The Project Portfolio Management Problem

Transcription

1 The Project Portfolio Management Problem Souvik Banerjee Wallace J. Hopp June 21, 2001 Abstract We consider the Project Portfolio Management Problem (PPMP) in which a limited resource must be allocated among a set of candidate projects over time so as to maximize expected net present value. We formulate this problem as a dynamic program but conclude that this approach is too computationally complex to be of value in supporting real-world project management. So, we investigate the structural properties of the optimal solution to the PPMP and demonstrate that the solution reduces to a simple form under certain environmental conditions. This simplified policy, which we term the index policy, sequences projects according to a simple ratio and then allocates resource up to each project s practical limit in the order given by this sequence. Through numerical tests we demonstrate that this policy performs robustly well on the general PPMP. Hence, we conclude that the index policy is a practical way to incorporate economic and timing issues into a multi-dimensional scoring model for addressing real-world project portfolio management situations. 1 Introduction One of the most critical problems facing most product-oriented firms is management of the research and new product development processes. A steady stream of new products is essential to the long term health of a company. Hence, the decision of how to allocate resources (money, people, space) to research, development and commercialization projects is of enormous strategic importance. The overall problem of investing in new product innovation is far too complex and multidimensional to be reduced to a single model. In real-world situations, managers must consider 1

2 economic, technical feasibility, marketing, competitive positioning, regulatory and many other issues. Because it is not possible to incorporate all of these issues in a detailed optimization model, such models are rarely used in practice. Instead, simple scoring models that rate projects according to many criteria are often used. In this paper we examine the Project Portfolio Management Problem (PPMP), which considers how to allocate a limited budget to a set of candidate projects over time with the objective of maximizing expected net present value. However, because we recognize that the many intangibles associated with product innovation are paramount to decisions in practice, we do not feel that a complex algorithmic solution to this piece of the problem would provide practical value. So instead we develop a simple procedure for ranking projects according to their economic attractiveness, which considers timing and resource interactions, but is transparent enough to incorporate into the scoring methods used by practitioners. By doing this, our work partially spans the gap between research and practice in the project management field. The remainder of the paper is organized as follows: In Section 2 we review the literature on the PPMP. In Section 3 we formulate a DP to support the resource allocation decision, but note that solution of this DP is too cumbersome to be of much use in practice. In Section 4 we characterize the structure of the optimal policy and show that it is still complex in general. In Section 5 we show that the optimal policy is much simpler for some special cases of the PPMP. In Section 6 we consider one of these simplified policies, the index policy, as a heuristic for the general PPMP and show that it performs robustly well. We conclude the paper in Section 7. 2

3 2 Literature Survey A vast amount of literature exists on the PPMP, most of which can be classified into: (a) scoring models, where each project is assigned an index based on various criteria (e.g., by a set of experts); (b) mathematical programming models, where an objective is maximized over a set of constraints using linear programming, integer programming, dynamic programming, goal programming or other optimization techniques; (c) economics or financial models, relying on cost/benefit analysis, payback period, NPV/IRR or other portfolio methods; d) decision analysis techniques such as decision trees, PERT/CPM and Monte Carlo simulation. Most early literature concentrated on analytical techniques for selecting and scheduling projects. However, it was soon pointed out, (for example in [1], [2], [3]and [23]), that these methods were virtually ignored by industry. Baker [2] reported that decision theory models were seldom used, scoring methods were somewhat more popular, but the most prevalent method actually used by managers was traditional capital budgeting. He conjectured that the trend in application appears to be away from decision models and toward decision information systems. A decade later, Liberatore [16] reported that even though managers were familiar with mathematical programming models they almost completely avoided them. Financial techniques and decision analysis techniques were more prevalent. Another decade later, Schmidt et al. [22], pointed out again that there has been a overall mismatch between modeling efforts and modeling needs. They argued that too much attention has been devoted to modeling the problem focusing on outcomes as opposed to developing schemes that help the decision makers gain insight into the decision process. The authors claimed that the failure to use mathematical models is not merely the lack of training or resistance to change on behalf of the managers, but rather the lack demonstrable proof that these models are indeed more effective in solving the problem. Gupta et al. [8] echoed most of the above conclusions. 3

