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The Engineering Economist

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Prioritizing Project Selection

Ali Koç a; David P. Morton a; Elmira Popova a; Stephen M. Hess b; Ernie Kee c; Drew Richards c a The University of Texas at Austin, Austin, Texas, USA b Electric Power Research Institute, West Chester, Pennsylvania, USA c South Texas Project Nuclear Operating Company, Wadsworth, Texas, USA Online publication date: 23 November 2009

To cite this Article Koç, Ali, Morton, David P., Popova, Elmira, Hess, Stephen M., Kee, Ernie and Richards, Drew(2009)

'Prioritizing Project Selection', The Engineering Economist, 54: 4, 267 — 297 To link to this Article: DOI: 10.1080/00137910903338545 URL: http://dx.doi.org/10.1080/00137910903338545

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The Engineering Economist, 54: 267–297, 2009 Copyright © 2009 Institute of Industrial Engineers ISSN: 0013-791X print / 1547-2701 online DOI: 10.1080/00137910903338545

PRIORITIZING PROJECT SELECTION

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Ali Koc¸,1 David P. Morton,1 Elmira Popova,1 Stephen M. Hess,2 Ernie Kee,3 and Drew Richards3 1 The University of Texas at Austin, Austin, Texas, USA Electric Power Research Institute, West Chester, Pennsylvania, USA 3 South Texas Project Nuclear Operating Company, Wadsworth, Texas, USA 2

We consider capital investments under uncertainty. A typical approach to this problem, when the problem parameters are assumed known, is via a multi-knapsack model. This model takes as input annual budgets as well as the cost streams and profit—i.e., net present value (NPV)—of each project. Its output is a portfolio of projects with the highest total NPV, observing yearly budget constraints. We argue that such a portfolio fails to hedge against uncertainties in the budgets, the cost streams, and the profits. As an alternative, we propose a model that forms an optimal priority list of projects, incorporating multiple scenarios for these input parameters. We apply our approach to two sets of example projects from the South Texas Project Nuclear Operating Company.

INTRODUCTION When practitioners plan for capital budgeting they often form a priority list of candidate projects, by scoring the projects individually, using economic measures like net present value, benefit-investment ratio, payback period, internal return rate, etc. The academic literature frequently points out (e.g., Brown et al., 2006; Savage et al., 2006) that priority lists built on such simple ranking measures are inferior to allocating funds to capital projects using variants of a multi-knapsack model. The multi-knapsack approach to capital budgeting (e.g., Bierman and Smidt 1980; Kellerer et al., 2004; Weingartner, 1966) takes as input a budget forecast, along with the stream Address correspondence to David Morton, Graduate Program in Operations Research and Industrial Engineering, 1 University Station C2200, The University of Texas at Austin, Austin, TX 78712. E-mail: [email protected]

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of liabilities and the profit of each project. The multi-knapsack model is an integer program that has: a binary decision variable for each project to indicate whether it is selected; a budget constraint for each time period (e.g., year); and the objective of maximizing total profit. The output of the multi-knapsack model is a collection of projects to be carried out, assuming the point forecasts for these input parameters are correct. We refer to this selected collection of projects as a project portfolio. If the costs and profits of the candidate projects as well as the budgets in coming years are known with certainty, the multi-knapsack model provides an attractive tool for selecting a project portfolio. However, how should we practice capital budgeting when we have uncertain forecasts for these parameters? One approach is to re-solve a multi-knapsack model when refined forecasts for costs, profits, and budgets become available. Unfortunately, this is not always viable. Capital projects typically are implemented in phases over time and usually some irreversible decisions have been made. Thus, it is not always practical to fully revise a project portfolio whenever better forecasts become available. Additionally, the process of obtaining and analyzing the data, performing required reviews, and obtaining necessary approvals typically is very time consuming and resource intensive. As a result, practitioners use either simplistic approaches or intuition and experience to address the impact of emerging events and conditions. To illustrate how uncertainty can cause a problem in a practical setting, we discuss the following common scenario. Imagine that a multi-knapsack model has been used in the capital budgeting process to form a project portfolio. Then, over the course of the year, the available budget decreases due to some reason such as an external event or because one or more projects experience cost overruns. As a practical matter, some other projects are then forced out of the portfolio; i.e., not carried out. Unfortunately, the multi-knapsack model is not designed to address such a scenario. Experience over many years of capital budgeting practice indicates that decision-makers often have the right intuition in seeking a priority list that is robust with respect to changes in budget values as well as project costs and profits. However, it is well known that priority lists formed by scoring projects individually fail to capture dependencies between projects, and we demonstrate this by considering heuristics based on scoring projects using net present value (NPV) and benefit-investment ratio (BIR). For more on these and other commonly-used investment criteria see, e.g., chapter 9 of Ross et al., 2008. Thus, it would be beneficial to have an approach to prioritizing that does capture dependencies between projects. Associated analyses then could be used to provide priority lists to decision-makers to support better risk-informed decisions. The path of investigation described in this article is as follows:

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r We first investigate whether the optimal solution to the integer programming multi-knapsack model naturally yields a prioritized list that is robust to the uncertainties described above. We show it does not. r Next, we heuristically alter the multi-knapsack approach and force it to produce a prioritized list. We describe a class of ways to do so depending on the initial budget value, and we call these greedyheuristic priority lists. We also build heuristic priority lists using NPV and BIR, as is commonly done in practice. r Then, we ask whether we can build a priority list that outperforms heuristic priority lists, at least when we assume a probabilistic forecast for the uncertain parameters. For the two sets of candidate projects we examine, this question is answered affirmatively. We formulate a model that explicitly incorporates multiple budget, cost, and profit scenarios and forms an optimal priority list, which maximizes the expected NPV of the project portfolio ultimately implemented. Our model for prioritizing projects in capital budgeting is a two-stage stochastic integer program. Its inputs include those described above for the multi-knapsack model, except that we have a probabilistic description of the uncertain parameters; i.e., the yearly costs and profit of each project as well as the annual budgets. The two-stage stochastic program forms a priority list as its first-stage decision and then forms a corresponding project portfolio for each scenario as its second-stage decision. We assume that the uncertain parameters are revealed for the entire planning period after having committed to the priority list. (It would be possible to develop a more time-dynamic model, but this is a topic for future research.) When forming the optimal second-stage project portfolio under a specific scenario, our model ensures that the portfolio is consistent with the first-stage prioritization; i.e., a project can be selected only if all higher-priority projects are also selected. Thus, the portfolios of projects corresponding to different scenarios are nested. Projects can span multiple years over say, a 5-year planning period. With stochastic cost streams, our model can capture the fact that the completion time of a project can be random. That said, the numerical examples we consider assume that the number of years in which a project incurs nonzero costs is deterministic. The next section provides further background specific to our motivating capital budgeting problem using data from the South Texas Project Nuclear Operating Company (STPNOC), a two-unit nuclear power plant in Wadsworth, Texas. Then, we formulate a multi-knapsack model for capital budgeting under a point forecast (i.e., a deterministic forecast) of the problem’s input parameters. The solution yields an optimal set of projects

