1. Slides by
John
Loucks
St. Edward’s
University

2. Chapter 14 Multicriteria Decisions
Goal Programming
Goal Programming: Formulation
and Graphical Solution
Goal Programming: Solving More
Complex Problems
Scoring Models
Analytic Hierarchy Process (AHP)
Establishing Priorities Using AHP
Using AHP to Develop an Overall
Priority Ranking

3. Goal Programming
Goal programming may be used to solve linear programs with multiple objectives, with each objective viewed as a "goal".
In goal programming, di+ and di- , deviation variables, are the amounts a targeted goal i is overachieved or underachieved, respectively.
The goals themselves are added to the constraint set with di+ and di- acting as the surplus and slack variables.

4. Goal Programming
One approach to goal programming is to satisfy goals in a priority sequence. Second-priority goals are pursued without reducing the first-priority goals, etc.
For each priority level, the objective function is to minimize the (weighted) sum of the goal deviations.
Previous "optimal" achievements of goals are added to the constraint set so that they are not degraded while trying to achieve lesser priority goals.

5. Goal Programming Formulation
Step 1: Decide the priority level of each goal. Step 2: Decide the weight on each goal. If a priority level has more than one goal, for each goal i decide the weight, wi , to be placed on the deviation(s), di+ and/or di-, from the goal.

6. Goal Programming Formulation
Step 3: Set up the initial linear program.
Min w1d1+ + w2d2-
s.t. Functional Constraints,
and Goal Constraints
Step 4: Solve the current linear program.
If there is a lower priority level, go to step Otherwise, a final solution has been reached.

7. Goal Programming Formulation
Step 5: Set up the new linear program. Consider the next-lower priority level goals and formulate a new objective function based on these goals. Add a constraint requiring the achievement of the next-higher priority level goals to be maintained. The new linear program might be: Min w3d3+ + w4d4- s.t. Functional Constraints, Goal Constraints, and w1d1+ + w2d2- = k Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.)

8. GP Example: Conceptual Products
Conceptual Products is a computer company that produces the CP400 and CP500 computers. Many of the components used in the two computer models are produced in abundant supply by the company. However, the memory modules, external hard drives, and cases are bought from suppliers. The CP400 model uses two memory modules and no external hard drive, whereas the CP500 uses one memory module and one external hard drive. Both models use one case.

9. GP Example: Conceptual Products
Suppliers can provide Conceptual Products with 1000 memory modules, 500 external hard drives, and 600 cases on a weekly basis. It takes one hour to manufacture a CP400 and its profit is $200 and it takes one and one-half hours to manufacture a CP500 and its profit is $500.

10. GP Example: Conceptual Products
The company has four goals: Priority 1: Meet a state contract of 200 CP400 machines weekly. (Goal 1) Priority 2: Make at least 500 total computers weekly. (Goal 2) Priority 3: Make at least $250,000 weekly. (Goal 3) Priority 4: Use no more than 400 man-hours per week. (Goal 4)

11. GP Example: Formulation
Variables
x1 = number of CP400 computers produced weekly
x2 = number of CP500 computers produced weekly
di- = amount the right hand side of goal i is deficient
di+ = amount the right hand side of goal i is exceeded
Functional Constraints
Availability of memory modules: x1 + x2 < 1000
Availability of external hard drives: x2 < 500
Availability of cases: x1 + x2 < 600

12. GP Example: Formulation
Goals
(1) 200 CP400 computers weekly:
x1 + d1- - d1+ = 200
(2) 500 total computers weekly:
x1 + x2 + d2- - d2+ = 500
(3) $250(in thousands) profit:
.2x1 + .5x2 + d3- - d3+ = 250
(4) 400 total man-hours weekly:
x x2 + d4- - d4+ = 400
Non-negativity:
x1, x2, di-, di+ > 0 for all i

13. GP Example: Formulation
Objective Functions
Priority 1: Minimize the amount the state contract is not met: Min d1-
Priority 2: Minimize the number under computers produced weekly: Min d2-
Priority 3: Minimize the amount under $250, earned weekly: Min d3-
Priority 4: Minimize the man-hours over 400 used weekly: Min d4+

14. GP Example: Formulation
Formulation Summary
Min P1(d1-) + P2(d2-) + P3(d3-) + P4(d4+)
s.t x x < 1000
+x < 500
x x < 600
x d1- -d = 200
x x d2- -d = 500
.2x1+ .5x d3- -d = 250
x1+1.5x d4- -d4+ = 400
x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+ > 0

15. GP Example: Graphical Solution
Iteration 1
To solve graphically, first graph the functional constraints. Then graph the first goal: x1 = Note on the next slide that there is a set of points that exceed x1 = 200 (where d1- = 0).

