Anderson
Sweeney
Williams
QUANTITATIVE
METHODS FOR
BUSINESS 8e
Slides Prepared by JOHN LOUCKS
© 2001 South-Western College Publishing/Thomson Learning
Slide 1
Chapter 18
Multicriteria Decision Problems



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

Goal Programming
Goal Programming: Formulation and Graphical
Solution
Scoring Models
The Analytic Hierarchy Process
Establishing Priorities Using AHP
Using AHP to Develop an Overall Priority Ranking
Slide 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.
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.
Slide 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.
Slide 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+ + w2d2s.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.
Slide 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+ + w4d4s.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.)
Slide 6
Example: Conceptual Products
Conceptual Products is a computer company that
produces the CP400 and the CP500 computers. The
computers use different mother boards produced in
abundant supply by the company, but use the same
cases and disk drives. The CP400 models use two
floppy disk drives and no zip disk drives whereas the
CP500 models use one floppy disk drive and one zip
disk drive.
Slide 7
Example: Conceptual Products
The disk drives and cases are bought from vendors.
There are 1000 floppy disk drives, 500 zip disk drives,
and 600 cases available to Conceptual Products 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.
Slide 8
Example: Conceptual Products
The company has four goals which are given below:
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)
Slide 9
Example: Conceptual Products
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 floppy disk drives: 2x1 + x2 < 1000
Availability of zip disk drives:
x2 < 500
Availability of cases:
x1 + x2 < 600

Slide 10
Example: Conceptual Products

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:
x1 + 1.5x2 + d4- - d4+ = 400
Non-negativity:
x1, x2, di-, di+ > 0 for all i
Slide 11
Example: Conceptual Products

Objective Functions
Priority 1: Minimize the amount the state contract is
not met: Min d1Priority 2: Minimize the number under 500
computers produced weekly: Min d2Priority 3: Minimize the amount under $250,000
earned weekly: Min d3Priority 4: Minimize the man-hours over 400 used
weekly: Min d4+
Slide 12
Example: Conceptual Products

Formulation Summary
Min P1(d1-) + P2(d2-) + P3(d3-) + P4(d4+)
s.t.
2x1
+x2
+x2
+x2
< 1000
< 500
x1
< 600
x1
+d1- -d1+
= 200
x1 +x2
+d2- -d2+
= 500
.2x1+ .5x2
+d3- -d3+
= 250
x1+1.5x2
+d4- -d4+ = 400
x1, x2, d1-, d1+, d2-, d2+, d3-, d3+, d4-, d4+ > 0
Slide 13
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).
Slide 14
Example: Conceptual Products

Functional Constraints and Goal 1 Graphed
x2
1000
2x1 + x2 < 1000
800
Goal 1: x1 > 200
x2 < 500
x1 + x2 < 600
600
400
Points Satisfying
Goal 1
200
x1
200
400
600
800
1000
1200
Slide 15
Example: Conceptual Products

Graphical Solution, Iteration 2
Now add Goal 1 as x1 > 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).
Slide 16
Example: Conceptual Products

Goal 1 (Constraint) and Goal 2 Graphed
x2
1000
2x1 + x2 < 1000
800
Goal 1: x1 > 200
x2 < 500
x1 + x2 < 600
600
400
Points Satisfying Both
Goals 1 and 2
200
Goal 2: x1 + x2 > 500
200
400
600
800
1000
1200
x1
Slide 17
Example: Conceptual Products

Graphical Solution, Iteration 3
Now add Goal 2 as x1 + x2 > 500 and Goal 3:
.2x1 + .5x2 = 250. 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.
Slide 18
Example: Conceptual Products

Goal 2 (Constraint) and Goal 3 Graphed
x2
1000
2x1 + x2 < 1000
800
Goal 1: x1 > 200
x2 < 500
x1 + x2 < 600
600
(200,400)
400
Points Satisfying Both
Goals 1 and 2
200
Goal 2: x1 + x2 > 500
Goal 3: .2x1 + .5x2 = 250
200
400
600
800
1000
1200
x1
Slide 19
A Scoring Model for 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
Slide 20
A Scoring Model for 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 multicriteria decision problem.
Slide 21
A Scoring Model for 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.
Slide 22
A Scoring Model for Job Selection

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.
Slide 23
A Scoring Model for Job Selection

Mathematical Model
Sj = S wi rij
i
where:
rij = rating for criterion i and decision alternative j
Sj = score for decision alternative j
Slide 24
A Scoring Model for Job Selection

Step 1: List the criteria (important factors).
• Career advancement
• Location
• Management
• Salary
• Prestige
• Job Security
• Enjoyable work
Slide 25
A Scoring Model for Job Selection

