Q 4 – 1 a. Let T = number of TV advertisements R = number of radio advertisements N = number of newspaper advertisements Max 100,000 T + 18,000R + 40,000N s.t . 2,000T + 300R + 600N ≦ 18,200 Budget T ≦ 10 Max TV R ≦ 20 Max Radio N ≦ 10 Max News -0.5T + 0.5R - 0.5N ≦ 0 Max 50% Radio
Q 4 – 1 a. LetT = number of TV advertisements R = number of radio advertisements N = number of newspaper advertisements. Q 4 – 1 a. cont’d. Optimal Solution: T = 4, R = 14, N = 10 Allocation: TV 2,000(4) = $8000 Radio 300(14) = $4,200 - PowerPoint PPT Presentation
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Q 4 – 1 a.Let T = number of TV advertisements
R = number of radio advertisements
N = number of newspaper advertisements
Max 100,000T + 18,000R + 40,000N
s.t.
2,000T + 300R + 600N ≦ 18,200 Budget
T ≦ 10 Max TV
R ≦ 20 Max Radio
N ≦ 10 Max News
-0.5T + 0.5R - 0.5N ≦ 0 Max 50% Radio
0.9T - 0.1R - 0.1N ≧ 0 Min 10% TV
T, R, N, 0≧
Optimal Solution: T = 4, R = 14, N = 10
Allocation: TV 2,000(4) = $8000
Radio 300(14) = $4,200
News 600(10) = $6,000
Objective Function Value
(Expected number of audience):
100,000(4) + 18,000(14) + 40,000(10)
=1,052,000
Q 4 – 1 a. cont’d
Q 4 – 1 b. Computer Results
OPTIMAL SOLUTION
Objective Function Value = 1052000
Variable Value Reduced Costs
T 4.000 0.000
R 14.000 0.000
N 10.000 0.000
Constraint Slack/Surplus Dual Prices
1 0.000 51.304
2 6.000 0.000
3 6.000 0.000
4 0.000 11826.087
5 0.000 5217.391
6 1.200 0.000
Q 4 – 1 b. cont’d
The dual price for the budget constraint is 51.30. Thus, a
$100 increase in budget should provide an increase in
audience coverage of approximately 5,130. The RHS range
for the budget constraint will show this interpretation is
correct.
RIGHT HAND SIDE RANGE
Constraint Lower Limit Current Value Upper Limit
1 14750.000 18200.000 31999.996
2 4.000 10.000 No Upper Limit
3 14.000 20.000 No Upper Limit
4 0.000 10.000 12.339
5 -8.050 0.000 2.936
6 No Lower Limit 0.000 1.200
Q 4 – 10 a.
Let S = the proportion of funds invested in stocksB = the proportion of funds invested in bondsM = the proportion of funds invested in mutual fundsC = the proportion of funds invested in cash
Max 0.1S + 0.03B + 0.04M + 0.01C
s.t.
1S + 1B + 1M + 1C = 1
0.8S + 0.2B + 0.3M ≦ 0.4
1S ≦ 0.75
- 1B + 1M ≧ 0
1C ≧ 0.1
1C ≦ 0.3
S, B, M, C ≧ 0
Q 4 – 10 a. cont’dOPTIMAL SOLUTION
Objective Function Value = 0.054
Variable Value Reduced Costs
S 0.409 0.000
B 0.145 0.000
M 0.145 0.000
C 0.300 0.000
Constraint Slack/Surplus Dual Prices
1 0.000 0.005
2 0.000 0.118
3 0.341 0.000
4 0.000 -0.001
5 0.200 0.000
6 0.000 0.005
Q 4 – 10 a. cont’d
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
S 0.090 0.100 No Upper Limit
B 0.028 0.030 0.036
M No Lower Limit 0.040 0.042
C 0.005 0.010 No Upper Limit
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
1 0.800 1.000 1.900
2 0.175 0.400 0.560
3 0.409 0.750 No Upper Limit
4 -0.267 0.000 0.320
5 No Lower Limit 0.100 0.300
6 0.100 0.300 0.500
Q 4 – 10 a. cont’d
From computer results, the optimal allocation among the four
investment alternatives is
Stocks 40.0%
Bonds 14.5%
Mutual Funds 14.5%
Cash 30.0%
The annual return associated with the optimal portfolio is 5.4%
The total risk = 0.409(0.8) + 0.145(0.2)
+ 0.145(0.3) + 0.300(0.0) = 0.4
Q 4 – 10 b.
