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 resourcemust be allocated among a set of candidate projects over time so as to maximize expected netpresent value. We formulate this problem as a dynamic program but conclude that this approachis 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 demonstratethat the solution reduces to a simple form under certain environmental conditions. This simplifiedpolicy, which we term the index policy, sequences projects according to a simple ratio and thenallocates resource up to each project’s practical limit in the order given by this sequence. Throughnumerical 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 issuesinto a multi-dimensional scoring model for addressing real-world project portfolio managementsituations.

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 multi-

dimensional to be reduced to a single model. In real-world situations, managers must consider

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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 op-

timization 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.

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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 program-

ming, 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.

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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.

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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 wi,j expressed in terms of total resource-hours required to complete the task.

Hence, the total work requirement of project i is given by wi =∑n

j=1 wi,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. Task j of project

i is assumed to succeed with a predetermined probability pi,j, which is independent of all

other problem parameters including the resource rate, and success or failure becomes known

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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 pi =∏n

j=1 pi,j. Upon completion,

each project i yields an expected profit of αi.

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 ti(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∑

i=1

piαiE[e−βti(γ)] (1)

where β is the continuous discount rate and the expectation is taken over all possible com-

binations of task completion events. Because the objective function is monotonically non-

increasing 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.

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limitResource

Linear

Completion rate

Approximation

Expected

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 li,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∑i=1

li,j > B, (2)

If this is not true, we can simply allocate each project resource up to its efficient level and

no further optimization is necessary.

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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 xi(t) and represents a valid resource allocation as long as∑n

i=1 xi(t) ≤ B and

0 ≤ xi(t) ≤ li(t), for all i, where li(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 xi

(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 t1 and t2 be two task completion points

that result from a given policy along a given sample path. Suppose during the interval [t1, t2]

work wi is done on project i with a possibly variable rate, so that∫ t2

t1xi(t)dt = w1. Using

a constant resource allocation xi such that xi(t2 − t1) = wi 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, where n ∈ Nm is the number of tasks left for each

of the projects, and w ∈ Rm is the amount of work left in the current task of the projects.

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The control variable is the vector x ∈ Rm, where xi denotes the allocation to project i, at

the first decision epoch. The allocation vector x must satisfy the following constraints:

xi ≤ li, i ∈ M (3)

where li is the resource limit for the current task of project i,

m∑i=1

xi ≤ B (4)

where B is the total budget, and

xi ≥ 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 χc[x] be the indicator that completion c ∈ C[s] occurs due to allocation x. For any

c ∈ C[s], let τc(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] denote the

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] = maxx∈A

∑c∈C[s]

χc[x]e−βτc(x)∑

y∈Y [s,c]

(ζ(y)V [y] + r[s,y]) (6)

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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 Kr(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 xri,k(σ) be the

allocation to project i at the kth epoch under sample path r for sequence σ. Let τ rk (σ)

denote the corresponding length of the task completion interval. Notice, that τ rk (σ) can be

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 rk (σ). We define tri (σ) = ∞

if project i fails under subsequence (σ, r). Let lri,k(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 lri,k(σ) = 0. Observe that at any epoch k, the full

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

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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∈ΣZσ (7)

where

Zσ = maxx∈C[σ]

∑r∈R

Pr[r]m∑

i=1

αie−tri (σ), σ ∈ Σ (8)

and C[σ] consists of the following set of constraints defined for each σ ∈ Σ:

(i) the budget constraints,

m∑i=1

xri,k(σ) ≤ B, k = 1, . . . , Kr(σ), r ∈ R (9)

(ii) the resource limit constraints,

0 ≤ xri,k(σ) ≤ lri,k(σ), i = 1, . . . ,m, k = 1, . . . , Kr(σ), r ∈ R (10)

(iii) the sequence determining constraints,

τ rk (σ) > 0, k = 1, . . . , Kr(σ), r ∈ R (11)

Each problem Zσ 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

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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 tri [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)] =∑r∈R

Prrm∑

i=1

αie−βtri [γ] (12)

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

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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 ΩC and the class of priority policies as ΩP .

The class of concurrent-priority policies is the union of these two sets and we denote it by

Ω = ΩC ∪ ΩP . 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 ΩP . Unfortunately, this is not optimal in general. For example, consider the

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Project α wi,1 li,1 pi,1 wi,2 li,2 pi,2 wi,3 li,3 pi,3

Project 1 20 0.1 0.4 1.0 0.3 0.4 1.0 2.0 0.7 1.0Project 2 1 0.675 0.75 1.0 – – – – – –

Table 1: Example Portfolio

following two project portfolio whose optimal solution is a concurrent policy in ΩC . 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 ΩP . 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

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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 alloca-

tions 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 ΩP .

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

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to ΩC . Hence, points in ΩC cannot be optimal, which from Theorem 4.1 implies that the

optimal policy lies in ΩP .

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, li,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 ΩP .

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

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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 = 0 as θi = αipie−βwi , for all i, and let Ai

j represent

the event that exactly j tasks of project i complete. The probability of the event Aij is given

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by

qi,j = Pr[Aij] =

(1 − pi,j)∏j−1

k=1 pi,k, j = 1, . . . , ni − 1

∏ni−1k=1 pi,k, j = ni

(13)

Notice, that under a priority policy, whenever a project i completes j tasks it introduces an

additional delay of∑j

k=1 wi,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

γi,j = e−β∑j

k=1 wi,k , i = 1, . . . ,m, j = 1, . . . , ni (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

Ii =θi

1 −(∑ni

j=1 qi,jγi,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, it is optimal

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 Ii 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

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inequalities. These inequalities ultimately reduce to a condition that implies that project i

is prioritized over project j iff quantity Ii > Ij.

Observe that index Ii 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. Theo-

rem 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 Γi for the discount factor induced by project i on all the projects with lower priorities,

as measured from the time project i starts. The term(∑ni

j=1 qi,jγi,j

)in the denominator of

the index can be interpreted as E[Γi]. This allows us to write the index as

Ii =θi

1 − E[Γi], i = 1, . . . ,m (16)

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

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 γi,j as derived in Equation (14) as follows:

γ′i,j = e

−β∑j

k=1

wi,kli,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

Ii =Π∗

i

1 −(∑ni

j=1 qi,jγ′i,j

) , i = 1, . . . ,m (18)

We now investigate the performance of the above index.

20

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.

6.1.1 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

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.

αi 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,

αi/αj ∈ [0.001, 1000]. Hence, we let αi ∼ U [1, 1000] for i = 1, . . . ,m.

wi,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 wi,j ∼ U [1, 100] for i = 1, . . . ,m

and j = 1, . . . , ni.

Further we restrict the efficient limits and probabilities as follows:

li,j We allow the efficient limits for the tasks to vary from 5% of the available funding to

100% of the available funding, or li,j ∼ U [0.05, 1.0], where the budget is fixed at 1.

pi,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 pi,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

2 Projects 3 Projects 4 Projects 5 Projects% Error CDF Std. Dev. CDF Std. Dev. CDF Std. Dev. CDF Std. Dev.

0 0.7729 2.09E-05 0.585 5.74E-05 0.486 3.79E-04 0.47 2.07E-031 0.9037 1.36E-05 0.825 6.92E-05 0.784 3.98E-04 0.77 2.13E-032 0.9442 1.04E-05 0.898 9.98E-05 0.875 5.82E-04 0.87 3.00E-033 0.9657 8.13E-06 0.936 1.12E-04 0.921 6.62E-04 0.92 3.52E-034 0.9785 6.44E-06 0.958 1.20E-04 0.949 7.00E-04 0.95 3.78E-035 0.9863 5.15E-06 0.972 1.24E-04 0.966 7.27E-04 0.97 3.89E-03

