The Priority Scaling allocation method translates a priority scale into an allocation of financial resources, using quadratic and dynamic programming.

The basic Priority Scaling allocation method essentially involves computer calculation of the distribution of a given available total budget (or balance) across all budget items so as to logically reflect the priority scale, together with any other variable or constraint (e.g. past budgets, demand and supply levels, unit costs, etc.). Alternatively from an allocation of resources across items, the implied priorities implicit in this allocation are calculated.

Allocation - mathematical programming techniques

Once the priority scale has been determined, it is often necessary to make allocations of resources (increments, decrements or reallocations) which logically reflect this scale. The priority weights guide the amount of allocation per budget item in the light of other relevant factors and constraints. In general, initial allocations are compromises between 'soft' subjective valuations (e.g. felt or planned priorities) and estimates (e.g. estimated demand), and more objective 'hard' data (e.g. last year's expenditure).

Quadratic and Dynamic Programming

The decision-makers' values and desiderata are obtained from their judgement analysis exercises in the form of priorities and criteria, and may be additionally specified by explicit allocation constraints. Information on other variables is also obtained, preferably from empirical data but alternatively from estimation exercises using judgement analysis and allied techniques. This enables a set of equations to be derived which describe in a integrated mathematical model the complex mix of priorities, criteria, variables and constraints involved in the resources to be allocated, as well as the varying interdependency relationships between these variables. The optimal feasible allocation of resources is then calculated, that is the allocation which best fits all the decision-makers' priorities, criteria and constraints given the particular disposition of the other variables.

The calculation is made by quadratic programming (Shapiro, 1979; Holt et aI, 1954). The decision-maker judges the relative influence of external factors (e.g. past budget allocation, level of demand, performance indicators) vis-à-vis internal factors (e.g. priorities), using judgement analysis to determine the influence weights and check consistency and coherence. The quadratic programme calculates the optimal solution which minimizes the quadratic function of the relative influence, subject to any linear constraints in the other relevant variables (e.g. stipulated minimal or maximal size of any budget items). Dynamic programming (Bellman, 1954) is also used to allow calculus of variations for a range of assumptions or scenarios concerning operating conditions.

The reliability tests used in basic judgement analysis may also be used to refine and check budget allocations themselves.

Allocating a given budget balance

A given available total budget (or balance) is allocated across all budget items such that the individual allocations sum to the given total, while reflecting both the priority weights of each item as elicited in the priority scale, and the available data on other relevant variables. The other variables or factors may be past budget size, demand and supply levels, unit costs, profitability or return-on-investment, product or service performance indicators, or indeed any criterion which would have been included in the criteria analysis.

The variables or factors are expressed in appropriate financial ratios (Weston, 1962). The variables or factors may themselves be weighted to reflect their relative differential influence, the variable weightings obtainable by the same methods as used to obtain priority weights between items. Quadratic or dynamic programming techniques are used to make the calculations.

Incremental historical budgeting

Under the incremental historical budgeting procedure (Braybrook and Lindblom, 1963; Etzioni, 1969), past budgets of items are assigned significant influence or weighting relative to other variables. If the influence of the past budget is weighted far higher than any other variables, this effectively reduces the budgeting procedure to incremental priority and allocation decisions at the margin, with the budget of the past period as the base. In this case, priorities and allocations may as well be divided between a 'shopping list' of new budget items (and/or between additional increments or decrements of past budget items) as usually happens in capital budgeting. If by contrast, the influence of the past budget is reduced to nil, then a zero-base budgeting procedure is being used. Clearly the higher the influence of the past budget relative to priorities, the less will be the change in the existing distribution of resources (all things else being equal), though the extent of change will be offset to the extent that decision-makers' priorities reflect the past situation. However, even where incremental budgeting is not the order of the day, past budgets in conjunction with past supply levels may under some circumstances function as surrogate indicators of unit costs for the past planning period.

Zero-budgeting

Under zero-base budgeting (Phyrr, 1973), the priorities and total available budget (together with unit costs) are the main variables used in calculating the allocations. The influence or effect of past budget size of items is thereby omitted or reduced in the new budget allocation.

Mixed Scanning

When a mixed scanning approach is used (Etzioni, 1969), the incremental and zero-based budgeting procedures are alternated. The zero budgeting procedure is used to obtain a base allocation based on a priority exercise in which local short term exigencies are discounted. The incremental budgeting plan is 'based on a priorities exercise in which local short term exigencies are recognized as significant criteria, and full account of past budgets is taken into account in the quadratic programme.

Cost Benefit Analysis

The priority scaling allocation procedure may also be used for cost-benefit or cost-utility analysis (Hitch and McKean, 1960; McKean, 1959). This involves eliciting priorities solely in terms of benefits or utilities (omitting cost considerations) which are then balanced against budget item costs in a single cost-benefit equation.

Determining a required budget balance

Once a priority scale of budget items has been elicited, and information or estimates are obtained for unit costs and forecast demand levels, the budget required for each item to attain a particular profit level (in the private sector), or to attain a particular level of service or supply (in the public sector), can be calculated using a variant of the Priority Scaling quadratic or dynamic programming model. This procedure is usually undertaken as part of the Operations Budgeting described below.

Calculating Implications of a given budget allocation

Decision-makers may make budget allocations directly without first deciding priorities. They may do this either by intuitive or negotiated decision as conventionally happens, or by using a variant of judgement analysis or 'fuzzy scaling' directly applied to allocations rather than to priorities. From these allocations, decision-makers' implied priorities (i.e. the priorities implicit in their allocations) can be calculated by applying the Priority Scaling quadratic and dynamic programming. This allows future budget allocations td be calculated using managerial coefficients analysis (Dawes, 1977), and this provides a further check on judgement analysis results.

Adjusting Priorities and Allocations

In the light of the initial allocations, decision­makers may choose to adjust their priorities, criteria and allocations. This is equivalent to adjusting their judgements in the light of experience. The adjustments are fed into the model, and everything is recalculated accordingly so as to preserve all other priorities, criteria, variables and constraints and their interrelationships so far as this is possible in the light of the adjustments.

Scenario Writing

By asking 'what if?' questions, decision-makers may review the alternative scenarios produced when they vary any variable. Thus, for example, they may vary their priorities to determine how allocations vary or vice versa, and similarly with other variables. The model calculates each alternative scenario.

Adapting Allocations to Reality

The accuracy of the model first generated is potentially limited by the 'softness' of any data that was estimated, and by the likelihood that decision-makers' priorities and criteria as revealed in daily decision-making and action may often be at variance with their consciously formulated priorities and criteria in the planning forum (e.g. when they were completing judgement analysis and allied exercises).

However, from quite minimal empirical data fed into the model, recalculations can be made which 'harden' the estimated data and adjust the decision-maker's planned priorities and criteria to the priorities and criteria implied by their decision and action. Decision-makers then decide whether they want to continue to work towards planned priorities and criteria, or whether they will adjust these plans in the light of revealed reality.