A robust R&D project portfolio optimization model for pharmaceutical contract research organizations

Pharmaceutical drug Research and Development (R&D) outsourcing to contract research organizations (CROs) has experienced a significant growth in recent decades and the trend is expected to continue. A key question for CROs and firms in similar
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  A robust R&D project portfolio optimization model for pharmaceuticalcontract research organizations Farhad Hassanzadeh a, n , Mohammad Modarres b , Hamid R. Nemati a ,Kwasi Amoako-Gyampah a a Department of Information Systems and Supply Chain Management, Bryan School of Business and Economics, University of North Carolina at Greensboro,Bryan Building, Greensboro, NC 27412, USA b Department of Industrial Engineering, Sharif University of Technology, Tehran 11365-9363, Iran a r t i c l e i n f o  Article history: Received 6 May 2013Accepted 1 July 2014Available online 9 July 2014 Keywords: Pharmaceutical contract researchorganization (CRO)Research and Development (R&D)Project selection and schedulingUncertaintyRobust optimization a b s t r a c t Pharmaceutical drug Research and Development (R&D) outsourcing to contract research organizations(CROs) has experienced a signi fi cant growth in recent decades and the trend is expected to continue. Akey question for CROs and  fi rms in similar environments is which projects should be included in the fi rm's portfolio of projects. As a distinctive contribution to the literature this paper develops andevaluates a business support tool to help a CRO decide on clinical R&D project opportunities and reviseits portfolio of R&D projects given the existing constraints, and  fi nancial and resource capabilities. A newmathematical programming model in the form of a capital budgeting problem is developed to helprevising and rescheduling of the project portfolio. The uncertainty of pharmaceutical R&D cost estimatesin drug development stages is captured to mimic a more realistic representation of pharmaceutical R&Dprojects, and a robust optimization approach is used to tackle the uncertain formulation. An illustrativeexample is presented to demonstrate the proposed approach. &  2014 Elsevier B.V. All rights reserved. 1. Introduction Americans spent over $2.6 trillion on healthcare last year. Thisrepresents about 17.9% of the total US GDP. The World HealthOrganization estimates that the healthcare share of US GDP couldclimb to 34% by 2040 and warns of adverse consequences for theworld economy if the current cost trajectory is not corrected. Acloser look at the healthcare expenditure shows that pharmaceu-ticals accounts for over 12.9% of total expenditure and it isprojected to be the fastest growing portion of healthcare spending.This is due to high prices of prescription drugs, particularly brandname and specialty drugs, and rising costs associated with theResearch and Development (R&D) of new drugs (CMS Report,2011). The pharmaceutical industry has long argued that theprocess of drug discovery through R&D is very expensive andrequires substantial capital expenditure. For example, an averagecancer drug costs around $1.75 billion to research and develop andmay take up to 10 years to test and market. In 2008, Americanpharmaceutical companies spent over $45 billion on developingnew drugs or modifying existing drugs. According to the Scienceand Engineering Indicators 2012 report published by the NationalScience Foundation, pharmaceuticals and medicines are the high-est R&D intensive industries in the world after semiconductor andcommunication industries. The average R&D intensity in thepharmaceutical industry — the ratio of total R&D spending to totalsales revenue — is 12.2%, which is more than three times that of theaverage manufacturing  fi rm in the US.R&D expenditures in the pharmaceutical industry might beas high as 40% of the cost of a newly developed drug (Gassmannet al., 2008). A signi fi cant contributor to the high R&D costs inpharmaceutical drug development projects is the high prevalenceof technological and market uncertainties (Rogers et al., 2002).Technological uncertainties are related to the ef  fi cacy and safety of the drugs being developed while market uncertainties are