The Allocation of Costs to Projects in a Multiproject Matrix Environment

The Use of a Matrix Cost Allocation Model

James E. Burcsu

Burroughs Wellcome Company

The research and development units of many firms utilize ad hoc committees for new product development projects. Such projects are by their nature temporary, special-purpose activities. The responsibilities and tasks that are established in this way overlap those of the permanent functional departments. This type of organization is referred to as a matrix organization (For example, see [2]). In multiproject matrix organizations, there exists the problem of accurately associating costs with the specific projects. A solution to this problem, suggested here, is the extension to projects of the matrix cost allocation/input-output model as originally described by Livingstone [1],

Problems exist in cost allocation to projects due to the fact that in most R&D organizations the operating budgets are the responsibility of the functional departments. Further complications arise because, in carrying out their functions, these departments expend some part of their total effort and/or funds directly for any of many specific projects. They may also expend some of their effort and funds in service to other departments in ways only indirectly related to the specific output efforts, the projects themselves. The activities of some functional departments may consist entirely of providing services to other departments and, hence, benefit projects only indirectly.

A common method for project cost allocation requires the maintenance of detailed accounting records of all activities and expenses, with the intent of allocating costs to each project, as well as to any or all other output efforts [3]. This approach can create numerous records and be quite expensive. The method, however, still requires the use of approximations for the allocation of costs, particularly the indirect costs, that tend to make precise recordkeeping superfluous.

Problems notwithstanding, there remains a need for a method of cost allocation that reasonably approximates the true costs of projects in any multiproject matrix environment. Such information can be vitally important in the management, planning, and monitoring of projects. The matrix cost allocation/input-output model suggested here provides a relatively easy means for allocating all budgeted funds of the input departments to the various projects, the output efforts. One of its most notable features is the ability to allocate indirect costs as well as direct costs.

The model consists of expressing effort allocation as a set of simultaneous equations that are solved as matrices. The equations consist of the fractions of total effort (or effort and funds) allocated by each department to the other departments or to specific outputs and the total departmental budgets. Where the overriding costs are personnel costs, only the fraction of total effort (time allocation) need be considered. Otherwise, direct expenses, expressed as a fraction of total departmental budget, are assigned to the outputs. The principal assumption of this model is that all units of effort of a particular department have equivalent value.

An example of a cost allocation problem, and its results, is given below. An explanation of the calculation procedure follows that discussion. Note that the effort and dollar values used in this problem and in the calculation need not be only retrospectively derived. The values used in the calculation can be alternate planned values. In such a planning process, iterative calculations would allow the evaluation of alternate courses of action for resource use optimization.

The table below gives the allocations of effort, as fractions of total department effort for the accounting period, by the functional departments to the other departments and to the final output projects. The separate far right column gives the budgets of the departments for the period. For the purposes of this example, there are six functional departments, 1-6, and three output projects, R, P and M. In a pharmaceutical company, Departments 1-6 could be departments such as Pharmaceutical Chemistry, Toxicology, etc. Outputs R, P and M could be general output categories such as support for Research, Projects, and Marketed Products or new product development projects, or even a mixture of both specific and general types of effort. The values in this table indicate that Department 3 expends 20% of its effort and budget helping Department 1 perform its task, 40% of its effort helping Department 4, 10% helping Department 6, and 30% of its effort directly involved in the projects. These values can be derived from time allocation and specific expenditure records. The period budget for Department 3 is $6,000. Note that there is no allocation of effort from R, P or M to the input departments nor of any one department to itself.


After calculation, the funds budgeted to the functional departments are allocated to the outputs:

Output Project Allocated Costs
R $  9,903
P   17,391
   M         11,705   
Total $39,000 (with rounding error)

In addition, the actual costs of each functional department are obtained.

Functional Department Actual Costs
1 $  5,634
2     4,672
3     2,011
4     7,244
5     8,732
    6          10,705   
Total $39,000 (with rounding error)

These Actual Cost values represent the true cost of the functions. These values include the amounts budgeted directly to the departments, plus the service costs that are allocated by the calculation from other departments’ budgets, less those costs allocated to other departments for services provided.


The calculation involves solving a set of simultaneous equations. In matrix terms, if the allocation matrix (the table above) is designated A, then:

(I – A) X = B

where I is the identity matrix, X is the column vector of calculated costs, and B is the vector of budgeted amounts. Since we need to determine X:

X = (I – A)-1B

For this problem:


Solving this equation gives:

X1  =  $ 7,043
X2  =     6,674
X3  =     6,704
X4  =   10,349
X5  =     9,702
X6  =   10,705
XR  =     9,903
XP  =   17,391
XM  =  11,705

Note that the sum of XR, Xp, and XM equals $39,000 (with a rounding error). This equals the total of the original budgets, hence all costs have been allocated to the outputs. Obtaining the actual costs of each of the six departments requires multiplying the value of X for the department by the fraction of effort the department spends performing its own direct output tasks.

This cost allocation method can be a useful management tool because it simplifies cost allocation to projects by eliminating maintenance of detailed cost accounting records and the interpretation of how specific effort and costs are being directed to final outputs on the part of departments aiding others.

1. Livingstone, J.L., “Input-Output Analysis for Cost Accounting, Planning and Control,” The Accounting Review, January, 1969.

2. McCarney, K.E., “Implementing Matrix Management in a Research and Development Environment, “Project Management Quarterly, September, 1980.

3. Moder, J.J. and C.R. Phillips, Project Management with CPM and PERT, New York: Van Nostrand Reinhold, 1970, pp. 236-270.

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