1. 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
Max News
-0.5T
0.5R -
0.5N
Max 50% Radio
0.9T
0.1R
0.1N ≧
Min 10% TV
T, R, N, ≧ 0

2. Q 4 – 1 a. cont’d Optimal Solution: T = 4, R = 14, N = 10
Allocation: TV ,000(4) = $8000
Radio (14) = $4,200
News (10) = $6,000
Objective Function Value
(Expected number of audience):
100,000(4) + 18,000(14) + 40,000(10)
=1,052,000

3. Q 4 – 1 b. Computer Results OPTIMAL SOLUTION
Objective Function Value =
Variable
Value
Reduced Costs T
4.000
0.000 R
14.000 N
10.000
Constraint
Slack/Surplus
Dual Prices 1
51.304 2
6.000 3 4 5 6
1.200

4. Q 4 – 1 b. cont’d
RIGHT HAND SIDE RANGE
Constraint
Lower Limit
Current Value
Upper Limit 1 2
4.000
10.000
No Upper Limit 3
14.000
20.000 4
0.000
12.339 5
-8.050
2.936 6
No Lower Limit
1.200
The dual price for the budget constraint is 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.

5. Q 4 – 10 a. Let S = the proportion of funds invested in stocks
B = the proportion of funds invested in bonds
M = the proportion of funds invested in mutual funds
C = 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
0.75 - ≧
0.1
0.3
S, B, M, C ≧ 0

6. Q 4 – 10 a. cont’d OPTIMAL SOLUTION Objective Function Value = 0.054
Variable
Value
Reduced Costs S
0.409
0.000 B
0.145 M C
0.300
Constraint
Slack/Surplus
Dual Prices 1
0.005 2
0.118 3
0.341 4
-0.001 5
0.200 6

7. 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
RIGHT HAND SIDE RANGES
Constraint 1
0.800
1.000
1.900 2
0.175
0.400
0.560 3
0.409
0.750 4
-0.267
0.000
0.320 5
0.300 6
0.500

8. Q 4 – 10 a. cont’d
From computer results, the optimal allocation among the four investment alternatives is
Stocks 40.0%
Bonds %
Mutual Funds 14.5%
Cash %
The annual return associated with the optimal portfolio is 5.4%
The total risk = 0.409(0.8) (0.2)
(0.3) (0.0) = 0.4

9. 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 %
Bonds %
Mutual Funds 36.0%
Cash %
The annual return associated with the optimal portfolio is 2.52%
The total risk = 0.0(0.8) (0.2)
+ 0.36(0.3) (0.0) = 0.18

10. 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 %
Mutual Funds 15.0%
Cash %
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) (0.0) = 0.65

11. 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 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.

12. 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.

13. A – 1 (a) & (b)

14. A – 1 (c)

15. 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

16. A – 2 (a) Min 10S + 8C 25V 15A 6T s.t. 50S 100C 120A 80T ≧ 380 ≦ 420-
24A
14T
20S
50V 2T 50 1S 2A 3S 1V
25T = 15
All variables ≧0

17. A – 2 (b) ≤ Max 380u1 - 420u2 + 0u3 50u4 0u5 15u6 s.t. 50u1 50u2 9u310
100u1
100u2
55u3
8u6 8
1u6 25
120u1
120u2
24u3
2u5
2u6 15
80u1
80u2
14u3
2u4
25u6 6
u1 ~ u5 ≧ 0, u6 : URS

18. A – 2 (c)
Since u1*, u5*, u6* > 0, the 1st, 5th, and 6th constraints are binding.

19. Data Envelopment Analysis
The Langley County School District is trying to
determine the relative efficiency of
its three high schools. In particular,
it wants to evaluate Roosevelt High.
The district is evaluating
performances on SAT scores, the
number of seniors finishing high
school, and the number of students
who enter college as a function of the
number of teachers teaching senior
classes, the prorated budget for senior instruction,
and the number of students in the senior class.

20. Roosevelt Lincoln Washington
Input
Roosevelt Lincoln Washington
Senior Faculty
Budget ($100,000's)
Senior Enrollments

21. Roosevelt Lincoln Washington
Output
Roosevelt Lincoln Washington
Average SAT Score
High School Graduates
College Admissions

22. Data Envelopment Analysis
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

23. Data Envelopment Analysis
Objective Function
Minimize the fraction of Roosevelt High School's input resources required by the composite high school:
MIN E

24. 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:
(2) 800w w w3 > (SAT Scores)
(3) 450w w w3 > (Graduates)
(4) 140w w w3 > (College Admissions)

25. Input Constraints:
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) 37w w w3 < 37E (Faculty)
(6) 6.4w w w3 < 6.4E (Budget)
(7) 850w w w3 < 850E (Seniors)
Non-negativity of variables:
E, w1, w2, w3 > 0

26. Data Envelopment Analysis
OBJECTIVE FUNCTION VALUE =
VARIABLE VALUE E W W W

27. 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.)