4 A key shortcoming in early PPMP models which may have affected their usefulness, was that they failed to account for uncertainty. Many recent models have tried to correct this by using some sort of stochastic analysis. For instance, Heidenberger [9] used a network model to represent a project and considered different types of probabilistic nodes controlling the progress of the project. Some nodes were critical, go-no-go type, (that is, a failure at these nodes terminates the project), while other nodes directed the outcome of the project based on a probability governed by the amount of resource allocated to the activity. Others, for example Tavares [24], considered uncertainty in the task duration, where the distribution is based on the level of resource allocation. A second challenge in developing models that approximate reality is representing the dependencies of tasks on resource allocation. Most literature uses a static approach. That is, it is assumed that each task has a constant requirement for each resource, which has to be satisfied fully in order for the task to proceed. This assumption frequently leads to 0-1 integer programs. While it may be valid with respect to some discrete resources (e.g., allocation of a machine to a job), there are many practical resources (e.g., capital or manpower) where partial allocation is possible. Gerchak [6] considered the effect of partial funding where the level of funding affects the achievement level of the task and developed a mathematical model which required numerical solution to find the optimal allocation of resources. Ulvila et al [25] also mentioned the effect of partial funding on the level of benefit obtained from a project and postulated that the benefit increase rapidly and monotonically with funding level in the initial stages, but above a certain efficient level of funding it displays diminishing marginal returns and eventually shows no further increase in benefit at all. Madey [19] similarly considered the success probability of a task being a monotonic nondecreasing function of the funding level. More recent approaches, e.g. [4], have modeled resource allocation dynamically, where at each decision current resources can be shifted among the ongoing projects. 4

5 A third area where PPMP models often fall short of reality is with respect to the fact that most project portfolio decisions involve more than a single criterion. Frequently, these criteria are non-quantifiable and highly subjective, for example long term strategic fit and company image. Notable work in this area includes [7], [11], [13], [17] and [19]. Related research has addressed the organizational structure and how to integrate multiple criteria via a Decision Support System (for example, [12], [20] and [26]). In this paper, we address the PPMP and explicitly model task uncertainty and dynamic resource allocation. To enable our model to be used in multi-criteria contexts (e.g., as part of a larger scoring model), we seek a simple, intuitive solution, rather than a detailed algorithmic approach. Our work is most closely related to that of Kavadias et. al [10]. However, while they consider a similar setting and provide related structural insights, they do not provide a way of finding an optimal policy or a method for adapting their results to a decision making framework. 3 Problem Formulation The PPMP involves allocating a budget over time to a portfolio of M = {1,...,m} candidate projects. (We consider the case where all candidate projects are available at the start of the problem, but discuss how our results can be adapted to situation with new projects arrival later on.) Each project i consists of a sequence of n tasks. Task j of project i has a work requirement w i,j expressed in terms of total resource-hours required to complete the task. Hence, the total work requirement of project i is given by w i = n j=1 w i,j. The time required to complete a task is a function of the resource rate allocated to it. The total resource rate available at any point of time, or the budget is given by some constant B. Taskj of project i is assumed to succeed with a predetermined probability p i,j, which is independent of all other problem parameters including the resource rate, and success or failure becomes known 5

6 only when the task is completed. If a task of a project fails then the entire project fails. Hence, in order for projects to be successful all tasks must be completed successfully, which implies that the probability of project success is given by p i = n j=1 p i,j. Upon completion, each project i yields an expected profit of α i. The objective is to find a policy that allocates the available budget so as to maximize the expected net present value. Such a policy must specify an allocation for any reachable state, where the state is defined by the work requirements and probabilities for all tasks (and partial tasks) of unfailed projects. Let the random variable t i (γ) be the time when project i completes all its tasks successfully under policy γ. Then the net present value for the policy can be expressed as Π= m p i α i E[e βti(γ) ] (1) i=1 where β is the continuous discount rate and the expectation is taken over all possible combinations of task completion events. Because the objective function is monotonically nonincreasing in the completion times it is easy to verify that without loss of optimality we can restrict attention to efficient policies, which are policies that always utilize as much of the available budget as possible while there are unfinished tasks. Note, that the implicit assumption is that all projects will be eventually undertaken, even though at any point in time the available budget might restrict work to only a few of them. 3.1 Modeling Resource Allocation In general, we expect the completion time of a task to be inversely proportional to the resource rate allocated to it. The simplest possibility is a linear relationship. That is, if we allocate resource at rate x to a task that has a work content of w, the time to complete the task will be w/x. However, a linear decrease in completion time is not sustainable forever since eventually there will be diminishing returns to additional resource allocation. 6

