Lee Merkhofer Consulting Priority Systems
Mathematical theory for prioritizing projects

A Mathematical Model of the Capital Allocation Problem

The standard, mathematical model of the capital allocation problem is constructed as follows. Assume there are m potential projects. For now, assume that the projects are independent; that is, it is reasonable to select any combination of projects and the cost and benefit of any project do not depend on what other projects are selected. Define, for each project i = 1, 2,,..., m the zero-one variable xi. The variable xi is one if the project is accepted and zero if it is rejected. Let bi be the incremental value (benefit) of the i'th project and ci be its cost. Let C be the total available budget. The goal is to select from the available projects the subset of projects with a total cost less than or equal to C that produces the greatest possible total value.

The problem may be expressed mathematically as:

Eq. 1

This is a formulation of the so-called knapsack problem, a constrained optimization that requires selecting from possibilities a subset that will maximize the achievement of some objective while satisfying some constraint.

As a concrete example, suppose we have $10,000 to spend on four possible projects:


Project ID Cost Benefit
1 $3571 $5714
2 $2857 $4286
3 $5000 $7857
4 $2143 $2857

The optimal allocation is the solution to:


Eq. 2

Mathematical optimization problems that are constrained to have only integer values (i.e. whole numbers such as -1, 0, 1, 2, etc.) at the optimal solution are known as integer (or pure integer) programming problems. The above formulation is a special case called a zero-one integer programming problem. As illustrated, zero-one variables are useful for capital budgeting because they can be used to represent yes/no decisions, specifically, whether to include a particular project in a selected project portfolio. Methods for solving the zero-one integer programming problem are described later below. First, though, some useful extensions to the basic model are indicated.


Extensions to the Basic Model

Although the basic capital allocation model is simplistic, it can be extended to handle many complexities that must be addressed in the real world. For example, the model can be extended to deal with projects whose costs and benefits are spread over time, the "costs" of not doing projects, interdependencies among projects, soft budget constraints, multi-period budget constraints, and differences in the degree to which projects can be delayed.

Benefits Spread Over Time

Although the above formulation assumes benefits are paid in the current period, the model can be applied to the case of projects that provide benefits spread out over future periods by simply defining the current benefit of each projects to be the present value of its future benefits. The time horizon for estimating project benefits should be the duration over which the project can be expected to provide value. In the case of a project that creates some new physical asset, for example, the time horizon might be set the lifespan of that asset. Typically, however, because of the discounting and the fact that project benefits usually decline over time, it is rare to count project benefits that occur beyond about 40 years.

Future Project Costs

Oftentimes, doing a project in a planning period effectively commits the organization to paying future year costs. For example, a new asset may need to be maintained, otherwise the benefits derived from that asset will cease. The future year costs can be treated as project "dis-benefits," that must be subtracted in the computation of the present value of the project.

"Costs" of Not Doing Projects

Oftentimes, projects are not done so much to make things better as to prevent things from getting worse. This can be handled by simply defining the benefit of a project as the difference between what happens if the project is done compared to what happens if the project is not done. In other words, project value is the incremental benefit relative to what would happen if the project is not conducted. For example, if not doing a project necessitates some expense, avoiding that expense is a legitimate benefit of the project.

Mutually Exclusive and Sequential Projects

By adding constraints to the basic model, mutually exclusive projects can be handled. Suppose S denotes a subset of the projects that represent different ways of doing the same thing (e.g., choosing competing technologies or alternative vendors for achieving the same goal). Then, the constraint


Eq. 3

ensures that no more than one project is selected from the set S.

Oftentimes, mutually exclusive projects represent different ways of addressing the same need, and the project choice affects the benefits obtained. This can be handled by changing the objective function so that benefit bij is obtained if the i'th need is addressed via the j'th project. Expressed mathematically, the problem is then:


Eq. 4

In these equations,

  • 1,..., nj are the indices of the mutually exclusive projects for addressing need i,
  • xij equals 1 if the need i is achieved through project j; otherwise it is zero for all i = 1,..., , m and j = 1,..., , nj,
  • cij is the cost of achieving the i'th need with the j'th project,
  • bij is the benefit obtained if the i'th need is achieved with the j''th project.

Adding constraints also works in situations where precedence relationships apply; that is where one project cannot be chosen unless another is also chosen. For example, suppose there are contingent projects (e.g., a software project) that can be selected only if another project (e.g., the necessary hardware) is also funded. Suppose project i is contingent on project k. The constraint

Eq. 5

ensures that i cannot be selected (xi cannot be 1) without k being selected (without xk being 1).

As a concrete example, suppose that, instead of being independent, the four projects from the previous example are dependent as follows:

  • Project 4 requires Project 2
  • Project 3 can't be conducted if Project 1 is conducted

The constraints that must be added to the previous formulation are:


Eq. 6
(If x4 is 1, then x2 must be 1.)
Eq. 7
(If x1 is 1, then x3 must not be 1.)

Inter-Dependent Projects and Optimal Project Portfolios

The most common type of project interdependency is that described above; unless another project is conducted, the benefits of the project won't be realized. However, sometimes the interdependencies among projects is partial; doing another project will enhance the attractiveness of a project, but it may still be worthwhile to do the project even if the projects that would enhance its value are not conducted. For example, combining multiple projects in strategic ways can produce economies of scale and resource sharing that can lower the total costs of the related projects. Similarly, there may be synergies among projects such that total benefits increase if they are all conducted together.

One type of benefit synergy is diversification of risks. Portfolio theory is a mathematical theory for understanding diversification and the relation between risk and return as it applies to portfolios of stocks and other financial investments. Just as a portfolio of stocks can lower risks through diversification, a portfolio of well-chosen projects can be more attractive because it creates a more desirable balance between near-term versus long-term payoffs, "sure things" versus "gambles," and risk versus return.

