WHO used fixed weights (0.25, 0.25, 0.125, 0.125, 0.25) to aggregate five health system outputs (respectively, the level of population health, inequality in the distribution of health, the level of health system responsiveness, inequality in the distribution of responsiveness, and fairness in financial contributions) into a scalar health system attainment index. Overall attainment ranged from 35.7 (Sierra Leone) to 93.4 (Japan) on a [0–100] scale.

We propose here an analytical framework that reduces to the WHO fixed-weight methodology as a special case, but that allows varying degrees of freedom for weights to be defined that – in the sense of Melyn and Moesen – implicitly take into account individual country circumstances. For shorthand, we say countries "choose" such weights, which in reality are determined as solution values of a linear program. If the linear program is a fair representation of the objective function of and constraints faced by decision makers, the weights are indeed "chosen", but even if this condition is not necessarily met, the resulting weights may still be of interest.

The extent of freedom to choose weights in a linear program can be set by the analyst. The analyst can enforce fixed weights common to all countries, allow countries complete freedom to choose their own weights, or adopt a middle ground in which countries are granted limited freedom to choose weights within bounds thought to be sensible by experts.

A generic statement of the performance evaluation problem in primal-dual linear programming format is:

In these programs y is a country's output vector, x is its input vector, Y is the sample output matrix and X is the sample input matrix. In the present context y is (5 × 1), Y is (5 × 191), x is (n × 1) and X is (n × 191), with n to be specified below.

The primal program seeks the maximum radial expansion of a country's outputs, provided that it not exceed the standards established by a convex combination (λ ≥ 0, ∑λ = 1) of best-practice countries in the sample. The optimal value of φ provides a distance measure (i.e. an indication of how far a country has to go) to match best practice as observed in the sample. Since φ ≥ 1, the attainment of a country is evaluated as φ^{-1} ≤ 1. Best practice countries have φ^{-1} = 1, other countries have φ^{-1} < 1, so φ provides a basis for acomplete ranking of countries on their relative ability to deliver five health system outputs. The dual program seeks a set of nonnegative weights μ,υ attached to a country's outputs and inputs that maximize its attainment. Each country's output weights are normalized by μy = 1, but each is free to select its own set of nonnegative weights.

In constructing the overall attainment index, WHO identified five output indicators and no inputs, preferring to treat each country's health system as a "health output management unit". Consequently each country's input vector is represented as a scalar with unit value. Under these circumstances the performance evaluation problem simplifies to:

where ω = υ + υ_{0}. The modified primal program seeks the maximum feasible radial expansion of a country's outputs consistent with best practice observed in the sample. The modified dual program seeks a set of nonnegative weights μ for a country's outputs that put it in the best light. A country can be expected to assign relatively large weights to those outputs at which it excels relative to best practice, and relatively low, possibly zero, weights to those outputs at which it lags behind best practice, subject to the normalization μy = 1. (See Annex for a graphical explanation.)

By complementary slackness, μ_{m}(Y_{m}λ - φy_{m}) = 0, m = 1, ..., M, so slack in any element of a country's projected output vector φy implies that the country assigns a zero weight to that output. Since it is unreasonable to allow a country to assign a zero weight to any output deemed sufficiently important to have been included in the WHO performance evaluation exercise, it is desirable to restrict weights in some way. This can be accomplished most easily by appending constraints to the dual side of (2) of the form:

γ_{m} ≥ μ_{m}y_{m}/μy ≥ β_{m}, m = 1, ..., M. (3)

Restrictions (3) place lower and upper bounds on the relative importance of each output (as measured by μ_{m}) in total output.

Implementation of the weight-restricted linear program requires specifying the 2M = 10 parameters γ_{m}, β_{m}. One procedure is to ignore the upper bounds γ_{m} and set the lower bounds β_{m} > 0. This eliminates the possibility of a country assigning zero weights to those outputs at which it lags behind best practice. A less arbitrary procedure is to follow Takamura and Tone [16] by adapting Saaty's Analytical Hierarchy Process (AHP) [17]. This procedure exploits expert judgment, that could be provided for example by the above-mentioned survey of health system professionals, to set lower bounds β_{m} > 0 and upper bounds 1 > γ_{m}. Although these bounds are common to all countries, they allow countries limited freedom to select weights appropriate to their circumstances.