7 Completion rate Linear Approximation Expected limit Resource Figure 1: Variation of Task Completion Time with Funding Level Hence, we expect the behavior to be as shown by the dashed line in Figure 1. To approximate such behavior we model the resource-rate profile as shown by the solid line in Figure 1. We assume a linear relationship up to some efficient resource allocation limit l i,j for task j of project i. Beyond this efficient limit, no further increase in rate is observed. In general, we expect the efficient limit of a project will be less than the budget available at all times, although this is not required. Also, we expect the sum of the efficient limits for the projects at any point to be greater than the budget. That is, m l i,j >B, (2) i=1 If this is not true, we can simply allocate each project resource up to its efficient level and no further optimization is necessary. 7

8 3.2 Constant Rate Equivalence Our problem is to find a policy that allocates resource rate to each project in the portfolio in a manner that maximizes the expected NPV. The resource allocated to project i at time t is denoted by x i (t) and represents a valid resource allocation as long as n i=1 x i(t) B and 0 x i (t) l i (t), for all i, wherel i (t) is the efficient limit of the task of project i undertaken at time t. Any such resource allocation policy induces a sequence of task completion times. By computing the probabilities associated with the task completion times, we can use Equation 1 to compute the expected profit for a given policy. We can simplify the required calculations by using an equivalent constant resource rate x i (constant w.r.t time) for each of the projects between any two task completion times. To see that this can be done without loss of optimality, let t 1 and t 2 be two task completion points that result from a given policy along a given sample path. Suppose during the interval [t 1,t 2 ] work w i is done on project i with a possibly variable rate, so that t 2 t 1 x i (t)dt = w 1. Using a constant resource allocation x i such that x i (t 2 t 1 )=w i will result in exactly the same completion times and therefore the same expected profit. Hence, we can restrict attention to policies that allocate constant resource amounts to tasks in between task completion times. The resulting problem will be equivalent in terms of both completion times and expected profit. Our use of constant rates is for algebraic convenience only; it does not imply that one is restricted to use them in real life. Note, however, that this restriction is possible only because the budget is constant. 3.3 Dynamic Programming Formulation We now develop a DP for the general PPMP. The state space for the DP is given by the work content of the portfolio, s = {n, w}, wheren N m is the number of tasks left for each of the projects, and w R m is the amount of work left in the current task of the projects. 8

9 The control variable is the vector x R m,wherex i denotes the allocation to project i, at the first decision epoch. The allocation vector x must satisfy the following constraints: x i ˆl i, i M (3) where ˆl i is the resource limit for the current task of project i, m x i B (4) i=1 where B is the total budget, and x i 0, i M (5) Constraints 3, 4 and 5 define the action space A for the DP. Any feasible allocation causes the next completion event to consist of completion by one or more project tasks. Let C[s] denote the set of all possible combinations of project tasks that can complete (possibly simultaneously) due to a feasible allocation starting at state s. Let χ c [x] be the indicator that completion c C[s] occurs due to allocation x. For any c C[s], let τ c (x) denote the completion time of a task (or tasks) in c under allocation x. Let Y [s, c] denote the set of states reachable from state s under task completion c and let ζ(s, y), y Y [s, c] be the corresponding probability. Finally, let r[s, y] denotethe expected revenue from reaching state y in Y [s, c] from state s after task completion c. This expectation is positive only if the completion c corresponds to a successful completion of a project or projects. The value function V [s] for the PPMP is obtained by summing over all possible completion events c in C[s] and computing the expected value function from states y Y [s, c] reachable from state s under completion event c. This is written as: V [s] =max x A c C[s] χ c [x]e βτc(x) y Y [s,c] (ζ(y)v [y]+r[s, y]) (6) 9

10 The DP recursion (6) is terminated when we reach a state with only one task left. For such a state, the expected NPV is computed by allocating to the remaining task the minimum of the budget and its efficient limit. 3.4 Dynamic Programming Solution The state-space of the DP given by Equation (6) is continuous. In general, such problems are difficult to solve. However, since the number of possible sample paths for the DP is finite, we can devise an algorithm to solve the DP in finite time. This is done as follows. Any policy for the PPMP induces a task completion sequence for every possible sample path. Let Σ denote the set of all possible task completion sequences for a PPMP that occurs under the condition that all projects are successful. Let R denote the set of sample paths possible for the PPMP. Note that both Σ and R are finite and known in advance. For every sequence σ in Σ, each sample path r in R generates a subsequence. For each such subsequence (σ, r), let K r (σ) be the number of task completion epochs that are realized. Observe that these subsequences are mutually exclusive and collectively exhaustive. A policy for a PPMP should specify the allocation for each reachable subsequence of it (by ruling out unreachable subsequences, we can improve computational efficiency). Let x r i,k (σ) bethe allocation to project i at the kth epoch under sample path r for sequence σ. Let τ r k (σ) denote the corresponding length of the task completion interval. Notice, that τk r (σ) canbe computed as a function of the decision variable x at time 0. The completion time of project i under (σ, r) can therefore be computed at time 0 as a function of τ r k (σ). We define tr i (σ) = if project i fails under subsequence (σ, r). Let li,k r (σ) denote the resource limit for the task of project i that requires funding at that epoch. Further, if project i has already completed by this epoch under (σ, r), then we define li,k r (σ) = 0. Observe that at any epoch k, thefull task completion sequence and the sample path is not necessarily revealed. However, we are specifying a decision variable for all sequences and sample paths that are possible at epoch 10

