Measuring the capacity utilization of public district hospitals in Tunisia: Using dual data envelopment analysis approach

Background: Public district hospitals (PDHs) in Tunisia are not operating at full plant capacity and underutilize their operating budget. Methods: Individual PDHs capacity utilization (CU) is measured for 2000 and 2010 using dual data envelopment analysis (DEA) approach with shadow prices input and output restrictions. The CU is estimated for 101 of 105 PDH in 2000 and 94 of 105 PDH in 2010. Results: In average, unused capacity is estimated at 18% in 2010 vs. 13% in 2000. Of PDHs 26% underutilize their operating budget in 2010 vs. 21% in 2000. Conclusion: Inadequate supply, health quality and the lack of operating budget should be tackled to reduce unmet user’s needs and the bypassing of the PDHs and, thus to increase their CU. Social health insurance should be turned into a direct purchaser of curative and preventive care for the PDHs.

Theoretical foundation of CU is provided by Johansen7 and Morrison.14 Johansen referred to the production function and defined single output capacity technology as: “Production capacity is defined as the maximum that can be produced by a production unit with fixed and variable inputs for a given period and provided that the availability of variable factors of production is not restricted.” Nelson16 and Morrison14 provide the economic definition of capacity, where the optimal output measure is the tangency between the short-run and long-run average cost curve. Many Studies12,13,15,17 have focused on Johansen’s definition assessing individual hospitals CU. CU corresponds to output produced, given full and efficient variable input utilization and capacity base constraints imposed by ie, fixed factors, technology, environmental conditions, and resource stock. Frontier setting preference using distance function18 with the DEA approach10,13 as key method is reasoned as DEA does not require input price information and can incorporate multiple outputs. DEA is based on Koopmans19 and Farrell20 (economic axioms). Arrow and Debreu21 provides the production technology and frontier enabling the scores estimation of efficiency and CU. In frontier setting, the CU scores can be estimated using the distance function10,12 or the directional distance function.22,23 Färe12,13,15 has developed a primal DEA model estimating CU from output-oriented efficiency scores. DEA constructs a ‘‘best practice frontier’’ for maximum possible outputs for fixed input quantities. DEA has been extended examining sufficient capacity among hospitals and their CU. Following Färe,11 we adopt Johansen’s definition of plant CU: ‘‘The maximal amount that can be produced per unit of time with existing plant and equipment without restrictions on the availability of variable production factors.’’8 To estimate individual PDH CU, we formulate a dual DEA model including additional constraints on input and output shadow prices. Shadow price restrictions enrich the empirical CU measure by adding priorities in terms of input and output costs which have a significant economic meaning. It is well-known that DEA models are very flexible in the sense that they evaluate an observation in its best possible light. Therefore, no a priori ordering is made among the shadow prices of inputs or outputs and all positive shadow prices are allowed for the different inputs and outputs. However, in order to maximize the efficiency of the evaluated observation, extreme shadows prices can be selected by the linear program which can be counterintuitive from an economic point of view. We can limit this extreme flexibility of DEA models by constraining the ordering of shadows prices. While not knowing the ‘real’ or ‘market’ prices of each input and output, we can nevertheless order theses prices and impose the same order to the shadow prices. The objective can be thought of as adding an economic meaning into the linear program of DEA. Summarizing, CU scores are obtained in three steps determining (a) the maximum output obtainable from observed (fixed and variable) inputs; (b) the maximum amount of output obtained from observed fixed inputs alone assuming unconstrained variable inputs; and (c) take the ratio of the first two steps to obtain a CU measure.9 Steps (a) and (b) require a series of the linear programming steps. Assuming identical hospital production technology, the production technology set (P(x)) transforms a vector N inputs (x=(x1,….,xN)∈R+N) into a vector of M outputs (y=(y1,….,yM)∈R+M). Applying basic economic axioms,24 the production set provides a convex and freely disposable input and output technology. Technical efficiency and CU ratio (CUR) of observed input and output (x, y) vectors are derived from the production technology P(x). Output-oriented efficiency scores are estimated by the following output distance function: DEA programs estimate the distance function and CURs.8,9 Färe11,12 derived CU in a frontier setting using distance function while Ferrier23 employs the directional distance function to aggregate capacities of individual hospitals into a group. We formulate a dual DEA program with shadow prices constraints using the output distance. For each hospital, the dual DEA estimates CU using individual hospital