28. 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

29. Case : 1 input – 1 output
Table 1.1 : 1 input – 1 output Case

30. Efficiency Frontier E G
Output F C A H B D
Employees
Figure 1.1:Comparison of efficiencies in 1 input–1 output case

31. Efficiency Frontier E G
Output F C A
Regression Line H B D
Employees
Figure 1.2 : Regression Line and Efficiency Frontier

32. Table 1.2 : Efficiency
1 = C > G > A> B > E > D = F > H = 0.4

33. Efficiency Frontier
Output C D2 D1 D
Employee
Figure 1.3 : Improvement of Company D

34. Case : 2 inputs – 1 output
Table 1.3 : 2 inputs – 1 output Case

35. Production Possibility SetG C F
Offices/Sales A I E D B
Efficiency Frontier H
Employees/Sales
Figure 1.4 : 2 inputs – 1 output Case

36. C A
Offices/Sales A1 A2 B
Employees/Sales
Figure 1.5 : Improvement of Company A

37. Case : 1 input – 2 outputs
Table 1.4 : 1 input – 2 outputs Case

38. A1 B C A
Efficiency Frontier D F
Sales/Office
Production Possibility Set E1 G E
Customers/Office
Figure 1.6 : 1 input – 2 outputs Case

39. Case : Multiple inputs – Multiple outputs
Table 1.5 : Example of Multiple inputs–Multiple outputs Case

41. Ratio model

42. R : A Reference Set

43. Primal Problem

44. Dual Problem

45. Slack

46. Reference Set:

47. Example Problem
Table 1.6 : 2 inputs – 1 output Case

48. Primal Problem

49. Dual Problem

50. Efficiency Frontier E D A A1 F C
Figure 1.7 : Efficiency of DMU A

51. Original Ratio Model

52. Model Under Variable RTS

53. Dual Problem

54. Efficiency Frontier of Ratio model
(B) d c
Output
Efficiency Frontier of VRTS model b
(C) a
(A)
Input
Figure 2.1 : Efficiency Frontier and Production Possibility Set

55. Data Envelopment Analysis
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

56. Tim Dekker MEMGT New Mexico Institute of Mining and Technology
Final Project: International Competitiveness in the Semiconductor Industry: An Application of DEA
Tim Dekker
MEMGT
New Mexico Institute of Mining and Technology

57. DEA Efficiency Results From Data Rev 2

58. Weights used in the Analysis (2003)

59. Final Ranking for Firms in 2004

60. Home Work
Problem 5-2
Problem 5-3
Problem 5-4
Due Day: Sep 23

61. “Complementary slackness Conditions” are obtained from (4)
( c - y*A ) x* = 0
y*( b - Ax* ) = 0
xj* > y*aj = cj , y*aj > cj xj* = 0
yi* > aix* = bi , ai x* < bi yi* = 0
(5)
(6)

62. Fundamental Insight Z
RHS Z 1
Row0
Row1~N

63. 18-1 BV x1 x2 x3 s1 s2 s3
RHS 2 1
220 4 -2 7 80 -1 3 30 20
Max 5x1+6x2+4x3+0s1+0s2+0s3
CB = [0, 4, 5]

64. 18-1 a. Replace “5” by “c1”. Since x1 is basic,
then cBB-1A2 – c2 ≥ 0 becomes

65. 18-1 a. cont.
cBB-1As1 – cs1 ≥ 0 becomes

66. 18-1 a. cont.
cBB-1As2 – cs2 ≥ 0 becomes

67. 18-1 b. Replace “6” by “c2”. Since x2 is non-basic,
then cBB-1A2 – c2 ≥ 0 becomes

68. 18-1 c. Replace “4” by “cs1”. Since s1 is non-basic,
then cBB-1As1 – cs1 ≥ 0 becomes

69. 18-3 a, b, c BV x1 x2 x3 s1 s2 s3
RHS 2 1
220 4 -2 7 80 -1 3 30 20
The dual prices for the first, second, third constraints
= 1
= 2
= 0

70. 18-3 d Let , then x1 x2 x3 s1 s2 s3 RHS 2 1 220 4 -2 7 80 -1 3 30 20BV x1 x2 x3 s1 s2 s3
RHS 2 1
220 4 -2 7 80 -1 3 30 20

71. 18-3 d cont.