11 k. Therefore, we need not know either the full sequence or the sample path in advance. Using the above notation we write the following optimization problem which is equivalent to DP (6): Z =max σ Σ {Z σ} (7) where Z σ =max Pr[r] x C[σ] r R m α i e tr i (σ), σ Σ (8) i=1 and C[σ] consists of the following set of constraints defined for each σ Σ: (i) the budget constraints, m x r i,k(σ) B, k =1,...,K r (σ), r R (9) i=1 (ii) the resource limit constraints, 0 x r i,k(σ) l r i,k(σ), i =1,...,m, k =1,...,K r (σ), r R (10) (iii) the sequence determining constraints, τ r k(σ) > 0, k =1,...,K r (σ), r R (11) Each problem Z σ can be solved by standard NLP solution techniques. For example, since the objective and the constraints are differentiable, we can construct the Lagrangian and solve for the corresponding KKT conditions. The above algorithm is useful, for example, to verify our numerical tests that follows. However, such an approach is not practical for large problems and is not consistent with our goal of a simple method that can be combined with other factors in a decision support 11

12 system. 4 Concurrent-Priority Policy To develop a simple but effective heuristic for the PPMP, we now examine the optimal policy more clearly. Since we can restrict attention to policies with constant resource allocations between task completions, a policy need only specify the allocation at the beginning of each decision epoch. This allocation, in turn, determines the time of the next task completion (decision epoch). This process is repeated as long as there are projects to complete. Hence, policy γ induces a task completion sequence for each possible sample path. Therefore, in the same manner as Section 3.4, let t r i [γ] denote the completion time of project i under policy γ and sample path r. Then, the expected NPV for policy γ can be computed as E[Π(γ)] = m Pr{r} α i e βtr i [γ] (12) r R i=1 We begin by showing that we can restrict attention to a specific class of policies, which we call concurrent-priority policies. Under a concurrent-priority policy we either follow a priority policy or a concurrent policy, defined as follows: Priority Policy: Under a priority policy, projects are ordered in a list. At each decision epoch, the project at the top of the list is allocated its efficient limit or the available budget, whichever is less. Next, the remaining budget, if any, is allocated in a similar fashion to the second project on the priority list. This process is continued until the entire budget has been allocated. Note that, under a priority policy, at each decision epoch, we fully fund a set of projects and possibly fund one additional project at a partial level. Projects funded in this manner are said to be prioritized. However, the priority order of the projects need not be the same at all decision epochs. As each task 12

13 is completed, the information on its success or failure can be used to recompute project priorities. Concurrent Policy: Under a concurrent policy, two or more tasks finish simultaneously at some point in time for at least one sample path. Whether or not two tasks actually complete simultaneously depends on the task outcomes (success of failure). But in any concurrent policy there is a nonzero probability of simultaneous task completions. We denote the class of concurrent policies as Ω C and the class of priority policies as Ω P. The class of concurrent-priority policies is the union of these two sets and we denote it by Ω=Ω C Ω P. To illustrate what is ruled out, observe that for a policy γ not in Ω, the following holds true. No two tasks complete simultaneously under any sample path with nonzero probability. Further, at some decision epoch there exist two or more projects that receive non-zero funding below their efficient limits. We now prove that for any policy γ/ Ω there exists a policy in Ω whose expected profit is at least as large. That is, we prove Theorem 4.1 For the PPMP there exists an optimal policy that belongs to class Ω. Proof outline: (Full proofs appear in the Appendix.) To prove the theorem, we start with any policy not in Ω. For this policy it must be true that there exists an epoch where at least two projects, project i and j are funded at an intermediate level. For this epoch it is feasible to reallocate some resource from project i to j (or vice versa) without changing the task completion sequence if the change is sufficiently small. We show that such a perturbed policy does not decrease the expected NPV, which proves the result. Further, for a policy in Ω such a perturbation is not possible since any such change in the allocation will alter the task completion sequence. The PPMP would be much simpler to solve if we could restrict attention to priority policies in Ω P. Unfortunately, this is not optimal in general. For example, consider the 13

