Allocation of environmental burdens is a recognised methodological problem in Life Cycle Assessment (LCA). It is the process of assigning to each of the functions of a multiple-function system only those environmental burdens and impacts that each function generates. It is argued in this paper that allocation is an artifact of applying LCA to individual products rather than to the whole productive system. To solve this problem, a new "marginal allocation" approach is proposed, based on whole system modelling. Marginal allocation is applicable when marginal changes about some defined state of the product system are to be considered and when the functional outputs can be varied independently. The specific approach developed here is based on representing the system by a model in the Linear Programming (LP) format. The allocation coefficients are equivalent to the marginal values calculated at the solution of the LP model. Marginal values represent a realistic description of the causal relationships between burdens and functional outputs and thus reflect the behaviour of the system. Changes in the system behaviour can also be modelled by LP. The approach is illustrated on three simple examples of multiple-function systems: combined waste treatment, co-production and recycling.