relatedto the supply and demand factors in the marketplace. Despitethese uncertainties, pharmaceutical companies have increasinglygrown their expenditures on R&D in an effort to boost pro fi tabilitythrough the introduction of novel drugs for treatment of variousailments (Lowman et al., 2012). This rising expenditure in phar-maceutical R&D projects is due to increases in cost of discoveringContents lists available at ScienceDirect journal homepage: Int. J. Production Economics &  2014 Elsevier B.V. All rights reserved. n Correspondence to: Department of Information Systems and Supply ChainManagement, Bryan School of Business and Economics, University of NorthCarolinaat Greensboro, 386 Bryan Building, Greensboro, NC 27412, USA. Tel.:  þ 1 336 2560189; fax:  þ 1 336 334 5580. E-mail addresses: (F. Hassanzadeh), (M. Modarres), (H.R. Nemati), (K. Amoako-Gyampah).Int. J. Production Economics 158 (2014) 18 – 27  new drugs, as well as higher costs associated with conductingclinical trials for developing these drugs. It is also important tonote that recently discovered drugs with demonstrated higherlevels of sophistication may require an even more costly and time-consuming clinical trial phases to ensure their safety and ef  fi cacybefore they could be introduced to the market (Craig and Malek,1995; DiMasi and Grabowski, 2007).As shown in Fig. 1, pharmaceutical drug development pipelineincludes drug discovery, pre-clinical and clinical trials, FDA review,and production and marketing phases. Among these developmentphases, clinical trials are the most time-consuming andinvestment-intensive ones. Clinical trials take about 6 years tocomplete and represent more than 50% of total pharmaceuticalR&D spending (Zeller, 2002; Mirowski and Van Horn, 2005;Parexel, 2004).Given the cost and complexities associated with developingnew drugs, pharmaceutical companies typically outsource some of their drug development activities to specialized organizations inorder to better focus on their own core competencies. Thesespecialized organizations are referred to as contract researchorganizations (CROs). CROs were  “ born ”  in the late 1970s andquickly assumed a signi fi cant role in the pharmaceutical industryin order to help the industry balance the need to consolidate itsoperations while simultaneously address getting drugs quicklythrough the development pipeline (Piachaud, 2002). The Associa-tion of Clinical Research Organizations (ACRO) estimates that over46% of pharmaceuticals have outsourced their R&D projects toCROs. ACRO also estimates that CROs employ over 2 million peopleand are present in 115 countries (ACRO, 2013). The CRO industryrepresents over 33% of the total spending on pharmaceutical R&D(Lowman et al., 2012). It is estimated that in the near future, CRO'sshare of total pharmaceutical R&D will exceed 60% (ACRO, 2013).The leading CROs manage more than 45,000 clinical trials annuallywith revenues in excess of $20 billion (Milne and Paquette, 2004;Tufts Center, 2006a; Getz, 2007). The CRO industry is also veryconcentrated; the top 10 out of existing 1000 CROs in the globalmarket control more than 80% of the total market share(Shuchman, 2007; Getz, 2007). This indicates that the CRO marketconsists of a few big multinational companies with R&D depart-ments even larger than some of their pharmaceutical clients alongwith numerous small or medium sized companies with a niche innational or regional markets (Piachaud, 2002).The CRO market has expanded from drug discovery andpreclinical work to clinical trials, drug manufacturing, and evenmarketing (Tufts Center, 2006b; Mehta and Peters, 2007). While itseems that it is the increase in the quantity of R&D projects thathas promoted the need to outsource clinical trials, there arebasically other motivations for pharmaceutical companies to usemore outsourcing (Piachaud, 2002). These include higher costef  fi ciency (Huang et al., 2009), less time to market (Mahnke et al., 2006), increased opportunity to gain needed knowledge,availability of advanced skills and technologies (Coombs et al.,2003), and the increased globalization of drug development(Gassmann et al., 2010). The CRO of today is a key driver of drugdevelopment success (Lowman et al., 2012).Although the practice of drug development