16. GP Example: Graphical Solution
Functional Constraints and Goal 1 Graphed x2
1000
800
600
400
200
2x1 + x2 < 1000
Goal 1: x1 > 200
x2 < 500
x1 + x2 < 600
Points
Satisfying
Goal 1 x1

17. GP Example: Graphical Solution
Iteration 2
Now add Goal 1 as x1 > 200 and graph Goal 2:
x1 + x2 = Note on the next slide that there is still a set of points satisfying the first goal that also satisfies this second goal (where d2- = 0).

18. GP Example: Graphical Solution
Goal 1 (Constraint) and Goal 2 Graphed x2
1000
800
600
400
200
2x1 + x2 < 1000
Goal 1: x1 > 200
x2 < 500
x1 + x2 < 600
Points Satisfying
Both Goals 1 and 2
Goal 2: x1 + x2 > 500 x1

19. GP Example: Graphical Solution
Iteration 3
Now add Goal 2 as x1 + x2 > 500 and Goal 3:
.2x1 + .5x2 = Note on the next slide that no points satisfy the previous functional constraints and goals and satisfy this constraint.
Thus, to Min d3-, this minimum value is achieved when we Max .2x1 + .5x2. Note that this occurs at x1 = 200 and x2 = 400, so that .2x1 + .5x2 = 240 or d3- = 10.

20. GP Example: Graphical Solution
Goal 2 (Constraint) and Goal 3 Graphed x2
2x1 + x2 < 1000
1000
800
600
400
200
Goal 1: x1 > 200
x1 + x2 < 600
x2 < 500
(200,400)
Points Satisfying
Both Goals 1 and 2
Goal 2: x1 + x2 > 500
Goal 3: .2x1 + .5x2 = 250 x1

21. Goal Programming: More Complex Problems
Now we will formulate and solve a GP problem that involves multiple goals within the same priority level.
No tradeoffs occur in the achievement of goals with different priorities. However, tradeoffs can occur in the achievement of goals with the same priority and different weights.
A GP problem with p preemptive goal priorities can be solved as a series of p linear programming problems.
Solving a GP problem as a series of LPs, the objective functions (always to be minimized) will include only relevant deviation variables and not decision variables.

22. GP Example 2: Conceptual Products
Conceptual Products is a computer company that
produces the CP400, CP500, and CP600 computers.
Many of the components used in the three computer
models are produced in abundant supply by the
company. However, the memory modules, external
hard drives, and cases are bought from suppliers.
The CP400 model uses two memory modules and
no external hard drive, the CP500 uses one memory
module and one external hard drive, and the CP600
uses two memory modules and one external hard
drive . All three models use the same case.

23. GP Example 2: Conceptual Products
Suppliers can provide Conceptual Products with
1000 memory modules, 500 external hard drives, and
600 cases on a weekly basis. It takes one hour to
manufacture a CP400 and its profit is $200; it takes
one and one-half hours to manufacture a CP500 and
its profit is $500; and it takes two hours to manufacture
a CP600 and its profit is $900.

24. GP Example 2: Conceptual Products
The company has four goals:
Priority 1: Meet a state contract of 200 CP machines weekly. (Goal 1)
Priority 2: Make at least 500 total computers weekly. (Goal 2)
Priority 3: Make at least $250,000 profit
weekly. (Goal 3)
Priority 3: Use no more than 400 man-hours
per week. (Goal 4)
Each $1000 underachieved from its profit goal is five times as important as an extra man-hour.

