulng a set of projects whle smultaneously maxmzng return and mnmzng rsk. The model s especally mportant for publc companes who have the responsblty of effcently allocang resources to obtan the best possble results for the socety as a whole. The model provdes valuable nformaon for a company wth lmted resources or wth dffcules obtanng credt. The proposed model s able to program future fnancal requrements and uses endogenous cash generaon ntellgently. It s also possble to model legal, techncal or busness restrcons faced by most companes. A sample of these restrcons are early and tardy project starng dates, precedence relaons among projects, and lower and upper bounds on the number of projects n the bank of projects. The proposed mxed-nteger program provdes a lnearzed approach to a problem that orgnally s quadrac (see Equaon 1). Our prelmnary experments show that the model scales well computaonally, thus allowng us to REFERENCES Beged-Dov, A.G Opmal assgnment of R&D projects n a large company usng an nteger programmng model. IEEE Transacons on Engneerng Management. EM-1: Benhard, R.H Mathemacal programmng models for captal budgeng - a survey, generalzaon, and crque. Journal of Fnancal and Quantave Analyss. 4 (): Benjamn, C.O A lnear goal-programmng model for publc-sector project selecon. Journal of the Operaonal Research Socety. 36 (1): Benl, Ä O. and Yavuz, S.. Makng project selecon decsons: a mul-perod captal budgeng problem. Internaonal Journal of Industral Engneerng. 9 (3): Chlds, P.D and Ott, S. and Trans, A. J Captal budgeng for nterrelated projects: a real opons approach. Journal of Fnancal and Quantave Analyss. 33 (3): Ehe, I.C. and Benjamn, C.O. and Omurtag, I. and Clarke, L Prorzng development goals n low-ncome 163

7 developng countres. OMEGA Internaonal Journal of Management Scence. 18 (): Freeland, J.R. & Rosenblatt, M.J An analyss of lnear programmng formulaons for the captal raonng problem. The Engneerng Economst, 3: Gupta, S.K. and Kyparss, J. and Ip, Ch-Mng Project selecon and sequencng to maxmze net present value of the total return. Management Scence. 38 (5): Notes. Kangar, R. and Boyer, L.T Project selecon under rsk. Journal of the Construcon Dvson. 17 (CO4): Kyparss, G. and Gupta, S.K. and Ip, Ch-Mng Project selecon wth dscounted returns and mulple constrants. European Journal of Operaons Research. 94. Lore, J.H. and Savage, L.J Three problems n raonng captal. The Journal of Busness. 8 (4): Markovtz, H Portfolo selecon. Journal of Fnance. 7 (1):47-6. Medagla, A. L. 3. An evoluonary algorthm for project selecon problems based on stochasc mulobjecve lnearly constraned opmzaon. Chapter n Graves, S. B. and Rnguest, J. L., Models and methods for project selecon: concepts from management scence, fnance, and nformaon technology. Boston: Kluwer Academc Publshers. Medagla, A. L. and Mendeta, J. C. and Hueth, D.L and Sefar, J.A. 5. A Mulobjecve Model for the Evaluaon and Tmng of Publc Enterprse Projects. Workng Paper. Unversdad de los Andes. Mukherjee, K. and Bera, A Applcaon of goal programmng n project selecon decson: a case study from the Indan coal mnng ndustry. European Journal of Operaonal Research. 8:18-5. Myers, S.C A note on lnear programmng and captal budgeng. The Journal of Fnance. 7:89-9. Orman, M.M. and Duggan, T.E Applyng Modern Portfolo Theory to Upstream Investment Decson Makng. Journal of Petroleum Technology. 51(3):5-53. Rnguest, J. and Graves, S The lnear mulobjecve R&D project selecon problem. IEEE Transacons on Engneerng Management. 36 (1): Rnguest, J. and Graves, S The lnear R&D project selecon problem. An alternave to net present value. IEEE Transacons on Engneerng Management. 37 (): Steuer, R. E Mulple Crtera Opmzaon: Theory, Computaon and Applcaon, Wley Seres n Probablty and Mathemacal Stascs, John Wley, New York. Sefar Medagla Stone, B A lnear programmng formulaon of the general portfolo selecon problem. The Journal of Fnancal and Quantave Analyss. 8 (4): Szëgo, G. 5. Measures of rsk. European Journal of Operaonal Research. 163:5-19. Vonortas, N. and Hertzfeld, H Research and development project selecon n the publc sector. Journal of Polcy Analyss and Management. 17 (4): Wengartner, H.M Crtera for programmng nvestment project selecon. The Journal of Industral Economcs. 15 (1): AUTHOR BIOGRAPHIES JORGE A. SEFAIR C. receved a B.Sc. (5) n Industral Engneerng from Unversdad de los Andes. He s a B.A. student n Economcs at Unversdad de los Andes. He works at the Center for Studes n Economc Development (CEDE) snce, and at the Centro para la Opmzacón y Probabldad Aplcada (COPA) snce 4. He can be reached by e-mal at j-sefar@unandes.edu.co. ANDRÉS MEDAGLIA receved a B.Sc. (199) n Industral Engneerng from Unversdad Javerana and a M.Sc. (1995) n Industral Engneerng from Unversdad de los Andes. He holds a Ph.D. (1) n Operaons Research from North Carolna State Unversty. In 1-, he worked as a postdoctoral fellow n the Industral Engneerng Department at N.C. State, sponsored by SAS Instute Inc. (Cary, North Carolna). In, he was apponted Assstant Professor of the Industral Engneerng Department at Unversdad de los Andes. He s a foundng member of the Centro de Opmzacón y Probabldad Aplcada (COPA). He can be reached by e-mal at amedagl@unandes.edu.co and hs web address s 164