outsourcing to CROhas been the motivation of numerous researchers (Alexander andYoung, 1996; Rettig, 2000; Dickert et al., 2002; Piachaud, 2002;Quelin and Duhamel, 2003; Mirowski and Van Horn, 2005; Angell,2008; Hsuan and Mahnke, 2011; Zirpoli and Becker, 2011), themajority of conducted studies are descriptive in nature and havebeen studied from the perspective of the goal attainment for thepharmaceutical companies and not for the CROs. Within thepharmaceutical business, for example, short time to marketincreases the novelty of a potential blockbuster drug to achieve aprolonged competitive advantage (Arlington, 1997; Piachaud,2002). A prolonged clinical testing may signi fi cantly reduce thecommercial value of a drug or may even render the whole projectinfeasible (Bauer and Fischer, 2000). In fact, studies show that fi nancial and commercial reasons account for more than one thirdof research abandonment which often occurs during late clinicaltesting phases (DiMasi, 2001). Thus, the role of the CRO is verycritical in achieving the drug development goals of client organi-zations. A 2010 survey of about 400 drug manufacturers andbiotech companies showed a potential growth of about 4 – 8% inCRO R&D budgets, indicating that the number of outsourcingactivities is on the upward trend. Given this trend, the questionthat needs to be asked is whether CROs have the ability to absorball the demand from client organizations. And if they do not, howshould they balance their capabilities with the contract projectloads in order to sustain long term pro fi tability and growth? Inother words, CROs must decide on which R&D projects to includein their optimum mix of project portfolio given their capacityconstraints and pro fi tability goals. Selecting a wrong mix of projects not only adversely impacts the contractual and  fi nancialobligations, but may also reduce the ability to successfully executeother projects already in the portfolio.Realizing this necessity, the goal of this paper is to develop abusiness support tool to help CROs make their contract decisionseffectively by integrating project opportunities with existingtechnical,  fi nancial, and resource restrictions within a mathema-tical model. There is very limited research on CROs in this context.The closest body of literature that underpins such models isreferred to as the project selection and scheduling (PSS). Sincethe PSS literature is developed for generic projects, we need tomodify the problem de fi nition to account for pharmaceutical R&Dprojects as a special category of R&D projects. From a modelingperspective, the distinguishing characteristics of R&D projectsfrom those of generic projects occur on the highly uncertainnature of R&D projects in that R&D cost and revenue estimatesare very unreliable and the market outcome is very risky (DiMasi,2001). To account for this uncertainty we employ a recentlydeveloped approach, called robust optimization, to solve ourformulated model for pharmaceutical R&D project portfolio deci-sion making.The remainder of this paper is organized as follows. In the nextsection, we review the PSS literature, introduce the robust opti-mization concept, review its recent applications in project selec-tion and project scheduling, and discuss its relevance to R&D PSS.In Section 3, we formally de fi ne the problem and propose ournominal model. Section 4 is dedicated to the introduction of arobust optimization framework and formulation of the robustcounterpart model. In Section 5, we present the results of ourrobust CRO portfolio optimization approach using an illustrativeexample. Finally, Section 6 provides some managerial implicationsalong with the conclusions. Fig. 1.  