observed inputs and outputs according to Johansen’s definition. In the short-run, inputs need to be categorized as fixed (xf) and variable (xv); that is, x = (xf, xv) for each hospital k. We define Nf fixed inputs and Nv variable inputs such that Nf+Nv = N. The fixed inputs sub-vector (xf) refers to the existing plant and equipment. The activity vector Z is a weight assigned to the observed kth hospital in the linear convex technology combination. Suppose K (k = 1,…, K) hospitals in the data sample. Under variable returns to scale and strong disposability, the production set P(x) with fixed and variable inputs is: Output-oriented efficiency can be provided by the output distance function giving radial proportion scaled-up hospital output projections on the frontier.11,12 The value of the output distance function for the ith hospital is calculated using dual linear programming. From a practical point of view, the above radial efficiency measure Do (xf, xv, y) is computed relative to the previously defined reference technology P(x) by solving a linear program for each observation. This yields a primal DEA linear program. The dual DEA program is derived with shadow price constraints estimating the distance function Do (xf, xv, y). The value of the output distance function for the kth hospital is found by solving the following dual linear programming input and output shadow prices constraints problem (3). (3) where v1f is the shadow price of medical doctors v2f is the shadow price of surgical dentists v3f is the shadow price of midwives v4f is the shadow price of nurses v5f is the shadow price of beds u1 is the shadow price of outpatients visits in stomatology u2 is the shadow price of outpatients visits in emergency u3 is the shadow price of outpatients visits in external wards u4 is the shadow price of admissions in maternity wards u5 is the shadow price of admissions Fixed factor shadow prices restrictions impose an input values ordering. Fixed inputs are the number of medical doctors, surgical dentists, midwives, nurses, and beds in the empirical study. Assuming a higher shadow price of one physician compared to one surgical dentist, which is higher than the one of a midwife which is higher than the one of a nurse. The shadow price of a bed (proxy of capital) is assumed higher than those of all other inputs. Following the same line of thought, a shadow prices ordering is imposed among outputs (admissions, admissions in maternity wards, outpatient visits in stomatology, in emergency and in external wards). Admissions are higher valued than outpatient visits. These are expected relationships in the real world. It avoids DEA models to choose inappropriate shadow prices for the evaluated hospitals. That is why it is so important to complement DEA models with sound exogenous economic information. The second step in measuring the CUR is to determine each hospital’s capacity with constant fixed inputs, allowing variable inputs to be unrestricted (consistent with Johansen’s definition of capacity). Hospital k’s capacity is given by the following linear programming solution (4): (4) The only difference between the equations (3) and (4) is the treatment of variable inputs. In equation (3) hospital variable inputs are restricted to currently available levels; in equation (4) variable inputs are unrestricted and not constrained. It is assumed that a hospital has access to as many variable inputs as needed for full capacity. Variable inputs can be omitted from the specification. D0(yk,xfk,xvk)-1 and D0(yk,xfk) are the inverse of Shephard distance function18 and can be interpreted as Farrell’s efficiency measure.22 They represent the possible radial increase in outputs if hospital (k) operates efficiently. The last step in the CUR process is taking the ratio of the inverse of solutions given by equations (3) and (4) to determine hospital k’s CUR: This measure is devoid of any technical inefficiency since the latter appears in the numerator and denominator.1−CUR(yk,xkf,xkv) can be interpreted as the percentage of additional output to be produced at full capacity without variable input restrictions. Obviously, given the property of the output distance function necessarily: Using CUR has several advantages. First, by taking the ratio of the output distance functions any technical inefficiency is removed by definition. This means the measure is not downward biased, in contrast to most traditional CU measures. Second, capacity measures are computed at the frontier and defined relative to observed best practices. Third, it is able to accommodate a multiplicity of inputs and outputs. Input and output data describe the hospital’s production technology without the use of functional forms. The advantage of using a non-parametric approach – not requiring an explicit representation of the form of the technology – is that it envelopes the observed input and output data by forming facets around the frontier of the observed data. This type of modeling has been used extensively in hospital studies as well as in health economics more generally (see Hollingsworth25,26 for a review of efficiency measurement and Ferrier23 and Färe11 for the applied nonparametric measure of CU for health facilities). Public hospitals in Tunisia are regionally-based of PDH as