14 Project α w i,1 l i,1 p i,1 w i,2 l i,2 p i,2 w i,3 l i,3 p i,3 Project Project Table 1: Example Portfolio following two project portfolio whose optimal solution is a concurrent policy in Ω C. Project 1 has three tasks and project 2 has one task with parameters given in Table 1. The budget is normalized to 1 unit and the discount factor is β =1.0. The optimal policy for this PPMP is obtained by solving DP (6) which yields the following nonunique solution: In the first decision epoch we fund project 1 at any resource level in the range [0.37, 0.4]. This ensures that task 1 of project 1 finishes first. In the second epoch, we fund such that the second task of project 1 and the only task of project 2 finish together. At the third epoch, only the third task of project 1 is left and we fund it at its efficient limit until completion. For any policy in this set the optimal expected NPV is $0.73. Notice that, in the first two epochs, both projects may be funded in an intermediate level, so the policy is not in Ω P. Figure 2 (not to scale on the time axis.) illustrates the allocations for two cases: (i) allocate 0.37 at the first epoch and (ii) allocate 0.4 at the first epoch. Note that both projects complete at the same time under (i) and (ii) and therefore these policies yield the same NPV. The restriction to concurrent-priority policies reduces the solution space for the PPMP. For example it reduces the number of (σ, r) pairs to search over in the solution procedure described in Section 3.4. However, the problem of finding an optimal policy is still complex. 5 Priority and Index Policies While it is possible to find an optimal concurrent-priroity policy via numerical search, the solution from such a procedure may not be intuitive and therefore difficult to combine with 14

15 other criteria. A priority policy is fairly intuitive and easy to implement; a concurrent policy is not. We now show that for two simpler versions of the PPMP, we can restrict ourselves to priority policies. 5.1 Single Task PPMP First, we consider a PPMP where each project has a single task, which we call the Single Task PPMP, or STPPMP. We first show Lemma 5.1 For the STPPMP, we can restrict attention to policies under which the allocations to any project are non-decreasing in time until the completion of the project. Proof Outline: Whenever the condition of the result is violated there exists a project i whose funding decreases across epochs. For the policy to be efficient there must also exist a project j whose funding increases across the same epochs. An alternate policy is created by reallocating resource from project i to project j in the first epoch and doing the reverse in the next. The expected NPV for this policy is shown to be at least as high as the original policy, which proves the result. The intuition is that it is sub-optimal to postpone a project once resource has been invested in it unless additional information is obtained. In the multi-task PPMP, successful completion of a low probability task may warrant additional funding for a project. Such situations do not arise in the STPPMP. Hence, once a project is prioritized, it receives the same (or higher) priority until its completion. We use this result to prove the following property for the STPPMP: Theorem 5.1 For the STPPMP, there exists an optimal policy in Ω P. Proof Outline: We show that project completion times are convex in the work content of the projects. This allows construction of an improving sub-gradient for every point corresponding 15

16 to Ω C. Hence, points in Ω C cannot be optimal, which from Theorem 4.1 implies that the optimal policy lies in Ω P. In the next section we show that for a different simplification of the PPMP we can restrict attention to an even smaller, and more practical, action space. 5.2 Restricted Budget PPMP The main reason the optimal policy for the PPMP is complicated is that the contribution of a project depends not only on its own parameters but also on those of the other projects. It would be vastly simpler, and hence more feasible to combine economic analysis with other considerations, if we could evaluate each project independently. In this section we show that this is optimal for a certain class of PPMP. Specifically, we consider PPMPs where the efficient limit of all tasks of all projects is at least equal to the budget, that is, l i,j B for all i, j. We call the PPMP with this property, the Restricted Budget PPMP or RBPPMP. Although there are practical situations where such conditions exists for example, in smaller organizations that have only enough resources to fund a single project at a time the main goal of this simplification is to gain insight and identify factors that will help us develop a useful and practical heuristic for the general PPMP. First, we show that for the RBPPMP we can restrict attention to priority policies. In what follows, we normalize the constant budget B to 1 for notational convenience. Theorem 5.2 For the RBPPMP, there exists an optimal policy in Ω P. Proof Outline: We show that whenever multiple tasks are funded in any decision epoch we can construct an alternate policy where we fund only one of these tasks, say of project i, by reallocating resource from the other funded projects. As a result of this reallocation project 16

