Transcript Multicriteria Decision Models

Chapter 10
Multicriteria Decision-Marking
Models
1
Application Context
 multiple
objectives that cannot be put under
a single measure; e.g.,
 distribution:
cost and time
 as
a single objective function problem if time can be
converted into cost
 supply
cost
chain: customer service and inventory
多目標而目標沒有
共同的衡量方式。
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Chapter Summary
 10.0
Scoring Model 評點法
 10.1
Weighting Method 權重法
 10.2
Goal Programming 目標規劃
 10.3
AHP (Analytical Hierarchy Process)層
級分析法
我們只學每個方法
基本的概念。
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Motivation Problem
 dinner,
two factors
to consider: distance
and cost
 three
restaurants
(2, 3)
 B: (7, 1)
 C: (4, 2)
(distance from home, cost)
(7, $)
B
A
(2, $$$)
 A:
 which
C
home
(4, $$)
one to choose?
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Scoring Model 評點法
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Scoring Model 評點法
(7, $)
 a subjective method
(2, $$$)
B
A
C
assign weights to each criterion
home
(4, $$)
 assign a rating for each decision alternative on each
criterion
(distance from home, cost)


Restaurant Selection Example: Version 1

min w1 (distance) + w2 (cost)
w1
1
1
w2
1
3
A (2, $$$) B (7, $)
5
8
11
10
C (4, $$) choice
6
A
10
B, C
Version 1只需要決定各目標的權重。
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Scoring Model 評點法
 Restaurant
Selection Example: Version 2
 weights
of criteria, and ratings on criteria for
alternatives
 Example:
Tom dislikes walking and likes good
food (from expensive restaurants)
每個選擇在每一項目標中有點數
（ratings, scores），而目標有
各自的權重（weight）。
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Scoring Model 評點法
 Restaurant
 weights
Selection Example, Version 2
of walking and price by Tom: 1 to 1
 ratings
(scores) of each restaurant for walking
and price:
dislikes walking and likes expensive, good food
criterion
restaurant
distance
price
w1 = 1
w2 = 1
A (2, $$$)
10
8
B (7, $)
3
2
C (4, $$)
6
5
objective of Tom:
max w1 (rating of
distance)
+ w2 (rating of price)
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Scoring Model 評點法
 a subjective method on assigning
weights
 ratings

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Example 10-5 Product Selection

to expand the product line by adding one of the following:
microwave ovens, refrigerators, and stoves

decision criteria


manufacturing capability/cost

market demand

profit margin

long-term profitability/growth

transportation costs

useful life
assigning weights to the criteria and ratings to the three alternatives
for each criterion

maximizing the total score
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Example 10-5 Product Selection
manuf. cap./cost
market demand
profit margin
(long-term)
prof./growth
Transp. costs
useful life
weight
4
5
3
5
2
1
microwave
refers
stoves
4
8
6
3
9
1
3
4
9
6
2
5
8
2
5
7
4
6

Scoremicro = 4(4)+5(8)+3(6)+5(3)+2(9)+1(1) = 108

Scorerefer = 4(3)+5(4)+3(9)+5(6)+2(2)+1(5) = 98

Scorestove = 4(8)+5(2)+3(5)+5(7)+2(4)+1(6) = 106
any comments
on the relative
values?
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Weighting Method 權重法
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Weighting Method 權重法
a
form of scoring method
 transforming
a multi- to a singlecriterion objective function by finding the
weights of the criteria
以目標的權重（weight）
將多目標的問題轉化為
單目標的問題。
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Weighting Method 權重法
Z(x) = [z1(x), z2(x), …, zP(x)]
s.t. x  S
 max
 turning
into a single-criterion objective
function by weighting (with weights)
Z(x) = w1z1(x)+w2z2(x)+… +wpzP(x)
s.t. x  S
 max
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Weighting Method 權重法


criteria (i.e., objectives)
 max z1(x)
= 2x1+3x2x3
 min z2(x)
= 6x1x2
 max z3(x)
= 2x1+x3
constraints
 x1+x2+x3
 15
 x1+2x2+x3
 x3
 20
2
 x1, x2, x3
0
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Weighting Method 權重法

somehow got: w1 = 1, w2 = 2, w3 = 4
= (2x1+3x2x3)  2(6x1x2)
+ 4(2x1+x3) = 18x1+5x2+3x3,
 max z1(x)2z2(x)+4z3(x)
 s.t.
negative sign
x1+x2+x3  15 ; x1+2x2+x3  20; x3  2;
x1, x2, x3  0.
max z1(x) = 2x1+3x2x3, min z2(x) =
6x1x2,, max z3(x) = 2x1+x3,
s.t. x1+x2+x3  15 ; x1+2x2+x3  20;
x3  2; x1, x2, x3  0.
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Goal Programming 目標規劃
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1/4: Introducing the Ideas of Goal
Programming
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Goal Programming

GP: priority + goal

priority of the goals (i.e., of the criteria)