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to select, and we show that these portfolios fail to yield a prioritized list of projects. That is, we address the first of the three bulleted items listed above. The subsequent two sections then address the second and third bulleted items in turn, assuming a distributional forecast for the uncertain parameters. The article’s penultimate section then applies the optimal prioritization scheme to two larger problem instances; it also investigates the performance of the heuristics on these problem instances. The final section summarizes, concludes, and points to future research directions. We numerically consider capital budgeting problems based on two sets of candidate projects using data from STPNOC. The first set has 9 projects, the second set has 41 projects, and all of our problems span 5 years. Our first problem instance involves the smaller set of projects, and we consider uncertainty only in the yearly budgets with a total of 10 scenarios. We solve two problem instances with 41 projects, the first again with uncertainty only in the budgets and the second with uncertainty in the budgets and project costs. The former instance has 10 budget scenarios, and the latter a total of 90 scenarios. We solve these prioritization models optimally and also solve them with heuristics that mimic how one might use a deterministic multi-knapsack model in practice when faced with uncertain budgets and project costs. Sometimes our heuristics perform well but other times fail to form a good priority list. We also construct stylized examples to assess the performance of our heuristics with an eye toward showing that they can perform poorly; i.e., that performance guarantees for the heuristics are not assured. As a result, we recommend using the optimal prioritization scheme, instead of employing an approximation using a heuristic. BACKGROUND AND MOTIVATION As the operator of a large commercial nuclear power-generating station, STPNOC evaluates investment in numerous projects and aims to select projects that achieve the organization’s objectives. To do so, STPNOC annually develops a priority list of projects. This rank-ordered list specifies the highest priority project, the second highest priority project, and so forth. The current budget and project cost forecasts yield what STPNOC calls the blue line. Projects above the blue line are to be funded and those below it are not. Thus, the blue line serves as the demarcation where the available budget is exhausted. Over the course of the year, the blue line can shift due to a variety of reasons such as an external event or because a highpriority project experiences cost over runs. We note that this paradigm is not unique to STPNOC or to the nuclear power industry. Rather, similar capital budgeting practices are employed across a wide range of industrial and government applications. The optimization model we describe recognizes

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the fact that prioritizing is common practice and aims to build priority lists that are financially robust to the types of uncertainties described above. More specifically, we seek a priority list that maximizes the expected NPV of the project portfolio ultimately implemented. Our approach to forming an optimal priority list focuses on financial performance measures. However, it is recognized that financial goals alone do not drive capital planning decisions. The need to ensure regulatory compliance enters heavily into decision-making at STPNOC and throughout the commercial nuclear power industry. So, as in other optimization problems, forming an optimal priority list generally requires addressing multiple criteria including both financial and non-financial issues. At STPNOC, and many commercial nuclear plants, this results in application of a multiattribute utility theoretic approach to performing this integration (see, e.g., Keeney and Raiffa 1976). In demonstrating the approach we propose, we first consider a small set of example projects from STPNOC in which some projects have negative NPV estimates and hence would be rejected from a purely financial perspective. However, these projects are forced into the project portfolio by plant management because they are deemed necessary to address a safety or regulatory issue. In this example we show how this affects our approach and we further discuss how regulatory and safety issues often can be well-aligned with financial goals. A typical project at STPNOC is implemented over 1–5 years, and we may prioritize a project now even though its first costs are not incurred until a future year. Furthermore, STPNOC carries out project prioritization annually, and in this sense capital budgeting decisions are implemented using a rolling horizon. Current STPNOC practice in estimating the cost streams associated with the candidate projects is as follows: Optimistic, pessimistic, and most-likely cost streams are estimated for each project. Then, each project is categorized as being low risk, medium risk, or high risk. This categorization is based on answering multiple questions within each of 17 categories, which range from the project’s engineering design complexity to STPNOC’s level of experience with the proposed contractor to the nature of the radiation fields in the installation environment (i.e., anticipated personnel radiation exposure during installation), etc. The higher the risk of the candidate project, the more this weighted sum is skewed toward the pessimistic cost forecast. A point estimate for the project’s costs is formed by assigning normalized weights to each of the optimistic, pessimistic, and most likely cost streams and calculating the weighted sum. The manner in which the prioritization model we propose explicitly captures multiple budget, cost, and profit scenarios was motivated by this type of cost stream analysis at STPNOC. A simplistic ranking scheme scores projects individually—e.g., using their NPV, BIR, internal return rate, etc.— and then forms a priority list by

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sorting the projects based on their individual scores. Such an approach fails to recognize the structural and stochastic dependencies among the projects. Though our approach forms a priority list, it recognizes that the projects ultimately implemented, after the stochastic budgets, costs, and profits are realized, act as a portfolio. That is, the model captures dependencies among the projects. We emphasize that our model is appropriate only when irreversible decisions regarding project selection must be made before knowing budget, cost, and profit values with certainty. If we can wait until these quantities become known before committing to project selection decisions, we should do so and solve what is then a deterministic multiknapsack model. OPTIMAL PROJECT PORTFOLIO As indicated above, capital budgeting classically is formulated using variants of a multi dimensional knapsack problem (e.g., Bierman and Smidt, 1980; Brown et al., 2004; Meier et al., 2001; Weingartner, 1966). Specifically, given a set of candidate projects, and given (point estimates of) the NPV of each project, the cost of each project in each year, and the available yearly budgets, the goal is to find the subset of projects that maximizes the total NPV while staying within the budget in each year. When considering a single year, the problem can be visualized as packing a knapsack with items of different volume and utility such that the selected items fit in the knapsack and maximize total utility. When the time horizon includes multiple years and selecting a project can obligate funds in more than one year, there are multiple knapsack constraints— i.e., budget constraints—to satisfy; hence the name multi-knapsack. The knapsack problem, and its variations, such as multi-knapsack, have a rich history and have received significant attention in the literature (e.g., Kellerer et al., 2004). In this section, we first set notation and briefly describe a multi-knapsack formulation for the deterministic capital budgeting problem and then discuss the implications of instead having stochastic budget levels. For simplicity, we begin by only considering stochastic budget levels, but later we handle uncertain project costs and profits. The notation and formulation of the multi-knapsack model are as follows: Indices and sets: i ∈ I candidate projects t ∈ T time periods (years) Data: ai net present value of project i bt available budget in year t

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cit cost of project i in year t Decision variables: xi 1 if project i is selected; 0 otherwise

Formulation: max

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x

s.t.

(1a)

ai xi

i∈I

cit xi ≤ bt ,

t ∈T

(1b)

i ∈ I.

(1c)

i∈I

xi ∈ {0, 1},

Constraint (1b) ensures that the yearly cost of selected projects is within the budget for each year bt , t ∈ T . Yes/no restrictions on selecting projects are enforced by constraint (1c). The objective function (1a) sums the NPV contributions of all selected projects. The optimal solution to the multiknapsack model (1) gives the portfolio of projects to select that maximizes total NPV while staying within the yearly budgets. To understand the nature of solutions to model (1), we consider a numerical example with 16 projects (see Table 1) each having liabilities in some or all of the next 5 years. These projects are from STPNOC and were selected because they constituted a set of projects that were close to the budget cutoff point, with some being funded and others not. Thus, the subset of projects identified in Table 1 was selected to provide a useful validation of the applicability of the methods discussed in this article. Table 1 shows the project cost (cit ) and NPV (ai ) values for each of the projects, and the table orders projects by their BIR; i.e., by the ratio of the NPV of a project to its net present cost. Projects 10–16 have negative NPVs—i.e., ai < 0—and thus also have negative BIRs. We note that the optimization model (1) is driven by a purely financial goal, and, hence, if this were the sole basis for the investment decision, we would not choose any of these projects. However, projects 10–16 have been managerially mandated for inclusion in the portfolio for reasons beyond the scope of our analysis. Obviously, regulatory and safety goals are of foremost concern in the nuclear power industry. In some industries, inaction on regulatory or safety requirements may result in a fine or other regulatory consequence that is deemed to be “acceptable” to the organization; i.e., one that could be accepted as a profitable business decision. However, failure to meet regulatory or safety goals in commercial nuclear power can very easily result in significant revenue loss. For example, a year-long

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Table 1. Problem data Yearly costs ci t

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Project i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1

2

3

4

5

0.219

0.257

0.085 0.122

0.103

0.013

0.012

2.383

0.192

0.688 0.763

0.739

2.539

0.095 0.500

6.803

6.778

5.044 6.740 0.425 2.125 2.387 0.030 0.081 0.300 0.347 4.025 0.095 5.487

1.839 6.134 2.122 0.190 0.950 0.030 0.2 0.032

0.297 0.095 5.664

10.442

NPV (ai )