Pharmaceutical drug development pipeline. F. Hassanzadeh et al. / Int. J. Production Economics 158 (2014) 18 –  27   19  2. Literature review  In the project management literature, project decision isusually considered the same as the project selection. As a result,the majority of studies generally do not include project schedulingas part of the decision process (Cof  fi n and Taylor, 1996). As such,most of these studies present models that are usually designed toselect a subset of projects from a larger pool of candidate projectsto meet return objectives and budget constraints (Fox et al.,1984).In addition, these models sometimes consider resource allocation(Taylor et al.,1982) and project risk (Heidenberger,1996; Gabriel et al., 2006) in the selection process. Project scheduling becomes asubsequent follow-up activity after the selection process.There are a few mathematical models that include the schedul-ing aspect of projects as part of the decision process. The resultingPSS problem can be classi fi ed into two different categories. In the fi rst category each project is composed of a set of tasks requiringspeci fi c resources to complete, and precedence relations amongtasks (within a project or between projects) exist. The major focusof this category is on the scheduling aspect of tasks, and theproposed models and solution techniques are similar to what isgenerally referred to as the resource constrained project schedul-ing problem (Cof  fi n and Taylor,1996; Kolisch and Hartmann, 2006;Chen and Askin, 2009). In the second category it is assumedthat each project is composed of a set of stages which, uponselection of the project, must be sequentially accomplished. Themajor focus of this category is on selection rather than schedulingof projects.The  fi rst notable work in the second category, which ispertinent to R&D projects and hence our problem, is presentedin Ghasemzadeh et al. (1999) where a multiobjective binaryinteger linear model with additive objectives is developed. Theyconsider resource limitations and interdependences among pro- jects and comment on the issue of sensitivity of the resultingportfolio to problem parameters. To address the schedulingproblem resulting from the uncertainty in parameters, they re-optimize the model for sensitive parameter values through aninteractive process. Benli and Yavuz (2002) address this problemdifferently by presenting a binary integer linear model where totalnet present value of selected projects is maximized and capitalbudgets and precedence relations among projects are considered.Sefair and Medaglia (2005) extend this work by incorporatingexogenous cash  fl ow generation and considering risk minimiza-tion as the second objective function. Zuluaga et al. (2007) extendSefair and Medaglia (2005) by considering budget and bene fi tinterdependence among projects, but do not consider the secondobjective of risk minimization. Sun and Ma (2005) address thisproblem within the R&D setting of a company and attempt to  fi nda solution that maximizes the total priority of projects whilebudget restrictions in future time periods are respected. Theyformulate the problem as a simple packing multiple boxes modeland heuristically solve it in a sequence of binary integer linear(Knapsack) models. Medaglia et al. (2008) address the selectionand scheduling of public sector projects subject to resourcelimitations and precedence constraints, where endogenous projectcash  fl ow generations in addition to exogenous budgetary limitsexist. They develop a multiobjective mixed integer linear pro-gramming (MILP) model with additive objectives and performsensitivity analysis on various problem parameters. They alsoaccentuate the issue of solution robustness and illustrate howrobustness may be used as a negotiation tool in a politicalenvironment. Carazo et al. (2010) develop a multiobjective non-linear binary integer model to select and schedule a projectportfolio where all problem data are deterministic, and resourceconstraints, project interdependence, and the possibility of trans-ferring resources to the following period are taken into account.They assume no  a priori  information on the objectives' preferencesand propose a metaheuristic procedure based on scatter search todetermine the set of (Pareto) ef  fi cient portfolios.It is worth noting that previous studies have addressed the PSSproblem from a deterministic point of view. Some studies (e.g.Ghasemzadeh et al.,1999; Medaglia et al., 2008) have commentedon the uncertainty of parameters and used sensitivity analysis toinvestigate the stability of the solutions with regard to uncertaintyin the data. We emphasize that sensitivity analysis, as a  post-optimization  tool, measures sensitivity of the solution only regard-ing small changes in the problem data and does not measure theamount by which the solution