Chapter 15 Multicriteria Decision Problems

1
Chapter 15Multicriteria Decision Problems

• Goal Programming
• Goal Programming Formulation and Graphical
  Solution
• The Analytic Hierarchy Process
• Establishing Priorities Using AHP
• Using AHP to Develop an Overall Priority Ranking

2
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.
• An approach to goal programming is to satisfy
  goals in a priority sequence. Second-priority
  goals are pursued without reducing the
  first-priority goals, etc.

3
Goal Programming

• 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.

4
Goal Programming Approach

• 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.
• Step 3 Set up the initial linear program.
• Min w1d1 w2d2-
• s.t. Functional Constraints,
• and Goal Constraints
• Step 4 Solve this linear program.
• If there is a lower priority level, go to step
  5. Otherwise, a final optimal solution has been
  reached.

5
Goal Programming Approach

• 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.)

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Example Conceptual Products

• Conceptual Products is a computer company that
  produces the CP286 and the CP386 computers. The
  computers use different mother boards produced
  in abundant supply by the company, but use the
  same cases and disk drives. The CP286 models use
  two floppy disk drives and no hard disks whereas
  the CP386 models use one floppy disk drive and
  one hard disk drive.
• The disk drives and cases are bought from
  vendors. There are 1000 floppy disk drives, 500
  hard disk drives, and 600 cases available to
  Conceptual Products on a weekly basis. It takes
  one hour to manufacture a CP286 and its profit is
  200 and it takes one and one-half hours to
  manufacture a CP386 and its profit is 500.

7
Example Conceptual Products

• The company has four goals which are given below
• Priority 1 Meet a state contract of 200
  CP286 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)

8
Example Conceptual Products

• Variables
• x1 number of CP286 computers produced
  weekly
• x2 number of CP386 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 floppy disk drives 2x1
  x2 lt 1000
• Availability of hard disk drives
  x2 lt 500
• Availability of cases x1 x2 lt
  600

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Example Conceptual Products

• Goals
• (1) 200 CP286 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
• x1 1.5x2 d4- - d4 400
• Non-negativity
• x1, x2, di-, di gt 0 for all i

10
Example Conceptual Products

• Objective Functions
• Priority 1 Minimize the amount the state
  contract is not met Min d1-
• Priority 2 Minimize the number under 500
  computers produced weekly Min d2-
• Priority 3 Minimize the amount under
  250,000 earned weekly Min d3-
• Priority 4 Minimize the man-hours over 400
  used weekly Min d4

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Example Conceptual Products

• Formulation Summary
• Min P1(d1-) P2(d2-) P3(d3-) P4(d4)
• s.t. 2x1 x2
  lt 1000
• x2
  lt 500
• x1 x2
  lt 600
• x1 d1- -d1
  200
• x1 x2 d2-
  -d2 500
• .2x1 .5x2
  d3- -d3 250
• x11.5x2
  d4- -d4 400
• x1, x2, d1-, d1, d2-, d2,
  d3-, d3, d4-, d4 gt 0

12
Example Conceptual Products

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

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Example Conceptual Products

• Functional Constraints and Goal 1 Graphed

x2
1000 800 600 400 200
200 400 600 800 1000
1200
2x1 x2 lt 1000
Goal 1 x1 gt 200
x2 lt 500
x1 x2 lt 600
Points Satisfying Goal 1
x1
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Example Conceptual Products

• Graphical Solution, Iteration 2
• Now add Goal 1 as x1 gt 200 and graph Goal 2
• x1 x2 500. 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).

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Example Conceptual Products

• Goal 1 (Constraint) and Goal 2 Graphed

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Example Conceptual Products

• Graphical Solution, Iteration 3
• Now add Goal 2 as x1 x2 gt 500 and Goal 3
• .2x1 .5x2 250. Note on the next slide that
  no points satisfy the previous functional
  constraints and goals as 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.