Project interdependencies can be handled within the model by changing the decision unit from individual projects to alternative sets of projects. The approach involves defining clusters of inter-dependent projects such that each cluster is independent of every other cluster. A project cluster is independent of other project clusters if no project in the cluster depends on any project outside the cluster. The costs and benefits of conducting each possible subset from each cluster are determined, taking into account synergies and other interdependencies. The optimal portfolio is then obtained by choosing the subset from each cluster (a subset containing no projects is a candidate) such that total benefits are maximized without exceeding the constraints on total costs. The mathematical formulation is identical to the formulation for mutually exclusive and independent projects (since the sets from independent clusters are independent), except that the maximization involves choosing the best project sets (rather than individual projects) and the constraint is that at most one project set can be chosen from each project cluster.

Soft Budget Constraints

The basic formulation of the capital budgeting problem is somewhat unrealistic in that budget constraints are rarely completely hard. For this reason, it is often useful to solve the problem for alternative total budgets so as to measure the sensitivity of the solution and total benefit to the budget level.

Multi-Period Planning

The basic formulation can be extended to represent multi-period project planning with constraints. In this case, the basic decision problem is when; that is, in which planning period, should each project be started? The simplest formulation merely requires specifying the multi-period funding requirements for each project and including each period's budget constraint. To illustrate, suppose the planning is for T time periods. Let the total cost constraint for period t be Ct. Suppose once a project i starts, it requires a specified investment cit in each period t. Once completed, project i generates a net present value benefit of bi. The problem is to select the projects for funding across the T periods that will maximize the total benefit subject to the period-by-period cost constraints:

Eq. 8

Sensitivity to Delay

In some situations, there may be only a limited time window of opportunity to conduct some proposed projects. In such cases, the goal for each budget cycle of picking the project portfolio that maximizes benefit may not be appropriate. To illustrate, suppose you have a daily “budget” of $1. I offer you two “project” investment alternatives: With Alternative A, you invest $1 and immediately receive in return a financial benefit of $10. With Alternative B, you invest $1 and immediately receive a financial benefit of $5. Using the traditional approach of maximizing benefit you would, of course, choose Alternative A, because it creates greater total benefit.

Suppose, however, that I tell you that Alternative A will be available tomorrow, while Alternative B is only available today. You would then be wise to invest your $1 budget today in Alternative B and invest tomorrow's $1 budget in Alternative A. The table below shows that this choice maximizes the combined benefit derived from the today's and tomorrow's budgets:


Today Tomorrow Total
Choice 1 A: Cost = $1 Benefit = $10 Return = $10-$1 = $9
Choice 2 B: Cost = $1 Benefit = $5 A: Cost = $1 Benefit = $10 Return = $15-$2 = $13

Of course, the above example assumes that you will have a dollar to invest tomorrow and that there will be no other investments tomorrow that will generate a $10 return. If, for example, you knew that tomorrow someone else was going to offer you a $10 return for a $1 investment, you would be wise to take Alternative A today. Roughly speaking, it really only makes sense to delay high-value projects if you believe there won't be equally valuable investments available in the future.

To properly select projects that differ in the degree to which they can be delayed requires simultaneously optimizing current and future choices so as to maximize total, multi-period, project benefit. One approach is to expand the multi-period project planning formulation described above to make the costs and benefits of projects depend on the period in which the project is funded. As described above, this formulation requires knowing what all of the future potential projects and project budget constraints are. If there's uncertainty over future project characteristics and budget constraints, the possibilities can be represented probabilistically, but, as you can see, solving the problem by formulated it as a multi-period project selection problem is complex.

Because of problem complexity (actually, the large and difficult nature of the required problem inputs), it is generally not feasible to simultaneously optimize project portfolios for multiple budget cycles. There is, however, an approximate approach that involves thinking one budget cycle ahead. With this approach, each proposed project is viewed as presenting three possibilities: (1) the project could be funded now out of the current period budget, (2) the project could be deferred and funded in the next budget period, or (3) the project can be eliminated and never funded. To implement the approach, a minimum benefit/cost ratio is established as a threshold for determining whether a project is attractive. Any project whose B/C ratio is below that threshold is eliminated from further consideration. Any project above the threshold is assumed to be funded either this period or to be deferred and funded in the following budget period. The goal for allocating the current budget, then, is to pick the portfolio of projects that produces the minimum lost value from delaying attractive projects.

To express this mathematically, suppose that there are m “attractive” projects denoted i = 1,..., , m with benefit/cost ratios greater than some specified value R:

bi1/ci1 > R,

where bi1 and ci1 are the benefit and cost of the i'th project if it is conducted this (1st) budget period. Let bi2 and ci2 be the benefit and cost of the project if it deferred and conducted next period. Let xi1 be one if the project is funded this period and zero otherwise. Since bi2 - bi1 + ci2 - ci1 is the loss from delaying project i, to minimize the total loss from delay, we need to solve:

Eq. 9

This “minimize loss from delay” goal can be interpreted as reflecting an “real options” view of the capital budgeting problem. Projects represent options for organizations. An option is an opportunity, but not an obligation. Proposed projects represent options because the organization has the opportunity to fund them, but is not required to do so. Since options have value (which can be quantified using real options analysis), organizations should consider how their decisions impact option value. Deciding to fund a project converts the project from an option to a commitment. The option value is lost, but that value loss is (presumably) justified by the value derived from doing the project. Deciding not to fund a project may or may not preserve its option value. If the project can be delayed without adverse impact, its option value is preserved. Conversely, if delay causes project benefits to decrease or its cost to rise, then some of its option value is lost if the project is delayed. Thus, optimizing a project portfolio based on minimizing the losses from delay can be regarded as a “real options” solution to capital budgeting.