can violate constraints of theproblem. In addition, while sensitivity analysis quanti fi es localstability of the solution with respect to small data changes, it doesnot provide a tool to improve this stability. Finally, sensitivityanalysis requires the analysis of uncertain parameters one ata time.Robust optimization is a new approach that addresses uncer-tainty of all problem parameters concurrently. It guaranteesfeasibility and optimality of the solution for the worst instancesof the problem without jeopardizing performance of the optimalsolution. Robust optimization incorporates the random characterof problem parameters without making any assumptions abouttheir probability distributions. The use of robust optimization toaddress data uncertainty of pharmaceutical R&D project is perti-nent because development of drugs precludes the availability of historical data to simulate the probability or possibility distribu-tion of the R&D project parameters. As a result, the application of fuzzy or stochastic optimization approaches to pharmaceuticalR&D that deals with unexperienced uncertainties is questionable.Both fuzzy and stochastic approaches assume that possible out-comes of an event and their distributions are known withcertainty. While in robust optimization the possible outcomesare known, no distribution is associated with them. In addition,fuzzy or stochastic optimization approaches optimize the objectivefunction based on the preference degree of the decision makerwhereas robust optimization seeks a solution that remains  reason-ably good  under all possible realizations of the uncertainparameters.Robust optimization was  fi rst introduced by Soyster (1973). Inhis approach each uncertain parameter is considered at its worstpossible value within a range, resulting in solutions that are overlyconservative. El-Ghaoui et al. (1998) and Ben-Tal and Nemirovski (2002) took the next steps by addressing the overconservatismissue and considered ellipsoidal uncertainties which result in conicquadratic robust counterparts for uncertain linear formulations.Although these formulations are able to approximate more com-plicated uncertainty sets they lead to counterpart models that arenonlinear and hence, computationally less tractable and lesspractical than linear models. Bertsimas and Sim (2003) developedthe  “ budget of uncertainty ”  approach that has the advantage of retaining linearity over the robust counterpart. In addition, thisapproach provides full control over the degree of conservatism forevery constraint.A recent review by Gabrel et al. (2014) reveals that althoughthere are a number of studies that present the application of robust optimization in the context of either project selection orproject scheduling, no such application exists in the PSS domain.In the context of project selection, Liesiö et al. (2008) developed amultiobjective project selection model where a wide range of project interdependences and variable budget levels exist. Theyconsidered incomplete information on project costs and used analgorithm to determine all non-dominated solutions. Driouchiet al. (2009) provided a robustness framework for monitoring realoptions under uncertainty that can be used to make robust projectdecisions. Hassanzadeh and Modarres (2009) used the real options F. Hassanzadeh et al. / Int. J. Production Economics 158 (2014) 18 –  27  20  approach to valuate R&D projects and developed a model to selecta robust portfolio of R&D projects. Düzgün and Thiele (2010)considered a project selection problem where uncertainty inproject cash  fl ow depends on an underlying random variable.They modeled the uncertainty using multiple ranges for eachuncertain parameter and developed a mathematical optimizationmodel as well as a ranking heuristic to  fi nd the best portfolio of R&D projects. Hassanzadeh et al. (2014) considered a multiobjec-tive R&D project portfolio selection problem where bene fi ts andcosts are uncertain. They used the budget of uncertainty approachwithin an interactive Tchebycheff procedure to elicit preferenceinformation from the decision maker and  fi nd the preferredportfolio of projects. Lo Nigro et al. (2014) developed a biophar-maceutical R&D project selection model where future cash in fl owsof