Example 4.7 Data Envelopment Analysis (DEA)
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)
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?
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.
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 hospital’s outputs Value of hospital’s inputs
Solution -- continued • When we are trying to determine whether a hospital is efficient, it simplifies matters to scale input prices so that the value of the hospital’s 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 hospital’s outputs. • 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 hospital’s efficiency equals 1, then the hospital is efficient; if the hospital’s efficiency is less than 1, then the hospital is inefficient. • All input cost and output prices must be nonnegative.
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.
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 B6:C8 and F6:H8. • 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.
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,B6:C6) 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,F6:H6) in cell D14 and copying it to the rest of the OutputValues range.
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 hospital’s 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.
Developing the Model -- continued • Changing cells. Choose the UnitInputCosts and UnitOutputPrices ranges as the changing cells. • Selected hospital’s input cost constraint. Add the constraint SelectedInputCost=1. This sets the value of hospital 1 inputs equal to 1. • Efficiency constraint. Add the constraint SelectedInputCost=1. 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.
Developing the Model -- continued • The Solver dialog should appear as shown here.
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 1’s 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.
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.
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.
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.
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.
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.
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.