BIR

2.315 0.824 22.459 60.589 0.667 5.173 4.003 0.582 0.122 −2.870 −0.102 −0.278 −0.322 −3.996 −0.246 −20.155

4.405 4.328 3.338 2.871 1.569 1.272 0.883 0.669 0.192 −0.905 −0.925 −0.927 −0.928 −0.930 −0.940 −0.957

Note: Cost and NPV values are in $M.

regulatory-mandated shutdown could lead to revenue losses in the range of hundreds of millions of dollars in some cases and more than a billion dollars at a multi-plant site. We contend that if such opportunity costs were captured in the profit estimates for projects 10–16 their NPVs would be very large, dominating those of projects 1–9. However, for the purposes of this study because these projects are managerially mandated, we assume there is little reason to justify them financially. Thus, in our example, projects 10–16 are not included when solving model (1), except that they reduce the budget available for choosing among projects 1–9, and they do decrease overall NPV of the portfolio by almost $28M. We solve 10 instances of model (1) with bt = $11M, . . . , $20M for each of the 5 years in these respective instances and display the solutions in Table 2. For each budget level, the 1s and 0s indicate whether the corresponding project was selected (1) or not (0), and the final column gives the optimal NPV. For example, when bt = $16M we do not select projects 4, 5, 6, 7 but do select the others. The corresponding NPV for this budget is negative $1.67M. We solve model (1) for a range of budget values because we recognize that the budget is uncertain. Moreover, for reasons explained in the first

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Table 2. Solutions to 10 instances of model (1) with bt = $11M, . . . , $20M

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xi Budget, bt ($M)

1

2

3

4

5

6

7

8

9

NPV ($M)

11 12 13 14 15 16 17 18 19 20

1 1 1 1 0 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 1 1 0 0 0

0 0 0 0 0 0 0 1 1 1

1 1 0 1 0 0 1 1 1 1

0 0 1 1 1 0 0 0 0 1

0 0 0 0 1 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1 1 1

−23.58 −23.46 −18.95 −18.29 −17.27 −1.67 −1.00 37.13 37.13 42.30

two sections, we know STPNOC management seeks a priority list. If the sets of projects selected as we increase the budget from $11M to $20M are nested—i.e., the project portfolio at each budget level is a superset of all those at lower budget levels—then the multi-knapsack model yields a prioritized solution. However, as we notice from Table 2, some of the projects are part of the portfolio for a particular budget level but are absent from the portfolio at higher budget levels. For instance, as we parametrically range the budget level from $11M to $20M, projects 1, 3, 5, 6, and 7 alternate in and out of the portfolio. This is a typical situation in knapsack problems and, more generally, in resource-constrained combinatorial optimization problems. That is, when the problem data are slightly perturbed, the new optimal solution can be far from the original optimal solution. This phenomenon represents a significant issue to decision-makers in our setting because it can result in decreased confidence in the project selection decisions recommended by the multi-knapsack model. To understand why the solutions behave in this manner, it is instructive to compare projects 4 and 7. Project 4 has a large NPV compared to project 7 ($60M and $4M, respectively), but it is costly, at $23M (nominal) over 3 years (see Table 1). Hence, we would like to include project 4 in the portfolio if it fits within the available budget. This is exactly what happens for budget levels of $18M and higher. Project 7 enters the portfolio only when its relatively low cost allows it to “just fit” within the residual budget when other more profitable projects are too costly to do so, and this is what occurs at the $15M budget level. So, we can view project 7 as a “filler” project, funded when its cost profile happens to align well with the residual planned budget.

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Table 3. Solutions to the restricted problem that forms the heuristic priority list, H-11 Budget, bt ($M)

1

2

3

4

5

6

7

8

9

NPV ($M)

11 12 13 14 15 16 17 18 19 20

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1

0 0 0 1 1 1 1 1 1 1

0 0 0 0 0 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1 1 1

−23.58 −23.46 −23.46 −18.29 −18.29 −14.28 −14.28 −14.28 −14.28 −14.28

HEURISTIC PROJECT PRIORITIZATION Managers, including those at STPNOC, often seek a priority list as the solution to a capital budgeting problem. The “volatility” of the optimal portfolios obtained from model (1), with respect to budget changes, complicates our ability to extract a priority list from the portfolios in Table 2. More generally, this volatility may be disconcerting to decision-makers, and so we investigate an alternative that lends itself to building a priority list: As an initial simplistic approach, we begin by solving model (1) with bt = $11M. Then, we solve model (1) with bt = $12M under the additional requirement that all projects selected at the $11M budget level remain in the portfolio. We continue in this way to larger budget levels. The result is a nested collection of portfolios, shown in Table 3, from which we easily can extract a priority list. The associated heuristic priority list consists of the following. Projects in the group {1, 2, 5, 8} all receive top priority because they are funded for all budget levels we consider. Projects 9, 6, and 7 follow, prioritized in that order. Project 9 is funded if the budget is $12M or above, project 6 if the budget is $14M or above, and project 7 if the budget is $16M or above. Finally, projects 3 and 4 received lowest priority because they are not funded even with the highest budget level of $20M. We denote the resulting priority list L = [{1, 2, 5, 8}, {9}, {6}, {7}, {3, 4}]. The heuristic approach has placed additional restrictions on the multiknapsack model that were not present in the solutions obtained via the analysis presented in Table 2. Thus it is natural that the NPV values obtained for portfolios from the heuristic approach are not as large. The magnitude

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of the difference between the NPVs can be significant, particularly at the larger budget values. The intuition behind this result should be clear: As we incrementally raise the budget level we continue to add projects that fit within the new budget increment. Though these projects increase NPV, this incremental strategy never allows us to select the higher cost (but higher value) project 4. In practice, project 4 would likely be funded by management because its benefits are so clear. However, less extreme instances of this issue often arise for projects, and collections of projects, that fall near the cut off point. Without the type of tool we describe, the benefits of such projects easily may be missed. We can initialize the heuristic scheme just described at levels other than the lowest budget. Instead, we can begin by solving model (1) with the largest budget level, bt = $20M, then, decrement the budget to bt = $19M, and resolve model (1), subject to the restriction that we can only choose projects that were present in the portfolio for bt = $20M. (From the last row of Table 2, we see this excludes from consideration projects 3 and 7.) We then decrement the budget, i.e., repeat this with bt = $18M, and so on. This again leads to a nesting of project portfolios at different budget levels from which we can extract a priority list. Given the incremental and decremental techniques for forcing nested portfolios, we can therefore start by solving model (1) at any intermediate budget level—e.g., bt = $17M— and then decrement to bt = $16M, . . . , $11M, and increment from bt = $17M to bt = $18M, . . . , $20M and finally extract a priority list. We call these greedy heuristics. Algorithm 1 formalizes this class of heuristics, where H-ω corresponds to initializing the heuristic with budget scenario ω . Algorithm 1 assumes that = {ωmin , ωmin +1 , . . . , ωmax } is a set of consecutive integers, and in our numerical example = {11, 12, . . . , 20}. Of course, constructing heuristic priority lists need not be rooted in model (1). We can instead score the projects individually using, e.g., BIR or NPV, to form a priority list, and we label these ranking heuristics. We use the designations H-BIR and H-NPV to denote the ranking heuristic lists based on the projects’ benefit-investment ratios and net present values, respectively. In order to assess the quality of a priority list, we require a model of uncertainty governing the realizations of the projects’ profits, costs, and the yearly budgets. Our uncertainty model places a probability distribution on the realizations of these model parameters, and for the moment, we focus on budget uncertainty. The range of possible budgets used in the analyses provided in Tables 2 and 3 is large and management probably only approves capital budgets relatively close to some predefined target. To address this, we assign relatively low probabilities to budget realizations far from the target budget. For this example, these probabilities were assigned in an ad hoc manner but were selected so that they represented plausible values.