successfullycompleted projects can be used to partially fund thedevelopment stages of other ongoing projects. They allowed forthe option to develop a project either internally or through alliesand used the real options valuation approach to evaluate candi-date projects. Their proposed model maximizes the real optionsvalue of the portfolio less discounted value of the endogenous cash fl ows used to fund other projects. In the context of projectscheduling Yamashita et al. (2007) considered a formulation of project scheduling problem with resource availability cost whereuncertainty in activity durations is modeled as a set of scenarios,and a heuristic based on scatter search is proposed to solve theformulation. Artigues et al. (2013) also considered a projectscheduling problem with uncertain activity durations. They for-mulated the problem as a minimax-regret optimization model andused a scenario-relaxation algorithm to solve the problem. Theyalso commented on the computational intractability of theirapproach even for medium-sized instances of the problem andproposed a heuristic solution procedure which demands lessCPU time.In this paper, we develop a mathematical model to periodicallyselect and concurrently schedule a portfolio of R&D projects. Ourcontribution is that our proposed model and solution approachaccount for the critical distinctions between R&D project andgeneric project setting. First, early market introduction is a sig-ni fi cant factor in commercial success of an R&D project (Arlington,1997; Piachaud, 2002); it has the short-term bene fi t of higherpro fi ts (Musselwhite, 1990) as well as long-term bene fi t of highermarket share (Robinson and Fornell,1985) and more pro fi table druglife cycle (Bauer and Fischer, 2000). Therefore, R&D project returns(contract values) are highly time-dependent. Second, R&D projectsare very risky and hence, externally available funds for investmenthave much higher cost than internally available funds (Grabowskiand Vernon, 2000). This is in addition to transaction costs, taxadvantages,  fi nancial distress costs, and asymmetric informationthat make externally available funds generally more expensive thaninternally available funds (Hubbard, 1998). Our formulated modelallows for reinvestment of revenues generated from successfullydelivered R&D projects as an alternative to borrowing from a fi nancial institution. Third, R&D projects usually have long lifecycles and are very imprecise and uncertain in terms of cash  fl owestimates compared to non-R&D projects (Carlsson et al., 2007). Toaddress this issue, we employ the robust optimization method of Bertsimas and Sim (2003) as our solution approach to account forthis inherent uncertainty in R&D  fi nancial estimates. 3. Problem de fi nition In order to characterize the problem, consider a situationwhere a CRO faces several R&D project opportunities from variouspharmaceutical companies. Each project opportunity, if under-taken, will lead to a return equal to the contract value of theproject. Without loss of generality, we assume that this revenueoccurs as soon as the project is completed. In addition to newproject opportunities, we assume that the CRO is currently con-ducting a portfolio of ongoing R&D projects where each project isundergoing a particular clinical testing phase. So, the problem thatneeds to be addressed is which ongoing projects should beabandoned, which additional project(s) should be accepted intothe portfolio, and how should the new portfolio mix be scheduled.Every R&D project has a speci fi c number of development phases,eachof whichrequires speci fi c fi nancialaswellas human, laboratory,and several other resources. Due to the limited availability of theseresources, the CRO cannot initiate all promising projects simulta-neously. The CRO, however, may offer a delayed initiation to thepharmaceutical company which will highly degrade the contractvalue. To formally de fi ne the problem, we  fi rst observe that a clinicaltesting phase may span several time periods. To model clinical trialswe consider that each phase consists of several project stages whereeach stage lasts for a single time period, e.g. one year. We focus onthis problem overa planning horizonof   T   time periods.The CROcash fl ows are modeled as  