Changing the RHS value for constraint 2 to 0.18 and resolving
using computer, we obtain the following optimal solution:
Stocks 0.0%
Bonds 36.0%
Mutual Funds 36.0%
Cash 28.0%
The annual return associated with the optimal portfolio is 2.52%
The total risk = 0.0(0.8) + 0.36(0.2)
+ 0.36(0.3) + 0.28(0.0) = 0.18
Q 4 – 10 c.
Changing the RHS value for constraint 2 to 0.7 and resolving
using computer, we obtain the following optimal solution:
Stocks 75.0%
Bonds 0.0%
Mutual Funds 15.0%
Cash 10.0%
The annual return associated with the optimal portfolio is 8.2%
The total risk = 0.75(0.8) + 0.0(0.2)
+ 0.15(0.3) + 0.10(0.0) = 0.65
Q 4 – 10 d.
Note that a maximum risk of 0.7 was specified for this
aggressive investor, but that the risk index for the portfolio is
only 0.67. Thus, this investor is willing to take more risk than
the solution shown above provides. There are only two ways
the investor can become even more aggressive: increase the
proportion invested in stocks to more than 75% or reduce the
cash requirement of at least 10% so that additional cash could
be put into stocks. For the data given here, the investor should
ask the investment advisor to relax either or both of these
constraints.
Q 4 – 10 e.
Defining the decision variables as proportions means the
investment advisor can use the linear programming model for
any investor, regardless of the amount of the investment. All
the investor advisor needs to do is to establish the maximum
total risk for the investor and resolve the problem using the new
value for maximum total risk.
A – 1 (a) & (b)
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A – 2 (a)
Let S = Tablespoons of Strawberry
C = Tablespoons of Cream
V = Tablespoons of Vitamin
A = Tablespoons of Artificial sweetener
T = Tablespoons of Thickening agent
A – 2 (a)
Min 10S + 8C + 25V + 15A + 6T
s.t. 50S + 100C + 120A + 80T ≧ 380
50S + 100C + 120A + 80T ≦ 420
-9S + 55C - 24A + 14T ≦ 0
20S + 50V + 2T ≧ 50
1S - 2A ≧ 0
3S + 8C + 1V + 2A + 25T = 15
All variables 0≧
3125.58Total
0T,6042.1A,0V,2708.0C,2083.3S *****
A – 2 (b)
Max 380u1 - 420u2 + 0u3 + 50u4 + 0u5 + 15u6
s.t. 50u1 - 50u2 + 9u3 + 20u4 + 1u5 + 3u6 ≤ 10
100u1 - 100u2 - 55u3 + 8u6 ≤ 8
50u4 + 1u6 ≤ 25
120u1 - 120u2 + 24u3 - 2u5 + 2u6 ≤ 15
80u1 - 80u2 - 14u3 + 2u4 + 25u6 ≤ 6
u1 ~ u5 ≧ 0, u6 : URS
A – 2 (c)
8125.1u0u
1875.1u0u
0u2250.0u
*6
*3
*5
*2
*4
*1
Since u1*, u5
*, u6* > 0, the 1st, 5th, and 6th
constraints are binding.
The Langley County School District is trying to
determine the relative efficiency ofits three high schools. In particular,it wants to evaluate Roosevelt High.
The district is evaluating performances on SAT scores, thenumber of seniors finishing highschool, and the number of studentswho enter college as a function of thenumber of teachers teaching seniorclasses, the prorated budget for senior instruction,
and the number of students in the senior class.
Data Envelopment Analysis
• Input
Roosevelt Lincoln Washington
Senior Faculty 37 25 23
Budget ($100,000's) 6.4 5.0 4.7
Senior Enrollments 850 700 600
• Output
Roosevelt Lincoln Washington
Average SAT Score 800 830 900
High School Graduates 450 500 400
College Admissions 140 250 370
• Decision Variables
E = Fraction of Roosevelt's input resources required by the composite high school
w1 = Weight applied to Roosevelt's input/output resources by the composite high school
w2 = Weight applied to Lincoln’s input/output resources by the composite high school
w3 = Weight applied to Washington's input/output
resources by the composite high school
Data Envelopment Analysis
• Objective Function
Minimize the fraction of Roosevelt High School's input resources required by the composite high school:
MIN E
Data Envelopment Analysis
• Constraints
Sum of the Weights is 1:
(1) w1 + w2 + w3 = 1
Output Constraints:
Since w1 = 1 is possible, each output of the composite school must be at least as great as that of Roosevelt:
The input resources available to the composite school is a fractional multiple, E, of the resources available to Roosevelt. Since the composite high school cannot use more input than that available to it, the input constraints are:
(5) 37w1 + 25w2 + 23w3 < 37E (Faculty)
(6) 6.4w1 + 5.0w2 + 4.7w3 < 6.4E (Budget)
(7) 850w1 + 700w2 + 600w3 < 850E (Seniors)
Non-negativity of variables:
E, w1, w2, w3 > 0
OBJECTIVE FUNCTION VALUE = 0.765
VARIABLE VALUE E 0.765 W1 0.000 W2 0.500
W3 0.500
Data Envelopment Analysis
• Conclusion
The output shows that the composite school is made up of equal weights of Lincoln and Washington. Roosevelt is 76.5% efficient compared to this composite school when measured by college admissions (because of the 0 slack on this constraint (#4)). It is less than 76.5% efficient when using measures of SAT scores and high school graduates (there is positive slack in constraints 2 and 3.)