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Algorithm 1 Greedy heuristic starting at budget scenario ω : H-ω . Input: (ai , cit , btω ), i ∈ I, t ∈ T , ω ∈ = {ωmin , ωmin + 1, . . . , ωmax } and ω , the heuristic’s initial budget scenario. Output: Priority list for greedy heuristic H-ω , L = [L1 , . . . , LL ]. L is the number of priority levels, L1 denotes the highest priority projects, L2 denotes the second highest priority projects, etc. Solve model (1) with parameters (ai , cit , bt = btω ), i ∈ I , t ∈ T , to obtain solution x ∗ . ω ∗ S ← {i | xi = 1}. for ω = ω + 1 incremented to ωmax do Solve model (1) with parameters (ai , cit , bt = btω ), i ∈ I , t ∈ T , and with additional constraint set {x | xi = 1, i ∈ S ω−1 }, to obtain solution x ∗ . S ω ← {i | xi∗ = 1}. end for for ω = ω − 1 decremented to ωmin do Solve model (1) with parameters (ai , cit , bt = btω ) i ∈ I , t ∈ T , and with additional constraint set {x | xi = 0, i ∈ / S ω+1 }, to obtain solution x ∗ . S ω ← {i | xi∗ = 1}. end for j ← 0, S ωmin −1 ← ∅ for ω = ωmin incremented to ωmax do if S ω \ S ω−1 = ∅ then j ←j +1 Lj ← S ω \ S ω−1 end if end for if ∪ω∈ S ω = I then L ← j and L ← [L1 , . . . , LL ]. else L ← j + 1, LL ← I \ ∪ω∈ S ω and L ← [L1 , . . . , LL ]. end if

For our example application, we assume the most likely (target) budget value is $17M. Under some scenarios the budget realization is larger and the probabilities for realizations $18M, $19M, and $20M are assumed to drop off linearly. Conversely, the actual budget may be smaller than $17M, and the individual probability masses are assumed to drop exponentially from $17M to $11M with the specific values shown in Figure 1 and Table 4. The weights in the second column of Table 4 are normalized to yield the

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Table 4. Budget realizations and probabilities Budget, bωt ($M)

Weight

Probability, q ω

11 12 13 14 15 16 17 18 19 20

5e−3 5e−5/2 5e−2 5e−3/2 5e−1 5e−1/2 5 4 3 2

0.012 0.019 0.032 0.052 0.086 0.142 0.235 0.188 0.141 0.094

probabilities in the third column. And the 10 budget realizations of Table 4 are perfectly correlated over time; i.e., if the budget realization in year 1 is $13M, then it also takes that same value in the next four years; i.e., to the model’s time horizon. Under this probability distribution we obtain an expected NPV of −$15.42M by implementing the heuristic priority list, H-11. This expectation is computed by forming the weighted sum of the 10 NPV realizations in Table 3, using the probability mass function, q ω , from Table 4. We

Figure 1. Discrete probability mass function for budget realizations. The line only serves to illustrate exponential and linear drops in probability mass.

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Heuristic H-11 H-12 H-13 H-14 H-15 H-16 H-17 H-18 H-19 H-20 H-BIR H-NPV

Priority list

NPV ($M)

[{1, 2, 5, 8}, {9}, {6}, {7}, {3, 4}] [{1, 2, 5, 8}, {9}, {6}, {7}, {3, 4}] [{1, 2, 8, 9}, {6}, {5}, {7}, {3, 4}] [{1, 2, 8, 9}, {6}, {5}, {7}, {3, 4}] [{2, 8, 9}, {6}, {7}, {1, 5}, {3, 4}] [{1, 2, 8, 9}, {3}, {5}, {6}, {4, 7}] [{1, 2, 8, 9}, {3}, {5}, {6}, {4, 7}] [{1, 2, 5, 8}, {9}, {4}, {6}, {3, 7}] [{1, 2, 5, 8}, {9}, {4}, {6}, {3, 7}] [{1, 2, 5, 8}, {9}, {4}, {6}, {3, 7}] [{1, 2}, {3}, {4, 5, 6, 7, 8, 9}] [{4}, {1, 2, 3, 5, 6, 7, 8, 9}]

−15.42 −15.42 −15.29 −15.29 −15.50 −4.54 −4.54 2.59 2.59 2.59 −6.89 −2.40

compute the expected NPV under the other heuristics in a similar manner. The resulting heuristic priority lists and their NPVs are given in Table 5. Examining the results in Table 5, we see that initializing the greedy heuristic at a larger budget value in the range $11M to $20M allows selection of higher-cost projects that are also of higher NPV. As indicated above, project 4 is a project that provides a large NPV but also incurs high costs. Heuristics H-11 through H-17 give project 4 the lowest possible priority because they cannot feasibly choose that project with their initial budget, and as the budget is incremented, their existing commitments leave no room for project 4. Nominally, the ranking heuristics produce a fully ordered list. For example, H-BIR’s ordered list is simply projects 1-9, in that order. The H-BIR list given in Table 5 indicates that projects 1 and 2 can both be performed at the lowest budget level ($11M), that project 3 is included at a higher budget level (which is $16M), and that the remaining projects are unfunded, even at the highest budget level. The HNPV prioritization is obtained by simply sorting the projects from largest to smallest NPV. From Table 1 we see that under this scheme project 4 has the highest priority, project 3 is ranked second, etc. Table 5 indicates that project 4 is funded (eventually, under the $18M budget scenario) but that even under the highest budget realization, we cannot fund project 3. We also compute the expected NPV under perfect information by weighting the NPVs in Table 2 to obtain $11.90M. The expected NPV under perfect information is the value we would obtain if we could wait until the budget is realized before selecting our portfolio of projects. Comparing $11.90M with the values obtained using the heuristics in Table 5

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indicates that the value of information is significant in this problem. If we could improve the budget forecast—e.g., by further data collection; obtaining better estimates of project costs, revenue, and NPV; or performing a more detailed forecasting analysis—then there would be significant financial benefit. Some additional remarks regarding the greedy and ranking heuristics are in order. From Table 5, one may be tempted to conclude that greedy heuristics perform better if initialized at larger budget realizations. Comparing H-14 and H-15 already shows this temptation is vain, but the following example gives further insight. Example 1. Consider an instance of a single-knapsack model with 11 projects. Ten projects have profit and cost equal to 1 and the other project has a profit of 11 and a cost of 10. The budget takes values 1, . . . , 10 with probabilities q 1 , . . . , q 10 , which sum to one. For this problem, H-1 to H-9 ω all yield the same expected NPV of 10 ω=1 ωq , and H-10 has an expected NPV of 11q 10 . If the budget scenarios are equally likely, the H-10 heuristic is worse than the other heuristics by a factor of 5, and as the probability mass on the budget realization of 10 shrinks to zero the factor by which H-10 is worse grows without bound, regardless of the specific values of q 1 , . . . , q 9 .

As Example 1 indicates, the fact that the greedy heuristics ignore the probability distribution means we should not anticipate, in general, one heuristic to dominate the others with respect to expected NPV. As the example also suggests, the worst-case performance of the greedy heuristics can be arbitrarily poor. The ranking heuristics H-BIR and H-NPV ignore the budget, which points to a potential pitfall. Foremost, the highest priority project may fail to satisfy the budget under one or more (even all!) of the scenarios. In this case, until the budget scenario climbs to a level where that project can be funded, we cannot perform any other project. As described above, this is exactly what occurred for H-NPV in Table 5 where no project was funded for budget realizations $11-17M. Even when this feasibility issue does not arise, a potential pitfall remains, as the following example illustrates.