7.5 0.9954 2.98E-06 0.990 6.84E-05 0.987 4.04E-04 0.99 2.13E-0310 0.9985 1.67E-06 0.996 6.75E-05 0.995 4.03E-04 1.00 2.11E-0315 0.9999 4.10E-07 1.00 2.91E-05 0.999 1.94E-04 1.00 1.06E-0320 1.0000 ∼ 0.0 1.00 2.11E-05 1.00 2.10E-04 1.00 8.66E-0425 1.0000 ∼ 0.0 1.00 ∼ 0.0 1.00 1.64E-04 1.00 4.30E-09. . . . . . . . . . . . . . . . . . . . . . . . . . .100 1.0000 ∼ 0.0 1.00 ∼ 0.0 1.00 ∼ 0.0 1.00 ∼ 0.0

Sample Size 5.19 E08 1.26 E08 3.53 E06 1.25 E05Mean 0.003 0.006 0.008 0.008

Variance 0.0001 0.0002 0.0003 0.0003Std. Dev. 0.011 0.015 0.017 0.018

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

1A technical note on the sampling procedure: It is known that sampling from the Uniform distributionin higher dimensions results in “banding”, that is, “the k-tuples (Ui, . . . , Ui+k−1) will always lie on a finitenumber of hyper-planes in [0, 1]k” (for example see [21] pp. 22). To alleviate such problems, we use a fifthorder 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

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

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-9732868.

25

<1,1>prioritized

<2,1>

<1,2>

<2,1>

<1,3>prioritized

<1,1>

<2,1>

<1,2>prioritized

<2,1>

<1,3>prioritized

0.37

0.4

Simultaneous task completion

Simultaneous task completion

Allocate 0.4 to task 1 of project 1 at first epo

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

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6 8 10 12 14 16

Percent Error

Probability

2 Projects

3 Projects

4 Projects

5 Projects

Figure 3: Distribution of Errors for Index Policy for Single Task Portfolio

27

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20

Percentage Error

Probability

1 Task

2 Tasks

3 Tasks

4 Tasks

5 Tasks

Figure 4: Distribution of Errors for Index Policy for Multiple Task Portfolio

28

References

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[2] Baker, Norman R. (1974) “R&D project selection models: an assessment,” IEEE Trans-actions on Engineering Management, Vol. EM-21(4), pp. 165-171.

[3] Baker, J. R. and J. R. Freeland, (1975) “Recent advances in R&D benefit measurementand project selection methods,” Management Science, Vol. 21(6) 1164-1175.

[4] Chung, A., W. D. Cook and M. Kress (1998) “Resource allocation in R&D depart-ments,” INFOR Vol. 36(1/2), pp. 41-56.

[5] Dixit, A. K. and R. S. Pindyck, (1995) “The options approach to capital investment,”Harvard Business Review, Vol. 73(3), pp. 105-115.

[6] Gerchak, Y. (1998) “On allocating R&D budgets among and within projects” R&DManagement Vol. 28(4), pp. 305-309.

[7] Gandforoush, P., P. Y. Huang and L. J. Moore (1992) “Multi-project, multi-criteriaevaluation and selection model for R&D management,” Management of R&D and En-gineering, D. F. Kocaoglu (Ed.) Elsevier Science Publishers, pp 89-100.

[8] Gupta, D. K., T. Madakovic (1992) “Contemporary approaches to R&D project selec-tion, a literature survey”, Management of R&D and Engineering, D. F. Kocaoglu (Ed.)Elsevier Science Publishers, pp. 67-86.

[9] Heidenberger, K. (1996) “Dynamic project selection and funding under risk: A decisiontree based MILP approach,” European Journal of Operations Research, Vol. 95, pp.284-298.

[10] Kavadias and Loch, “Dynamic portfolio selection of NPD programs using marginalreturns”, Working Paper. INSEAD, France.

[11] Krawiec, F. (1984) “Evaluating and selecting research projects by scoring,” ResearchManagement, Vol. 27(2), pp. 21-25.

[12] Kocaoglu, D. F., M. G. Iyigun (1994) “Strategic R&D program selection and resourceallocation with a decision support system application,” Proceedings of the IEEE Inter-national Engineering Management Conference 1994, Dayton Ohio, pp. 225-232.

[13] Lai, K. K. and L. Li (1999) “A dynamic approach to multiple-objective resource allo-cation problem,” European Journal of Operations Research Vol. 117, pp. 293-309.

[14] Law, A. M and W. D. Kelton (1991) Simulation modeling & analysis, Second Edition,McGraw Hills.

[15] L’Ecuyer, P. (1999) “Good parameters and implementations for combined multiple re-cursive random number generators,” Operations Research, Vol. 47(1), pp. 159-164.

[16] Liberatore, M. J. and G. J. Titus (1983) “The practice of management science in R&Dproject management,” Management Science, Vol. 29(8), pp. 962-974.

29

[17] Lootsma, F. A., J. Meisner and F. Schellemans (1986) “Multi-criteria decision analysisas an aid to the strategic planning of energy R&D,” European Journal of OperationsResearch, Vol. 25, pp. 216-234. Management Science Vol. 29(8), pp. 962-974.

[18] Luehrman, T. A. (1998) “Investment opportunities as real options: getting started onthe numbers,” (1998) Harvard Business Review, Vol. 76(4), pp. 51-67. option

[19] Madey, G. R. and B. V. Dean (1992) “An R&D project selection and budgeting modelusing decision analysis and mathematical programming,” (1992) Management of R&DTechnology, D. F. Kocaoglu (Ed.) Elsevier Science Publishers. .

[20] Oral, M, O. Kettani and P. Lang (1991) “A methodology for collective evaluation andselection of industrial R&D projects,” Management Science Vol. 37(7), pp. 971-885.

[21] Ripley, B. D. (1987) Stochastic Simulation, John Wiley & Sons.

[22] Schmidt, R. L. and J. Freeland (1992) “Recent progress in modeling R&D project-selection processes,” IEEE Transactions on Engineering Management, Vol. 39(2), pp189-201.

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[24] Tavares, L. V. (1990) “A multi-stage non-deterministic model for project schedulingunder resource constraints,”European Journal of Operations Research, Vol. 64, pp 312-325.

[25] Ulvila, J. W. and J. O. Chinnis, Jr. (1992) “Decision analysis for R&D resource manage-ment,” Management of R&D and Engineering, D. F. Kocaoglu (Ed.) Elsevier SciencePublishers.

[26] Winkofsky, E. P., N. R. Baker and D. J. Sweeney (1981) “A decision process modelof R&D resource allocation in hierarchical organizations,” Management Science, Vol.27(3), pp. 268-283. European Journal of Operations Research, Vol. 49, pp. 93-101.

30

A Appendix

We first introduce the following notation, which is used throughout this Appendix.

We denote the allocation to project i during the kth task completion interval under policyγ and sample path r by xr

i,k(γ). The duration of the kth task completion interval under apolicy γ and sample path r is denoted by τ r

k (γ). The completion time of project i underpolicy γ and sample path r is denoted as tri (γ).

Theorem 4.1 For any PPMP there exists an optimal policy that belongs to class Ω.

Proof: We need to show, that for any policy γ /∈ Ω we can find a policy λ ∈ Ω which is asgood or better than policy γ.

For any policy γ the expected NPV is given by expression (12). Since γ is not in Ω,there must exist some epoch where two (or more) projects are funded at positive levelsbelow their efficient limits. We show that the expected NPV from such a policy canbe improved. We show this for policies where an intermediate allocation occurs at thefirst decision interval. This is sufficient to prove the result since such a an allocation isembedded in the DP given by Equation 6 for the general PPMP.