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Example Conceptual Products

• Goal 2 (Constraint) and Goal 3 Graphed

x2
1000 800 600 400 200
200 400 600 800 1000
1200
2x1 x2 lt 1000
Goal 1 x1 gt 200
x2 lt 500
x1 x2 lt 600
(200,400)
Points Satisfying Both Goals 1 and 2
Goal 2 x1 x2 lt 500
Goal 3 .2x1 .5x2 gt 500
x1
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3. The L. Young Sons Manufacturing Company
produces two products, which have the following
profit and resource requirement characteristics.
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• Last months production schedule used 350 hours
  of labor in department A and 1000 hours of labor
  in department B.
• Youngs management has been experiencing work
  force morale and labor union problems during the
  past 6 months because of monthly departmental
  work load fluctuations. New hiring, layoffs, and
  interdepartmental transfers have been common
  because the firm has not attempted to stabilize
  work-load requirements.
• Management would like to develop a production
  schedule for the coming month that will achieve
  the following goals.
• Goal 1 Use 350 hours of labor in department A.
• Goal 2 Use 1000 hours of labor in department B.
• Goal 3 Earn a profit of at least 1300.

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• a. Formulate a goal programming model for this
  problem, assuming that goals 1 and 2 are P1 level
  goals and goal 3 is a P2 level goal assume that
  goals 1 and 2 are equally important.
• b. Solve the model formulated in part (a) using
  the graphical goal programming procedure.
• c. Suppose that the firm ignores the work-load
  fluctuations and considers the 350 hours in
  department A and the 1000 hours in department B
  as the maximum available. Formulate and solve a
  linear programming problem to maximize profit
  subject to these constraints.

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• d. Compare the solutions obtained in parts (b)
  and (c). Discuss which approach you favor and
  why.
• e. Recosider part (a), assuming that the
  priority level 1 goal is goal 3 and the priority
  level 2 goals are goals 1 and 2 as before,
  assume that goals 1 and 2 are equally important.
  Solve this revised problem using the graphical
  goal programming procedure and compare your
  solution to the one obtained for the original
  problem.

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• 3. a Let
• x1 number of units of product 1 produced
• x2 number of units of product 2 produced
• Min P1(d1) P1(d1-) P1(d2) P1(d2-)
  P2(d3-)
• s.t.
• 1x1 1x2 - d1 d1- 350 Goal 1
• 2x1 5x2 - d2 d2- 1000 Goal 2
• 4x1 2x2 - d3 d3- 1300 Goal 3
• x1 , x2 , d1 , d1- , d2 , d2- , d3- , d3 ³ 0

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• 3. b
• In the graphical solution next, point A provides
  the optimal solution. Note that with x1 250
  and x2 100, this solution achieves goals 1 and
  2, but underachieves goal 3 (profit) by 100
  since 4(250) 2(100) 1200.

25
700
3. b
600
500
Goal 3
400
Goal 1
300
Goal 2
200
B (281.25, 87.5)
100
A (250,100)
0 100 200 300 400
500
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• 3. c
• Max 4x1 2x2
• s.t.
• 1x1 1x2 350 Dept. A
• 2x2 5x2 1000 Dept. B
• x1 , x2 ³ 0

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3. c (continued)
The graphical solution shown below indicates that
there are four extreme points. The profit
corresponding to each extreme point is as
follows
Thus, the optimal product mix is x1 350 and x2
0 with a profit of 1400.
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3. c (continued)
400 300 200 100
Department A
(4) (0,250)
Dept. B
(3) (250,100)
(1) (0,0)
0 100 200 300 400
500
(2) (350,0)
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• 3.
• d The solution to part (a) achieves both labor
  goals, whereas the solution to part (b) results
  in using only 2(350) 5(0) 700 hours of labor
  in department B. Although (c) results in a 100
  increase in profit, the problems associated with
  underachieving the original department labor goal
  by 300 hours may be more significant in therms of
  long-term considerations
• e Refer to the graphical solution in part (b).
  The solution to the revised problem is point B,
  with x1 281.25 and x2 87.5. Although this
  solution achieves the original department B labor
  goal and the profit goal, this solution uses
  1(281.25) 1(87.5) 368.75 hours of labor in
  department B, which is 18.75 hours more than the
  original goal.

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• 4. Durham Designs manufacturs home furnishings
  for department stores. Planning is underway for
  the production of items in the Wildflower
  fabric pattern during the next production
  periods.
• Bedspread Curtains Dustruffles
• Fabrics reqd (yds) 7 4 9 Time reqd (hrs)
  1.5 2 .5
• Packaging Matl 3 2 1
• Profit 12 10 8
• Inventory of the Wildflower fabric is 3000 yards.
  
