Signature
PI = E_F / E_V
| Inputs | Definition | Unit |
|---|---|---|
E_F | Output efficiency score of the unit against the constant returns frontier when only the fixed inputs constrain output and the variable inputs are unrestricted, the reciprocal of the largest proportional expansion of all its outputs in that programme; above zero and at most E_V | score, at most 1 |
E_V | Output efficiency score of the same unit against the constant returns frontier when all inputs constrain output, the reciprocal of the largest proportional expansion of all its outputs, equal to TE_O of HE-FM-EFF-001 for a single output; above zero and at most 1 | score, at most 1 |
PI | Output efficiency with only the fixed inputs constraining output, as a share of output efficiency with all inputs constraining output; above zero and at most 1, with values below 1 showing unused plant capacity | proportion |
|---|
Function
Capacity utilisation of a health service with plant capacity measured on a frontier
Maps a health service's use of its capacity to a capacity utilisation figure. The simple measure in the article, capacity used over capacity available in the same units and period times 100, is a plain proportion of the form of the first part of HE-FM-PR-001, P_1 = a / (a + b), with a the capacity used and b the spare capacity, and of HE-FM-PDC-001, so it is not restated. Its bed form, occupied over available bed-days, is the occupancy o defined in HE-VR-ALOS-006 and used in HE-FM-ALOS-002, HE-FM-BTO-003 and HE-FM-BTO-004, and the empty bed-days of a period are the numerator of HE-FM-BTO-002. The fixed cost of a bed spread over the patients it served is HE-FM-BTO-005, the break-even volume of a service with fixed costs is HE-FM-BEA-002, annualised capital cost is HE-FM-COST-003 and HE-CF-ANLS-001, and the utilisation of a single-server queue is rho in HE-FM-ARRP-005. The average cost form AC = F / Q + v of the article belongs to the average-cost concept. HE-FM-CAPU-001 is plant capacity utilisation in frontier analysis, which divides the output efficiency of a unit when only its fixed inputs constrain production by its output efficiency when all inputs do, so that technical inefficiency (HE-FM-EFF-001) is set aside before unused plant capacity is measured. It is built like scale efficiency HE-FM-EFF-003 but compares a different pair of frontiers. Notation follows the Capacity Utilisation article, which describes the plant capacity measure in words, with the symbols of Valdmanis and colleagues.
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Implementations
Excel
Plant capacity utilisation from two named efficiency cells
With the score from the fixed-input programme in a cell named OutputEfficiencyFixedInputs and the score from the all-input programme in a cell named OutputEfficiencyAllInputs, the formula returns PI in a cell named PlantCapacityUtilisation. The scores themselves come from the two linear programmes, solved with a data envelopment analysis tool or the Excel Solver. A blank OutputEfficiencyAllInputs cell is treated as 0 and the formula returns #DIV/0!.
=OutputEfficiencyFixedInputs/OutputEfficiencyAllInputs
Assumptions
Both plant capacity scores from one sample under constant returns to scale
E_F and E_V come from the same sample of units, with the same outputs, inputs and output orientation and a constant returns to scale frontier, as Valdmanis and colleagues restrict the plant capacity analysis. Only the treatment of the variable inputs differs. Removing the constraints on variable inputs cannot lower the largest attainable expansion of outputs, so E_F is at most E_V and PI is above zero and at most 1 (derived, and checked here on 1,200 unit scores from random data sets solved as linear programmes).
Fixed and variable inputs classified before the plant capacity analysis
The inputs that cannot change in the short run are named in advance. Valdmanis and colleagues class beds, doctors, nurses and other staff as fixed, because the ministry sets them, and allowance, drug and other operating expenditure as variable. PI measures unused capacity given that classification, and a different split of the same inputs gives a different value.
Proportional output expansion against best observed practice
All outputs are expanded in the same proportion, as in the output-based programmes of Valdmanis and colleagues, and capacity output is the best performance the frontier finds among comparable units (HE-AS-EFF-001), not an engineering maximum. Outputs are measured on the same basis for every unit, with inpatient care weighted for case mix as in that study.