Let projects 1 and 2 be two of the projects funded at an intermediate level at the firstdecision epoch. Therefore 0 < xr

i,1(γ) < li,1 for i = 1, 2 and all r (because the firsttask completion interval is common to all possible sample paths). Now, consider analternate policy γε defined as follows. In the first decision epoch, we allocate xr

1,1(γ)+ε toproject 1 and xr

2,1(γ)−ε to project 2, for some non-zero ε, such that, the task completionsequence for every possible sample path remains unchanged from those under γ. Allother allocations are kept exactly the same as in policy γ. Notice, that such a policy isguaranteed to be feasible for a sufficiently small ε because γ /∈ Ω. For any policy in Ω,if projects are prioritized at decision epoch 1, then any extra allocation will result inviolation of the efficient limit constraint. Alternately, if the policy induces simultaneoustask completions for some sample path, then perturbation by a non-zero ε may changethe completion sequence for at least one sample path. Since, γ /∈ Ω, these cases cannotoccur.

We denote the change in the completion time due to this perturbation by δri (γ, ε) =

tri (γε)− tri (γ) for all i and r. In Lemma A.1, we prove that δrk(γ, ε) = dr

k,jδrj (γ, ε), where

drk,j is some constant independent of ε, that is, δr

k(γ, ε) is a linear in δrj (γ, ε) for any

31

k, j ∈ M . Using this we can calculate the expected NPV for this policy to be:

E[Π(γε)] =∑

r

Prrm∑

i=1

αie−βtri (γε)

=∑

r

Pr[r]m∑

i=1

αie−β[tri (γ)+δr

i (ε)]

=∑

r

Pr[r]m∑

i=1

αie−β[tri (γ)+dr

i,1δr1(ε)]

(19)

where the last equality follows from Lemma A.1. Substituting δr1(ε) from Equation (31),

we get

E[Π(γε)] =∑

r

Prrm∑

i=1

αie−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

](20)

for some dri independent of ε.

We first consider the case when δri (ε) is non-zero for at least one (i, r). For this case

we show that E[Π(γε)] is maximized only at the extreme points of ε. DifferentiatingE[Π(γε)] once with respect to ε we get,

d E[Π(γε)]

d ε= −

∑r

Prr xr1,1(γ)β(

xr1,1(γ) + ε

)2[

m∑i=1

αidri e

−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

]](21)

Let this derivative vanish at ε = ε∗. Then, observing that in Equation (21) the termsoutside the parenthesis are positive, we can write

m∑i=1

αidri e

−β

[tri (γ)+dr

iε∗

xr1,1(γ)+ε∗

]= 0, for all r (22)

Now consider the second derivative with respect to ε,

d2 E[Π(γε)]

d ε2=∑

r

Prr xr1,1(γ)β(

xr1,1(γ) + ε

)2[m∑

i=1

αidri

(2(ε + xr

1,1(γ)) + dri βxr

1,1

)e−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

]](23)

32

which after rearranging can be written as:

d2 E[Π(γε)]

d ε2=∑

r

Prr xr1,1(γ)β(

xr1,1(γ) + ε

)4[2(ε + xr

1,1(γ))m∑

i=1

αidri e

−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

]

+βxr1,1(γ)

m∑i=1

αi(dri )

2e−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

]](24)

At ε = ε∗, we can use Equation (22) to write the expression for the second derivativeas,

d2 E[Π(γε)]

d ε2

∣∣∣∣ε∗

=∑

r

Prr (xr1,1(γ)β)2(

xr1,1(γ) + ε

)4m∑

i=1

αi(dri )

2e−β

[tri (γ)+dr

iε

xr1,1(γ)+ε

](25)

which is clearly positive. Therefore, any interior point is a minimum rather than amaximum. Hence, the maximum must occur at an extreme point of ε. An extremepoint of ε in the neighborhood of tr1(γ) is reached for an ε such that either the efficientlimit constraint is tight or at the point where two (or more) tasks along some samplepath (or paths) complete simultaneously. The latter case gives an extreme point sincewhen more than a single task complete together for some sample path r due to theabove allocation, the task completion sequence Sr(γε) = Sr(γ) and equation (19) is nolonger valid.

Finally, if all δri (ε) = 0, then changing ε does not have any effect on the completion

times of the projects. Hence, we can still change ε such that we reach a policy in Ω.This proves the result.

Lemma A.1 δrk(γ, ε) = dr

k,jδrj (γ, ε), where dr

k,j is some constant independent of ε, that is,δrk(γ, ε) is a linear in δr

j (γ, ε) for any k, j ∈ M .

Proof: To prove the result we need to find the dependence of the first period’s allocationon the completion times of the projects. This is done as follows.

Suppose policy γ /∈ Ω under sample path r induces a task completion sequence Sr(γ) =(ik, jk)|k=1,...,Kr(γ). Further, if s = Sr

k(γ), then s[1] and s[2] denote the project andtask that finish at the kth epoch under policy γ respectively. Also, Kr(γ) denotes thetotal number of task completion epochs under sample path r. Let θr

i,j(γ) denote theposition of the jth task completion of project i in Sr(γ); where we define θr

i,0(γ) = 0.Finally, let νr

i (γ) denotes the number of tasks of project i completed under sample pathr under policy γ.

33

We illustrate this notation via the following example. Suppose policy γ, under somesample path r induces the task completion sequence

Sr(γ) = (1, 1), (1, 2), (2, 1), (3, 1), (3, 2), (2, 2), (1, 3), (2, 3)

This means, for example, the first task to complete is task 1 of project 1, followed bytask 2 of project 1, followed by task 1 project 2 and so on. Then we for project 1 wehave θr

1,1(γ) = 1, θr1,2(γ) = 2, θr

1,3(γ) = 7, for project 2 we get θr2,1(γ) = 3, θr

2,2(γ) = 6,and so on. Also, since 3 tasks of project 1 complete, we have νr

1(γ) = 3. Similarlyνr

2(γ) = 3 and νr3(γ) = 2.

We now derive an expression for the completion times tri (γ), for a given sample path.Using the above notation, at the kth task completion interval, task j of project icorresponding to sr

k(γ) gets completed. The resource allocated to this task at thisepoch is xs[1],s[2] for s = Sr

k(γ). The original amount of work for this task is ws[1],s[2]

for s = Srk(γ), but at period k some work may have already been completed. This

is calculated as follows. Due to the precedence constraint, the first time any resourcemay be allocated to this task is after the completion of task j − 1. Task j − 1 for thisproject completes at epoch θr

i,j−1(γ). Therefore, this task may receive allocation fromepoch θr

i,j−1(γ) + 1 onward. Note, for task j = 1, θri,0(γ) is defined to be 0. Therefore,

the amount of work already done for task j of project i at the beginning of the epochwhere it completes is

∑k−1l=θr

i,j−1(γ)+1 τ rl (γ)xr

i,l(γ). This quantity must be subtracted from

the original amount of work for the task in order to compute the length of this epoch.This analysis allows us to write the expression for τ r

k (γ) as,

τ rk (γ) =

ws[1],s[2] −∑k−1

l=θri,j−1(γ)+1 τ r

l (γ)xrs[1],l(γ)

xs[1],s[2]

, s = Srk(γ), k = 1, . . . , Kr(γ) (26)

Therefore, the project completion times are obtained as,

tri (γ) =

θri,νr

i(γ)

(γ)∑k=1

τ rk (γ), i = 1, . . . ,m (27)

We are interested in the effect of changing the allocation during the first period. Letthe first task to complete under γ be project i, that is, cr

1(γ) = (i, 1). We first showthat τ r

k (γ) can be written as,

τ rk (γ) = br

k,0 +brk,i +

∑l∈M\i x

rl,1(γ)br

k,l

xri,1(γ)

, k = 1, . . . , Kr(γ) (28)

for some brk,l, constant with respect to the allocation in the initial period xr

j,1(γ), j ∈ M .We prove this via induction on k. Clearly for k = 1, τ1 = wi,1/xi,1 and the assertion istrue. Assuming that the hypothesis is true for an arbitrary k, we substitute Equation

34

(28) into Equation (26). Noting, that w(srk(γ)) and x(sr

k) are independent of xj,1, j ∈ Min (26) the result follows after some algebra.

