 Applied Mathematics, 2010, 1, 189-199 doi:10.4236/am.2010.13023 Published Online September 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Uncertainty Theory Based Novel Multi-Objective Optimization Technique Using Embedding Theorem with Application to R & D Project Portfolio Selection Rupak Bhattacharyya, Amitava Chatterjee, Samarjit Kar Department of Mathematics, National Institute of Technology, Durgapur, India E-mail: {mathsrup, amitavamath}@gmail.com, kar_s_k@yahoo.com Received June 6, 2010; revised July 19, 2010; accepted July 22, 2010 Abstract This paper introduces a novice solution methodology for multi-objective optimization problems having the coefficients in the form of uncertain variables. The embedding theorem, which establishes that the set of un-certain variables can be embedded into the Banach space C[0, 1] × C[0, 1] isometrically and isomorphically, is developed. Based on this embedding theorem, each objective with uncertain coefficients can be trans-formed into two objectives with crisp coefficients. The solution of the original m-objectives optimization problem with uncertain coefficients will be obtained by solving the corresponding 2 m-objectives crisp opti-mization problem. The R & D project portfolio decision deals with future events and opportunities, much of the information required to make portfolio decisions is uncertain. Here parameters like outcome, risk, and cost are considered as uncertain variables and an uncertain bi-objective optimization problem with some useful constraints is developed. The corresponding crisp tetra-objective optimization model is then devel-oped by embedding theorem. The feasibility and effectiveness of the proposed method is verified by a real case study with the consideration that the uncertain variables are triangular in nature. Keywords: Uncertainty Theory, Uncertain Variable, Embedding Theorem, α-Optimistic and α-Pessimistic Value, R & D Project Portfolio Selection 1. Introduction An important problem in topology is to decide when a space X can be embedded into another space Y, i.e., when there exists an embedding from X into Y. This problem is called embedding problem. Theorems asserting the em-bedding of a space into some other space which is more manageable than the original space are known as embed-ding theorems. On the other hand, a theorem which as-serts that a certain space cannot be embedded into some other space is known as non-embedding theorems. Non- embedding theorems are often quite deep and require methods well beyond the general topology. For example it is by no means trivial to prove that the 2-sphere (S2) cannot be embedded into the Euclidean space. The embedding theorems in crisp and fuzzy environ-ments are already established. The fuzzy embedding theorem shows that each fuzzy number can be identified isometrically and isomorphically with an element in C [0, 1] × C[0, 1] where C [0, 1] is the set of all real valued continuous functions on [0, 1]. Puri and Ralescu  and Kaleva  have proved that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Wu and Ma  provide a specific Ba-nach space, which shows that the set of all fuzzy num-bers can be embedded into the Banach space C[0, 1] × C[0, 1]. Wu  propose an (α, β)-optimal solution con-cept of fuzzy optimization problem based on the possi-bility and necessity measures. To do so, the fuzzy opti-mization problem is transformed into a bi-objective pro-gramming problem by applying the embedding theorem. Wu  shows that the optimal solution of the crisp opti-mization problem obtained from the fuzzy optimization problem by using embedding theorem is also an optimal solution of the original fuzzy optimization problem under the set of core values of fuzzy numbers. With increasing competition and limitations of finan-cial resources, the way of selection of R & D projects R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 