• 500 hours of production time have been scheduled.
• 400 units of packaging materials are available.
• Formulate and solve a linear programming problem
  to maximize profit subject to these constraints.
• .

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LP Formulation

• Let x1 the of bedspreads to make
• x2 the of curtains to make
• x3 the of dustruffles to make
• Max Z 12x1 10x2 9x3
• s.t. 7x1 4x2 9x3 lt3200
• 1.5x1 2x2 .5x3 lt500
• 3x1 2x2 1x3 lt 400
• x1, x2, x3 gt 0

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• Using the same data, Durham would like to
• Achieve a profit of 3200,
• Avoid purchasing more fabric or packaging
  material,
• Use all of the hours scheduled
• Formulate a goal programming model.

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• 4. a Let
• x1 number of bedspreads to make
• x2 number of curtains to make
• x3 number of dustruffles to make
• Min d1- d2 d3- d4
• s.t.
• 12x1 10x2 8x3 - d1 d1- 3200
  Profit
• 7x1 4x2 9x3 - d2 d2- 3000
  Fabric
• 1.5x1 2x2 .5x3 - d3 d3- 500
  Hours
• 3x1 2x2 1x3 - d4 d4- 400
  Packg
• x1 , x2 , x3, all deviation variables ³ 0

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2. DJS Investment Services must develop an
investment portfolio for a new client. As an
initial investment strategy, the new client would
like to restrict the portfolio to a mix of two
stocks
35

• The client has 50,000 to invest and has
  established the following two investment goals.
• Priority Level 1 Goal
• Goal 1 Obtain an annual return of at least 9.
• Priority Level 2 Goal
• Goal 2 Limit the investment in Key Oil, the
  riskier investment to no more than 60 of the
  total investment.
• a. Formulate a goal programming model for the
  DJS Investment problem.
• b. Use the graphical goal programming procedure
  to obtain a solution.

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• 2. a Let
• x1 number of shares of AGA Products purchased
• x2 number of shares of Key Oil purchased
• To obtain an annual return of exactly 9
• 0.06(50) x1 0.10(100) x2 0.09(50,000)
• 3x1 10x2 4500
• To have exactly 60 of the total investment in
  Key Oil
• 100x2 0.60(50,000)
• x2 300
• Therefore, we can write the goal programming
  model as follows

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• Min P1 (d1-) P2 (d2)
• s.t.
• 50x1 100x2 50,000 Funds
  Available
• 3x1 10x2 - d1 d1- 4,500 P1 Goal
• x2 - d2 d2- 300 P2 Goal
• x1, x2, d1, d1-, d2, d2- ³ 0
• 2. b In the graphical solution shown next, x1
  250 and x2 375.

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2. b (continued)
1000
Points that satisfy the funds available constraint
and satisfy the priority 1 goal
(250, 375) P2 Goal
P1 Goal
500
Funds Available
0 500
1000 1500
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• 4. Industrial Chemicals produces two adhesives
  used in the manufacturing process for airplanes.
  The two adhesives, which have different bonding
  strengths, require different amounts of
  production time the IC-100 adhesive requires 20
  minutes of production time per gallon of finished
  product, and the IC-200 adhesive uses 30 minutes
  of production time per gallon. Both products use
  1 pound of a highly perishable resin for each
  gallon of finished product. There are 300 pounds
  of the resin inventory, and more can be obtained
  if necessary. However, because of the shelf life
  of the material, any amount not used in the next
  2 weeks will be discarded.
• The firm has existing orders for 100 gallons of
  IC-100 and 120 gallons of IC-200. Under normal
  conditions the production process operates 8
  hours per day, 5 days per week. Management wants
  to schedule production for the next two weeks to
  achieve the following goals.

