Project Prioritization via Optimization

We consider a problem commonly faced in industry, involving annual selection of plant capital investments. A typical approach to such a problem uses a multi-knapsack formulation, which takes as input the available budget in each year, the stream of liabilities induced by selecting each project, and the profit, i.e., net present value, of each project. The goal is to select the portfolio of projects with the highest total net present value, while observing the budget constraint for each year, as well as any additional constraints. A portfolio selected in this manner can fail to hedge against uncertainties in the budget, the liability stream and the profit. So, we propose a model that forms an optimal priority list of projects, incorporating multiple scenarios for these input parameters. Our model is not a simplistic ranking scheme. Structural and stochastic dependencies among the projects are key to our approach. We apply our methods on a set of example projects from 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 payback period, internal rate of return, net present value (NPV), etc. The academic literature frequently points out (e.g., Ref. [1] and [3]) that priority lists built on such simple ranking measures are inferior to allocating funds to capital projects using variants of a multi-knapsack problem formulation (e.g., Ref. [2]). The multi-knapsack approach takes as input a budget forecast and selects a collection of projects to be carried out, assuming the point forecast for the budget is correct. We will refer to this selected collection of projects as a project portfolio. If the costs and profits of candidate projects as well as the budgets in coming years are known with certainty, we agree that multi-knapsack models can provide an attractive tool. However, how should we approach capital budgeting when we have uncer1 Copyright c © 2007 by ASME tain 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 are typically implemented in phases over time and usually, some irreversible decisions must be made. It is not always practical to fully revise a project portfolio whenever better forecasts for these parameters become available. We believe practitioners 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. It is well known that priority lists formed by scoring projects individually fail to capture dependencies between projects and we do not repeat such analyses here. Instead the path of this paper is as follows: • We first investigate whether the solution to a multi-knapsack model naturally yields a prioritized list. We show it does not. • Next, we heuristically alter the multi-knapsack approach, and force it to produce a prioritized list. We call this our heuristic priority list. • Finally, we ask whether we can build a priority list that outperforms the heuristic priority list, at least when we assume a probabilistic forecast for the uncertain parameters. We answer this question affirmatively. We formulate a model that explicitly incorporates multiple budget, cost and profit scenarios and forms an optimal priority list. And, we show that the optimal priority list can significantly outperform the heuristic priority list. As indicated, we begin with the recommended approach to capital budgeting when the budget, cost and profit parameters are known with certainty, i.e., the multi-knapsack model. In this setting, we have as input the available budget bt in each year t ∈ T , the stream of liabilities cit induced in each year t by selecting project i ∈ I, and the profit ai, i.e., NPV, of selecting project i. Further structural dependencies can involve mutually exclusive project selections, precedence relations, and other types of logical constraints between projects. The deterministic capitalbudgeting problem is to select the most profitable portfolio of projects in the sense of highest total NPV, while observing the budget constraint for each year in the planning horizon as well as any additional structural dependencies. When the latter constraints are dropped, the model is known as a multi-knapsack problem. The multi-knapsack approach outlined above has serious shortcomings because the parameters bt , ai, and cit are typically uncertain. When this is the case, in addition to the structural dependencies among projects mentioned above, stochastic dependencies can arise. To illustrate the flaw of the multi-knapsack approach, suppose we have used it to select a portfolio of projects. Then, over the course of the year, the available budget decreases due to external events or because a high-priority project experiences cost over-runs. As a practical matter, a low-priority project will now be forced out of the portfolio, i.e., it will not be carried out. Unfortunately, solutions to the multi-knapsack formulation can be fragile to such events. We develop an approach that better hedges against these types of future contingencies. In this paper, we utilize data provided by the South Texas Project Nuclear Operating Company (STPNOC) to illustrate our approach. As the operator of a large commercial nuclear generating station, STPNOC must evaluate investment in numerous projects and choose a portfolio that will achieve the objectives of the organization. As a result, 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 a “blue line.” Projects above the blue line are to be funded and those below it are not. Over the course of the year, the blue line can shift for reasons described above. This paradigm is not unique to STPNOC, and it is not unique to the nuclear industry. Rather, similar capital budgeting practices are employed across a wide range of industries and in government, too. The optimization model we propose recognizes that prioritizing is common practice and aims to build priority lists that are financially robust to the types of uncertainties described above. Our approach to forming an optimal priority list focuses on financial performance measures. However, financial goals alone do not drive capital planning decisions in the nuclear industry. The need to ensure regulatory compliance factors heavily into decision-making at STPNOC, and throughout the nuclear power industry. We note that this characteristic is not unique to STPNOC (or even to the nuclear industry). To address this issue, priority lists generally include an integration of both financial and non-financial aspects into the decision process. At STPNOC (and many other commercial nuclear plants) this results in application of a multi-attribute utility theoretic approach to performing this integration (see e.g., Ref. [2]). We consider an example problem 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 managerial dictate for safety and regulatory reasons. We show how this affects our approach and we further discuss how regulatory and safety issues are often well-aligned with financial goals. We explicitly model multiple budget, cost and profit scenarios in order to form an optimal priority list. Current STPNOC practice is to forecast optimistic, pessimistic and most-likely cost streams for each candidate project. These give us scenario values for cit . Project managers also assign a “risk score” for each project that relates to the likelihood of a cost over-run. So, a project with a high risk score receives larger weight (probability mass) on the pessimistic cost forecast than a low-risk project. Common or independent external factors can induce stochastic dependencies among the projects. 2 Copyright c © 2007 by ASME Our model is not a simplistic ranking scheme. Instead, key to our approach of prioritizing projects are the structural and stochastic dependencies among the projects. That is, the model recognizes that the projects which are ultimately implemented, after the stochastic budgets, costs and profits are realized, will act as a portfolio. Further, we emphasize that our model is appropriate only when irreversible decisions regarding project selection must be made before knowing cost, budget and profit values. If we can wait until these become known before committing to project selection decisions, we should do so and solve what is then a deterministic multi-knapsack model. DETERMINISTIC CAPITAL BUDGETING: MODEL (1) We first 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 will only consider stochastic budget levels, but our approach easily extends to 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 Data: ai net present value of project i cit cost of project i in year t bt available budget in period t Decision variables: xi 1 if project i is selected; 0 otherwise