CHAPTER 6: Sensitivity Analysis and Duality
hospital as the weight) we obtain the following:
Averaged Output Vector
9
4
5.785
.261538 64 .661538 9 7.785
16
13
12.785
Averaged Input Vector
.261538
14 .661538 12 11.600
5
7
15.938
Suppose we create a composite hospital by combining .261538 of hospital 1 with
.661538 of hospital 3. The averaged output vector tells us that the composite hospital produces the same amount of outputs 1 and 2 as hospital 2, but the composite hospital produces 12.785 10 2.785 more of output 3 (patient days for more than 65 patients).
From the averaged input vector for the composite hospital, we find that the composite hospital uses less of each input than does hospital 2. We now see exactly where hospital 2 is
inefficient!
By the way, the objective function value of .7730 for the hospital 2 LP implies that the
more efficient composite hospital produces its superior outputs by using at most 77.30%
as much of each input. Note that
Input 1 used by composite hospital
.7730 * (Input 1 used by hospital 2) 6.2186
and
Input 2 used by composite hospital .7730 * (Input 2 used by hospital 2) 11.6
An explanation of why the dual prices are needed to find a composite hospital that is superior to an inefficient hospital is given in Problems 5–7.
TA B L E
47
School
1
2
3
1
2
3
1
2
3
4
13
14
11
15
4
5
6
8
.05
.05
.06
.08
9
10
11
9
7
8
7
9
6
7
8
9
PROBLEMS
Inputs
Group A
1 The Salem Board of Education wants to evaluate the
efficiency of the town’s four elementary schools. The three
outputs of the schools are defined to be
Output 1 average reading score
Output 2 average mathematics score
Output 3 average self-esteem score
The three inputs to the schools are defined to be
Input 1 average educational level of mothers (defined
by highest grade completed—12 high
school graduate; 16 college graduate, and
so on).
Input 2 number of parent visits to school (per child)
Input 3 teacher to student ratio
The relevant information for the four schools is given in
340
CHAPTER
Outputs
Table 47. Determine which (if any) schools are inefficient. For
any inefficient school, determine the nature of the inefficiency.
2 Pine Valley Bank has three branches. You have been
assigned to evaluate the efficiency of each. The following
inputs and outputs are to be used for the study.
Input 1 labor hours used (hundreds per month)
Input 2 space used (in hundreds of square feet)
Input 3 supplies used per month (in dollars)
6 Sensitivity Analysis and Duality
340
CHAPTER 6: Sensitivity Analysis and Duality
TA B L E
48
49
TA B L E
Inputs
Outputs
Inputs
Outputs
Bank
1
2
3
1
2
3
Precinct
1
2
1
2
1
2
3
15
14
16
20
23
19
50
51
51
200
220
210
15
18
17
35
45
20
1
2
3
200
300
400
160
190
120
16
18
10
18.5
19.5
11.5
Output 1 loan applications per month
Output 2 deposits processed per month (in thousands)
Output 3 checks processed per month (in thousands)
The relevant information is given in Table 48. Use this data
to determine if any bank branches are inefficient. If any
bank branches are inefficient, determine the nature of the
inefficiency.
efficient precincts, determine the nature of the inefficiency.
4 You have been assigned by Indiana University to evaluate
the relative efficiency of four degree-granting units: Business;
Education; Arts and Sciences; and Health, Physical
Education, and Recreation (HPER). You are given the
information in Table 50. Use DEA to find all inefficient
units. Comment on the nature of the inefficiencies you found.
3 You have been assigned to evaluate the efficiency of the
Port Charles Police Department. Three precincts are to be
evaluated. The inputs and outputs for each precinct are as
follows:
Input 1 number of police officers
Input 2 number of vehicles used
Output 1 number of patrol units responding to service
requests (thousands per year)
Output 2 number of convictions obtained each year
(in hundreds)
You are given the data in Table 49. Use this information to
determine which precincts, if any, are inefficient. For any in-
TA B L E
Group B
5 Explain why the amount of each output produced by the
composite hospital obtained by averaging hospitals 1 and 3
(with the absolute value of the dual prices as weights) is at
least as large as the amount of the corresponding output
produced by hospital 2. (Hint: Price out variables t1, t2, and
t3, and use the fact that the coefficient of these variables in
row 0 of the optimal tableau must equal 0.)
6 Explain why the dual price for the 8w1 15w2 1
constraint must equal the optimal z-value for the hospital 2 LP.
7
a Explain why the amount of each input used by the
composite hospital is at most (efficiency of hospital 2) *
50
Business
Education
Arts and Sciences
HPER
Faculty
Support
Staff
Supply Budget
(in Millions)
Credit Hours
(in Thousands)
Research Publications
150
160
800
130
170
120
140
115
25
23
20
21
15.4
15.4
56.4
22.1
1225
1170
1,300
1140
SUMMARY
Graphical Sensitivity Analysis
To determine whether the current basis remains optimal after changing an objective function coefficient, note that the change affects the slope of the isoprofit line. The current basis remains optimal as long as the current optimal solution is the last point in the feasible region to make contact with isoprofit lines as we move in the direction of increasing
z (for a max problem). If the current basis remains optimal, then the values of the decision variables remain unchanged, but the optimal z-value may change.
To determine whether the current basis remains optimal after changing the right-hand
side of a constraint, find the constraints (possibly including sign restrictions) that are binding for the current optimal solution. As we change the right-hand side of a constraint, the
6 . 1 Summary
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341