Five-Point Scale Chosen for Step 2
Importance
Very unimportant
Somewhat unimportant
Average importance
Somewhat important
Very important
Weight
1
2
3
4
5
Slide 26
A Scoring Model for Job Selection

Step 2: Assign a weight to each criterion.
Criterion
Career advancement
Location
Management
Salary
Prestige
Job security
Enjoyable work
Importance
Weight
Very important
5
Average importance
3
Somewhat important
4
Average importance
3
Somewhat unimportant 2
Somewhat important
4
Very important
5
Slide 27
A Scoring Model for Job Selection

Nine-Point Scale Chosen for Step 3
Level of Satisfaction
Extremely low
Very low
Low
Slightly low
Average
Slightly high
High
Very high
Extremely high
Rating
1
2
3
4
5
6
7
8
9
Slide 28
A Scoring Model for Job Selection
Step 3: Rate how well each decision alternative satisfies
each criterion.
Decision Alternative
Analyst Accountant
Auditor
Criterion
Chicago
Denver
Houston
Career advancement
8
6
4
Location
3
8
7
Management
5
6
9
Salary
6
7
5
Prestige
7
5
4
Job security
4
7
6
Enjoyable work
8
6
5

Slide 29
A Scoring Model for Job Selection

Step 4: Compute the score for each decision alternative.
Decision Alternative 1 - Analyst in Chicago
Criterion
Weight (wi ) Rating (ri1)
Career advancement
5
x
8
=
Location
3
3
Management
4
5
Salary
3
6
Prestige
2
7
Job security
4
4
Enjoyable work
5
8
Score
wiri1
40
9
20
18
14
16
40
157
Slide 30
A Scoring Model for Job Selection

Step 4: Compute the score for each decision alternative.
Sj = S wi rij
i
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
Slide 31
A Scoring Model for Job Selection
Step 4: Compute the score for each decision alternative.
Decision Alternative
Analyst
Accountant
Auditor
Criterion
Chicago
Denver
Houston
Career advancement
40
30
20
Location
9
24
21
Management
20
24
36
Salary
18
21
15
Prestige
14
10
8
Job security
16
28
24
Enjoyable work
40
30
25
Score
157
167
149

Slide 32
A Scoring Model for Job Selection

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 4 of 7 criteria compared to only 2 of 7 for the
accountant position in Denver.
• But when the weights of the criteria are considered,
the Denver position is superior to the Chicago job.
Slide 33
A Scoring Model for Job Selection

Partial Spreadsheet Showing Steps 1 - 3
A
1
2
3
4
5
6
7
8
9
10
B
C
D
E
RATINGS
Analyst Accountant Auditor
Weight Chicago Denver Houston
Criteria
4
6
8
5
Career Advancement
7
8
3
3
Location
9
6
5
4
Management
5
7
6
3
Salary
4
5
7
2
Prestige
6
7
4
4
Job Security
5
6
8
5
Enjoyable Work
Slide 34
A Scoring Model for Job Selection

Partial Spreadsheet Showing Formulas for Step 4
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
A
B
SCORING CALCULATIONS
Criteria
Career Advancement
Location
Management
Salary
Prestige
Job Security
Enjoyable Work
Score
C
D
E
F
Analyst Accountant Auditor
Chicago
Denver
Houston
=B4*C4
=B4*D4
=B4*E4
=B5*C5
=B5*D5
=B5*E5
=B6*C6
=B6*D6
=B6*E6
=B7*C7
=B7*D7
=B7*E7
=B8*C8
=B8*D8
=B8*E8
=B9*C9
=B9*D9
=B9*E9
=B10*C10 =B10*D10 =B10*E10
=sum(C16:C22)
=sum(D16:D22)
=sum(E16:E22)
Slide 35
A Scoring Model for Job Selection

11
12
13
14
15
16
17
18
19
20
21
22
23
Partial Spreadsheet Showing Results of Step 4
A
B
SCORING CALCULATIONS
Criteria
Career Advancement
Location
Management
Salary
Prestige
Job Security
Enjoyable Work
Score
C
D
E
Analyst Accountant Auditor
Chicago
Denver
Houston
40
30
20
9
24
21
20
24
36
18
21
15
14
10
8
16
28
24
40
30
25
157
167
149
Slide 36
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.
Slide 37
Analytic Hierarchy Process

Step 1: List the Overall Goal, Criteria, and Decision
Alternatives
------- For each criterion, perform steps 2 through 5 ------
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
Slide 38
Analytic Hierarchy Process

Step 2: Pairwise 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
9
7
5
3
1
Slide 39
Analytic Hierarchy Process