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• Priority Level 1 Goals
• Goal 1 Avoid underutilization of the production
  process
• Goal 2 Avoid overtime in excess of 20 hours for
  the 2 weeks.
• Priority level 2 Goals
• Goal 3 Satisfy existing orders for the IC-100
  adhesive that is, produce at least 100 gallons
  of IC-100.
• Goal 4 Satisfy existing orders for the IC-200
  adhesive that is produce at least 120 gallons of
  IC-200
• Priority Level 3 Goal
• Goal 5 Use all the available resin
• a. Formulate a goal model for the Industrial
  Chemicals problem. Assume that both priority
  level 1 goals and that both priority level 2
  goals are equally important.
• b. Use the graphical goal programming procedure
  to develop a solution for the model formulated in
  part (a).

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• 4. a Let
• x1 number of gallons of IC-100 produced
• x2 number of gallons of IC-200 produced
• Min P1(d1-) P1(d2) P2(d3-) P2(d4-)
  P5(d5-)
• s.t.
• 20x1 30x2 - d1 d1- 4800 Goal
  1
• 20x1 30x2 - d2 d2- 6000 Goal
  2
• x1 - d3 d3- 100
  Goal 3
• x2 - d4 d4- 120
  Goal 4
• x1 x2 - d5 d5- 300
  Goal 5
• x1 , x2 , all deviation variables ³ 0

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4. b
In the graphical solution shown below, the point
x1 120 and x2 120 is optimal.
Goal 3
Solution points that satisfy goals 1-4
300 200 100
Goal 5
Optimal solution (120, 120)
Goal 2
Goal 1
Goal 4
0 100 200
300
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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.

44
Analytic Hierarchy Process

• Step 1 List the Overall Goal, Criteria, and
  Decision Alternatives
• Step 2 Develop a Pairwise 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 5
45
Analytic Hierarchy Process

• Step 2 Pairwise Comparison Matrix (continued)
• Compared to the second
• alternative, the first alternative is
  Numerical rating
• extremely preferred
  9
• very strongly preferred
  7
• strongly preferred
  5
• moderately preferred
  3
• equally preferred
  1
• 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.

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Analytic Hierarchy Process

• Step 3 Develop a Normalized Matrix
• Divide each number in a column of the pairwise
  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.

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Analytic Hierarchy Process

• Step 5 Calculate a Consistency Ratio
• The consistency of the subjective input in the
  pairwise 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.
• 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.

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Analytic Hierarchy Process

• Step 7 Develop a Criteria Pairwise Development
  Matrix
• This is done in the same manner as that used to
  construct alternative pairwise 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).

49
Determining the Consistency Ratio

• Step 1
• For each row of the pairwise 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, lmax, of the results of
  step 2.

50
Determining the Consistency Ratio

• Step 4
• Compute the consistency index, CI, of the n
  alternatives by CI (lmax - 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)
• 3 0.58 6
  1.24
• 4 0.90 7
  1.32
• 5 1.12 8
  1.41
• Step 6
• Determine the consistency ratio, CR, as
  follows
• CR CR/RI.

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Example Gill Glass

• Designer Gill Glass must decide which of three
  manufacturers will develop his "signature"
  toothbrushes. Three factors seem 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. 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.

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Example Gill Glass

• Forming the Pairwise Comparison Matrix For 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.

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Example Gill Glass

• Pairwise Comparison Matrix for Cost
• Cornell
  Brush Pik Picobuy
• Cornell 1 1/3 6
• Brush Pik 3 1 7
• Picobuy 1/6 1/7 1

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Example Gill Glass

• Normalized Matrix for Cost
• Divide each entry in the pairwise comparison
  matrix by its corresponding column sum. For
  example, for Cornell the column sum 1 3 1/6
  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

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Example Gill Glass

• Priority Vector For 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 (18/25 21/31 7/14)/3
  .632
• Picobuy ( 1/25 3/31 1/14)/3
  .069

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Example Gill Glass

• Checking Consistency
• Multiply each column of the pairwise comparison
  matrix by its priority
• 1 1/3
  6 .923
• .298 3 .632 1 .069
  7 2.009
• 1/6 1/7
  1 .209
• Divide these number by their priorities to get
• .923/.298 3.097
• 2.009/.632 3.179
• .209/.069 3.029

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Example Gill Glass

• Checking Consistency
• Average the above results to get lmax.
• lmax (3.097 3.179 3.029)/3
  3.102
• Compute the consistence index, CI, for two terms
  by
• CI (lmax - n)/(n - 1) (3.102
  - 3)/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.

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The End of Chapter 15