190 that maximize some measure of utility or benefit to the organization has become a critical one. The purposes of project portfolio decision are to allocate a limited set of resources to projects in a way that balance risk, reward and alignment with corporate strategy. Poor selection of R & D projects could have a significantly negative im-pact on organizations for decades. The R & D project portfolio decision deals with future events and opportu-nities, much of the information required to make portfo-lio decisions is at best uncertain and at worst very unre-liable. Project selection is usually described in term of constraint optimization problem. Given a set of project proposals, the goal is to select a subset of projects to maximize some objective without violating the con-straints. An R & D project usually involves several phases. Therefore, each phase is an option that is contin-gent on earlier of other options. Some attempts already exist for R & D project portfolio selection. Rabbani et al. [6,7], Fang et al. , Riddell and Wallace , Eilat et al. , Stummer and Heidenberger , Linton et al. , Ringuest et al. , Schmidt  and others have done significant works in the field of R & D project portfolio selection. To model uncertainty and vagueness, fuzzy set theory is used by many to characterize uncertain R & D project information. Pereira and junior , Coffin and Taylor , Machacha and Bhattacharyya , Kuchta , Mohamed and McCowan , Hsu et al. , Wang and Hwang , Kim et al. , Karsak  have applied fuzzy set theory in the field of R & D project portfolio selection. Unfortunately, R & D project man-agers have been unable to adopt many of these mecha-nisms. But in reality, sometimes investors have to deal with the uncertainty which acts neither randomness nor fuzz- iness. In order to deal with such type of uncertainty, Liu  founds uncertainty theory as a branch of mathemat-ics. Subsequently, Liu  proposes uncertain process and uncertain differential equation to deal with dynamic uncertain phenomena. In addition, uncertain calculus is introduced by Liu  to describe the function of uncer-tain processes, uncertain inference is introduced by Liu  via the tool of conditional uncertainty and uncertain logic is proposed by Li and Liu  to deal with uncer-tain knowledge. Liu  proposes an uncertain pro-gramming including expected value model, chance con-strained programming and dependent-chance program-ming to model several optimization problems. Till now, several research works [29,30] have been done in this area, but none has considered the R & D project portfolio selection problem in the uncertain environment. Basi-cally, till date, no embedding theorem based optimization technique is proposed in uncertainty theory. In this paper, in Section 2, we provide some prelimi-naries required to develop the paper. In Section 3, an uncertain embedding theorem is proved and an uncertain single/multiple objective optimization method using the embedding theorem is established. In Section 4, we de-velop an uncertain linear bi-objective R & D project portfolio selection model. The objectives are 1) maximi-zation of project benefit and 2) minimization of project risk. The risk is defined as the maximum loss that the decision maker may face in the worst case. This is con-sidered as the projected maximum loss in case of failure of the project. Constraints on budget and resources are also considered. Using the embedding theorem estab-lished in Section 3, we convert the bi-objective uncertain optimization problem into a tetra-objective crisp optimi-zation problem which is further transformed into a de-terministic convex optimization model by global criteria approach. In Section 5 of this paper a real case study is provided to illustrate our method. The optimization software LINGO is used for the simulation. Finally in Section 6 some concluding remarks are presented. 