Substituting Equation (28) into Expression (27) for the completion time and after somealgebra we conclude that,

trk(γ) = crk,0 +

crk,i +

∑l∈M\i x

rl,1(γ)cr

k,l

xri,1(γ)

, k ∈ M (29)

for some crk,l, constant with respect to the allocation in the initial period xr

j,1(γ), j ∈ M .

Now consider the effect of policy γε on the completion times. To remind ourselves,under policy γε, we change the allocation to project 1 to xr

1,1(γ) + ε and the allocationto project 2 to xr

2,1(γ) − ε for some non-zero ε, such that, Sr(γ) = Sr(γε) for all r. Wekeep all other allocations as in policy γ.

Depending on which project completes its task in the first decision epoch, we have thefollowing three cases: (a) project 1 finishes first b) project 2 finishes first and 3) neitherproject 1 nor project 2 is first to finish. We prove the result explicitly for Case (a). Theresult is obtained for the other cases via similar analysis.

When project 1 is the first to have its task complete, the completion time for the projectsunder γε can be written by substituting xr

1,1(γ) = xr1,1(γε) + ε and xr

2,1(γ) = xr2,1(γε)− ε

in Equation (29). This yields,

trk(γ) = crk,0 +

crk,1 + (xr

2,1(γ) − ε)crk,2 +

∑l∈M\1∪2 xr

l,1(γ)crk,l

xr1,1(γ) + ε

, k ∈ M (30)

Using Equations (29) and (30) we can compute, after some algebra,

δrk(γ, ε) = trk(γε) − trk(γ)

= − ε

xr1,1(γ) + ε

[crk,1 + 2xr

1,1(γ)xr2,1(γ)cr

k,2 +∑

l∈M\1∪2 xrl,1(γ)cr

k,l

xr1,1(γ)

](31)

Finally, using (31) we compute the ratio,

δrk(γ, ε)

δrj (γ, ε)

=crk,1 + 2xr

1,1(γ)xr2,1(γ)cr

k,2 +∑

l∈M\1∪2 xrl,1(γ)cr

k,l

crj,1 + 2xr

1,1(γ)xr2,1(γ)cr

j,2 +∑

l∈M\1∪2 xrl,1(γ)cr

j,l

= drk,j (32)

where drk,j is a constant clearly independent of ε.

Lemma 5.1 For any STPPMP, we can restrict our attention to policies under which theallocations to any project is non-decreasing in time until the completion of the project.

Proof: We need to show for any policy γ for the STPPMP, xi,k+1(γ) ≥ xi,k(γ) for allprojects i and task completion epochs k, unless project i is completed at the end oftask completion epoch k.

35

WLOG we start at the first decision epoch. Suppose under a given policy γ, there existsa project, say project 1, such that x1,1(γ) > x1,2(γ) and project 1 does not complete atthe end of the first epoch.

At time τ1, by definition, some project or projects must complete. Hence, the availablebudget to allocate the remaining projects must increase. Since, x1,1(γ) > x1,2(γ), forthe allocation at the second epoch to be efficient, there must exist a set of projects J ,such that xj,1(γ) < xj,2(γ) for all j ∈ J . Since the total budget is constant, we musthave

x1,1(γ) − x1,2(γ) =∑j∈J

(xj,2(γ) − xj,1(γ)) (33)

We now show that it is possible to construct a modified policy γε which is feasible, hasat least the same expected NPV as γ and also maintains xi,1 ≤ xi,2 for all i ∈ J ∪ 1.Policy γε is constructed as follows. At epoch 1, project 1 receives x1,1(γe) = x1,1(γ)−ε1,1

and projects j ∈ J receives xj,1(γε) = xj,1(γ) + εj,1. In period 2, project 1 receivesx1,2(γε) = x1,2(γε) + ε1,2 and projects in J receives xj,2(γε) = xj,2 − εj,2. In order for γε

to satisfy the desired conditions we impose the following conditions on εi,j. To ensurethat the work done by the two policies at the end of the second epoch is the same forthe two policies, we require

2∑k=1

xi,k(γε)τk =2∑

k=1

xi,k(γ)τk, for all i ∈ J ∪ 1 (34)

To ensure that the budget constraint is not violated in the two epochs we require

ε1,k =∑j∈J

εj,k, for k = 1, 2 (35)

Finally, to ensure that the statement of the Lemma holds, we require

xi,1(γε) ≤ xi,2(γε), for all i ∈ J (36)

Converting the inequalities in (36) to equalities, that is

xi,1(γε) = xi,2(γε), for all i ∈ J (37)

we see, after some algebra, that the system of equations (34), (35) and (37) are consistentas long as x1,1(γ) − x1,2(γ) =

∑j∈J (xj,2(γ) − xj,1(γ)) holds. However, this is precisely

Condition (33) and hence is satisfied. Using this equality we can compute εi,j explicitly

36

as

ε1,1 =τ2

τ1 + τ2

(x1,1(γ) − x1,2(γ))

ε1,2 =τ1

τ1 + τ2

(x1,1(γ) − x1,2(γ))

εj,1 =τ2

τ1 + τ2

(xj,2(γ) − xj,1(γ)) , for all j ∈ J

εj,2 =τ1

τ1 + τ2

(xj,2(γ) − xj,1(γ)) , for all j ∈ J

Notice, that εi,j > 0 for all i, j since x1,1(γ) > x1,2(γ) and xj,2(γ) > xj,1(γ) for all j ∈ J .Finally, Equalities (37) imply for all j ∈ J , xj,1(γε) = xj,1(γ) + εj,1 = xj,2(γ) − εj,2 =xj,2(γε). Since εj,2 > 0, this implies xj,1(γε) < xj,2(γ) ≤ lj. Using the equality in reversewe get for all j ∈ J , xj,2(γε) > xj,1(γ) ≥ 0. Similar arguments show 0 ≤ x1,1(γε) ≤ l1.Therefore, the individual efficient limit conditions are also satisfied by the new policy,so γε is feasible. Equalities (34) ensure that γε finishes all tasks at least as early as doesγ. Notice, that under γε some projects in J can finish at τ1, which can improve theexpected NPV. Finally, Equations (37) guarantee that the statement of the Lemma issatisfied. Hence, the result is proved.

We use the following two technical Lemmas to prove Theorem 5.1.

Lemma A.2 Under a priority policy for the STPPMP, if the completion time for project iis ti then ∂ti

∂wjis nonincreasing in wj for any projects i and j.

Proof: WLOG, assume that the priority at the first decision epoch is 1, 2, . . . , m. FromLemma 5.1 we know that this order of priority is maintained in all epochs. LetB0(t) = B be the available budget for the PPMP. Therefore, project 1 receives rateρ1(t) = minB0(t), l1 until its completion and zero thereafter. In the remaining budget,B1(t) = (B0(t)−ρ1(t)) project 2 is allocated ρ2(t) = minB1(t), l2 until its completion.In general, the budget available after allocation of project i is

Bi(t) = Bi−1(t) − ρi(t) (38)

and the funding to project i is given by

ρi(t) =

minli, Bi−1(t), 0 ≤ ti

0, t > ti(39)

A simple induction argument shows that Bi(t) is a piece-wise constant, monotonousnon-decreasing function of t. Similarly, ρi(t) is a monotonous non-decreasing functionof t up to ti and drops to zero thereafter. Also, since the jumps (if any) in ρi(t) and

37

Bi(t) take place at task completion epochs, we can write for some Bki > Bk−1

i

Bi(t) = Bki , τk−1 < t ≤ τk, k = 1, . . . , Ki (40)

where Ki is the task completion epoch at which project i completes. Similarly, for someρk

i > ρk−1i , we can write

ρi(t) = ρki , τk−1 < t ≤ τk, k = 1, . . . , Ki (41)

We now investigate the effect of increasing the work content of project j, wj on com-pletion time ti of project i. We have the following three cases: (i) j = i, (ii) j > i and(iii) j < i and we show that the result is true for all these cases.