25. GP Example 2: Formulation
Variables
x1 = number of CP400 computers produced weekly
x2 = number of CP500 computers produced weekly
x3 = number of CP600 computers produced weekly
di- = amount the right hand side of goal i is deficient
di+ = amount the right hand side of goal i is exceeded
Functional Constraints
Availability of memory modules: x1 + x2 + x3 < 1000
Availability of external hard drives: x2 + x3 < 500
Availability of cases: x1 + x2 + x3 < 600

26. GP Example 2: Formulation
Goals
(1) 200 CP400 computers weekly:
x1 + d1- - d1+ = 200
(2) 500 total computers weekly:
x1 + x2 + x3 + d2- - d2+ = 500
(3) $250(in thousands) profit:
.2x1 + .5x2 + .9x3 + d3- - d3+ = 250
(4) 400 total man-hours weekly:
x x2 + 2x3 + d4- - d4+ = 400
Non-negativity:
x1, x2, x3, di-, di+ > 0 for all i

27. GP Example 2: Conceptual Products
Objective Functions
Priority 1: Minimize the amount the state contract is not met: Min d1-
Priority 2: Minimize the number under computers produced weekly: Min d2-
Priority 3: Minimize the amount under $250, earned weekly: Min 5d3-
Priority 3: Minimize the man-hours over 400 used weekly: Min d4+

28. GP Example 2: Conceptual Products
Priority 1 Formulation
Min d1-
s.t. 2x x2 +x < 1000
+x2 +x < 500
x x2 +x < 600
x d1- -d = 200
x x2 +x d2- -d = 500
.2x1+ .5x2 +.9x d3- -d = 250
x1+1.5x2 +2x d4- -d4+ = 400
x1, x2, x3, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+ > 0

29. GP Example 2: Conceptual Products
Computer Solution (First LP)
Objective Function Value =
Variable Value Reduced Cost x x x d d d d d d d d

30. GP Example 2: Conceptual Products
Priority 2 Formulation
Min d2-
( include the previous 7 constraints )
( add the constraint: d1- = 0 )

31. GP Example 2: Conceptual Products
Computer Solution (Second LP)
Objective Function Value =
Variable Value Reduced Cost x x x d d d d d d d d

32. GP Example 2: Conceptual Products
Priority 3 Formulation
Min 5d3- + d4+
( include the previous 8 constraints )
( add the constraint: d2- = 0 )

33. GP Example 2: Conceptual Products
Computer Solution (Third LP)
Objective Function Value =
Variable Value Reduced Cost x x x d d d d d d d d

34. GP Example 2: Conceptual Products
Final Solution
Thus the optimal recommendation is to produce CP400 computers, 0 CP500 computers, and weekly and CP600 computers weekly.
All goals will be met except goal extra man-hours or a total of man-hours will be used.

35. Scoring Models
A scoring model is a relatively quick and easy way to identify the best (or recommended) decision alternative for a multi-criteria decision problem.
A scoring model enables a decision maker to identify the criteria and indicate the weight or importance of each criterion.
By comparing the scores for each criterion, a decision maker can learn why a particular decision alternative has the highest score.

36. Steps Required to Develop a Scoring Model
Step 1: List the decision-making criteria.
Step 2: Assign a weight to each criterion.
Step 3: Rate how well each decision alternative satisfies each criterion.
Step 4: Compute the score for each decision alternative.
Step 5: Order the decision alternatives from highest score to lowest score. The alternative with the highest score is the recommended alternative.

37. Scoring Model Sj = S wi rij Mathematical Model for Step 4 i where:
wi = weight for criterion i
rij = rating for criterion i and decision alternative j
Sj = score for decision alternative j

38. Scoring Model Example: Job Selection
A graduating college student with a double major
in Finance and Accounting has received the following
three job offers:
financial analyst for an investment firm in Chicago
accountant for a manufacturing firm in Denver
auditor for a CPA firm in Houston

39. Scoring Model Example: Job Selection
The student made the following comments:
“The financial analyst position provides the best opportunity for my long-run career advancement.”
“I would prefer living in Denver rather than in Chicago or Houston.”
“I like the management style and philosophy at the Houston CPA firm the best.”
Clearly, this is a multi-criteria decision.

40. Scoring Model Example: Job Selection
Considering only the long-run career advancement criterion:
The financial analyst position in Chicago is the best decision alternative.
Considering only the location criterion:
The accountant position in Denver is the best decision alternative.
Considering only the style criterion:
The auditor position in Houston is the best alternative.