Step 2: Pairwise 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.
Slide 40
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.
Slide 41
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.
Slide 42
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).
Slide 43
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, max, of the results of
step 2.
Slide 44
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)
3
0.58
6
1.24
4
0.90
7
1.32
5
1.12
8
1.41
Step 6:
Compute the consistency ratio: CR = CR/RI.
Slide 45
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.
Slide 46
Example: Gill Glass

Hierarchy for the Manufacturer Selection Problem
Overall Goal
Select the Best Toothbrush Manufacturer
Criteria
Cost
Reliability
Delivery Time
Cornell
Brush Pik
Picobuy
Cornell
Brush Pik
Picobuy
Cornell
Brush Pik
Picobuy
Decision
Alternatives
Slide 47
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.
Slide 48
Example: Gill Glass

Pairwise Comparison Matrix for Cost
Cornell Brush Pik Picobuy
Cornell
Brush Pik
Picobuy
1
3
1/6
1/3
1
1/7
6
7
1
Slide 49
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
Brush Pik
Picobuy
6/25
18/25
1/25
7/31
21/31
3/31
6/14
7/14
1/14
Slide 50
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
Slide 51
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
Slide 52
Example: Gill Glass

Checking Consistency
• Average the above results to get max.
max = (3.097 + 3.179 + 3.029)/3 = 3.102
• Compute the consistence index, CI, for two terms
by:
CI = (max - 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.
Slide 53
Example: Gill Glass
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.
Slide 54
Example: Gill Glass

Pairwise Comparison Matrix for Reliability
Cornell Brush Pik Picobuy
Cornell
Brush Pik
Picobuy
1
1/7
1/2
7
1
1/5
2
5
1
Slide 55
Example: Gill Glass

Normalized Matrix for Reliability
Divide each entry in the pairwise 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
Brush Pik
Picobuy
14/23
2/23
7/23
35/41
5/41
1/41
2/8
5/8
1/8
Slide 56
Example: Gill Glass

Priority Vector For Reliability
The priority vector is determined by averaging the
row entries in the normalized matrix. Converting to
decimals we get:
Cornell: (14/23 + 35/41 + 2/8)/3 = .571
Brush Pik: ( 2/23 + 5/41 + 5/8)/3 = .278
Picobuy: ( 7/23 + 1/41 + 1/8)/3 = .151

Checking Consistency
Gill Glass’ responses to reliability could be checked
for consistency in the same manner as was cost.
Slide 57
Example: Gill Glass
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.
Slide 58
Example: Gill Glass

Pairwise Comparison Matrix for Delivery Time
Cornell Brush Pik Picobuy
Cornell
Brush Pik
Picobuy
1
1/8
1
8
1
8
1
1/8
1
Slide 59
Example: Gill Glass

Normalized Matrix for Delivery Time
Divide each entry in the pairwise comparison
matrix by its corresponding column sum.
Cornell Brush Pik Picobuy
Cornell
Brush Pik
Picobuy
8/17
1/17
8/17
8/17
1/17
8/17
8/17
1/17
8/17
Slide 60
Example: Gill Glass

Priority Vector For 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 = .471
Brush Pik: (1/17 + 1/17 + 1/17)/3 = .059
Picobuy: (8/17 + 8/17 + 8/17)/3 = .471

Checking Consistency
Gill Glass’ responses to delivery time could be
checked for consistency in the same manner as was
cost.
Slide 61
Example: Gill Glass
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.
Slide 62
Example: Gill Glass

Pairwise Comparison Matrix for Criteria
Cost
Cost
Reliability
Delivery
1
1/7
1/9
Reliability Delivery
7
1
1/7
9
7
1
Slide 63
Example: Gill Glass

Normalized Matrix for Criteria
Divide each entry in the pairwise comparison
matrix by its corresponding column sum.
Cost
Cost
Reliability
Delivery
63/79
9/79
7/79
Reliability Delivery
49/57
7/57
1/57
9/17
7/17
1/17
Slide 64
Example: Gill Glass

Priority Vector For Criteria
The priority vector is determined by averaging the
row entries in the normalized matrix. Converting to
decimals we get:
Cost:
Reliability:
Delivery:
(63/79 + 49/57 + 9/17)/3 =
( 9/79 + 7/57 + 7/17)/3 =
( 7/79 + 1/57 + 1/17)/3 =
.729
.216
.055
Slide 65
Example: Gill Glass

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
[ .729
.216
.055 ]
Cost Reliability Delivery
Cornell
.298
.571
.471
Brush Pik
.632
.278
.059
Picobuy
.069
.151
.471
Slide 66
Example: Gill Glass

Overall Priority Vector (continued)
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.
Slide 67
The End of Chapter 18
Slide 68