Case (i): Suppose the work content of project i increases from wi to w′i = wi+∆i. Since

under a priority policy project i utilizes all available resource until its current completiontime ti, the extra task must be done with available resource after ti. Therefore, if t′i is

the new completion time for the project, we have∫ t′ii

tiρi(t) dt = ∆i. Using the fact that

ρi(t) is piece-wise constant and monotonous non-decreasing, we can write

K′i∑

k=1

ρki τk = wi + ∆i = w′

i (42)

where K ′i is the new task completion epoch of project i. Differentiating above with

respect to w′i we see that ∂ti

∂w′i

= 1/ρK′i. Since ρk

i is monotonous nondecreasing, this

shows ∂ti∂wi

is monotonous nonincreasing and the result holds.

Case (ii): Since j < i, project j has a lower priority than project i. Hence, the allocationto project i takes place before allocation to project j. Therefore, any change in the wj

cannot affect ti and hence ∂ti∂wj

= 0 and the result holds.

Case (iii): We have the following two sub-cases. a) tj ≥ ti and b) tj < ti.

a) If tj ≥ ti, any increase in wj does not affect ti since, project i receives allocation onlyafter project j is allocated. Hence, ∂ti

wj= 0 and the result holds.

b) If tj < ti we proceed as follows. We first examine the effect of increasing wj onthe completion time of project j + 1. If wj increases by ∆j, then using the samereasoning as in Case (i) we see that if the new completion time of project j is t′j,

then the available budget after allocating project j, which is∫ t′j

0 Bj(t) dt, decreases by∆j. If tj+1 < tj using similar reasoning as in Case (ii) above, the allocation rate toproject j + 1 and consequently the completion time of project j + 1 is unchanged.

However, because of the increase in wj,∫ t′j

0 B2(t) dt is decreased by ∆j. If however,tj+1 > tj, then the available budget for project j + 1 decreases and hence completion

time increases to t′j+1. Therefore, the available budget∫ t′j+1

0 B2(t) dt decreases by ∆j.

Combining the two cases, we see that∫ maxt′j ,t′j+10 B2(t) dt decreases by ∆j. The effect

38

on tj+2 is computed similarly, so that we get that∫ maxt′j ,t′j+1,t′j+2

0 B2(t) dt decreases

by ∆j. Continuing this way we see that∫ maxt′j ,t′j+1,...,t′i−1

0 Bi−1 dt is decreased by ∆j.Consequently, ρi(t) also decreases proportionally. Hence, the completion time of projecti increases to compensate for this loss and the increase is calculated similar to Equation(42) and therefore using identical reasoning, ∂ti

∂wiis monotonous nonincreasing and the

result holds.

Therefore the Lemma is proved.

For the following technical lemma we assume a budget function that satisfies the following

Assumption A.1 The budget B(t) is a function of time t with the following restrictions:

1. B(t) is a non-decreasing function of time t,

2. It is integrable,

3. It is differentiable except at a finite number of non-differentiable points, b1, . . . , bn.

Note, that we do not require the function to be continuous at the non-smooth points, eventhough the non-decreasing assumption restricts the ’jumps’ to be positive.

Lemma A.3 For a two project portfolio, each project with a single task, the completion timeof one project is a concave function of the completion time of the other project, under anyefficient policy and budget B(t) satisfying Assumption A.1.

Proof: For an arbitrary policy let the completion time of project i be ti for i = 1, 2. Wefirst consider the case where t1 ≤ t2. We denote the allocation to project i by ρi(t),i = 1, 2. Therefore, since the work done under ρ1(t) in [0, t1] is w1, we have∫ t1

0

ρ1(t) dt = w1 (43)

Since the budget not spent on project 1 is available for project 2, ρ2(t) = minl2, B(t)−ρ1(t) over the interval [0, t1]. After time t1, since project 2 is the only project left,ρ2(t) = minB(t), l2 over the interval (t1, t2]. Therefore, we can write∫ t1

0

minB(t) − ρ1(t), l2 dt +

∫ t2

t1

minl2, B(t) dt = w2 (44)

Let τ1 = mint | l2 ≤ B(t), that is, the first point in time when the budget exceeds l2.Let τ2 = mint | l1 + l2 ≤ B(t), that is, the first point in time when the budget exceedsthe sum of the efficient limits for the two projects. Clearly, τ1 < τ2 for l1 > 0. Hence,t1 must be in one of three mutually exclusive regions: (i) t1 ≤ τ1, (ii) τ1 < t1 ≤ τ2 and(iii) t1 > τ2. We consider these cases separately:

39

Case (i): Since l2 < B(t1), we can write Equation (44) as∫ t1

0

(B(t) − ρ1(t)) dt +

∫ t2

t1

l2 dt = w2

or,

∫ t1

0

(B(t) − ρ1(t)) dt + l2(t2 − t1) = w2

(45)

Equation (43) and (45) yields

t2 = t1 +w1 + w2

l2− 1

l2

∫ t1

0

B(t) dt (46)

which givesdt2dt1

= 1 − B(t1)

l2(47)

If t1 /∈ b1, . . . , bn, that is, B(t) is differentiable at t1, then Equation (47) yields

d2t2dt21

= −B′(t1)l2

(48)

Since B′(t) ≥ 0, t2 is a concave function of t1. On the other hand, if t1 ∈ b1, . . . , bn,that is, B(t1) is not differentiable, then we proceed as follows. Consider the subgradients

at the time t1. To prove concavity we need to show that the subgradient at t−1∆= t1 − ε

is larger than the subgradient at t+1∆= t1 + ε, for some ε > 0. From Equation (47),

the subgradient at t−1 is dt2dt1

|t−1 = 1 − B(t−1 )/l2 and the subgradient at t+1 is dt2dt1

|t+1 =

1 − B(t+1 )/l2. Since, from Assumption A.1, B(t+1 ) > B(t−1 ), the required inequalityfollows.

Case (ii): For this case, if τ2 < t2 we can write Equation (44) as:∫ t1

0

(B(t) − ρ1(t)) dt +

∫ τ1

t1

B(t) dt + l2(t2 − τ2) = w2 (49)

Equation (43) and (49) implies

t2 = τ2 +w1 + w2

l2− 1

l2

∫ τ1

0

B(t) dt (50)

which implies,dt2dt1

= 0 (51)

andd2t2dt21

= 0 (52)

40

Hence, t2 is a concave function of t1 as before. If τ2 ≥ t2, using a similar analysis weget the same dt2

dt1and d2t2

dt21as in Equation (51) and (52).

Case (iii): For this case, both projects 1 and 2 receive allocation up to their efficientlimits from time τ2 onwards. Hence, we can write Equation 43) as∫ τ2

0

ρ1(t) dt + l1(t1 − τ2) = w1 (53)

Similarly, we can write Equation (44) as∫ τ2

0

(B(t) − ρ1(t)) dt + l2(t2 − τ2) = w2 (54)

Equations (53) and (54) implydt2dt1

= − l1l2

(55)

andd2t2dt21

= 0 (56)

Hence t2 is concave in t1 as before.

We have shown that t2 is a concave w.r.t. t1 whenever t1 < t2. Identical analysis showsthe same result when t1 > t2. However, the value of the derivative is different, that is,Equation (47) becomes

dt1dt2

= 1 − B(t2)

l1(57)

Which yields, if t2 /∈ b1, . . . , bn,d2t1dt22

= −B′(t2)l1

(58)

We have now shown t2 is convex in t1 if t1 > t2 or t1 < t2. Therefore, to complete

the proof we need to show that t2 is concave at the point t∆= t1 = t2. We consider

the subgradients at this point under the three cases as before, namely (i) t1 ≤ τ1, (ii)τ1 < t1 ≤ τ2 and (iii) t1 > τ2.