41. Scoring Model Example: Step 1
List of Criteria
Career advancement
Location
Management
Salary
Prestige
Job Security
Enjoyable work

42. Scoring Model Example: Step 2
Five-Point Scale Chosen
Importance Weight
Very unimportant 1
Somewhat unimportant 2
Average importance 3
Somewhat important 4
Very important 5

43. Scoring Model Example: Step 2
Assigning a Weight to Each Criterion
Criterion Importance Weight
Career advancement Very important 5
Location Average importance 3
Management Somewhat important 4
Salary Average importance 3
Prestige Somewhat unimportant 2
Job security Somewhat important 4
Enjoyable work Very important 5

44. Scoring Model Example: Step 3
Nine-Point Scale Chosen
Level of Satisfaction Rating
Extremely low 1
Very low 2
Low 3
Slightly low 4
Average 5
Slightly high 6
High 7
Very high 8
Extremely high 9

45. Scoring Model Example: Step 3
Rate how well each decision alternative satisfies each criterion.
Decision Alternative
Analyst Accountant Auditor
Criterion Chicago Denver Houston
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work

46. Scoring Model Example: Step 4
Compute the score for each decision alternative.
Decision Alternative Analyst in Chicago
Criterion Weight (wi ) Rating (ri1) wiri1
Career advancement x = 40
Location
Management
Salary
Prestige
Job security
Enjoyable work
Score

47. Scoring Model Example: Step 4
Compute the score for each decision alternative.
S1 = 5(8)+3(3)+4(5)+3(6)+2(7)+4(4)+5(8) = 157
S2 = 5(6)+3(8)+4(6)+3(7)+2(5)+4(7)+5(6) = 167
S3 = 5(4)+3(7)+4(9)+3(5)+2(4)+4(6)+5(5) = 149

48. Scoring Model Example: Step 4
Compute the score for each decision alternative.
Decision Alternative
Analyst Accountant Auditor
Criterion Chicago Denver Houston
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Score

49. Scoring Model Example: Step 5
Order the decision alternatives from highest
score to lowest score. The alternative with the highest
score is the recommended alternative.
The accountant position in Denver has the highest score and is the recommended decision alternative.
Note that the analyst position in Chicago ranks first in 3 of 7 criteria as does the accountant position in Denver.
But when the weights of the criteria are considered, the Denver position is superior to the Chicago job.

50. Analytic Hierarchy Process
The Analytic Hierarchy Process (AHP), is a procedure designed to quantify managerial judgments of the relative importance of each of several conflicting criteria used in the decision making process.

51. Analytic Hierarchy Process
Step 1: List the Overall Goal, Criteria, and Decision Alternatives
Step 2: Develop a Pair-wise Comparison Matrix
Rate the relative importance between each pair of decision alternatives. The matrix lists the alternatives horizontally and vertically and has the numerical ratings comparing the horizontal (first) alternative with the vertical (second) alternative.
Ratings are given as follows:
. . . continued
For each criterion, perform steps 2 through

52. Analytic Hierarchy Process
Step 2: Pair-wise Comparison Matrix (continued)
Compared to the second
alternative, the first alternative is: Numerical rating
extremely preferred
very strongly preferred
strongly preferred
moderately preferred
equally preferred

53. Analytic Hierarchy Process
Step 2: Pair-wise Comparison Matrix (continued)
Intermediate numeric ratings of 8, 6, 4, 2 can be assigned. A reciprocal rating (i.e. 1/9, 1/8, etc.) is assigned when the second alternative is preferred to the first. The value of 1 is always assigned when comparing an alternative with itself.

54. Analytic Hierarchy Process
Step 3: Develop a Normalized Matrix
Divide each number in a column of the pair-wise comparison matrix by its column sum.
Step 4: Develop the Priority Vector
Average each row of the normalized matrix. These row averages form the priority vector of alternative preferences with respect to the particular criterion. The values in this vector sum to 1.