Example 2. Consider a single-knapsack problem instance with two projects. The first project’s cost and profit are 2 and 4, respectively. The second project’s cost and profit are M, a large number. There are two budget scenarios: M and M + 1. The expected NPV obtained under H-BIR is 4,

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whereas the optimal solution has an expected NPV of M. As M grows, the ratio of the optimal priority list’s expected NPV to that of H-BIR grows without bound.

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These simple examples show that commonly employed heuristics that ignore either the budgets or their likelihood of occurrence can lead to arbitrarily poor priority lists. The next section formulates a new model that yields an optimal priority list by incorporating a probability distribution governing the budget, cost, and profit realizations. OPTIMAL PROJECT PRIORITIZATION The deterministic capital budgeting model (1) assumes that we know the problem data (i.e., project costs and revenues and the budget) in advance with certainty. And, as was demonstrated, the model does not naturally produce a priority list. In the previous section, we used the deterministic model to deal with uncertain budgets, but that analysis was admittedly ad hoc and is why we referred to the results as heuristic priority lists. In this section we build a model that explicitly incorporates multiple budget, cost, and revenue scenarios. The need to deal with these uncertain parameters motivates extending model (1) to form a priority list with the goal of maximizing the expected NPV of the projects we can implement after the uncertain parameters are revealed. The notation and formulation of the optimal prioritization model are as follows: Indices and sets: i, i ∈ I candidate projects p ∈ P priorities; P = {1, 2, . . . , |I |} t ∈ T time periods (years) ω ∈ scenarios Data: aiω net present value of project i under scenario ω btω available budget in period t under scenario ω citω cost of project i in period t under scenario ω qω probability of scenario ω Decision variables: xiω 1 if project i is selected under scenario ω; 0 otherwise yii 1 if project i has higher priority than i ; 0 otherwise zip 1 if project i is assigned priority level p; 0 otherwise

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Formulation: max x,y,z

s.t

qω

ω∈

aiω xiω

(2a)

i∈I

citω xiω ≤ btω ,

t ∈ T, ω ∈

(2b)

zip = 1,

p∈P

(2c)

zip = 1,

i∈I

(2d)

i∈I

i∈I

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p∈P

|P |yii ≥

(|P | − p)(zip − zi p ),

i = i , i, i ∈ I

(2e)

p∈P

yii + yi i = 1,

i < i , i, i ∈ I

(2f)

xiω ≥ xiω + yii − 1,

ω ∈ , i = i , i, i ∈ I

(2g)

i ∈ I, ω ∈

(2h)

yii ∈ {0, 1},

i = i , i, i ∈ I

(2i)

zip ∈ {0, 1},

i ∈ I, p ∈ P .

(2j)

xiω

∈ {0, 1},

Model (2) is a two-stage stochastic integer program. The first-stage decision variables, z and y, form the priority list and establish the precedence between projects based on their respective ranking in the list. The secondstage decision variable, x, selects the portfolio of projects to implement under each scenario. The objective function (2a) captures the expected NPV, forming the weighted sum of NPVs over all scenarios. Constraint (2b) ensures that the implemented projects stay within budget under each scenario, for each year. Constraint (2c) assigns exactly one project to each priority level, and constraint (2d) assigns exactly one priority level to each project. These constraints assume that all projects are assigned a priority; i.e., |P | = |I |. Given z, constraints (2e) and (2f) define y, ensuring that each pair of projects is properly ordered. Constraint (2g) requires that the projects implemented by x ω , under each scenario, are consistent with the priority list’s ordering. The last three sets of constraints are binary restrictions; for example, either a project is selected for execution or not under scenario ω (variable xiω ). The timing of decisions and observations of uncertainty are key to understanding the optimal prioritization model (2). First, the priority list is formed via y and z. Next, the values of the budget, cost, and profit

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Figure 2. Illustration of project prioritization.

parameters (a ω , bω , cω ) are realized. We then effectively work down the priority list performing the projects, using x ω , until the budget is exhausted. These dynamics are illustrated in Figure 2. Like model (1), model (2) can be solved with commercially available integer programming software. All of the problem instances in this article have been solved with ILOG’s software CPLEX version 10.1 (CPLEX, 2007). Although model (2) prioritizes a relatively simple multi-knapsack model, we can similarly prioritize any resource-constrained combinatorial optimization problem with binary activity selection decisions (see Koc¸, 2010). In the context of capital budgeting, a more detailed model would capture important issues such as: selecting one project can yield a synergistic opportunity for other projects; selection of a project may require selection of one or more prerequisite projects; a collection of projects may represent mutually exclusive alternatives; some projects are implemented in phases, with opportunities for acceleration or delay; colors, beyond yearly availability, can distinguish types of money; both fixed and variable costs play a role in asset replacement models; resources in addition to monetary budgets can limit project selection; and demand constraints can drive project selection. See Brown et al. (2004) and Hartman (2000) for a discussion of these and related issues. For the purpose of prioritization via a model like model (2), enriching a multi-knapsack model in these ways typically requires introducing new second-stage variables and constraints—i.e., increasing the number of variables indexed by ω and increasing the number of constraints of type (2b)—but does not alter the prioritization constructs. For capital budgeting under uncertainty, Meier et al. (2001) couple a knapsack model with contingent-claim analysis, in

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place of traditional cash-flow analysis. Through a proper partitioning of the uncertainties, their model is also amenable to prioritization. To illustrate the optimal prioritization model, we continue our example from the previous two sections in which only the budget is random, and we have 9 projects with positive NPV. Though our illustrative example only has uncertainty in the budget level, model (2) also handles uncertainty in the profits and costs of the projects. The next section considers a larger problem with 41 projects, including an instance in which the budget, cost, and profit parameters are all random. We use the same values for ai and cit as given in Table 1. As in the previous section, we use the budget realizations, btω , and associated probabilities, q ω , given in Table 4. The priority list solving this instance of model (2) is given in Table 6. The priority lists for a number of the greedy heuristics are also given for reference. As is shown in the table, the list obtained by the heuristics H-18, H-19, and H-20 finds the optimal priority list, confirmed by the solution of model (2). The other heuristics yield substantially inferior lists; i.e., they yield a substantially lower expected NPV. It is, perhaps, not surprising that when prioritizing a small set of candidate projects (in this case consisting of only 9 projects) using multiple heuristics (in this case 12 different heuristics; see Table 5) that some of those heuristics find the optimal solution. This result is also not surprising given the dominant nature of project 4 and the fact that the heuristics H-18, H-19, and H-20 have sufficient budget to include that project. We can obtain further insight by comparing the optimal priority list to that obtained by the heuristics H-11 and H-12. Comparing these two lists in Table 6, the only difference is that in the optimal list, the largest NPV project 4 has higher priority. Despite this simple difference, it is interesting to note that priority lists obtained by intermediate heuristics—e.g., H-15 and H-17—have a somewhat different structure. In H-15, projects 1 and 5 move substantially down the list. H-17 is arguably a natural heuristic to run in this setting because it first solves model (1) under the most likely budget scenario. Still, its expected NPV is $7.13M short of optimal. The performance of the optimal priority list under each budget scenario is given in Table 7. Comparing this with the results of the H-11 heuristic for each budget scenario in Table 3, we see that the optimal priority list underperforms the priority list of the greedy heuristic (by a relatively small amount) for the budget scenarios of $14M–17M. However, for larger realizations of the budget, the optimal priority list significantly outperforms the heuristic’s priority list, as seen by the resultant portfolio NPVs obtained for the budget realizations between $18M and 20M. The optimal expected NPV from the prioritization problem of $2.59M is (necessarily) at least as large as that of all the heuristics in Table 5 and, of course, smaller than that under perfect information ($11.90M). Figure 3 compares the NPVs