Case (i): For t+1 = t+ ε, for some ε > 0, we have corresponding t+2 < t+1 and hence fromEquation (57),

dt2dt1

∣∣∣∣t+1

=l1

l1 − B(t+2 )(59)

Similarly, for t−1 = t − ε, we have t−2 > t−1 and hence from Equation (47) we have

dt2dt1

∣∣∣∣t−1

= 1 − B(t−1 )/l2 (60)

41

To show, that the curve is concave at t, we need to show, dt2dt1

|t−1 ≥ dt2dt1

|t+1 , as ε → 0. Note

that, as ε approaches 0, t−1 , t+1 , t−2 and t+2 all approach t, and both dt2dt1

|t−1 and dt2dt1

|t+1 are

less than 0. Therefore, using Equation (59) and (60) we need to show,

limε→0

dt2dt1

∣∣∣t+1

dt2dt1

∣∣∣t−1

=l1

l1 − B(t)

l2l2 − B(t)

=l1l2

l1l2 − B(t)(l1 + l2 − B(t))> 1 (61)

The last inequality is true since l1 + l2 > B(t) for this case.

Case (ii): For this case dt2dt1

|t−1 = 0 from (51). Therefore, dt2dt1

|t−1 ≥ dt2dt1

|t+1 , since dt2dt1

|t+1 is atmost 0. Hence the result holds.

Case (iii): For this case dt2dt1

|t+1 = dt2dt1

|t+1 = −l1/l2 from (55) and hence the result holds.

This completes the proof.

Theorem 5.1 For any STPPMP, there exists an optimal policy in ΩP .

Proof: First we prove the result for a two project portfolio. Assume WLOG that the NPVfrom prioritizing project 1 is greater than equal to the profit from prioritizing project 2.That is, if tji , i, j = 1, 2 is the completion time of project i when project j is prioritized,then α1p1e

−βt11 + α2p2e−βt12 > α1p1e

−βt21 + α2p2e−βt22 . We show that prioritizing project

1 is the optimal policy for this case.

Since project 1 receives the maximum resource possible when it is prioritized t11 isthe minimum possible value of t1. For the same reason t22 is the minimum possiblecompletion time for project 2. Similarly, t21 and t12 are the maximum completion timesfor projects 1 and 2, respectively, for any efficient policy. We can write t21 = t11 + δ1

for some δ1 > 0 and t22 = t12 − δ2 for some δ2 > 0. Any intermediate policy generatescompletion time of project 1 in [t11, t

21] and a completion time of project 2 in [t22, t

12].

Suppose an intermediate policy results in t1 = t11 + µδ1 for some µ ∈ (0, 1). If, for thispolicy were t2 = t12−µδ2, that is, there is a corresponding linear change in t2, then, sinceα1e

−β(t11+µδ1) + α2e−β(t12−µδ2) is convex in µ, it follows that the corresponding objective

function for this policy must be less than α1e−βt11 +α2e

−βt12 . Hence, for an intermediatenon-priority policy to be optimal, starting from t11 a µδ1 increase in t1 must result in alarger than µδ2 decrease in t2 for the NPV to increase. However, by Lemma A.3, t2 isa concave function of t1. Hence, increasing t11 by µδ1 must decrease t12 by less than µδ1

and therefore the result is true for the two project case.

We use induction to prove the result for the general case with m projects. Assume thatthe result is true for m − 1 projects. Suppose a policy γ for the m project STPPMPallocates intermediate funding to two or more tasks at the some decision epoch. Asbefore, WLOG we assume that such allocation takes place at the first epoch. Letproject 1 and project 2 be two of the intermediate funded tasks. We show that we canconstruct a policy in ΩP which is at least as good as γ. We have the following two

42

cases. Case (i): one of the non-prioritized project, say project 1, is the first to finish.Case (ii): A prioritized project finishes first. We prove the theorem for the first case.Similar analysis can be used to show the result for the other case.

First observe, that after the first task completion interval, since it is also a projectcompletion interval for the STPPMP, there are less than m projects to complete. Hence,under the induction hypothesis, a priority policy is optimal from then on. Hence, γis a priority policy after τ1. Consider the following policy γε which allocates x1,1 + εto project 1 and x2,1 − ε to project 2 during [0, τ1(γε)], for some |ε| > 0 and leaves allother allocations unchanged. We know from Theorem 4.1 that E[Π(γε)] is maximizedat an extreme point of ε. Further, the extreme points occur either when project 1 or2 is prioritized or when the task completion sequence changes for some sample path.We show, for the STPPMP, a maximum cannot occur when the extreme point is of thelatter kind. That is, E[Π(γε)] is maximized only when either project 1 or project 2 isprioritized.

First note that, since the STPPMP, has only one single task per project, there is onlyone path. Hence, we can write Equation (6) for the NPV of policy γε as:

E[Π(γε)] = e− w1,1

x1,1+ε

(Vγ

[w2,1 − (x2,1 − ε)

w1,1

x1,1 + ε, w3,1 − x3,1

w1,1

x1,1 + ε,

. . . , wm,1 − xm,1w1,1

x1,1 + ε]

+ p1,1α1

)(62)

where Vγ is the cost-to-go function from the priority policy followed under γ after τ1(γε).Let the time required to complete project i as measured from time τ1(γε) under γε beti(W (ε)), where W (ε) = w2,1 − (x2,1 − ε) w1,1

x1,1+ε, w3,1 − x3,1

w1,1

x1,1+ε, . . . , wm,1 − xm,1

w1,1

x1,1+ε,

is the vector of tasks left for the projects remaining at time τ1(γε) as shown in Equation(62). Therefore, we can write Equation (62) as:

E[Π(γε)] = e− w1,1

x1,1+ε

( m∑i=2

pi,1αie−ti(W (ε)) + p1,1α1

)

=m∑

i=1

pi,1αie−(

ti(W (ε))+w1,1

x1,1+ε

) (63)

where we have defined t1(W (ε)) = 0. Using the same argument as in the proof ofTheorem 4.1, the maximum of E[Π(γε)] lies at an extreme point of ε. If the maximizingextreme point ε0 is such that either project 1 or 2 is prioritized at that point then weare done. Therefore, consider ε0 where the task completion sequence changes. We showthat such a point cannot be optimal. We first show the result for the case when ε0 ≥ 0.

Since ε0 is the maximizing point (within the interval where the task completion sequence

is the same as under γ), it must be true that for δ > 0, limδ→0dE[Π(γε0−δ)]

dε> 0. We show

43

that on increasing ε further the NPV continues to increase, implying ε0 is not optimal.

That is, we show that for δ > 0, limδ→0dE[Π(γε0+δ)]

dε> 0.

The derivative ofdE[Π(γε0+δ)]

dεexists for sufficiently small non-zero δ and is computed as

dE[Π(γε)]

dε=

m∑i=1

−

d(ti(W (ε)) + w1,1

x1,1+ε

)dε

pi,1αie

−(

ti(W (ε))+w1,1

x1,1+ε

)(64)

As δ → 0, limδ→0 ti(W (ε − δ)) = limδ→0 ti(W (ε + δ)) = ti(W (ε0)), from continuity ofthe completion times. Therefore, to prove our claim it is sufficient to show that

limδ→0

−d(ti(W (ε0 − δ)) + w1,1

x1,1+ε0−δ

)dε

≤ limδ→0

−d(ti(W (ε0 + δ)) + w1,1

x1,1+ε+0+δ

)dε

, i = 1, . . . ,m

or equivalently, after some algebra

limδ→0

d ti(W (ε0 − δ))

dε≥ lim

δ→0

d ti(W (ε0 + δ))

dε, for all i (65)

Denote wi(ε) as the component of the ith project in W (ε), which can be obtained fromEquation (62) as

wi(ε) =

w2,1 − (x2,1 − ε) w1,1

x1,1+ε, i = 2

wi,1 − xi,1w1,1

x1,1+ε, i > 2

(66)

We can compute d ti(Wi(ε))dε

as,

d ti(Wi(ε))

dε=

m∑j=2

∂ti(W (ε))

∂wi(ε)

wi(ε)

∂ε(67)

From Equation (66)

∂wi(ε)

∂ε=

w1,1(x1,1+x2,1)

(x1,1+ε)2, i = 2

w1,1(x1,1)

(x1,1+ε)2, i > 2

(68)

and is seen to be positive on inspection. Therefore, it is sufficient to prove that

limδ→0

∂ti(W (ε0 − δ))

∂wj(ε0 − δ)≥ lim

δ→0

∂ti(W (ε0 + δ))

∂wj(ε0 + δ), for all i and j (69)

From Equation (68), wj(ε0+δ) > wj(ε0−δ). Therefore, to prove the result it is sufficientto show that ∂ti

∂wjis nonincreasing in wj for any projects i and j. Lemma A.2 proves

exactly this and hence the theorem is proved for the case where ε > 0.

