OR II.ppt

1
Slide
© 2005 Thomson/South-Western
Lesson 10
Multicriteria Decisions within LP Framework
 Goal Programming
 Goal Programming: Formulation
and Graphical Solution
 Scoring Model for Job Selection

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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.

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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.

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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.

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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 5.
Otherwise, a final solution has been reached.

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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 CP400 and 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.

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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.

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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.

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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)

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 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: 2x1 + x2 < 1000
Availability of external hard drives: x2 < 500
Availability of cases: x1 + x2 < 600
Goal Programming: Formulation

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 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

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 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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Slide
 Formulation Summary
Min P1(d1
-) + P2(d2
-) + P3(d3
-) + P4(d4
+)
s.t. 2x1 +x2 < 1000
+x2 < 500
x1 +x2 < 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

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 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).
Goal Programming:
Graphical Solution

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 Functional Constraints and Goal 1 Graphed
2x1 + x2 < 1000
Goal 1: x1 > 200
x1 + x2 < 600
x2 < 500
Points
Satisfying
Goal 1
x1
x2
Goal Programming:
Graphical Solution
1000
800
600
400
200
200 400 600 800 1000 1200

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Slide
 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).
Goal Programming:
Graphical Solution

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 Goal 1 (Constraint) and Goal 2 Graphed
2x1 + x2 < 1000
Goal 1: x1 > 200
x1 + x2 < 600
x2 < 500
Points Satisfying
Both Goals 1 and 2
x1
x2
Goal 2: x1 + x2 > 500
Goal Programming:
Graphical Solution
200 400 600 800 1000 1200
1000
800
600
400
200

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 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.
Goal Programming:
Graphical Solution

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 Goal 2 (Constraint) and Goal 3 Graphed
2x1 + x2 < 1000
Goal 1: x1 > 200
x1 + x2 < 600 x2 < 500
Points Satisfying
Both Goals 1 and 2
x1
x2
Goal 2: x1 + x2 > 500
Goal 3: .2x1 + .5x2 = 250
(200,400)
Goal Programming:
Graphical Solution
200 400 600 800 1000 1200
1000
800
600
400
200

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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

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 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.

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 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.

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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.

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 Mathematical Model
Sj = S wi rij
i
where:
rij = rating for criterion i and decision alternative j
Sj = score for decision alternative j

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Scoring Model: Step 1
 List of Criteria
•Career advancement
•Location
•Management
•Salary
•Prestige
•Job Security
•Enjoyable work

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 Five-Point Scale Chosen
Importance Weight
Very unimportant 1
Somewhat unimportant 2
Average importance 3
Somewhat important 4
Very important 5

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 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

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 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

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 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

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 Compute the score for each decision alternative.
Decision Alternative 1 - Analyst in Chicago
Criterion Weight (wi ) Rating (ri1) wiri1
Career advancement 5 x 8 = 40
Location 3 3 9
Management 4 5 20
Salary 3 6 18
Prestige 2 7 14
Job security 4 4 16
Score 157

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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
 
j i ij
i
s w r

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Slide
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
Score 157 167 149

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 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.

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 Partial Spreadsheet Showing Steps 1 - 3
A B C D E
1 RATINGS
2 Analyst Accountant Auditor
3 Criteria Weight Chicago Denver Houston
4 Career Advance. 5 8 6 4
5 Location 3 3 8 7
6 Management 4 5 6 9
7 Salary 3 6 7 5
8 Prestige 2 7 5 4
9 Job Security 4 4 7 6
10 Enjoyable Work 5 8 6 5

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 Partial Spreadsheet Showing Formulas of Step 4
A B C D E
12 SCORING CALCULATIONS
14 Criteria Chicago Denver Houston
15 Career Advance. =B4*C4 =B4*D4 =B4*E4
16 Location =B5*C5 =B5*D5 =B5*E5
17 Management =B6*C6 =B6*D6 =B6*E6
18 Salary =B7*C7 =B7*D7 =B7*E7
19 Prestige =B8*C8 =B8*D8 =B8*E8
20 Job Security =B9*C9 =B9*D9 =B9*E9
21 Enjoyable Work =B10*C10 =B10*D10 =B10*E10
22 Score =sum(C16:C22) =sum(D16:D22) =sum(E16:E22)

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 Partial Spreadsheet Showing Results of Step 4
A B C D E
12 SCORING CALCULATIONS
14 Criteria Chicago Denver Houston
15 Career Advance. 40 30 20
16 Location 9 24 21
17 Management 20 24 36
18 Salary 18 21 15
19 Prestige 14 10 8
20 Job Security 16 28 24
21 Enjoyable Work 40 30 25
22 Score 157 167 149

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End of Lesson 10

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