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Table 6. Priority lists from greedy heuristics and model (2), with their expected NPVs H-11, H-12 Priority

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1–4 5 6 7 8–9

H-15

H-17

H-18—H-20, Optimal

Project

Priority

Project

Priority

Project

Priority

Project

{1, 2, 5, 8}

1–3 4 5 6–7

{2, 8, 9} 6 7 {1,5}

1–4

{1, 2, 8, 9}

1–4

{1, 2, 5, 8}

9 6 7 {3, 4} −$15.42M

8–9 {3, 4} −$15.50M

5 6 7 8–9

3 5 6 {4, 7} −$4.54M

5 6 7 8–9

9 4 6 {3, 7} $2.59M

for each budget scenario for H-11, the optimal priority list, and perfect information. The expected NPV obtained under H-11 (−$15.42M), the optimal prioritization ($2.59M), and the perfect information ($11.90M) can be obtained from Figure 3 by weighing the respective points for each budget realization by the probabilities from Figure 1 and summing. Again, as the figure shows, the heuristic priority list outperforms the optimal priority list under some budget scenarios ($14M–17M) but not in the overall expected value of the NPV due to its large underperformance for budget realizations $18M–20M. With the small number of projects in this example, we likely could have obtained the optimal priority list by trial-and-error, or even brute force, because the total number of possible priority lists is modest (i.e., there are 9! = 362,880 possible orderings of the 9 projects). This small-sized example is useful because the behavior of the optimal solution is transparent. However, as we show in the next section, the prioritization model can produce similar results when the number of projects is larger and it is impossible to exhaustively examine all such alternatives. Before turning to this larger numerical example, we note that from Table 6 one may be tempted to infer that the best of the greedy heuristics obtains an optimal, or near-optimal solution, as H-18–H-20 do in this example. However, the following example shows that the factor by which the expected NPV of the optimal prioritization outperforms that of the best of the greedy heuristics can be arbitrarily large. In other words, we do not have any such performance guarantee even for the best of these greedy heuristics.

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Table 7. Solution to the 9-project prioritization problem Budget Level ($M)

1

2

3

4

5

6

7

8

9

NPV ($M)

11 12 13 14 15 16 17 18 19 20

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1 1 1

1 1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1

0 1 1 1 1 1 1 1 1 1

−23.58 −23.46 −23.46 −23.46 −23.46 −23.46 −23.46 37.13 37.13 42.30

Figure 3. Obtained NPVs for each budget realization under perfect information, the H-11 heuristic priority list, and the optimal priority list. The y-axis is NPV ($M) and x-axis is budget level ($M).

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Example 3. Let k be a positive integer and consider an instance of a twodimensional knapsack problem with projects indexed by I = {1, 2, . . . , 2k} and scenarios indexed by = {1, 2, . . . , k}. Projects i = 1, . . . , k have profit/cost streams of M/(1, 1), 2M/(2, 1), . . . , kM/(k, 1), where the pair (i, 1) represents the first- and second-year project costs, respectively. The remaining projects i = k + 1, . . . , 2k are all identical and have a profit of M − 1 and costs of (1, 1/k). The only uncertainty lies in the budget. In scenario ω there is a budget of (ω, 1) for ω = 1, . . . , k. The probabilities of 1 these scenarios are q ω = ω(ω+1) for ω = 1, . . . , k − 1, and the last scenario 1 has a probability of k . When M is sufficiently large, it is not difficult to see that the H-ω greedy heuristic, for ω = 1, . . . , k, begins by choosing project i = ω because the optimal solution of the two-dimensional knapsack problem with budget (ω, 1) is project i = ω. In all scenarios ω < ω, no project is selected under H-ω. And for scenarios ω > ω, project i = ω remains the only project selected because it exhausts the second-year budget. Hence, the objective function value under the H-ω heuristic is given by: k ω =ω

(ωM)q

ω

k−1 1 1 + = (ωM) k ω =ω ω (ω + 1) 1 1 1 + − = M. = (ωM) k ω k

On the other hand, the solution to the optimal prioritization model (2) selects project k + 1 under the first scenario, selects projects k + 1 and k + 2 under the second, and so forth until it selects projects k + 1, . . . , 2k under the last scenario. This optimal solution’s objective function value is k

k−1 k 1 1 ω(M − 1) = (M − 1) . ω(M − 1)q ω = k(M − 1) + k ω(ω + 1) ω ω=1 ω=1 ω=1

In this example, the H-ω heuristics all perform identically. Thus the ratio of the expected NPV from the best of the H-ω heuristics to that of the optimal prioritization is (M−1)Mk 1 . As M grows large, this ratio converges to ω=1 ω ( kω=1 ω1 )−1 . And finally, as k grows large, this ratio shrinks to zero, meaning that the factor by which the optimal prioritization outperforms the best of the H-ω heuristics grows without bound.

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A PROBLEM WITH MORE PROJECTS In this section, we consider a larger problem with 41 projects, all of which have positive NPV. As in the smaller problem considered in the previous sections, all of the project data, including NPVs and cost streams, are from STPNOC, with the cost stream estimates covering the next 5 years. Point forecasts for the project NPVs and cost streams (nominal) over the next 5 years are given in Table A1 in the Appendix, along with the benefitinvestment ratio for each project. The NPV and cost-stream values are of similar magnitude to those reported in Table 1, but Table A1 reports both the cit and ai values as percentages of the total NPV summed over all 41 projects. So, the sum of the 41 entries in the NPV (ai ) column is 100. As before, a blank entry for a cit value means that project i did not incur a cost in year t. We analyze two problem instances based on these projects. In the first instance, only the budget is uncertain and we consider 10 budget scenarios over the range of $2.5M to $7M in increments of $0.5M. As in the previous sections, if the budget scenario takes a value, say $4.5M, in year 1, then it takes that same value in years 2–5. These yearly budget realizations range from about 2% to 6% of the total NPV. The probability weights we place on these 10 scenarios are those given in the third column of Table 4. That is, the probability of having a budget of $2.5M is 0.012, that of having $3M is 0.019, and so on. Solving this instance of model (2), we obtain the desired priority list. For comparison we also obtain priority lists using the greedy heuristics H-2.5 through H-7. Finally, we also use the H-BIR and H-NPV heuristics. Figure 4 plots the NPV of the selected projects as a function of the budget realization for this problem instance for the prioritization model (2), the best (H-4.5) and worst (H-2.5) performing greedy heuristic procedures, the BIR heuristic, and the NPV under perfect information. These NPV results are reported as a percentage of total NPV, summed over all projects. The expected NPV of each method is given in Table 8, again as a percentage of total. In our earlier small test problem the heuristics initialized at the largest budget realizations performed best. Here, however, the heuristic initialized at either extreme (H-2.5 or H-7) performs poorly relative to the optimal list and relative to the heuristic initialized near the most likely budget scenario (H-5.5). We note that this emphasizes the conclusion presented in the previous section that use of the greedy heuristics is not guaranteed to obtain an optimal, or near-optimal, solution. We also note that the H-NPV has an expected NPV of zero because the highest NPV project is so costly that it cannot be implemented even under the highest budget scenario we consider. Finally, we turn to a problem instance in which the cost streams also are uncertain. As described in the background section, for each project

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Table 8. Procedures for 41-project problem instance and their expected NPVs (%) Procedure

Perf. Info.