2. Preliminaries Before discussing the embedding theorem and its rele-vance in uncertain optimization problem we would like to discuss some basic concepts related with metric space, topology and uncertainty theory. Definition 2.1 (Metric Space) A non-empty set X is said to be a metric space if to every pair of elements x, y of this set, there corresponds a non-negative real number ρ(x, y) for which the following conditions hold. 1) ρ(x, y) > 0 and ρ(x, y) = 0 if and only if x = y 2) ρ(x, y) = ρ(y, x) 3) for any three elements x, y, z in X, ρ(x, y) ≤ ρ(x, z) + ρ(z, y). The number ρ(x, y) is called the difference or metric between the elements x, y and the above three conditions are called metric axioms and a metric space is sometimes written as (X, ρ). Definition 2.2 A sequence {xn} of elements of a met-ric space X is called a Cauchy sequence if for every ε > 0 there exists a positive integer N such that ρ (xn, xm) < ε whenever n, m ≥ N. If every Cauchy sequence of a metric space X has a fi-nite limit in X then X is called complete. By Cauchy’s general principle of convergence it can be shown that the real line and the complex plane with usual metric are the complete metric spaces. Definition 2.3 A set E is called a normed linear space if 1) E is a linear space with real or complex numbers as scalars and 2) to every element x of E there is associated a unique R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 191real number, called the norm of the element x and de-noted by ||x||. The norm of an element x has to satisfy the following axioms. 1) ||x|| > 0 and ||x|| = 0 if and only if x = θ 2) ||αx|| = |α| ||x|| where α is a scalar 3) ||x + y|| ≤ ||x|| + ||y|| for every x, y in E. Note: In a normed linear space we can introduce a metric by ρ(x, y) = ||x - y||. The metric axioms are fulfilled as 1) ρ(x, y) > 0 and ρ(x, y) = 0 if and only if ||x - y|| = 0,i.e., if and only if x - y = , i.e., if and only if x = y. 2) ρ(x, y) = ||x – y|| = ||(–1)(y – x)|| = |–1| ||y – x|| = ||y – x|| = ρ(y, x) 3) ρ(x, y) = ||x – y|| = ||(x – z) + (z – x)|| ≤ ||x – z|| + ||z – y|| = ρ(x, z) + ρ(z, y). Definition 2.4 (Banach Space) If a normed linear space is complete in the sense of the convergence in norm, then it is called a Banach space. Every finite dimensional normed linear space E is complete (that is a Banach space) and bounded. Every finite dimensional linear space can be made a Banach space. Definition 2.5 Two objects A and B are said to be con-gruent (or isometric) if there exists a bijection f: A→ B which preserves all distances in the sense that d(x, y) = d (f(x), f(y)) for all pairs (x, y) of points in A, d being used to denote the distance between points. Such a bijection, when it exists, is called congruence (or an isometry). Definition 2.6 Let (X, τ), (Y, Ч) be topological spaces. An embedding (or imbedding) theorem of X into Y is a function e: X → Y which is a homeomorphism when considered as a function from (X, τ) onto (e(X), Ч/e(X)). Definition 2.7 A function e: X → Y is an embedding function if and only if it is continuous and one-one and for every open set V in X there exists an open subset W of Y such that e(V) = W ∩ Y. Definition 2.8 The space C[0, 1] is the set of all real valued continuous functions f on [0, 1], such that f is left-continuous for any t (0, 1] and right-continuous at 0, and f has a right limit for any t [0, 1). The norm is defined as t [0,1] = sup() .fft Definition 2.9 Let Γ be a non-empty set and Å a σ-al-gebra over Γ. Each element Λ  Å is called an event. Let M be a set function over Å. Then M is called an un-certain measure (Liu, ) if it satisfies the following four axioms. Axiom 1. (Normality) M{Γ} =1; Axiom 2. (Monotonicity) 12{}{ }MM  whenever 12; Axiom 3. (Self-Duality) {}{} 1cMM for any event Λ; Axiom 4. (Countable Subadditivity) 11{}iiiiMM, for every countable sequence of events {Λi}. The triplet (Γ, Å, M) is called an