17 i s task finish earlier, thereby improving the expected NPV for the project. We then show that such reallocation does not delay any of the other projects hence the expected NPV of the other projects are unaltered. A priority policy is much easier to implement and understand than a concurrent-priority policy. Since a priority policy can be described as a sequence (that is, of projects listed in order of priorities), for a m-project PPMP there are at most m! possible priority policies. We now show that, for the PPMP, we can identify the optimal priority policy; in polynomial time. For the general PPMP, the optimal allocation must be computed at each task completion interval to incorporate the information obtained from the current task completion events. However, we can show that for the RBPPMP such computation is unnecessary. Observe, that for the RBPPMP, a prioritized project absorbs the entire budget and hence only one project is funded at any given time. Therefore, if a funded project completes successfully, its work content decreases and probability of successful completion is increased, and hence it becomes even more attractive to fund. The projects that are not funded, do not undergo any change of state and hence their relative attractiveness is unchanged. Therefore, if the task of the funded project succeeds, we continue with it. If it fails, then we switch to the project that had the next highest priority at the beginning of the previous epoch. Therefore, the priority policy has to be computed only once, at the beginning of the decision process. We will prove this formally in Theorem 5.3. This structure of the optimal policy for the RBPPMP allows us to identify the optimal sequence very easily. To do this, we introduce the following notation. Denote the expected NPV of prioritizing project i at time t =0asθ i = α i p i e βw i, for all i, andleta i j represent the event that exactly j tasks of project i complete. The probability of the event A i j is given 17

18 by (1 p i,j ) j 1 q i,j =Pr[A i k=1 j]= p i,k, j =1,...,n i 1 (13) ni 1 k=1 p i,k, j = n i Notice, that under a priority policy, whenever a project i completes j tasks it introduces an additional delay of j k=1 w i,k to all the projects that have a lower priority than project i. Therefore, the extra discount factor introduced by project i to the expected NPV of all the lower priority projects is γ i,j = e β j k=1 w i,k, i =1,...,m, j =1,...,n i (14) Using this notation we prove Theorem 5.3 For the RBPPMP, an optimal policy is obtained by computing the optimal priority sequence at the beginning of the decision process, and then fully funding the highest priority project until it completes successfully or fails. Upon completion, we prioritize the next project in the original sequence and continue in this manner until all projects complete. Further, the optimal priority sequence is computed according to the non-increasing order of the quantity I i = θ i 1 ( ni j=1 q i,jγ i,j ), i =1,...,m (15) Proof Outline: First we show that for an RBPPMP after a successful completion of a project we do not switch. This is done by showing that if for two projects i and j, itisoptimal to switch to project j even after a successful task completion of project i, then it must be optimal to fund project j in the previous epoch, hence contradicting the optimality of funding project i in that epoch. The optimality of prioritizing according to I i is proven as follows. Theorem 5.2 is used to write the expected NPV for a priority policy in closed form. The optimality condition for priority policies can then be written in terms of a set of 18

19 inequalities. These inequalities ultimately reduce to a condition that implies that project i is prioritized over project j iff quantity I i >I j. Observe that index I i given by (15) is completely determined by the parameters of project i and hence can be computed independently of the other projects in the portfolio. Theorem 5.3 states that in order to obtain the optimal policy for the RBPPMP, we can compute and sort the indices for all projects in nonincreasing order and then execute the projects in this order. We term this simplified version of a priority policy an index policy. We can interpret the index in Equation (15) more intuitively by defining a random variable Γ i for the discount factor induced by project i on all the projects with lower priorities, ( ni ) as measured from the time project i starts. The term j=1 q i,jγ i,j in the denominator of the index can be interpreted as E[Γ i ]. This allows us to write the index as I i = θ i, i =1,...,m (16) 1 E[Γ i ] Thus, the index for a project is proportional to the best expected NPV that can be obtained from a project and is inversely proportional to the delay it causes to other projects in the portfolio. It indicates, that shorter projects should be prioritized over longer projects if they have comparable expected NPV, since they introduce less delay on the projects subsequently undertaken. An index policy provides a simple solution to the RBPPMP. The index is computationally inexpensive and offers insight into the relative worth of the projects. Moreover, it provides a cardinal ranking. As such, it is well-suited to a multi-objective framework, in which this economic index is combined with other criteria. Further, since the index of a project is computed from parameters of that project only, it is easy to incorporate new project arrivals. Since the indices of the existing projects are unaffected by such arrivals, new projects can be inserted into the sequence according to 19