For the case where ε < 0 the same argument as above eliminates all simultaneous task

44

completion times as possible maxima except the point where project 1 and 2 finishestogether, since Equation (64) is not valid at that point. We consider this case separately.

The task completions for projects 1 and 2 under this allocation take place at timeτ1(γε) = w1,1

x1,1(γ)+ε= w2,1

x2,1(γ)−ε= w1,1+w2,1

x1,1(γ)+x2,1(γ). Now consider the budget defined as follows:

B(t) =

x1,1(γ) + x2,1(γ), 0 ≤ t ≤ τ1(γ)

B, t > τ1(γ)(70)

Clearly, this budget satisfies Assumption A.1. We know from previous arguments thatthe optimal policy for two projects under a budget of this type is a priority policy.Suppose it is optimal to prioritize project 1 under B(t). Then, since under this policyproject 1 must finish before project 2 (since they finish simultaneously for an interme-diate allocation), we have already considered this case when discussing ε > 0 and knowthat the result is true. Hence, we consider the case where we prioritize project 2 overproject 1 under B(t). This allocation improves the contribution of projects 1 and 2relative to their contribution under the original policy γ. Hence, it is sufficient to showthat by allocating project 1 and 2 in such a manner we do not decrease the contributionfrom projects 3 to m.

Consider a policy γp as follows. We prioritize project 2 over project 1 under budgetB(t). Then, since x2,1(γp) > x2,1(γ), project 2 completes at time τ1(γp) < τ1(γ). Intime [0, τ1(γ)] the allocations of projects 3 to m are kept the same as in policy γ.This is possible, since the new allocations for projects 1 and 2 in this interval use thesame resource they utilized under policy γ. In the interval (τ1(γp), τ1(γ)], project 1is prioritized under B(t). Depending on the efficient limit of project 1, we have thefollowing two cases: (i) l1 ≥ x1,1(γ) + x2,1(γ) and (ii) l1 < x1,1(γ) + x2,1(γ).

Under Case (i) project 1 utilizes all of its available budget, namely x1,1(γ) + x2,1(γ),and therefore finishes exactly at time τ1(γ). Therefore, from τ1(γ) onwards projects 2to m have budget B available. Since, this is the same budget available to these projectsunder policy γ, the original allocation remains feasible. Hence, the contribution fromthese projects does not increase. Since the contribution from projects 1 and 2 does, theresult is shown for this case.

For Case (ii) we proceed as follows. Since, project 1 can no longer utilize all of the budgetavailable to it in (τ1(γp), τ1(γ)], its completion time must now exceed τ1(γ). Using Equa-tions (26, 27) and the fact that project 2 is prioritized in [0, τ1(γp)] we compute the work

left for project 1 at time τ1(γ) as w1,1 = w1,1− w2,1

l2

((x1,1(γ) + x2,1(γ) − l2) + l2−x2,1(γ)

l1x2(γ)

).

Since, we prioritize project 1 under B(t), this work is completed by funding the project

at level l1 in (τ1(γ), τ1(γ)+ w1,1

l1]. Therefore, project 1 preempts resource of amount w1,1

that was allocated to projects 3 to m during this time. However, it frees up resource in(τ1(γp), τ1(γ)] is exactly the same. Hence, projects 3 to m, due to the lack of precedenceconstraints for single task projects, can now utilize this resource to offset the preemptioneffects. That is, whatever work for these projects gets done in (τ1(γ), τ1(γ) + w1,1

l1] can

45

now be done in (τ1(γp), τ1(γ)]. Therefore none of the completion times of these projectsincrease (they could decrease however). Therefore, the contribution of projects 3 to mto the expected NPV does not decrease under the new policy. Since the contributionfrom projects 1 and 2 can improve, the result is proved.

Theorem 5.2 For any RBPPMP, there exists an optimal policy in ΩP .

Proof: For the RBPPMP any project can utilize the entire budget. Therefore, under apriority policy, only a single project is funded at each decision epoch. Therefore, toprove our result we need to show that for any policy which funds more than one projectin a decision epoch, we do as well or better by funding only one project at a time.

Consider a policy γ for the RBPPMP which funds multiple projects at the first decisionepoch and follows the optimal policy from then on. As in the proof of Theorem 4.1proving the sub-optimality of this policy is sufficient to prove the sub-optimality of ageneral policy for the RBPPMP. Suppose under γ tasks of projects 1 to k ≤ m finishat the end of the first task completion interval. The expected NPV for this policy canbe expressed as

E[Π(γ)] = e−β

w1,1x1,1(γ)

∑

s∈Cll=1,...,k

∏s∈S

ps,1

∏f∈F

(1 − pf,1)V∗[W (S, F )]

(71)

Under this policy, at time τ1(γ) = w1,1

x1,1(γ)= w2,1

x2,1(γ)= . . . =

wk,1

xk,1(γ), the first task of

projects 1 to k is completed and projects i = k + 1, . . . ,m has wi,1 − xi,1(γ)τ1 amountof work for its first task remaining.

Now consider the policy µ where only project 1 is funded at the first epoch. At τ1(µ) =

w1,1 < τ1(γ), we allocate xi,2(µ) = x2,1(γ)τ1(γ)

τ1(γ)−τ1(µ)for i = 2, . . . , k and xi,2(µ) = τ1(γ)xi,1

for i = k + 1, . . . ,m. Simple algebra shows that 0 ≤ xi,2(µ) ≤ 1 and∑m

i=2 xi,2(µ) = 1and hence this policy is feasible. Under this allocation work done for projects 2 to mat time τ1(γ) is exactly the same as under policy γ. Therefore, the contribution to theexpected NPV from these tasks remains unchanged from that under γ. However, ifproject 1 has only task, the expected NPV from project 1 increases since it completesat w1,1 < w1,1

x1,1(γ). Therefore, E[Π(µ)] ≥ E[Π(γ)]. This is sufficient to prove the theorem.

Further, we can write the expected NPV of policy µ as

Π(ν) = e−βw1,1 [p1,1Vµ[W (S, F )] + (1 − p1,1)V

µ[W (S, F )]

where V µ[W (S, F )] is the expected NPV under policy µ. This is clearly less than equalto

Π(ν) = e−βw1,1 [p1,1V∗[W (S, F )] + (1 − p1,1)V

∗[W (S, F )]

since V ∗[W (S, F )] is the maximum over all policies possible at time τ1(µ), including µ.

46

Theorem 5.3 For the RBPPMP, an optimal policy is obtained by computing the optimalpriority sequence at the beginning of the decision process, and then fully funding thehighest priority project until it completes successfully or fails. Upon completion, weprioritize the next project in the original sequence and continue in this manner until allprojects complete.