uncertainty space. Definition 2.10 An uncertain variable is a measurable function , from an uncertain space (Γ, Å, M) to the set of all real numbers, i.e ., for any Borel set B of real num-bers, the set {}{ ()}BB  is an event. Definition 2.11 (Liu, ) An uncertain variable is said to have a first identification function λ if 1) λ(x) is a non-negative function on such that sup(( )())1;xy xy 2) for any set B of real numbers, we have, sup( )sup( )0.5{}1sup()sup()0.5.cxB xBxBxBxif xMB xif x Definition 2.12 A rectangular uncertain variable is de-fined to be the uncertain variable which is fully deter-mined by the pair (a, b) of crisp numbers with a < b, and whose first identification function is ()0.5, axb.x Definition 2.13 A triangular uncertain variable is de-fined to be the uncertain variable which is fully deter-mined by the triplet (a, b, c) of crisp numbers with a < b < c, and whose first identification function is axb2( )()if bxc.2( )xa ifbaxcxcb Definition 2.14 A trapezoidal uncertain variable is de-fined to be the uncertain variable which is fully deter-mined by the quadruplet (a, b, c, d) of crisp numbers with a < b < c < d, and whose first identification func-tion is axb2( )()0.5 bxcd cxd.2( )xa ifbaxifxifdc Definition 2.15 (Liu, ) An uncertain variable  is said to have a second identification function ρ if 1) ρ(x) is a nonnegative and integrable function on  such that () 1;xdx 2) for any set B of real numbers, we have, () ()0.5{}1() ()0.5.cBBBBxdx ifxdxMB xdx ifxdx R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 192 Definition 2.16 An exponential uncertain variable is defined to be the uncertain variable having the second identification function 1()exp, 0,xxx and is denoted by EXP(α, β), where α, β are real numbers with α ≥ β > 0. Note that () 1.xdx  Definition 2.17 A bell-shaped uncertain variable is defined to be the uncertain variable having the second identification function 221()()exp, ,xmxx  and is denoted by B(m, α, β), where α, β are real numbers with α ≥ β > 0. Note that 01(x)dx . Definition 2.18 (Liu, ) The uncertain variables 12,,,n  are said to be independent if  1 1nMBminMBii iiini  for Borel sets B1, B2,…, Bn of real numbers. Definition 2.19 (Liu, ) The uncertainty distribu-tion Ф: → [0, 1] of an uncertain variable  is de-fined by (x)=M{ x}. Definition 2.20 An uncertain variable  is called normal if its distribution function Ф is given by 1()()1exp,.3mxxx  It is then denoted by N(m, σ), where m, σ(>0) are real numbers. Definition 2.21 (Chen ) Let  be an uncertain variable and α(0, 1]. Then sup | {}opt rM r is called the α – optimistic value of ξ, and inf| {}pes rM r is called the α – pessimistic value of . Example 2.22 Let (,)ab be a rectangular uncer-tain variable. Then its α–optimistic and α–pessimistic values are 0.5 0.5 0.5, 0.5.opt pesbifbifaif aif Example 2.23 Let (,,)abc be a triangular un-certain variable. Then its α–optimistic and α–pessimistic values are 2(12) 0.5(21)(22) 0.5,(12)2 0.5(22)(21) 0.5.optpesbcifabifabifbcif        Example 2.24 Let (,,, )abcd be a trapezoidal uncertain variable. Then its α–optimistic and α–pessi-mistic values are 2(12) 0.5(21)(22) 0.5,(12)2 0.5(22)(21) 0.5.optpescdifabifabifcdif       Example 2.25 Let (,)EXP a b be an exponential uncertain variable. Then its α–optimistic and α–pessi-mistic values are .ln 0.5.ln 0.5,.ln 0.5.ln 0.5.(1 )optpesaaifabaaifbbaaifbaaifab   Example 2.26 Let (, , )eab be a bell–shaped uncertain variable. Then its α–optimistic and α–pessi-mistic values are 1111 < 0.52(1 ) 0.5,2 < 0.52(1 ) 0.5.2optpesabeifaaa beifaaabeifaabeifa Example 2.27 Let (,)mbe a normal uncertain variable. Then its α–optimistic and α–pessimistic values are 33ln, ln.11opt pesmm      Theorem 2.28 Let  be an uncertain variable. Then opt is an increasing and left continuous function of α. R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 193Also pes is a decreasing and left-continuous function of α (Liu ). 