a first reference level receiving patients that go beyond the primary healthcare centers. These hospitals regulate access to referral regional and university hospitals with a mission to provide healthcare services for the whole territory. They consume a high proportion of the overall MoH budget (25% in the 2013 budget year). In addition, these hospitals account for about 66% of total public hospitals. PDH were created the decade following the independence in 1956. It focused on building health infrastructure, when medical treatment uses less specialized technology for high prevalence of communicable diseases. More PDH were created in the 1980s to support primary healthcare, especially maternal and child health. Their mission as defined by the sanitary law 1991 was limited to provide ambulatory and emergency care, maternal and child care and short day’s hospitalizations. PDH offer many healthcare services such as outpatient, inpatient and emergency of internal medicine, surgery, as well as pediatric, dental, and maternity care. Outpatient and inpatient care are the first services delivered while the second set essentially consists of emergency. Medical consultations of the PDH represent more than 40% of total medical visits of all public health facilities. Overall, the volume of medical services increases over the period. The size of the PDH is relatively small with on average of 38 beds and 16 physicians. Large disparities can be detected by the variation in the number of beds ranging from 20 to 128 and in the number of physician ranging from 8 to 26 (Table 1), Overall, the size of PDH, as measured by human and physical resources, increased over the study period. Indeed, of the 166 of total public’s hospitals in the country, 105 are PDH with 15% of total public beds in 2010 vs. 12% in 2000.2,3 PDH are managed and subsidized MoH. One of the most striking observations is that social health insurance contributes indirectly to the finance of PDH; contrary to the billing system implemented for the regional and university hospitals. Abbreviation: SD, standard deviation‏. Relevant data of the PDH is limited in Tunisia. In this study, data was collected from various MoH reports for 20102 and from a survey for 2000.27 The CU is estimated for 101 of 105 PDH in 2000 and 94 of 105 PDH in 2010. Inputs and outputs were selected according to the literature on nonparametric DEA-based measurement of hospital efficiency.25,26,28,29 PDH production technology was represented by five fixed inputs, one variable input and five outputs, common to all hospitals. The input variables are broadly classified into labor, capital and technological input. The input variable of ‘staff’ consists of total staff of a particular hospital. The break-down of the total staff in terms of the number of doctors, number of surgical dentist, number of midwives and number of nurses was available. Two measures of the capital input were used, one based on the number of beds per hospital as proxy for net capital assets (see Färe11) and one being the operating budget. The operating budget includes expenditure on drugs, maintenance of medical equipment, machinery, vehicles, infrastructure, etc as a proxy of the quantity of capital investment. In general, hospitals provide three major services: outpatient, inpatient, and laboratory services. Ideally, health output should be measured as an increment to patient health status as final products of hospitals. However, since this is technically impossible to measure, in all hospital CU and efficiency studies intermediate outputs of various kinds are used instead. Outputs disaggregated into inpatient and outpatient output is used in many studies.30,31 Given the complex definition of hospital output due to limited data availability in PDH, the number of cases treated in outpatient and inpatient services handled in five categories was chosen as a representative measure of the hospital’s output, since these were assumed to have significant implications for the use of resources. The selected outputs are services that supposedly improve health status. Descriptive statistics of input and output variables are provided in Table 1. The availability of data on various indicators in the PDH was limited in Tunisia and, due to this constraint, we had to restrict our analysis to the above-mentioned input and output variables. Even we could not get data for the selected variables for 4 PDH in 2000 and for 11 in 2010; therefore, we could only include the remaining PDH out of 105 in our study. The data included in the study were reliable as we conducted checks and found that they were of good quality. A DEA model for the measurement of CUR was run after feeding the input and output variables into program equation 5. The CUR has economic relevance in the short term. The physical inputs are fixed (physicians, surgical dentists, midwives, nurses, and beds). Considering the operating budget as the only variable input, 1 minus CUR can be interpreted as the additional output if the budget were increased given the existing fixed inputs. In the short term, a low CUR is due to an insufficient budget. In the long run, the size (number of beds) and personnel categories can be considered as variable too.

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