Data Envelopment Analysis
• (1) Relative Comparison
• (2) Multiple Inputs and Outputs
• (3) Efficiency Measurement (0%-100%)
• (4) Avoid the Specification Error between Inputs and Outputs
• (5) Production/Cost Analysis
Table 1.1 : 1 input – 1 output Case
Company A B C D E F G HEmployees 4 3 3 2 8 6 5 5Output 3 2 3 1 5 3 4 2Output/Employee 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4
Case : 1 input – 1 output
0
Out
put
Employees
D
B
A
G
H
F
E
C
Efficiency Frontier
Figure 1.1:Comparison of efficiencies in 1 input–1 output case
0
Out
put
Employees
C
Efficiency Frontier
Figure 1.2 : Regression Line and Efficiency Frontier
Regression Line
D
B
A
G
H
F
E
Table 1.2 : Efficiency
Company A B C D E F G HEfficiency 0.75 0.667 1 0.5 0.625 0.5 0.8 0.4
1 of employeeper Sales
another of employeeper Sales0
C
1 = C > G > A> B > E > D = F > H = 0.4
0
Out
put
Employee
D
C
Efficiency Frontier
Figure 1.3 : Improvement of Company D
D2
D1
Table 1.3 : 2 inputs – 1 output Case
Company A B C D E F G H IEmployees 4 4 2 6 7 7 3 8 5Offices 3 2 4 2 3 4 4 1 3Sales 1 1 1 1 1 1 1 1 1
Case : 2 inputs – 1 output
0
Off
ices
/Sal
es
Employees/Sales
DB
A
G
H
F
E
C
Efficiency Frontier
Figure 1.4 : 2 inputs – 1 output Case
I
Production Possibility Set
0
Off
ices
/Sal
es
Employees/Sales
B
C
Figure 1.5 : Improvement of Company A
AA1
A2
C and B :A of set referenceR
OAOAA of efficiency The 2
Table 1.4 : 1 input – 2 outputs Case
Company A B C D E F GOffices 1 1 1 1 1 1 1Customers 1 2 4 4 5 6 7Sales 6 7 6 5 2 4 2
Case : 1 input – 2 outputs
0
Sal
es/O
ffice
Customers/Office
B
F
C
Figure 1.6 : 1 input – 2 outputs Case
G
A
A1
D
E1
E
Efficiency Frontier
Production Possibility Set
1
1
OEOEE of efficiency The
OAOAA of efficiency The
Table 1.5 : Example of Multiple inputs–Multiple outputs Case
Company A B C D E F G H I J K LEmployees 10 26 40 35 30 33 37 50 31 12 20 45Offices 8 10 15 28 21 10 12 22 15 10 12 26Customers 23 37 80 76 23 38 78 68 48 16 64 72Sales 21 32 68 60 20 41 65 77 33 36 23 35
Case : Multiple inputs – Multiple outputs
1.1,
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Figure 2.1 : Efficiency Frontier and Production Possibility Set
d
(A)
(C)
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Efficiency Frontier of VRTS model
MIN E
s.t. Weighted outputs > Unit k’s output (for each measured output)
Weighted inputs < E [Unit k’s input] (for each measured input)
Sum of weights = 1
E, weights > 0
Data Envelopment Analysis
Final Project:International Competitiveness in the
Semiconductor Industry: An Application of DEA
Tim DekkerMEMGT
New Mexico Institute of Mining and Technology
DEA Efficiency Results From Data Rev 2No. DMU Score 2002 Score 2003 Score 2004