20 their indices. If a new project is attractive enough it may even replace the currently funded project. Since the RBPPMP is only valid for environments where the entire budget can be devoted to one project at a time, it is not directly applicable to most realistic situations. However, we can use the insights from the index policy to develop a heuristic for the general PPMP. 6 Index Policy for the General PPMP The solution to the RBPPMP suggests that the primary drivers of profitability of a project are its revenue and the delay it causes to other projects in the portfolio when it is undertaken. Since this is likely the case for the general PPMP as well, we now consider using a modified version of the index in Equation (15) for the general PPMP. First we modify the index to allow for the delay caused by tasks with arbitrary resource limits. We modify γ i,j as derived in Equation (14) as follows: γ i,j = e β j w i,k k=1 l i,j, i =1,...,m, j =1,...,ni (17) Note, that with this modification it can no longer be interpreted as the exact discount factor introduced for the lower priority projects, since other projects can receive funding while project i does. However, it does gives a lower bound on it. Using this γ i,j, the index for the general PPMP becomes I i = We now investigate the performance of the above index. Π i ( ni ), i =1,...,m (18) 1 j=1 q i,jγ i,j 20

21 6.1 Average Performance of Index Policy First we show that the worst case error from using Equation (18) can be large. Theorem 6.1 The worst case percentage error from using Index (18) for the PPMP is 100%. Proof Outline: We first argue that the maximum percentage error occurs when the indices of the projects are equal. We then maximize the error from a two project portfolio and show that 100% error can be achieved asymptotically. This is obtained when the work content of one project reaches infinity while that of the other reaches zero (while their index remains the same). Since 100% is the maximum percentage error possible and the addition of further projects does not reduce the maximum error the result is proved. Even though the worst case error for the index policy is not encouraging, the proof of Theorem 6.1 shows that the maximum error occurs when allocating vastly dissimilar projects (with respect to work content). In practice this implies that the project with small work content could be something like a one day improvement effort, while the larger project is a long-term exploratory research project. It is highly unlikely that two such dissimilar projects would be funded from the same resource pool. So, the cases where the index policy performs poorly are unlikely to be of practical interest. Therefore it is worthwhile to explore the average error that results from using an index policy in a general PPMP. We do this numerically below Simulation of Index Policy Errors To evaluate the performance of the index policy for the general PPMP over the range of practical interest we consider the following ranges for the problem parameters. We feel that these reasonably represent most project portfolio management situations. From this range, 21

22 we select points representing individual cases of the PPMP. All points are considered equally likely, so we select the parameters according to the uniform distribution. α i In a typical portfolio management decision we expect the ratio of the project revenues to rarely exceed a factor of a thousand. That is, we expect for any two projects i and j, α i /α j [0.001, 1000]. Hence, we let α i U[1, 1000] for i =1,...,m. w i,j With similar reasoning as above, we expect the ratio of the work content of the projects to rarely exceed a factor of a hundred. Hence we let w i,j U[1, 100] for i =1,...,m and j =1,...,n i. Further we restrict the efficient limits and probabilities as follows: l i,j We allow the efficient limits for the tasks to vary from 5% of the available funding to 100% of the available funding, or l i,j U[0.05, 1.0], where the budget is fixed at 1. p i,j We assume that each task has at least a 5% chance of success in order for a project to be considered for funding. Hence, we let p i,j U[0.05, 1] for all i and j. We fix the discount rate β at 50%. In most practical situations the discount rate is far less than this value. Therefore, we expect to get a conservative estimate of the error from this choice since the errors increases monotonically with the discount rate. The methodology for analyzing the error is as follows. We sample the project parameters from the respective distribution to generate the project portfolio. We then find the optimal expected NPV for the portfolio. Next, we compute the index policy and the expected NPV for that portfolio. Finally, we tabulate the percent difference between the two. Note, that the sampling procedure may generate projects which are not resource constrained, that is, the sum of the efficient limits may be less than the available budget. In such cases, Theorem 5.3 guarantees that the index policy will be optimal. In order not to bias our results, we ignore such portfolios. Thus, we tabulate errors only when there is a chance of making an error. 22