3. Uncertain Multiple Objective Optimization Method Using Embedding Theorem In this section, we introduce a solution methodology for multiple objective programming problems in uncertain environment by using the concept of optimistic and pes-simistic values of uncertain variables. Definition 3.1 Let ,  be two uncertain variables. We write  if and only if ,[0,1].optopt pespes We also write  if and only if . On the other hand, we write that  if and only if , [0,1]optopt pespes   or, , [0,1]optopt pespes or, , [0,1].optopt pespes  We also write  if and only if . Definition 3.2 Let  be an uncertain variable. Then the norm of  is defined by .E Justification: 0001) ={ }[{2} + {2}]==2 + 2EMrdrMrdr MrdrEE    (Triangular inequality as defined by Liu ). 2) ==.cEccE c  Theorem 3.3 (Embedding theorem) Let the function :()[0,1] [0,1]UCC is defined by (, ),0.5()(,),0.5opt pespes optifif    Then the following properties hold good. 1)  is injective. 2) ((1{s})+(1{t})) = s() + t(), ()U, s ≥ 0, t ≥ 0. 3) (,)() ().Ud That is, ()Ucan be embedded into C[0,1]  C[0,1] isometrically and isomorphically. Proof. Let 0.5. 1) Let, if possible, ,  be two distinct uncertain variables such that () =().Then (,) = (,).pes optpesopt    Since the two real open intervals are equal, the corre-sponding boundary points must be the same. Then = pes pesand = opt opt, which contradicts the fact that . Hence our assumption is wrong and conse-quently the mapping  is injective. 2) We have the bottom equation. 3) We have, (,)=(,) =((),())=(,) =() =()().Upes optpesoptpes optdEEEE      For 0.5the proof is similar. Proposition 3.4 (Order preserving) Let π be the function defined in theorem 3.3 and let ,()U. Then if and only if ()().We also have if and only if () (). Proof. We note that , [0,1]optopt pespes   1(,)(, ) [0,],21 (,)(, ) [,1]2opt pesoptpespes optpesopt     () (). Similarly it can be shown that  if and only if () (). Definition 3.5 Let 121 2,,,ffgg be real valued func-tions defined by 12 12,,, :(),ffggV U  where V is a real vector space. We say that 112 2(, )(,)fgfg if and ((1{})(1{})) =([(1{})(1{})], [((1{})(1{ })])= ([(1{})(1{ })], [(1{})(1{})])= (([1{}]. [1{ }].), ([1{}]. [1{}]pes optpespesopt optpespespespesopt optostst stst stst st     .)= (. + ., . + .) = () + ().pt optpespes optoptststs t  R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 194 only if 121 2, ffgg. We say that 112 2(,)( ,)fgfg if and only if [0,1], 1122()(), ()()optopt pespesfgf g  or, 112 2()(), ()()optopt pespesfgf g  or, 112 2()(), ()().optopt pespesfgfg Let f be a function defined by :().fV U Let (), 1,2,...,igxi m be real-valued functions defined on the same real vector space V and X be any subspace of V. Then let us now consider the following optimization problem as follows. min( ) to()0,1,2,...,,.ifxsubjectg ximxX (3.1) Then x* is an optimal solution of the problem (3.1) if there exists no x (≠ x*) such that *() ().fxfx Let π be the function defined in theorem 3.3. Now we consider the following optimization problem by applying the embedding function π to problem (3.1). min(( )) to()0,1,2,...,..ifxsubjectgximxX (3.2) Then x* is an optimal solution of the problem (3.2) if there exists no x (≠ x*) such that *(()) < (()).fxfx Therefore, x* is an optimal solution of the problem (3.2) if there exists no x (≠ x*) such that **(((),( ())(( (),( ()),pes optpesoptfx fxfxfx  (follow defi-nition (3.1). Proposition 3.6 x* is an optimal solution of the prob-lem (3.1) if and only if x* is an optimal solution of the problem (3.2). Proof. Proposition 3.4 states that *()( )fxfx if and only if *(())(()),fxfxand so, the proof is ob-vious. Therefore, the optimal solution of problem (3.1) is same as the optimal solution of the following problem: min{( ),( )}to()0,1, 2,...,..pes optifxfxsubjectgximxX (3.3) Proposition 3.7 If x* is a Pareto optimal solution of the problem (3.3) for some *[0,1], then