fi nancial transactions between the CRO and abank. Out fl ows include costs to be paid for drug developmentpurposes and in fl ows involve external budgets and revenuesreceived as a result of delivering a completed drug project to theclient, as shown in Fig. 2.Our proposed R&D PSS model encompasses the followingparameters: n  number of projects (ongoing and candidate) k i  duration of project  i  in terms of stages or periods(project life) t   i  early start time of project  it  þ i  tardy start time of project  i  ( t   i  r t  þ i  ) c  kit   time-dependent cost of undertaking the  k th stage of project  i  in period  t   ð k r k i Þ c  k i  þ 1 it  contract value of project  i  if completed at thebeginning of period  t r  kim amount of resource type  m  required to handle  k thstage of project  ir  t   single-period interest rate in period  t   Δ  premium paid over the interest rate for single-periodborrowing; strictly positive  β  t   external budget available in period  t B  liability (borrowing) limit R mt   amount of resource type  m  available in period  t T   planning horizon (in time periods)  g  ip  precedence gap between project  i  and project  p the following sets:  Ω  set of precedence relations Θ M  set of mandatory projects Θ O set of ongoing projects that should not be interruptedand the following decision variables: l t   amount of cash in account at the beginning of period  t  (earning interest at rate  r  t  ) R&D RequirementsCROPharmaceutical CompaniesBank Other Flows Cash Flows Fig. 2.  Flows in the problem setting. F. Hassanzadeh et al. / Int. J. Production Economics 158 (2014) 18 –  27   21  b t   total liability at the beginning of period  t   (with interestrate  r  t   þ  Δ Þ  x it   ¼ 1 if project  i  is initiated at the beginning of period  t  0 otherwise  In addition, we de fi ne possible precedence relations in thefollowing sense: if project  i  precedes project  p , then  ð i ;  p Þ A  Ω  andtheir precedence gap is an integer number,  i.e .,  g  ip A Z . If   g  ip  ¼ 0,then project  i  must be over before project  p  can begin; if   g  ip o 0, anoverlap of at most  ð  g  ip Þ  periods is allowed;  fi nally, if   g  ip 4 0,project  i  must be over at least  g  ip  periods before project  p  canbegin. We also view every ongoing project as a new projectopportunity with the related parameters modi fi ed accordingly. Inaddition, without loss of generality, we assume that  fi nancialtransactions, cost spending, and resource consumptions occur atthe beginning of periods. The nominal R&D PSS problem isformulated similar to a capital budgeting problem as max  ð l T   þ 1   b T   þ 1 Þ s : t  :  (1) ∑ i ∑ min  f t  ;  t  þ i  g  j  ¼  max  f t   i  ;  t   k i g c  t    j þ 1 it   x ij  þ l t   þ b t   1 ð 1 þ r  t   1  þ  Δ Þ r l t   1 ð 1 þ r  t   1 Þþ b t   þ  β  t  ;  t   ¼ 1 ; :::; T  þ 1 (2) ∑ t  þ i  j  ¼  t   i  x ij r 1 ;  i ¼ 1 ; :::;  n  (3) ∑ i ∑ min  f t  ;  t  þ i  g  j  ¼  max  f t   i  ;  t   k i  þ 1 g r  t    j þ 1 im  x ij r R mt   8 m ;  t   ¼ 1 ; :::; T   (4)  x  pt  r  ∑ min  f t  þ i  ;  t   k i    g  ip g  j  ¼  t   i  x ij ;  ð i ;  p Þ A  Ω ;  t   ¼ t    p  ; :::; t  þ  p  (5) ∑ t  þ i  j  ¼  t   i  x ij  ¼ 1 ;  i A Θ M  (6)  x i 1  ¼ 1 ;  i A Θ O (7) b t  r B ;  t   ¼ 1 ; :::;  T   (8)  x ij  ¼ ð 0 ;  1 Þ ;  i ¼ 1 ; :::;  n ;  j ¼ t   i  ; :::;  t  þ i  (9) b t  ; l t  Z 0 ;  t   ¼ 1 ; :::; T  þ 1 :  (10)The bank account periodically receives cash from completedprojects and potential external budgets in order to  fi nance theongoing projects. At the beginning of period  t  , the account gainsinterest  r  t   l t  , pays interest  ð r  t   þ  Δ Þ  b t  , and rolls over to the nextperiod. The bank account balance at the beginning of period  t  ,represented by  l t    b t  , encompasses all revenues gained fromcompleted project, investment costs paid to undertake projectphases, and interests gained from or paid to the bank up to period t   (apparently,  l t   b t   ¼ 0 for all  t  ). We therefore formulate theobjective function (1) to maximize the account balance at theend of the planning horizon  T  . The cash balance constraint isshown in (2), where cash in period  t   comes from the principal andinterest from lending in period  t   1, the amount borrowed inperiod  