55. Analytic Hierarchy Process
Step 5: Calculate a Consistency Ratio
The consistency of the subjective input in the pair-wise comparison matrix can be measured by calculating a consistency ratio. A consistency ratio of less than .1 is good. For ratios which are greater than .1, the subjective input should be re-evaluated.
For each criterion, perform steps 2 through

56. Analytic Hierarchy Process
Step 6: Develop a Priority Matrix
After steps 2 through 5 has been performed for all criteria, the results of step 4 are summarized in a priority matrix by listing the decision alternatives horizontally and the criteria vertically. The column entries are the priority vectors for each criterion.

57. Analytic Hierarchy Process
Step 7: Develop a Criteria Pair-wise Development Matrix
This is done in the same manner as that used to construct alternative pair-wise comparison matrices by using subjective ratings (step 2). Similarly, normalize the matrix (step 3) and develop a criteria priority vector (step 4).
Step 8: Develop an Overall Priority Vector
Multiply the criteria priority vector (from step 7) by the priority matrix (from step 6).

58. Determining the Consistency Ratio
Step 1:
For each row of the pair-wise comparison matrix, determine a weighted sum by summing the multiples of the entries by the priority of its corresponding (column) alternative.
Step 2:
For each row, divide its weighted sum by the priority of its corresponding (row) alternative.
Step 3:
Determine the average, max, of the results of step 2.

59. Determining the Consistency Ratio
Step 4:
Compute the consistency index, CI, of the n alternatives by: CI = (max - n)/(n - 1).
Step 5:
Determine the random index, RI, as follows:
Number of Random Number of Random
Alternative (n) Index (RI) Alternative (n) Index (RI)
Step 6:
Compute the consistency ratio: CR = CR/RI.

60. AHP Example: Gill Glass
Designer Gill Glass must decide which of three manufacturers will develop his "signature“ toothbrushes. Three factors are important to Gill: (1) his costs, (2) reliability of the product, and (3) delivery time of the orders. The three manufacturers are Cornell Industries, Brush Pik, and Picobuy. Cornell Industries will sell toothbrushes to Gill Glass for $100 per gross, Brush Pik for $80 per gross, and Picobuy for $144 per gross.

61. AHP Example: Gill Glass
Hierarchy for the Manufacturer Selection Problem
Overall Goal
Select the Best Toothbrush Manufacturer
Criteria
Cost
Reliability
Deliv. Time
Cornell
Brush Pik
Picobuy
Cornell
Brush Pik
Picobuy
Cornell
Brush Pik
Picobuy
Decision
Alternatives

62. Example: Pairwise Comparison Matrix - Cost
Gill has decided that in terms of price, Brush
Pik is moderately preferred to Cornell and very
strongly preferred to Picobuy. In turn Cornell is
strongly to very strongly preferred to Picobuy.

63. Example: Pairwise Comparison Matrix - Cost
Since Brush Pik is moderately preferred to Cornell, Cornell's entry in the Brush Pik row is 3 and Brush Pik's entry in the Cornell row is 1/3.
Since Brush Pik is very strongly preferred to Picobuy, Picobuy's entry in the Brush Pik row is 7 and Brush Pik's entry in the Picobuy row is 1/7.
Since Cornell is strongly to very strongly preferred to Picobuy, Picobuy's entry in the Cornell row is 6 and Cornell's entry in the Picobuy row is 1/6.

64. Example: Pairwise Comparison Matrix - Cost
Cornell Brush Pik Picobuy Cornell 1 1/3 6 Brush Pik Picobuy 1/6 1/7 1

65. Example: Normalized Matrix - Cost
Divide each entry in the pair-wise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = /6 = 25/6. This gives: Cornell Brush Pik Picobuy Cornell 6/25 7/31 6/14 Brush Pik 18/25 21/31 7/14 Picobuy 1/25 3/31 1/14

66. Example: Priority Vector - Cost
The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cornell: ( 6/25 + 7/31 + 6/14)/3 = .298 Brush Pik: (18/ /31 + 7/14)/3 = .632 Picobuy: ( 1/25 + 3/31 + 1/14)/3 = .069

67. Example: Checking Consistency
Multiply each column of the pair-wise comparison matrix by its priority: / =
1/ /
Divide these number by their priorities to get:
.923/.298 =
2.009/.632 =
.209/.069 =

68. Example: Checking Consistency
Average the above results to get max.
max = ( )/3 = 3.102
Compute the consistence index, CI, for two terms.
CI = (max - n)/(n - 1) = ( )/2 = .051
Compute the consistency ratio, CR, by CI/RI, where RI = .58 for 3 factors:
CR = CI/RI = .051/.58 = .088
Since the consistency ratio, CR, is less than .10, this is well within the acceptable range for consistency.