x* is an optimal solution of the problem (3.2). Proof. Since x* is a Pareto optimal solution of the problem (3.3), x* is a feasible solution of the problem (3.2). If possible, let x* is not an optimal solution of problem (3.2). Then there exists a feasible solution x (≠ x*) such that *(())(())[0,1].fx fx Then [0,1], we have either **()(),()()pespesoptoptfxfxfxfx  or, **()( ),()( )pespes optoptfxfx fxfx  or, **()(), ()().pespes optoptfxfxfxfx Since *[0,1], we then should have, either ******()(),()()pespesoptoptfxfxfxfx s or, ******()( ),()( )pespes optoptfxfx fxfx  or, ******()(), ()().pespesoptoptfxfx fxfx This shows that x* is not an optimal solution of the problem (3.3); a contradiction with the assumption that it is a pareto-optimal solution of (3.3). So, our assumption is wrong and we are with the theorem. Theorem 3.8 If x* is a Pareto optimal solution of the problem (3.3) for some *[0,1], then x* is an optimal solution of the problem (3.1). Proof. The theorem is obvious from proposition 3.6 and 3.7. Now we consider the uncertain multi-objective opti-mization problem as follows: 12min((), (),.....,())subject to ()0, = 1,2,..., njfx fxfxgxj mxX (3.4) 1212112 21min{((), (),....., ())}min{(()), ( ()),......, (())}min[{() ,()}, {() ,()},......., {() ,()}]min[()nnpes optpesoptnpes n optpesfx fxfxfx fxfxfx fxfxfxfx fxfx    122 ,(), () ,(),......., () ,()]subject to ()0, =1,2,....,, 01optpesoptnpes n optjfxfx fxfxfxgx jmxX     (3.5) R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 195 We say that x* is an optimal solution of problem (3.4) if there exists no x ≠ x* such that ()(().iifxfx Let  be the embedding function defined in theorem 3.3. Then we consider the multi-objective optimization problem (3.5) by applying the embedding function  to problem (3.4). We say that x* is an optimal solution of problem (3.5), if there exists no x ≠ x* such that *(()) ((()).iifxfx Theorem 3.9 If x* is a Pareto optimal solution of (3.5) for some*[0,1], then x* is an optimal solution of the uncertain multi-objective optimization problem (3.4). Note: To solve uncertain multi-objective problem by using embedding theorem we first have to transform the uncertain multi-objective optimization problem into the crisp multi-objective optimization problem (3.5). The Pareto optimal solution of this problem is the optimal solution of the original uncertain multi-objective prob-lem. 4. Uncertain R & D Project Portfolio Selection model In this section, we first describe the notations used in the construction of the R & D project portfolio selection model. Then the objective function of the models will be constructed in the second subsection. In the third subsec-tion we will discuss the constraints used in our portfolio selection model. The next subsection will include final mathematical model. 4.1. Notations N = Number of candidate projects. T = Number of periods. I = Interest rate. 1 if project i is selected in period ,0 otherwiseittx itNTxx=decision matrix. itv = Projected uncertain outcome of project i in period t. itr = Projected uncertain risk of implementing project i in period t. itc = Expected uncertain cost required by ith project in period t. Bt = Budget available for stage t. sitR = Amount of resource of type s required for imple-mentation of project i individually in period t. 'stR = Amount of available resources of type s in period t. sR = Total amount of available resources of type s. 4.2. Formulation of Objective Functions In this R & D project portfolio selection problem we have considered two objectives: maximization of the benefit and minimization of the project risk. 4.2.1. Maximization of Benefit The total outcome from the projects will be obtained by considering the total individual. If the interest rate for each period is I, the total outcome is 111() .(1 )TNOititttiZxvxI The total cost will be obtained by the total individual costs for each project. Then the total cost is 11() .TNCitittiZxcx Thus the benefit of the project portfolio is 111()() ()[].