Optimal

H-2.5

H-3

H-3.5

H-4

H-4.5

NPV (%)

61.20

60.25

54.69

54.69

60.16

60.16

60.17

Procedure

H-5

H-5.5

H-6

H-6.5

H-7

H-BIR

H-NPV

NPV (%)

60.17

60.14

58.83

57.77

57.77

55.25

0.00

STPNOC forecasts a pessimistic, optimistic, and most likely cost stream, and these in turn yield three forecasts for each project’s NPV. In the 41 projects we consider, there are two types of projects, those labeled low risk and those labeled medium risk (i.e., none of the projects were classified as high risk). For a low-risk project, its cost and NPV are assigned the pessimistic NPV value with probability 1/6, the optimistic value with probability 1/6, and the most likely value with probability 4/6. For mediumrisk projects these three respective probability masses are instead 2/6, 1/6, and 3/6. (For completeness we note that for high-risk projects, these

Figure 4. Replication of Figure 3 for 41-project problem under budget uncertainty.

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respective probability masses are 3/6, 1/6, and 2/6.) These weights reflect estimates based on STPNOC’s experience. The point forecasts for the cost streams and the NPVs are the same as those used above; i.e., those given in Table A1. The respective multipliers that yield the pessimistic, optimistic, and most likely cost streams and NPVs are given in Table A2, in the Appendix. That table also contains the low-risk or medium-risk label for each project. In our analysis, we assume that the project’s cost and NPV are perfectly correlated; e.g., if the cost stream takes the pessimistic realization, then so does the NPV. We further assume that all projects with the same risk label are perfectly correlated. So, all medium-risk projects either take the pessimistic, optimistic, or most likely realization. The same holds for projects within the low-risk category. However, the low-risk and medium-risk projects are assumed to behave independently. Each risk group thus has three realizations and the two groups are independent; hence there are a total of 9 scenarios governing the cost-profit uncertainty. In addition to this uncertainty, we also have budgetary uncertainty, which is modeled as unfolding independently of the cost-profit uncertainty. We use the same 10 budget scenarios described previously. This results in a total of || = 90 scenarios. The greedy heuristics must be altered slightly to deal with a problem instance that contains cost uncertainty. When there is only budgetary uncertainty, the scenarios can be ordered, and hence we can naturally produce a nested set of portfolios that, in turn, yields a priority list. This is not possible when costs, profits, and budgets are all uncertain. So, we instead use the average cost estimate and then form the greedy priority lists. Still, as is shown in Tables 5 and 6, these heuristics produce a partially ordered priority list. For example, under a number of the greedy heuristics for the 9-project problem, projects {1, 2, 5, 8} all receive top priority. We break the ties within this partial ordering using BIR, and this allows us to produce the fully ordered priority lists that are required to compute the expected NPV under cost uncertainty. The problem instance with cost uncertainty is almost an order of magnitude larger than the 41-project problem with just 10 budget scenarios. We solve this stochastic integer program to within 1% of optimality, again using ILOG’s software CPLEX version 10.1 (Cplex, 2007). The near-optimal priority list we find has an expected NPV of 60.18%. The respective expected NPV results for the heuristics H-2.5, H-5.5, and H-7 are 54.65%, 60.04%, and 57.04%. (The expected NPVs of these priority lists are given as percentages of the total NPVs summed over all 41 candidate projects.) We report H-5.5 because it is the heuristic initialized at the most likely scenario and because, of all the greedy heuristics, it yielded the highest expected NPV. The H-BIR heuristic yielded an expected NPV of 55.27%, and H-NPV again could not implement any projects because of the top

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priority being given to an excessively costly project. Finally, the expected NPV under perfect information is 61.19%. As in our smaller computational example using data from STPNOC, sometimes a heuristic performs well. However, predicting which, if any, heuristic will perform well can be difficult. One can avoid the need for our prioritization model (2) if the gap between a heuristic’s expected NPV and the expected NPV under perfect information is small. This gap is a posterior bound in that it can be computed only once the problem data are known. In the 90-scenario instance just considered, the relative gap between the expected NPV of the H-5.5 heuristic and that under perfect information is 1.9%, and if that is deemed sufficiently small, we can employ the priority list from the H-5.5 heuristic. However, in other problem instances (e.g., our 9-project instance) this gap is large. In such cases, we recommend applying our optimal prioritization model (2). SUMMARY In practice, it is common to use performance measures like payback period, internal return rate, benefit-investment ratio and net present value for individual capital projects to form a priority list. Such an approach fails to recognize that the selected projects act as a portfolio, and ignoring this fact can lead to suboptimal results. It is recommended in the literature that capital budgeting be done using variants of a multi-knapsack problem formulation. In this article we have demonstrated that such solutions can be volatile with respect to changes in budget values. This can be disconcerting to decision-makers, especially because it is almost certain that budget allocations need to respond to emerging events. In this setting, a properly formed priority list can provide a valuable approach to hedging against such uncertainties, at least when irreversible decisions must be made prior to knowing the budget realizations. In this article we have discussed an optimal prioritization model that explicitly incorporates multiple budget, cost, and profit scenarios to develop an optimal priority list that is more robust to external events that impact these parameters. We showed that this priority list can outperform priority lists formed through heuristic means. We investigated heuristic rankings using NPV and BIR, and we investigated a class of heuristics that relies on solving multiple instances of the multi-knapsack model. The value of forming an optimal priority list was validated on problem instances derived from two sets of projects from STPNOC. Our smaller example involves 9 projects with positive NPV, whereas the larger example involves 41 such projects. Moreover, in the larger example, we consider a problem instance with uncertainty in the yearly budgets, as well as the cost streams and profits associated with projects.

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Sometimes our heuristics performed well, but in other problem instances they did not. A stylized example showed that even the best of our greedy heuristics can perform arbitrarily poorly. So, it is not possible to obtain a constant-factor performance guarantee; e.g., that the expected NPV obtained by the best of the greedy heuristics is within, say, 50% of optimal. For this reason, unless the upper bound on expected NPV obtained via perfect information is close to the expected NPV obtained by a heuristic, we recommend use of our optimal prioritization model. There are multiple important directions to be pursued as future work to further the development of our approach and to permit its use by decisionmakers in budget planning and allocations. At this stage, our approach has been demonstrated using two sets of actual data provided by a commercial nuclear power plant. However, the multi-knapsack model we have described here ignores details often present in real-world capital budgeting problems. For example, a collection of projects can represent mutually exclusive alternatives, selecting one project can yield a synergistic opportunity for another project, and projects can be implemented in phases with opportunities for delayed implementation. Fortunately, the prioritization approach we have described is amenable to incorporating such constructs. We assumed that all projects are to be prioritized; i.e., in the notation of model (2) that |P | = |I |. However, often the total cost of performing all of the candidate projects exceeds even the most optimistic budget forecast. This, along with potential gains in computational efficiency, motivates future consideration of the case when |P | < |I |. Finally, in the future we intend to generalize our prioritization model with respect to time dynamics so that we need not commit to prioritizing all projects at the beginning of the planning period. ACKNOWLEDGMENTS The authors thank two anonymous referees for helpful comments that improved the article. This research was supported by the Electric Power Research Institute under contract EP-P22134/C10834, STPNOC under grant B02857, and the National Science Foundation under grants CMMI0457558, CMMI-0653916, and CMMI-0855577. Abbreviated versions of this article have appeared in Koc¸ et al. (2007, 2008). REFERENCES Bierman, H., and Smidt, S. (1980). The Capital Budgeting Decision: Economic Analysis of Investment Projects, 5th ed. Macmillan Publishing Co., New York.

