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Data Envelopment Analysis(DEA) Page 193

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- DEA is an application of Linear Programming (LP) to measure the relative efficiency of operating units with the same goals and objectives.
- In each application, the performance of each institution or organization is measured relative to
the performance of all operating units in the same system

- DEA is particularly useful in an organization where there are multiple inputs and multiple outputs
- In many cases, the goal of DEA is to identify the inefficient unit(s) that should be targeted for further study and, if necessary, corrective action.

- DEA creates a fictitious composite unit made up of an optimal weighted average (W1, W2,…) of existing units.
- An individual unit, k, can be compared by determining E, the fraction of unit k’s input resources required by the optimal composite unit.
- If E < 1, unit k is less efficient than the composite unit and be deemed relatively inefficient.
- If E = 1, there is no evidence that unit k is inefficient, but one cannot conclude that k is absolutely efficient.

- The DEA Model
MIN E

s.t.1. Sum of weights = 1

2. Weighted outputs > Unit k’s output

(for each measured output)

3. Weighted inputs <E [Unit k’s input]

(for each measured input)

E, weights > 0

- The case of using DEA to improve performance at four hospitals is used as an example
- To evaluate the use of DEA, 3 input and 4 output measures are identified

- Input Measures (page 195)
1.The number of full-time equivalent (FTE) nonphysician personnel

2. The amount spent on supplies

3. The number of bed-days available

- Output Measures
1.Patient-days of service under Medicare

2.Patient-days of service not under Medicare

3.Number of nurses trained

4.Number of interns trained

- In this case,a linear programming model is developed for each hospital whose efficiency has to be evaluated.
- Then a composite hospital is created based on the outputs and inputs for all the four hospitals
- For each output, an output for the composite hospital is determined by computing a weighted average of corresponding output for all the four hospitals.

- Similarly, input for the composite hospital is calculated using the same weighted average method for all the inputs.
- Constraints in the LP model require all outputs for the composite to be greater than or equal to the outputs of the county hospital, the one which is being evaluated.
- If the inputs are less than that of the county hospital, the hospital being evaluated is having more output for the same input.

- wg= weight applied to inputs and outputs for General Hospital
- wu = weight applied to inputs and outputs for University Hospital
- wc = weight applied to inputs and outputs for County Hospital
- ws = weight applied to inputs and outputs for State Hospital

- The DEA approach requires that the sum of these weights equal 1. Thus, the first constraint is
wg + wu + wc + ws = 1 (page 200)

- Medicare Patient-Days = (Medicare Patient-Days for (composite hospital) General Hospital )*wg + (Medicare Patient-Days for County Hospital)*wc + (Medicare Patient-Days for University hospital)*wu +
(Medicare Patient-Days for state hospital)*ws

- Medicare Patient-Days =48.l4wg + 34.62wu + (compositehospital) 36.72wc + 33.l6ws
- The other output measures for the composite hospital are computed in a similar fashion.
Figure 4.25 (page 197) provides a summary of the results.

General format for the output constraints:

Output for the composite hospital >= Output for county hospital

All 4 constraints (page 197):

- Medicare Patient days: 48.14 wg + 34.62 wu + 36.72 wc + 33.16 ws >= 36.72
- Non medicare patient days: 43.10 wg + 27.11 wu + 45.98 wc + 56.46 ws >= 45.98
- Nurses: 253 wg+148 wu+175 wc+l60 ws >= 175
- Interns 41 wg+27 wu+23 wc+84 ws >= 23
Open page 200

General format for the input constraints:

(Input for the composite hospital) <= (resources available for the composite hospitals)

FTE Non physicians for composite hospital =

(FTE Non physicians for general hospital) * wg

+ (FTE Non physicians for university hospital) * wu

+(FTE Non physicians for county hospital) *wc

+ (FTE Non physicians for state hospital) * ws

The number of full time equivalent non physicians for the hospital is:

285.20wg + 162.30wu + 275.70wc + 210.40ws

- E = the fraction of County Hospital’s input available to the composite hospital
- The expression for the number of FTE non physicians available to the composite hospital has to be evaluated
- E is also known as efficiency index
- The input constraint corresponding to FTE non physicians:
285.20wg+l62.30wu+275.70wc+2l0.40ws <=275.70E

Two other input constraints are

- 123.80wg + 128.70wu + 348.50wc + l54.10ws <= 348.50E (Supplies)
- 106.72wg + 64.21wu + lO4.l0wc ± l04.04ws <= 104.10E (Bed-days)

- Minimize E
- The decision rule is:
- If E = 1, the composite hospital requires as much input as County Hospital does. There is no evidence that County Hospital is inefficient.
- If E< 1, the composite hospital requires less input to obtain the output achieved by County Hospital. The composite hospital is more efficient; thus, County Hospital can be judged relatively inefficient

- Wg + wu + wc + ws =1
- 48.14wg+ 34.62wu +36.72wc+33.16ws >=36.72
- 43.10wg+ 27.11wu +45.98wc+33.16ws >=45.98
- 253wg + 148wu +175wc + 160ws >=175
- 41wg + 27wu +23wc + 84ws >=23
- 123.80wg+ 128.70wu +348.50wc+154.10ws <=348.50E
- 285.20wg+ 162.30wu +275.70wc+210.40ws <=275050E
- 106.72wg + 64.21wu + 104.10wc + 104.04ws<= 104.10E
- E, wg, wu, ws, wc >= 0

- Step 1: Define decision variables or weights (one for each operating unit) that can be used to determine the inputs and outputs for the composite operating unit.Step 2: Write a constraint that requires the weights to sum to 1.
- Step 3: For each output measure write a constraint that requires the output for the composite operating unit to be greater than or equal to the corresponding output for the jth operating unit.
- Step 4: Define a decision variable, E, which determines the fraction of the jth operating unit’s input available to the composite hospital.

- Step 5: For each input measure write a constraint that requires the input for the composite operating unit to be less than or equal to the resources available to the composite operating unit.
- Step 6: Write the objective function as Min E.

- 1.Remember that the goal of data envelopment analysis is to identify operating units that are relatively inefficient. The method does not necessarily identify the operating units that are relativelyefficient. Just because the efficiency index is E = 1, we cannot conclude that the unit being analyzed is relatively efficient. Indeed, any unit that has the largest output on any one of the output measures cannot be judged relatively inefficient.
- 2.DEA could show all but one unit to be relatively inefficient. Such would be the case if a unit producing the most of every output also consumes the least of every input. Such cases are extremely rare in practice.

- 3.In applying data envelopment analysis to problems involving a large group of operating units, practitioners have found that roughly 50% of the operating units can be identified as inefficient. Comparing each relatively inefficient unit to the units contributing to the composite unit may be helpful in understanding how the operation of each relatively inefficient unit can be improved.