(1 )TNBOCit itit itttiZxZxZx vxcxI Our objective will be to maximize the benefit. 4.2.2. Minimization of Project Risk For successfully implementation of R&D project portfo-lio, the risk attached with the projects must be as less as possible. Here, we have defined risk as the opposite of expected profit. As the futures of all the projects are un-certain, implementation of a project may or may not yield us success. In case of failure, the decision maker may loose their money and time and resource. Let rit ≥ 0 is the amount the decision maker may loose in worst case for the ith project at period t. Then the total risk involved in the project portfolio is 11.TNit ittirx Therefore, the objective is to minimize total risk 11() .TNRit ittiZxrx Thus we are with the following bi-objective optimiza-tion problem R()Z().BMaxZ xMin x 4.3. Formulation of the Constraints In this subsection we will formulate the constraints re-quired to model the problem realistically. R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 196 4.3.1. Outcome Constraints As the minimum expected outcome for the projects in period t is tV, we have, 1 ,Nit ittivx Vt i.e., 11(),() .NNitpesit titoptit tiivxVvxVt 4.3.2. Resource Constraints The projects are implemented by using limited amount of resources. As the available resources are always finite, the required resource with particular type should be within the resource available of that type for each period. Thus we have '1 ,.Nsit itstiRx Rst The total amount of resources available is limited. So, the amount of resource required should not be more than the total resource available for each type of resources. Thus we have, 11 .TNsit itstiRxR s 4.3.3. Budget Constraints The project expenses during the planning horizon should not exceed the predetermined budget for each stage or period. So, we have 1 ,Nit itticxB t i.e., 11(),() .NNitpesit titoptit tiicxB cxBt 4.4. The Set of Feasible Solutions In this subsection we construct the set X of feasible solu-tions x = (xit)NT. Then we have problem (4.1). 4.5. The R & D Project Portfolio Selection Mode l Keeping in mind the objectives and constrained obtained in the last two subsections, the R & D project portfolio selection problem is modeled as ()Z().. .BRMaxZ xMin xstxX (4.2) By applying embedding theorem, the uncertain mul-tiob jective optimization problem (4.2) is converted into the crisp multi-objective problem {(), ()}{(), ()}..,01.Bpes BoptRpes RoptMaxZ xZ xMinZxZ xstx X (4.3) The global criteria methods developed in the context of multi-objective optimization problem are really handy for obtaining the Pareto optimal solution. Let, max{ ()},max{ ()},min{( )},min{( )},min{( )},min{( )},max{( )},max{( )}.pesBpesoptB optpesBpesoptB optpesBpesoptB optpesBpesoptB optBZxBZxRRxRRxBZxBZxRRxRRx Then the problem (4.3) is further converted into the following single objective convex programming prob-lem. 221222() ()() () that ,01. pesBpesoptB optpespesopt optBpespesBoptoptpespesoptoptBZxBZxMin BB BBRxRRx RRR RRsuchx X    (4.4) 5. Case Study In this section a model is developed and solved based on data from the large scale organization B. M. Enterprise, Berhampore, West Bengal, India. The R & D wing of this organization is involved in different structural works in civil, mechanical and electrical fields. During the year 2009 the organization gets 10 project proposals from private as well as public sectors. All the proposals ac-company data on the estimated outcome, estimated cost, funds, workers, budget and risk. After first round of '1111111 (): (),(),, (), (), ,NNNsitNTitpesit titoptit tititstiiiNNTNsitpesit titoptit tititsiitiXxx vxVvxVRxRcxB cxBRxRst    (4.1) R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 197 scrutiny 5 project proposals are short listed. All the five projects are