Data Envelopment Analysis (DEA)

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Example 4.7

• Data Envelopment Analysis (DEA)

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Background Information

• Consider a group of three hospitals. To simplify
  matters, we assume that each hospital converts
  two inputs into three different outputs. (in a
  real DEA, there might be many more inputs and
  outputs.)
• The two inputs used by each hospital are
• Input 1 capital (measured by hundreds of
  hospital beds)
• Input 2 labor (measured by thousands of labor
  hours used in a month)

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Background Information -- continued

• The outputs produced by each hospital are
• Output 1 hundreds of patient-days during month
  for patients under age 14
• Output 2 hundreds of patient-days during month
  for patients between 14 and 65
• Output 3 hundreds of patient-days during month
  for patients over 65
• The inputs and outputs for these hospitals are
  given. Which of these hospitals is efficient in
  terms of using its inputs and producing outputs?

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Solution

• The idea is that if we focus on any particular
  hospital, we want to show it in the best
  possible light.
• That is, we want to value the inputs and outputs
  in such a way that this hospital looks as good as
  possible relative to the other hospitals.
• More specifically, to determine whether a
  hospital is efficient, we define a price per unit
  of each output and a cost per unit of each input.

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Solution -- continued

• Then the efficiency of a hospital is defined to
  be

• The DEA approach uses the following four ideas to
  determine whether a hospital is efficient.
• No hospital can be more than 100 efficient.
  Therefore, the efficiency of each hospital is
  constrained to be less than or equal to 1. To
  make this a linear constraint, we express it in
  this form Value of hospitals outputs ? Value
  of hospitals inputs

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Solution -- continued

1. When we are trying to determine whether a
   hospital is efficient, it simplifies matters to
   scale input prices so that the value of the
   hospitals inputs equals 1. Any other value would
   suffice, but by using 1, the efficiency of the
   hospital is the equal to the value of the
   hospitals outputs.
2. If we are interested in evaluating the efficiency
   of a hospital, we attempt to choose input and
   output prices that maximize this hospital's
   efficiency. If the hospitals efficiency equals
   1, then the hospital is efficient if the
   hospitals efficiency is less than 1, then the
   hospital is inefficient.
3. All input cost and output prices must be
   nonnegative.

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HOSPITALDEA.XLS

• This file contains the DEA spreadsheet model used
  to determine the efficiency of hospital 1.
• The spreadsheet is shown on the next slide.

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Developing the Model

• To develop this model, proceed as follows.
• Input given data. Enter the input and output
  information for each hospital in the ranges B6C8
  and F6H8.
• Selected hospitals. Enter 1, 2, or 3 in the cell
  B3, depending on which hospital you want to
  analyze.
• Unit input costs and output prices. Enter any
  trial values for the input costs and output
  prices in the UnitInputCosts and UnitOutputPrices
  ranges.

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Developing the Model -- continued

• Total input costs and output values. In the
  InputCosts range, calculate the cost of the
  inputs used by each hospital. To do this, enter
  the formula SUMPRODUCT(UnitInputCosts,B6C6) in
  cell B14 for hospital 1, and copy this to the
  rest of InputCosts range for the other hospitals.
  Similarly calculate the output values by entering
  the formula SUMPRODUCT(UnitOutputPrices,F6H6)
  in cell D14 and copying it to the rest of the
  OutputValues range.

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Developing the Model -- continued

• Total input cost and output value for selected
  hospitals. In row 19 we want to constrain the
  total input cost of the selected hospital to be
  1. To do this, enter the formula
  VLOOKUP(SelectedHospital,Ltable,2) in the
  SelectedInputCost cell, and enter a 1 in cell
  D19. Similarly, enter the formula,
  VLOOKUP(SelectedHospital,Ltable,4) in the
  SelectedOutputValue cell. Remember that by
  constraining the selected hospitals input cost
  to be 1, its output value in cell B22 is
  automatically its efficiency.
• Using Solver To see whether hospital 1 is
  efficient, use Solver as follows.
• Objective. Select the SelectedOutputValue cell as
  the target cell to maximize. Because the cost of
  hospital 1 inputs is constrained to be 1, this
  will cause Solver to maximize the efficiency of
  hospital 1.

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Developing the Model -- continued

• Changing cells. Choose the UnitInputCosts and
  UnitOutputPrices ranges as the changing cells.
• Selected hospitals input cost constraint. Add
  the constraint SelectedInputCost1. This sets the
  value of hospital 1 inputs equal to 1.
• Efficiency constraint. Add the constraint
  SelectedInputCost1. This sets the value of
  hospital 1 inputs equal to 1.
• Specify nonnegativity and optimize. Under
  SolverOptions, check the nonnegativity box, and
  use the LP algorithm to obtain the optimal
  solution as shown.

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Developing the Model -- continued

• The Solver dialog should appear as shown here.

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Solution

• The 1 in cell B22 of this solution means that
  hospital 1 is efficient. In words, we have been
  able to find a set of unit costs for the inputs
  and the unit price for the outputs such that the
  total value of hospital 1s output equals the
  total cost of its inputs.
• To determine whether hospital 2 is efficient, we
  simply replace the value in cell B3 by 2 and
  rerun Solver. The Solver settings do not need to
  be modified.
• The optimal solution appears on the next slide.
  From the value of .773 in cell B22, we see that
  hospital 2 is not efficient.

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Solution -- continued

• Similarly, we can determine that hospital 3 is
  efficient by replacing the value in cell B3 by 3
  and rerunning Solver.
• This solution appears on the next slide.
• In summary, we have found that hospitals 1 and 3
  are efficient, but hospital 2 is inefficient.

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Efficient or Inefficient?

• A hospital is efficient if we can price the
  inputs and outputs in such a way that this
  hospital gets all of the value out that it puts
  in.
• The pricing scheme will depend upon the hospital.
• Each hospital will try to price inputs and
  outputs so as to put its operations in the best
  possible light.
• If DEA finds that a hospital is inefficient, then
  there is no pricing scheme where that hospital
  can recover its entire input costs in output
  values.

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Efficient or Inefficient? -- continued

• Actually, it can be shown that if a hospital is
  inefficient, then a combination of the
  efficient hospitals can be found that uses no
  more inputs than the inefficient hospital, yet
  produces at least as much of each output as the
  inefficient hospital.
• To see how this combination can be found,
  consider the spreadsheet model shown on the next
  slide.

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Efficient or Inefficient? -- continued

• We begin by entering any positive weights in the
  Weights range.
• For any such weights, we consider the combination
  hospital as a fraction of hospital 1 and another
  fraction of hospital 3.
• When we combine these in row 28 with the
  SUMPRODUCT function, we find the quantities of
  inputs this combination hospital uses and the
  qualities of outputs it produces.

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Efficient or Inefficient? -- continued

• To find weights where the combination hospital is
  better than hospital 2, we find any feasible
  solution to the inequalities indicated in rows
  28-30 by using the Solver setup shown below.

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Efficient or Inefficient? -- continued

• Furthermore, we know there will be a feasible
  solution because we have already identified
  hospital 2 as being inefficient.
• In reality, once DEA analysis identifies an
  organizational unit as being inefficient, this
  unit should consider benchmarking itself relative
  to its competition to see where it might make
  more efficient use of its inputs.