Seminar 11:
Linear fractional programming, Data envelopment analy-
sis
Problem 1: Minimize the objective function of linear fractional program
f(x1, x2, x3) = 2x1+x2+3x3
x1+x2+x3
subject to
x1 − x2 + x3 ≤ 5
x2 ≥ 3
x1, x3 ≥ 0
a) Find suitable substitution for linearization of the problem
b) Solve linearized problem
c) Check the solution with Solver
Problem 2: Consider the DEA model for 8 hospital departments whose performance
is characterized by the following values:
Unit O1 O2 O3 O4 O5 O6 O7 O8
employees 7 6 6 8 10 5 4 5
outpatients 21 24 42 16 50 45 40 60
inpatients 63 36 48 40 40 15 24 10
a) Represent the problem graphically. Consider constant returns to scale and
find an efficient frontier.
b) Determine the efficiency of the 5th department using graphical method and
find its peer units (use the output orientation).
c) Formulate an input-oriented CCR model for the 5th department and find
the solution in the Solver (GRG nonlinear method must be selected)
d) Linearize the input-oriented CCR model for the 5th department and find
the solution in the Solver (Simplex LP method)
Problem 3: (BCC model of J. and M. Zouhar) The following table shows
the number of lecturers (in tens) and the number of successful graduates (in
hundreds) in the last year for certain A to D colleges:
college A B C D
lecturers [10] 1 3 7 5
graduates [100] 3 5 7 4
• In the chart with the input and output axes, capture the efficient frontier for
variable returns to scale. Compare the number of efficient units assuming
VRS with the CRS case.
• Find input, output, and reference units for the virtual unit D1 in the inputoriented
model, calculate the efficiency of D for this model.
• Find input, output, and reference units for the virtual unit D1 in the
output-oriented model, calculate the efficiency of D for this model.
Problem 4: (Problem of J. Kalˇcevov´a)
Consider DEA with 2 inputs, 3 outputs and 10 units:
X (inputs)
3 2,5 4 2,3 4 7 3 5 5 2
5 4,5 6 3,5 6,5 10 5 7 7 4
Y (outputs)
40 45 55 28 48 80 45 70 45 45
55 50 45 50 20 65 64 65 65 40
30 40 30 25 65 57 42 48 40 44
Find all efficient units, if you consider
a) input-oriented CCR model
b) output-oriented CCR model
c) input-oriented BCC model
d) output-oriented BCC model
Find the peer units for the 3rd unit if you consider
a) input-oriented CCR model
b) output-oriented BCC model
Use DEA Add-in for the calculations (download from https://webhosting.vse.cz/jablon/)