t  , and exogenous budget in period  t  . Cash is spent onlending in period  t  , the principal plus interest related to theamount borrowed in period  t   1 at rate  ð r  t   1  þ  Δ Þ , and the costof undertaking stage  t    j þ 1 of projects initiated at time period  j .Note that we have chosen  c  k i  þ 1 it   o 0 as the negative contract valuemerely to simplify the writing of this equation. Constraint (3)ensures that each R&D project, if selected, starts only once duringthe planning horizon. Constraint set (4) is established for eachrenewable resource, such as staff and laboratory to ensure thateach resource type consumption remains within the availableresource level of each period. Constraint (5) presents technicalinterdependence between projects; if   x  pt   ¼ 1 and project  i  is thepredecessor of project  p , then project  i  must start no earlier than t   i  and no later than min  f t  þ i  ; t   k i    g  ip g . Mandatory projects mayalso exist in the portfolio decision process. These are projects that,based on certain considerations, must be de fi nitely included in theportfolio. Constraint set (6) guarantees the inclusion of theseprojects in the portfolio. Moreover, at periodic revisions of theportfolio, it is normal for many or all of the ongoing projects to becontinued. Constraint set (7) does not allow such ongoing projectsto be interrupted. Constraint set (8) speci fi es the liability limit ineach period. Finally, constraints (9) and (10) identify decisionvariables of the problem. In the next section, we show how toincorporate the uncertainty of R&D  fi nancial estimates in theabove formulation. 4. Robust CRO portfolio optimization model (RoCROP) 4.1. Robust optimization for uncertain linear programming problems Consider the following MILP problem: max ∑  j c   j  x  j s : t  :  ∑  j a ij  x  j r b i ;  i ¼ 1 ;  … ;  ml L   j r  x  j r l U  j  ;  j ¼ 1 ;  … ;  n x  j A Z ;  j ¼ 1 ;  … ; n 0 Model (1)The problem has  n  decision variables (  x  j ), the  fi rst  n 0 of which areintegral.  c   j ,  a ij , and  b i  are objective function coef  fi cients, technicalcoef  fi cients, and right-hand side values, and  l L   j  and  l U  j  are lower andupper bounds on decision variables, respectively. Without any loss of generality, we assume that data uncertainty affects only  a ij (Bertsimas and Sim, 2003). To capture uncertainty we only assumethat each  a ij  is known to belong to an interval that is centered at thenominal value  a ij  and has half-interval length of   ^ a ij , i.e. a ij A ½ a ij    ^ a ij ;  a ij  þ  ^ a ij   and no distribution is associated with theuncertain parameter in its support. As much as it is unlikely that allcoef  fi cients are equal to their nominal value, it is also unlikely thatthey are all equal to their worst-case value. For this reason, the “ safest ”  approach where all parameters are taken equal to their worstbound leads to severe deterioration of the objective function withoutnecessarily being justi fi ed inpractice. Hence, the conservatism degreeof thesolutionneedstobeappropriatelyadjustedsothatareasonabletrade-off between robustness and performance is achieved.To quantify this concept in mathematical terms, the absolutevalue of the scaled deviation of parameter  a ij  from its nominalvalue is de fi ned as  z  ij  ¼ j a ij   a ij j = ^ a ij . Obviously,  z  ij  takes values ininterval  ½ 0  ;  1  . We now impose a  budget of uncertainty  in thefollowing sense: the total absolute value of the scaled deviation of the parameters in the  i th constraint cannot exceed some, notnecessarily integer, threshold  Γ  i , i.e.  ∑ n j  ¼  1  z  ij r Γ  i  8 i . By taking Γ  i  ¼ 0 ( Γ  i  ¼ n ) we obtain the nominal (worst) case formulations.Bertsimas and Sim (2003, 2004) show that letting the threshold Γ  i vary in  ð 0 ;  n Þ  makes it possible to build a robust model withgreater  fl exibility without excessively affecting the optimal objec-tive function. Intuitively, the budget of uncertainty rules out largedeviations in ∑  j a ij  x  j  which plays a predominant role in worst-caseanalysis but actually occurs with negligible probability. In F. Hassanzadeh et al. / Int. J. Production Economics 158 (2014) 18 –  27  22
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