scheduled over two periods and each period lasts one year. They are renamed as projects I, II, III, IV and V due to privacy. The estimated outcome, risk and projected cost are considered in the form of triangular uncertain variables. Interest rate is 5%. The estimated data for outcome, costs, risks, funds and workers are given in Table 1. Constraints on fund, workers and budget for each pe-riod are given in Table 2. As discussed in Section 4, the uncertain optimization model (4.2) is constructed which is converted into the model (4.3) by using the embedding theorem. The model (4.4) is then constructed as 22122216.000( )18.475()5.375 7.375( )2.075( )2.2 1.325 1.4 that , 01. BpesB optBpesB optZx ZxMinRx Rxsuchx X(5.1) The solution of the model (5.1) is done by using the LINGO software and the obtained solution is as follows: x11 = 0, x21 = 1, x31 = 0, x41 = 1, x51 = 1; x12 = 0, x22 = 1, x32 = 0, x42 = 0, x52 = 1. It means that B. M. Enterprise should select the 2nd, 4th and 5th projects in 1st stage and 2nd and 5th projects in the second stage to get the optimum result. For this portfolio, the benefit is estimated as (6.5, 11.4, 13.7) million rupees and the corresponding risk is (1.95, 2.4, 3.2) million rupees. 6. Conclusions This paper introduces the concept of multiple objective uncertain optimization problems. In particular, this paper concentrates on the problems where the coefficients of the decision variables are uncertain variables. To do so, we propose and prove the uncertain embedding theorem from the space of uncertain variables to the Banach space C [0, 1] × C [0, 1]. By applying embedding theorem, each uncertain objective function can be converted into two deterministic objectives functions. The Pareto opti-mal solution of both the deterministic objectives is the optimal solution of the uncertain objective. Table 1. Estimated project data. Projects I II III IV V 1st period (4, 7, 9) (1, 3, 5) (0.2, 1.4, 2.8) (0, 1, 1.4) (5, 6, 7) Outcome (in Million Rupees) 2nd period (7, 10, 12) (2, 3, 4) (2.5, 4, 5.2) (1, 2, 3) (6, 7.5, 9) 1st period (1, 2.2, 3) (0.4, 1.2, 2) (0.8,1, 1.2) (0.1, 0.3, 0.8 ) (2, 2.5, 2.9) Cost (in Million Rupees) 2nd period (2, 2.8, 3.7) (1.0, 1.1, 2) (1.5, 1.7, 2) (1.6, 1.8, 2.1) (3, 4, 5) 1st period (0.6, 0.8, 1) (0.1, 0.2, 0.4) (0.5, 0.7, 0.9) (0.4, 0.5, 0.6) (0.35, 0.4, 0.5) Risk (in Million Rupees) 2nd period (0, 0.4, 0.5) (0.6, 0.7, 1) (0.4, 0.5, 0.6) (0.4, 0.5, 0.6) (0.5, 0.6, 0.7) 1st period 0.6 0.4 0.25 0.11 0.21 Fund (in Million Rupees) 2nd period 0.9 0.2 0.21 0.09 0.2 1st period 31 15 20 21 16 Workers (in numbers) 2nd period 10 12 17 18 9 Table 2. Constraints. Category 1st period 2nd period Total Outcome (in Million Rupees) > 10.0 > 10.0 - Budget (in Million Rupees) < 8.0 < 9.0 - Fund (in Million Rupees) < 0.9 < 0.8 < 1.5 Workers (in numbers) < 85 < 80 < 150 R. BHATTACHARYYA ET AL. Copyright © 2010 SciRes. AM 198 This paper also introduces a new model of R & D pro-ject portfolio selection by identifying project information like estimated future outcome, risk or estimated cost of the projects as uncertain variables. An uncertain bi-ob-jective optimization model, that maximizes the benefit and minimizes the risk, is constructed. Constraints on budget, resources and outcomes are also included in the model to make it more realistic. The uncertain optimiza-tion method by embedding theorem is used to solve it. A real case study is provided for illustration. In future, we will use the uncertain optimization ap-proach to other real optimization problems like portfolio selection problem, supply chain management problem, poverty management problem etc. For large data sets, meta-heuristic algorithms such as tabu search, simulated annealing, ant-colony optimiza-tion, and particle swarm optimization may be employed to solve the non-linear programming problem (4.4). 7. 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