Worked examples
Illustrative hospital with technical inefficiency and little unused plant capacity
A hospital produces 80% of the frontier output for its inputs, so E_V is 0.80, and 76% of the output its fixed inputs could support with variable inputs unrestricted, so E_F is 0.76. Plant capacity utilisation is 0.76 divided by 0.80, or 0.95, so only 5% of capacity output is unused once its technical inefficiency is set aside, in line with the article's point that a hospital can lack technical efficiency while having little spare capacity. The scores are illustrative.
E_F = 0.76; E_V = 0.80; PI = 0.95
Illustrative technically efficient hospital with idle plant
A hospital on the all-input frontier, with E_V equal to 1, reaches only 75% of the output its beds and staff could support, so E_F is 0.75 and PI is 0.75. The hospital is technically efficient, yet a quarter of its plant capacity output is unused. The scores are illustrative.
E_F = 0.75; E_V = 1; PI = 0.75
Limiting case of full plant capacity utilisation
When freeing the variable inputs allows no further expansion, the two scores are equal, here both 0.88, and PI is 1. The hospital is technically inefficient, but once on the frontier it could not raise its outputs in proportion without more fixed inputs. The scores are illustrative and the case checks an implementation.
E_F = 0.88; E_V = 0.88; PI = 1
Common errors
Inverting the plant capacity utilisation ratio
Dividing E_V by E_F gives the factor by which output on the frontier could rise if the unused plant were brought into use, not the share of capacity in use. In the first worked example it returns 1.0526, which read as utilisation would claim that the hospital works above its capacity output (derived).
Reading the fixed-input score alone as capacity utilisation
E_F on its own is observed output over capacity output and equals PI times E_V (derived), so it mixes technical inefficiency with unused plant. Valdmanis and colleagues factor out inefficiency first because they measure plant capacity and not inefficient production. In the first worked example E_F of 0.76 would suggest that 24% of capacity output is unused, when 5% is due to idle plant and the rest of the gap to technical inefficiency (derived).
Taking the all-input score from a variable returns frontier
Both scores must come from constant returns frontiers. If E_F is from the constant returns programme but E_V is taken from a variable returns frontier, the ratio also carries scale efficiency (HE-FM-EFF-003). With E_F 0.76 and constant and variable returns scores for E_V of 0.80 and 0.90, PI falls from 0.95 to 0.844, and the scale effect is reported as idle plant (illustrative, derived).
Treating a bed occupancy ratio as plant capacity utilisation
An occupancy ratio such as HE-VR-ALOS-006 sets one used input against its capacity and says nothing about what was produced, as the article notes. A hospital can show high bed occupancy while its fixed inputs could support more output with the same variable spending, and a ratio of used to available beds cannot reveal that. PI compares outputs with what the fixed inputs could produce on the frontier.
Sources
Valdmanis and colleagues on plant capacity utilisation in Thai public hospitals
Valdmanis V, Kumanarayake L, Lertiendumrong J. Capacity in Thai public hospitals and the production of care for poor and nonpoor patients. Health Services Research. 2004;39(6 Pt 2):2117-2134. doi:10.1111/j.1475-6773.2004.00335.x (full text read in PubMed Central, PMC1361115). The Model subsection on capacity to provide increased services, the plant capacity approach, adopts the Johansen definition of plant capacity, states that the plant capacity measure PI is the ratio of the output efficiency where some inputs are variable and others fixed (E_F) to the output efficiency where all inputs are treated as variable (E_V), that a value below 1 indicates unused or underutilised plant capacity, that inefficiencies are factored out first and that the analysis is restricted to constant returns to scale, citing the methods of Färe, Grosskopf and Kokkelenberg (1989) and others. The Appendix gives the two output-based programmes, the second with the fixed inputs restricted and the variable inputs unrestricted. The Data section classes beds, doctors, nurses and other staff as fixed inputs and the three expenditure inputs as variable, and Table 4 reports a sample mean of 0.954 for 1999.
Canonical Identity
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