(saving) money is most important: B

(shortest) distance is most important: A

(best) food is the most important: A
(7, $)
B
A
(2, $$$)
C
home
(4, $$)
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Goal Programming
a
goal
 an
objective with a desirable quantity
 no good to be over and under this quantity
short, not enough exercise,
distance 
too tiring.
 long,

 low, not tasty,
money 
high, expensive.
v()
over
u()
under

goal
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General Idea of Goal Programming
 suppose
the goals are: 3 units for distance,
and 2 units (i.e., $$) for price
A(2, 3)
B(7, 1)
C(4, 2)
distance
u(d,)
v(d,)
1
0
0
4
0
1
price
u(p,)
v(p,)
0
1
1
0
0
0
(7, $)
B
A
(2, $$$)
C
home
(4, $$)
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General Idea of Goal Programming
 priority
A
B
C
P1 > P2 > P3 > …
distance
u(d,) v(d,)
1
0
0
4
0
1
price
u(p,) v(p,)
0
1
1
0
0
0
P1up > P2vd > P3ud > P4vp
P1up,
P2vd,
P3ud,
P 4v p
P1up,
P2ud,
P3vd,
P 4v p
P1vp,
P2ud,
P3vd,
P4up
A
C
C
B is dominated by C, i.e., C is optimal for
any priority that B is optimal.
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2/4 : A More General
Goal Programming Approach
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General Idea of Goal Programming

a goal program
 parts
like a linear program
 with
decisions variables
 with
hard constraints
 parts
unlike a linear program
 with
soft constraints

expressed as goals to be achieved

co-existence of constraints such as x1  10 and x1  7 in a
GP if they are soft constraints
 with
the objective function in LP replaced by the
priorities of goals in GP
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Deviation Variables
for a Soft Constraints

example: a soft constraint on labor hour
 x1
units of product 1, each for 4 labor hours
 x2
units of product 2, each for 2 labor hours
 goal:
100 labor hours
人世間有不少soft
constraints （可以
斟酌的限制式）
a
soft constraint: 4x1+2x2  100
2
deviation variables u and v: 4x1+2x2 + u  v = 100
 u:
under utilization of labor
 v:
over utilization of labor
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Example 10-1: Formulation of a GP

three products, quantities to produce x1, x2, and x3

objectives in order of priority
Suppose that the material availability
is a hard constraint, i.e., there is no
way to get more material.

min overtime in assembly

min undertime in assembly

min sum of undertime and overtime in packaging
product
material
(lb/unit)
assembly
(min. unit)
packaging
(min/unit)
x1
x2
x3
availability
2
4
3
600 pounds
9
8
7
900 minutes
1
2
3
300 minutes
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Example 10-1: Formulation of a GP
 GP
 min
P1v1, P2u1, P3(u2+v2),
 s.t.
2x1 + 4x2 + 3x3
 600 (lb., hard const.)
 9x1 + 8x2 + 7x3 + u1  v1 = 900 (min., soft const.)
 1x1 + 2x2 + 3x3 + u2  v2 = 300 (min., soft const.)
 all variables  0

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3/4 : Solution of a Goal Program
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Example: Solution of a GP

min P1u1, P2u2, P3u3,

s.t.

5x1 + 3x2
 150 (hard const.)
(A)

2x1 + 5x2 + u1  v1 = 100 (soft const.)
(1)

3x1 + 3x2 + u2  v2 = 180 (soft const.)
(2)

x1
+ u3  v3 = 40 (soft const.)
(3)

all variables  0
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Example: Solution of a GP
x2
min P1u1, P2u2, P3u3,
s.t.
5x1 + 3x2  150
2x1 + 5x2 + u1  v1 = 100
3x1 + 3x2 + u2  v2 = 180
x1
+ u3  v3 = 40
all variables  0
50
5x1 + 3x2 = 150
feasible
solution
space
x2
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(A)
(1)
(2)
(3)
u1 = 0, v1 > 0
x1
20
direction of
improvement in
P1
2x1 + 5x2 = 100
u1 > 0,
v1 = 0
50
x1
30
Example: Solution of a GP
x2
50
20
P1
30
Actually at this point we know 50
that the point is optimal even
with the third constraint added
and the third goal considered.
Why?
20
x2
60
50
x1
50
optimal with
(A), (1), (2),
and (3)
optimal with
(A), (1), and (2)
x2
60
region with
u1 = 0
P2
20
P2
P1
30
50 60x1
Soft (3)
P3
P1
30
50 60x1
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Example 10-2
x2
region
with v1 = 0
50
P1
40
x1
min P1v1, P2u2, P3v3,
s.t.
5x1 + 4x2 + u1  v1 = 200
2x1 + x2 + u2  v2 = 40
2x1 + 2x2 + u3  v3 = 30
all variables  0
x2
x2
50
50
40
40
(1)
(2)
(3)
P1
P1
15
P2
P2
P3
20
40
x1
1520
40
optimal, with
v1 = u2 = v3 = 0
x1
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4/4 : Another Form of
Goal Programming
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Another Form of GP:
Weighted Goals
 goals
 u1
 the
with weights
min P1u1, P2u2, P3u3,
s.t.
5x1 + 3x2  150
2x1 + 5x2 + u1  v1 = 100
3x1 + 3x2 + u2  v2 = 180
x1
+ u3  v3 = 40
all variables  0
(A)
(1)
(2)
(3)
= 30, u2 = 20, v2 = 20, u3 = 20, v3 = 10
GP expressed as LP
min 30u1+20u2+20v3 +20u3 + 10v3
s.t.
5x1 + 3x2  150
(A)
2x1 + 5x2 + u1  v1 = 100
(1)
3x1 + 3x2 + u2  v2 = 180
(2)
x1
+ u3  v3 = 40 (3)
all variables  0
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Assignment #4
 #1.
Chapter 8, Problem 16
 (a).
Find the maximal flow for this network.
Show all the steps.
 (b). Formulate this problem as a linear
program.
 #2.
Chapter 10, Problem 1
 #2.
Chapter 10, Problem 4
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