69. Example: Pairwise Comp. Matrix - Reliability
Gill Glass has determined that for reliability, Cornell is very strongly preferable to Brush Pik and equally preferable to Picobuy. Also, Picobuy is strongly preferable to Brush Pik.

70. Example: Pairwise Comp. Matrix - Reliability
Cornell Brush Pik Picobuy Cornell Brush Pik 1/7 1 5 Picobuy 1/2 1/5 1

71. Example: Normalized Matrix - Reliability
Divide each entry in the pair-wise comparison matrix by its corresponding column sum. For example, for Cornell the column sum = 1 + 1/7 + 1/2 = 23/14. This gives: Cornell Brush Pik Picobuy Cornell 14/23 35/41 2/8 Brush Pik 2/23 5/41 5/8 Picobuy 7/23 1/41 1/8

72. Example: Priority Vector - Reliability
The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get:
Cornell: (14/ /41 + 2/8)/3 =
Brush Pik: ( 2/ /41 + 5/8)/3 =
Picobuy: ( 7/ /41 + 1/8)/3 =
Checking Consistency
Gill Glass’ responses to reliability could be checked for consistency in the same manner as was cost.

73. Example: Pairwise Comp. Matrix - Delivery Time
Gill Glass has determined that for delivery time, Cornell is equally preferable to Picobuy. Both Cornell and Picobuy are very strongly to extremely preferable to Brush Pik.

74. Example: Pairwise Comp. Matrix - Delivery Time
Cornell Brush Pik Picobuy Cornell Brush Pik 1/8 1 1/8 Picobuy 1 8 1

75. Example: Normalized Matrix - Delivery Time
Divide each entry in the pair-wise comparison matrix by its corresponding column sum. Cornell Brush Pik Picobuy Cornell 8/17 8/17 8/17 Brush Pik 1/17 1/17 1/17 Picobuy 8/17 8/17 8/17

76. Example: Priority Vector - Delivery Time
The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get:
Cornell: (8/17 + 8/17 + 8/17)/3 =
Brush Pik: (1/17 + 1/17 + 1/17)/3 =
Picobuy: (8/17 + 8/17 + 8/17)/3 =
Checking Consistency
Gill Glass’ responses to delivery time could be checked for consistency in the same manner as was cost.

77. Example: Pairwise Comparison Matrix - Criteria
The accounting department has determined that in terms of criteria, cost is extremely preferable to delivery time and very strongly preferable to reliability, and that reliability is very strongly preferable to delivery time.

78. Example: Pairwise Comparison Matrix - Criteria
Cost Reliability Delivery Cost Reliability 1/7 1 7 Delivery 1/9 1/7 1

79. Example: Normalized Matrix - Criteria
Divide each entry in the pair-wise comparison matrix by its corresponding column sum.
Cost Reliability Delivery
Cost / / /17
Reliability 9/ / /17
Delivery 7/ / /17

80. Example: Priority Vector - Criteria
The priority vector is determined by averaging the row entries in the normalized matrix. Converting to decimals we get: Cost: (63/ /57 + 9/17)/3 = .729 Reliability: ( 9/79 + 7/57 + 7/17)/3 = .216 Delivery: ( 7/79 + 1/57 + 1/17)/3 = .055

81. Example: Overall Priority Vector
The overall priorities are determined by multiplying the priority vector of the criteria by the priorities for each decision alternative for each objective. Priority Vector for Criteria [ ] Cost Reliability Delivery Cornell Brush Pik Picobuy

82. Example: Overall Priority Vector
Thus, the overall priority vector is: Cornell: (.729)(.298) + (.216)(.571) + (.055)(.471) = .366 Brush Pik: (.729)(.632) + (.216)(.278) + (.055)(.059) = .524 Picobuy: (.729)(.069) + (.216)(.151) + (.055)(.471) = .109 Brush Pik appears to be the overall recommendation.

83. End of Chapter 14