72. 18-3 e

73. The optimal simplex tableauBV x1 x2 x3 s1 s2 s3
RHS
2.5
7.5 15 75 1 4
0.25
-0.25
0.5
0.75
-0.75
-0.5
1.5
The optimal solution: x1 = 4, x2 = 0.5
The value of the objective function: 75

74. The optimal simplex tableau
18-15 c
The optimal simplex tableau BV x1 x2 x3 s1 s2 s3
RHS
2.5
7.5 15 75 1 4
0.25
-0.25
0.5
0.75
-0.75
-0.5
1.5
If a dual variable is positive, then its corresponding constraint is binding. So, the first and second constrains binding (due to CSC).

75. 18-15 d The redundant (non-binding) is Constraint 3.
x1+x2+2x3+s3=6 is replaced by x1=4, x2=0.5, x3=0 as follows:
(0)+s3=6.
Hence, s3=1.5.

76. The optimal simplex tableauBV x1 x2 x3 s1 s2 s3
RHS
2.5
7.5 15 75 1 4
0.25
-0.25
0.5
0.75
-0.75
-0.5
1.5
Dual prices:
= 7.5
= 15
= 0

77. 18-15 e Increasing the RHS of constraint 2 would have the
greatest positive effect on the objective function.

78. 18-15 f Replace “15” by “c1”. Since x1 is basic,
then cBB-1A3 – c3 ≥ 0 becomes

79. 18-15 f. cont.
cBB-1As1 – cs1 ≥ 0 becomes

80. 18-15 f. cont.
cBB-1As2 – cs2 ≥ 0 becomes

81. 18-15 f cont. Replace “30” by “c2”. Since x2 is basic,
then cBB-1A3 – c3 ≥ 0 becomes

82. 18-15 f cont.
cBB-1As1 – cs1 ≥ 0 becomes

83. 18-15 f cont.
cBB-1As2 – cs2 ≥ 0 becomes

84. 18-15 f cont. Replace “20” by “c3”. Since x2 is non-basic,
then cBB-1A3 – c3 ≥ 0 becomes

85. 18-15 f cont.
The optimal solution will not change as long as the objective function coefficients stay in these intervals.

86. 18-15 g

87. 18-15 g For b1 4 + ∆b1(1) ≥ → ∆b1 -4 0.5 ∆b1(-0.25) ≤ 2 1.5 ∆b1(-0.75)→
∆b1 -4
0.5
∆b1(-0.25) ≤ 2
1.5
∆b1(-0.75)
Therefore -4 ≤ ∆b1 ≤ 2
Range: (4-4 ≤ b1 ≤ 4+2) So, 0 ≤ b1 ≤ 6

88. 18-15 g

89. 18-15 g cont. For b2 4 + ∆b2(0) ≥ → no restriction 0.5 ∆b2(0.5) ∆b2 -1→
no restriction
0.5
∆b2(0.5)
∆b2 -1
1.5
∆b2(-0.5) ≤ 3
Therefore -1 ≤ ∆b2 ≤ 3
Range: (3-1≤ b2 ≤ 3+3) So, 2 ≤ b2 ≤ 6

90. 18-15 g

91. 18-15 g cont. For b3 4 + ∆b3(0) ≥ → no restriction 0.5 1.5 ∆b3(1) ∆b3→
no restriction
0.5
1.5
∆b3(1)
∆b3
-1.5
Therefore -1.5 ≤ ∆b3
Range: (6-1.5≤ b3) So, 4.5 ≤ b3

92. 18-17 a
Min
550y1 +
700y2
200y3
s.t.
1.5y1
4y2
2y3 ≥ 4
2y1
1y2
3y3 6
4y1
2y2
1y3 3
3y1 23 1
y1, y2 , y3 ≥ 0

93. The optimal simplex tableauBV y1 y2 y3 s1 s2 s3 s4
RHS
425 25
125
525 1
-0.4
0.2
1.8
-3.25
-0.65
-0.05
0.05
0.5
-0.5
3.5
0.1
-0.3
0.3
Optimal solution: y1 = 0.3, y2 = 0, y3 = 1.8
x1 = 0, x2 = 25, x3 = 125, x4 = 0.

94. 18-17 c
Dual variables y1 = 0.3, y2 = 0, y3 = 1.8 indicate the increase level of the objective function by one unit increase in each RHS.
If we increase Machine A by one hour, the profit will be increased by 0.3 ($/hour).
Similarly, a profit increase of Machine C will be 1.8 ($/hour).