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´ J., and Wood, R.K. (2006). Defending critical Brown, G.G., Carlyle, M., Salmeron, infrastructure. Interfaces, 36, 530–544. Brown, G.G., Dell, R.F., and Newman, A.M. (2004). Optimizing military capital budgeting. Interfaces, 34 (6), 415–425. CPLEX. (2007). CPLEX 10.1 Reference Manual. ILOG, S.A., Gentilly Cedex, France, Available at: http://www.ilog.com. Hartman, J.C. (2000). The parallel replacement problem with demand and capital budgeting constraints. Naval Research Logistics, 47, 40–56. Keeney, R.L., and Raiffa, H. (1976). Decisions with Multiple Objectives: Preferences and Value Tradeoffs. John Wiley & Sons, New York. Kellerer, H., Pferschy, U., and Pisinger, D. (2004). Knapsack Problems. SpringerVerlag, Heidelberg. Koc¸, A., (2010). Prioritization via Stochastic Optimization. Ph.D. thesis, The University of Texas at Austin, forthcoming. Koc¸, A., Morton, D., Popova, E., Hess, S., Kee, E., and Richards, D. (2008). Optimizing project prioritization under budget uncertainty. Paper read at ICONE 16: ASME 2008 International Conference on Nuclear Engineering, 11–15 May, Orlando, Florida. Koc¸, A., Morton, D., Popova, E., Kee, E., Richards, D., Sun, A., and Hess, S. (2007). Project prioritization via optimization. Paper read at PVP 2007/CREEP 8 ASME PVP 2007/CREEP 8 Conference, 22–26 July, San Antonio, Texas. Meier, H., Christofides, N., and Salkin, G. (2001). Capital budgeting under uncertainty—an integrated approach using contingent claims analysis and integer programming. Operations Research, 49 (2), 196–206. Ross, S.A., Westerfield, R.W., and Jordan, B.D. (2008). Fundamentals of Corporate Finance, 8th ed. McGraw-Hill, Boston. Savage, S., Scholtes, S., and Zweidler, D. (2006). Probability management. ORMS Today, 33, 20–28. Weingartner, H.M. (1966). Capital budgeting of interrelated projects: survey and synthesis. Management Science, 12, 485–516.

AUTHOR BIOGRAPHICAL SKETCHES Ali Koc¸ is a Ph.D. student in the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin. His research interests are in the areas of stochastic modeling and programming, algorithms for high-performance computing, capital budgeting and project scheduling, and production planning and control. David Morton is Engineering Foundation Professor in the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin. His research interests include developing models and algorithms for optimization problems that explicitly incorporate uncertainty.

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Elmira Popova is an Associate Professor in the Graduate Program in Operations Research & Industrial Engineering at The University of Texas at Austin. Her research interests include modeling of uncertainty.

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Stephen Hess is a Senior Project Manager—Safety and Asset Risk Management for the Electric Power Research Institute. His research interests include modeling and analysis of nonlinear dynamical systems, risk and decision analysis methods, and system reliability and safety analysis. Drew Richards is the supervisor, Station Applications and Optimization, in the Risk Management Department of the STP Nuclear Operating Company located near Wadsworth, Texas. His current interest areas are in deployment of applications within STP that would use results of risk models to help improve plant operation and maintenance (including outage maintenance). He has been instrumental in the design and development of the current U.S. industry’s approach to on-line risk management, working closely and providing guidance to the NRC and the industry. Ernie Kee is a senior consulting engineer in the Risk Management Department of the STP Nuclear Operating Company located near Wadsworth, Texas. His current primary interest areas are in (electrical) production risk management emphasizing quantitative operational risk models used in on-line risk applications by plant operators and work planners.

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APPENDIX Table A1. Data for 41 projects. Both the cost and NPV values are given as percentages of total NPV summed over all 41 projects Yearly costs ci t

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Project i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

1

2

3

4

5

NPV (ai )

BIR

0.07 0.09

0.08

0.01

0.01

0.33 0.08 0.06

0.22 0.08

0.29

0.45

0.07 0.12 0.29 0.12 6.46

0.15

0.17 0.27 0.17 0.29 0.13

0.78 4.54 0.17

0.07 13.96 0.17

0.13 0.10

0.13 0.29

0.17

0.13

0.13 0.52

0.13 0.75 0.01

0.61 0.36 0.04

1.08

0.42

0.66

0.45

0.81

12.71 3.10 0.40 0.66 3.12 2.86 3.35 4.63 2.11 0.36 0.26 1.17 2.21 4.32 1.94 1.71 7.37 1.11 0.77 1.02 0.76 17.72 0.56 4.34 0.98 16.26 0.75 0.59 0.44 0.21 0.49 0.23 0.32 0.33 0.07 0.04 0.50 0.10 0.03 0.01 0.09

96.612 38.769 20.855 16.009 14.950 10.006 9.973 9.499 9.267 6.973 6.237 6.016 5.350 5.090 4.484 3.159 3.055 2.661 2.342 2.339 2.315 1.735 1.568 1.272 1.231 1.204 1.196 1.141 0.784 0.740 0.669 0.662 0.568 0.359 0.339 0.301 0.222 0.208 0.201 0.158 0.101

0.02 0.04 0.21 0.29 0.34 0.08 0.04 0.11 0.22 0.27 0.18 0.93

0.09 0.21 0.53 0.21 1.30 0.12 0.08 0.08

0.36 1.78

0.08 0.13

0.11 0.07 0.35 0.31 0.19 0.13 0.19 2.24

4.81

1.78

0.17 0.29 0.13 0.80

0.34 0.13 0.20 0.13 0.15 0.12 0.04

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Table A2. Multiplier factors for 41 projects under pessimistic, optimistic, and most likely scenarios

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Cost factors Project i

Risk level

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

L M L L L L M L L L L M L L L M L M M M M M L M M M L L L M L L L L L L L M L L M

NPV factors

Pessimistic

Optimistic

Most likely

Pessimistic

Optimistic

Most likely

1.131 1.104 1.171 1.100 1.000 1.050 1.200 1.131 1.050 1.100 1.000 1.134 1.046 1.000 1.082 1.104 1.127 1.104 1.104 1.104 1.104 1.104 1.000 1.067 1.104 1.086 1.000 1.138 1.000 1.104 1.050 1.000 1.000 1.131 1.050 1.000 1.140 1.073 1.157 1.000 1.104

0.880 0.912 0.927 0.900 1.000 0.950 0.720 0.934 0.950 0.900 1.000 0.898 0.961 1.000 0.984 0.912 0.872 0.912 0.912 0.912 0.912 0.912 1.000 0.932 0.912 0.897 1.000 0.933 1.000 0.912 0.950 1.000 1.000 0.934 0.950 1.000 0.893 0.927 0.801 1.000 0.912

0.997 0.960 0.976 1.000 1.000 1.000 0.960 0.984 1.000 1.000 1.000 0.945 0.998 1.000 0.984 0.960 1.000 0.960 0.960 0.960 0.960 0.960 1.000 0.978 0.960 0.977 1.000 0.982 1.000 0.960 1.000 1.000 1.000 0.984 1.000 1.000 0.992 0.976 1.010 1.000 0.960

0.999 0.997 0.993 0.994 1.000 0.995 0.980 0.987 0.995 0.986 1.000 0.979 0.992 1.000 0.983 0.968 0.961 0.963 0.958 0.957 0.957 0.943 1.000 0.949 0.919 0.931 1.000 0.891 1.000 0.866 0.928 1.000 1.000 0.670 0.857 1.000 0.435 0.663 0.306 1.000 0.067

1.001 1.002 1.004 1.007 1.000 1.005 1.036 1.007 1.006 1.015 1.000 1.017 1.007 1.000 1.003 1.029 1.046 1.034 1.039 1.039 1.039 1.052 1.000 1.055 1.074 1.091 1.000 1.059 1.000 1.122 1.077 1.000 1.000 1.184 1.152 1.000 1.508 1.360 2.112 1.000 1.851

1.000 1.001 1.001 1.000 1.000 1.000 1.001 1.001 1.000 1.000 1.000 1.008 1.000 1.000 1.003 1.011 0.998 1.014 1.015 1.015 1.016 1.021 1.000 1.016 1.029 1.016 1.000 1.012 1.000 1.048 0.999 1.000 1.000 1.036 0.998 1.000 1.014 1.105 0.895 1.000 1.338

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