Further, the optimal priority sequence is computed according to the non-increasing orderof the quantity

Ii =θi

1 −(∑ni

j=1 qi,jγi,j

) , i = 1, . . . ,m

Proof: First we prove that for a RBPPMP we do not switch priorities after a successfulcompletion of a task. We prove this as Lemma A.4 below. This lemma along withTheorem 5.2 proves the first part of the theorem.

To prove that Ii gives the optimal priority sequence we proceed as follows. FromTheorem 5.2 we know that the optimal policy is a priority policy for the RBPPMP,which implies that we fund only one task at a time for this case. We wish to find theoptimal priority policy. Clearly, if (say) 1, 2, . . . , m is the optimal priority sequencethen it must satisfy

Π(1, 2, . . . , m) ≥ Π(P1, 2, . . . , m) for all P1, 2, . . . , m (72)

where Π(A) is the expected NPV from priority sequence A and P1, 2, . . . , m is apermutation of 1, 2, . . . , m.We now compute the expectation of a general priority sequence. For notational simplic-ity we will compute this for the sequence 1, 2, . . . , m. However, the same reasoningcan be applied to any arbitrary sequence. First observe, that under a priority policy,for this sequence, the starting time of project i is independent of all projects j > i sincenone of the projects j > i receive any allocation before the completion of project i. Ingeneral, the completion time of a project is independent of any project with a lowerpriority. However, notice that the starting time of project 2 depends on the number oftasks of project 1 that get completed. In general, the starting time of project i dependson the number of tasks of projects 1 to i − 1 that get completed. Therefore, usingEquations (13) and (14) we can write the expected contribution to the portfolio NPVfrom each project i, measured at time t = 0, as

E[NPVi] =

θ1 for i = 1

θi

∑n1

j1=1 · · ·∑ni−1

ji−1=1

∏k−1l=1 ql,jl

γl,jlfor i = 2, . . . ,m

(73)

The total NPV of the portfolio is given by summing Equation (73) over all i. Therefore,

47

the expected NPV resulting from the priority sequence i1, i2, . . . , im is

Π(1, 2, . . . , m) = θ1 +m∑

k=2

θk

n1∑

j1=1

. . .

nk−1∑jk−1=1

k−1∏l=1

ql,jlγl,jl

(74)

Substituting Equation (74) in (72) and after some algebra we obtain the followinginequalities

θi

1 −∑ni

k=1 qi,kγi,k

≥ θj

1 −∑nj

k=1 qj,kγj,k

for all i > j (75)

The above inequalities are the conditions for the optimality of priority sequence 1, 2, . . . , m.As mentioned before, this particular sequence is completely arbitrary, and thus the re-sult holds for any sequence c1, . . . , cm. Inequality set (75) implies that the optimalpolicy at any decision epoch is to fund project i if its corresponding Ii, as given byEquation (15) is the highest. This proves the result.

Lemma A.4 For the RBPPMP, under an optimal policy if a project completes with tasksuccess, we do not switch to other projects unless the completed task is the last of the project.

Proof: For simplicity of notation we prove the result for a two project portfolio. The samereasoning can be applied for a portfolio with an arbitrary number of projects. Also, wescale the budget B to 1.

We know from Theorem 5.2, that the optimal policy for this problem is to eitherprioritize project 1 or to prioritize project 2 at any decision epoch. WLOG, assumethat it optimal to prioritize project 1 at the first epoch. Therefore,

V [n1, n2, w1,1, w2,1] = maxe−βw1,1 (p1,1V

∗[n1 − 1, n2, w1,2, w2,1] + (1 − p1,1)V∗[∞, n2, ∞, w2,1]) ,

e−βw2,1 (p2,1V∗[n1, n2 − 1, w1,1, w2,2] + (1 − p2,1)V

∗[n1,∞, w1,1,∞])

= e−βw1,1 (p1,1V∗[n1 − 1, n2, w1,2, w2,1] + (1 − p1,1)V

∗[∞, n2, ∞, w2,1])(76)

If the task of project 1 succeeds we reach the state n1−1, n2, w1,2, w2,1. Suppose,contrary to the statement of the theorem, it is now optimal to prioritize project 2 at thisstate. Then, since V (n1, n2, w1, w2) decreases with wi and ni, for any i, it must betrue that for the state n1, n2, w1,1, w2,1, where there is one more task of project1 to complete and the same number of tasks for project 2, we must prioritize project2 at this state too. Hence, a contradiction. Therefore, one must continue to prioritizeproject 1 at this state. If project 1 fails, for the two project case, we prioritize 2 bydefault, but for higher number of projects it is easy to see that the original prioritiesdo not change and the theorem holds.

48

Proof of Theorem 6.1: To show the worst case error that can result from using the indexpolicy defined by Equation (18) we consider a two project portfolio where each projecthas only one task. Further, we assume that the probability of success pi,1 = pi = 1 fori = 1, 2. For such a case, the index for project i reduces to

Ii = αie−βwi/li

1 − e−βwi/li=

αi

eβwi/li − 1=

αi

eβti − 1(77)

where we have written ti = wi/li and li = li,1 for simplicity. For two projects wehave two possible policies in the first decision epoch: (i) prioritize project 1 with thecorresponding NPV Π(1) and (ii) prioritize project 2 with corresponding profit Π(2).Also, since the projects have only one task we do not have any more decision epochs;at the end of the first task completion we simply fund the remaining project as fullyas possible. If the optimal policy is to prioritize project 1 and the index suggests toprioritize project 2, then the error for such a policy is (Π(2) − Π(1))/Π(2). In general,the error ε for the policy is given by

ε =|Π(1) − Π(2)|

maxΠ(1), Π(2) (78)

Thus the error occurs when Ii ≥ Ij while Π(i) < Π(j) for i, j = 1, 2 and j = i.

It is easy to see that the maximum error from the index policy occurs when the indicesfor the two projects, as given by Equation (77), are equal, that is, Ii = Ij. For if Ii > Ij

and Π(i) < Π(j) at the same time, then we can increase αj to increase the percentageerror ε as given in Equation (78) as long as Ij is less than Ii.

Therefore, to find the worst case error we solve

max ε (79)

s.t. I1 = I2

Note, that the maximum value of ε is 1, corresponding to an error of 100%. We showthat the index policy reaches that bound. WLOG assume I1 = I2 while Π(1) > Π(2).Using (77) the first equality implies

α1 = α2eβt1−1

eβt2−1

The second equality implies we can write our maximization problem as

max ε =Π(1) − Π(2)

Π(1)

s.t. α1 = α2eβt1−1

eβt2−1

49

or equivalently

min ε =Π(2)

Π(1)(80)

s.t. α1 = α2eβt1−1

eβt2−1

Substituting the value of α1 and denoting ∆i as the delay caused by project i on theother when prioritized we can write

ε =α1e

−β(t1+∆1) + α2e−βt2

α1e−βt1 + α2e−β(t2+∆2)=

eβt1−1eβt2−1 e−β(t1+∆1) + e−βt2

eβt1−1eβt2−1 e−βt1 + e−β(t2+∆2)

(81)

which simplifies to

ε =e−β∆2 − e−β(t1+∆2) + 1 − e−βt1

1 − e−βt2 + e−β∆1 − et2+∆1(82)

To minimize ε we let ∆1 → 0 and ∆2 → ∞, which is achieved for example when w1 → 0and w2 → ∞. This also implies that t1 → 0 and t2 → ∞. Using the above

lim∆1→0,∆2→∞

ε = 0 (83)

Hence, the percentage error ε = 1 − ε = 1 as claimed.

To see that this bound is achieved for projects with higher number of tasks we canintroduce tasks with zero work content. Since the revenues and completion times areunaffected by such tasks the above error bound holds. Similarly, we can introduceprojects with zero revenue and work content to see that this error is achieved for anarbitrary PPMP. Finally, since the expected NPV is always non-negative, 100% is themaximum possible percentage error. Thus the result is true.

50