23 2 Projects 3 Projects 4 Projects 5 Projects % Error CDF Std. Dev. CDF Std. Dev. CDF Std. Dev. CDF Std. Dev E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Sample Size 5.19 E E E E05 Mean Variance Std. Dev Table 2: Percentage Errors from using Index Policy for Projects with Single Task So, in reality, the average errors we report below are higher than the true average across the entire sample space. 1 The computation time required to find the optimal policy becomes prohibitively expensive if the number of projects or the number of tasks per project is increased. Therefore, we first study the effect of increasing the number of projects in the portfolio. Then we look at the effect of increasing the number of tasks per project. First we show results for portfolios where projects have only one task. Table 2 shows the result of the computations for up to a 5 project portfolio. As can be seen, the average error 1 A technical note on the sampling procedure: It is known that sampling from the Uniform distribution in higher dimensions results in banding, that is, the k-tuples (U i,...,u i+k 1 ) will always lie on a finite number of hyper-planes in [0, 1] k (for example see [21] pp. 22). To alleviate such problems, we use a fifth order multiple recursive random number generator with two components as provided by P. L Ecuyer in [15]. Further, we use antithetic variables (see [14] pp. 628) to help reduce the variance. 23

24 from using the index policy is under 1% for all cases. Further, the distribution of the error is also shown, with the associated standard deviation for each of the probability estimates. Note, that while the standard deviation of the estimates are quite low, they become higher as number of projects grow. This is because less samples were generated for these projects since the procedure gets computationally intensive. The distribution of the errors are plotted in Figure 3. We see that a large percentage of the portfolios give no error at all and virtually no error above 20% was observed. We also note that while errors increase in the number of projects, the distribution of the 4 project portfolio and the 5 project portfolio are almost identical. Therefore, we expect that portfolios with a higher number of projects will also converge to this distribution of errors for this parameter range. However, due to the impractical computation time required to obtain the results for higher number of projects we are unable to verify this statement. We now look at the effect of increasing the number of tasks per projects. Figure 4 shows the distribution of error for a 2 project portfolio with between 1 and 5 tasks. Again, while errors increase in the number of tasks, they are small and appear to be converging to the distribution for the 5 task case. Again, errors above 10% are very rare and no error above 20% was observed. These analysis indicate that the index policy should perform extremely well in practical settings. Given that the input data is very difficult to estimate to an accuracy greater than 10%, the results in practice should be virtually indistinguishable from an optimal policy. 7 Conclusion and Future Research We have shown that the index policy is a simple and effective way to evaluate the payoff and timing effects of projects in a limited-resource portfolio. Because it allows projects to be rated according to an index that can be computed for each project independently, it is 24

25 simple to use and well-suited to scoring models that assign ratings for projects along various dimensions. It is also suited to environments where the set of candidate projects evolves over time, since new projects can be inserted into an existing sequence according to their index. But while the index policy approach captures a useful piece of the innovation management process in a practical way, it leaves out a number of important issues that might be able to be incorporated into a mathematical model. In particular, further work is needed to address the following: 1. Risk: The treatment in this paper only considers expected values but not the variance of return or the likelihood of cash flow problems. Some authors, for example Dixit et. al [5] and Luehrman [18], have suggested a real options approach for incorporating risk into the financial analysis of projects. It remains to be seen whether such an approach could be usefully combined with the optimization framework of this paper. 2. Project Interactions: In many environments the payoffs from projects are dependent on one another. For instance, a pharmaceutical company would not want to fund only projects aimed at developing anti-depressants because the resulting products would be competitors in the marketplace. Further work is needed to extend the approach of this paper to situations where such interactions are important. 3. Information Feedback: In many environments success or failure of one project has an impact on other projects. For instance, a breakthrough on a fuel-cell automotive power plant might substantially reduce the potential returns on an all-electric car. How to represent information accumulation in a framework more sophisticated than the simple 0-1 model of success and failure used in this paper is an interesting and challenging topic. Acknowledgement: This work supported in part by the National Science Foundation under grant DMI

26 Simultaneous task completion <2,1> <2,1> 0.4 <1,1> prioritized <1,2> <1,3> prioritized Allocate 0.4 to task 1 of project 1 at first epo Simultaneous task completion <2,1> <2,1> 0.37 <1,1> <1,2> prioritized <1,3> prioritized Allocate 0.37 to task 1 of project 1 at first ep Figure 2: Optimal allocation of example portfolio when task 1 of project 1 succeeds 26

27 Probability Projects 3 Projects 4 Projects 5 Projects Percent Error Figure 3: Distribution of Errors for Index Policy for Single Task Portfolio 27

28 Probability Task 2 Tasks 3 Tasks 4 Tasks 5 Tasks Percentage Error Figure 4: Distribution of Errors for Index Policy for Multiple Task Portfolio 28

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