Målprogrammering (Goal Programming –GP)
Introdusert av Charnes og Cooper i 1961
Flere mål som ønskes å optimeres eller tas hensyn til
samtidig i motsetning til for eksempel
profittmaksimeringsmodell med ett mål
Irwin/McGraw-Hill
11.1
© The McGraw-Hill Companies, Inc., 2003
Eksempler på mål:
1. Opprettholde stabil profitt
2. Øke eller opprettholde gitte markedsandeler
3. Diversifisering av produkter
4. Holde arbeidsstokken på et gitt nivå
5. Forurense minst mulig
• Målene kan ofte ikke sammenlignes eller kombineres
direkte
• Ulike mål er ofte i konflikt med hverandre
McGraw-Hill/Irwin
11.2
© The McGraw-Hill Companies, Inc., 2003
To hovedtilnærminger
• Vektet målprogrammering - målene er grovt sett
sammenlignbare
• Leksikografisk målprogrammering - Hierarki av
prioritetsnivåer for de ulike målene
McGraw-Hill/Irwin
11.3
© The McGraw-Hill Companies, Inc., 2003
Tradisjonelt rammeverk for analyse
av beslutningstaking forutsetter:
1. Èn beslutningstaker
2. En begrenset rekke av mulige valg
3. Et veldefinert kriterium (F.eks. nytte eller profitt)
Kritikk mot tradisjonell tilnærming
• Beslutningstakeren er vanligvis ikke interessert i å ordne
de mulige beslutningene i henhold til ett enkelt kriterium,
men prøver å finne et optimalt kompromiss blant flere
ulike mål
McGraw-Hill/Irwin
11.4
© The McGraw-Hill Companies, Inc., 2003
MULTIPLE OBJECTIVES
In many applications, the planner has more than one
objective. The presence of multiple objectives is
frequently referred to as the problem of “combining
apples and oranges.”
Consider a corporate planner whose long-range
goals are to:
1. Maximize discounted profits
2. Maximize market share at the end of the
planning period
3. Maximize existing physical capital at the end
of the planning period
McGraw-Hill/Irwin
11.5
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These goals are not commensurate (i.e., they cannot
be directly combined or compared).
It is also clear that the goals are conflicting (i.e.,
there are trade-offs in the sense that sacrificing the
requirements on any one goal will tend to produce
greater returns on the others.
These models, although not applied as often in
practice as some of the other models (such as linear
programming, forecasting, inventory control, etc.),
have been found to be especially useful on
problems in the public sector.
McGraw-Hill/Irwin
11.6
© The McGraw-Hill Companies, Inc., 2003
Several approaches to multiple objective models
(also called multi-criteria decision making) have
been developed:
Multi-attribute utility theory
Search for Pareto optimal solutions via
multi-criteria linear programming
Analytic Hierarchy Process (AHP)
Developed by Thomas Saaty, AHP helps
managers choose between many decision
alternatives on the basis of multiple criteria.
Goal Programming (GP)
Introduced by A. Charnes and W.W. Cooper. GP
is a heuristic approach to the multipleobjectives model.
11.7
Only Goal Programming will
be discussed.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
GOAL PROGRAMMING
Goal Programming is an extension of Linear
Programming that enables the planner to come as
close as possible to satisfying various goals and
constraints.
It allows the decision maker, at least in a heuristic
sense, to incorporate his or her preference system
in dealing with multiple conflicting goals.
GP is sometimes considered to be an attempt to put
into a mathematical programming context, the
concept of satisficing.
Coined by Herbert Simon, it communicates the idea
that individuals often do not seek optimal solutions,
but rather solutions that are “good enough” or
11.8
“close enough.”
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Weighted Goal Programming
•
A common characteristic of many management science models (linear
programming, integer programming, nonlinear programming) is that they have
a single objective function.
•
It is not always possible to fit all managerial objectives into a single objective
function. Managerial objectives might include:
–
–
–
–
–
–
–
•
Maintain stable profits.
Increase market share.
Diversify the product line.
Maintain stable prices.
Improve worker morale.
Maintain family control of the business.
Increase company prestige.
Weighted goal programming provides a way of striving toward several
objectives simultaneously.
McGraw-Hill/Irwin
11.9
© The McGraw-Hill Companies, Inc., 2003
Weighted Goal Programming
•
With weighted goal programming, the objective is to
– Minimize W = weighted sum of deviations from the goals.
– The weights are the penalty weights for missing the goal.
•
Introduce new changing cells, Amount Over and Amount Under, that will
measure how much the current solution is over or under each goal.
•
The Amount Over and Amount Under changing cells are forced to maintain
the correct value with the following constraints:
Level Achieved – Amount Over + Amount Under = Goal
McGraw-Hill/Irwin
11.10
© The McGraw-Hill Companies, Inc., 2003
The Dewright Company
•
The Dewright Company is one of the largest producers of power tools in the
United States.
•
The company is preparing to replace its current product line with the next
generation of products—three new power tools.
•
Management needs to determine the mix of the company’s three new products
to best meet the following three goals:
1. Achieve a total profit (net present value) of at least $125 million.
2. Maintain the current employment level of 4,000 employees.
3. Hold the capital investment down to no more than $55 million.
McGraw-Hill/Irwin
11.11
© The McGraw-Hill Companies, Inc., 2003
Data for Contribution to the Goals
Unit Contribution of Product
Factor
1
2
3
Goal
Total profit (millions of dollars)
12
9
15
≥ 125
Employment level (hundreds of employees)
5
3
4
= 40
Capital investment (millions of dollars)
5
7
8
≤ 55
McGraw-Hill/Irwin
11.12
© The McGraw-Hill Companies, Inc., 2003
Penalty Weights
Goal
Factor
Penalty Weight for Missing Goal
1
Total profit
5 (per $1 million under the goal)
2
Employment level
4 (per 100 employees under the goal)
2 (per 100 employees over the goal)
3
Capital investment
3 (per $1 million over the goal)
McGraw-Hill/Irwin
11.13
© The McGraw-Hill Companies, Inc., 2003
Data for Contribution to the Goals
Unit Contribution of Product
Factor
1
2
3
Goal
Total profit (millions of dollars)
12
9
15
≥ 125
Employment level (hundreds of employees)
5
3
4
= 40
Capital investment (millions of dollars)
5
7
8
≤ 55
McGraw-Hill/Irwin
11.14
© The McGraw-Hill Companies, Inc., 2003
Weighted Goal Programming Formulation for
the Dewright Co. Problem
Let
Pi = Number of units of product i to produce per day (i = 1, 2, 3),
Under Goal i = Amount under goal i (i = 1, 2, 3),
Over Goal i = Amount over goal i (i = 1, 2, 3),
Minimize W = 5(Under Goal 1) + 2Over Goal 2) + 4 (Under Goal 2) + 3 (Over Goal 3)
subject to
Level Achieved
Deviations
Goal
Goal 1: 12P1 + 9P2 + 15P3
– (Over Goal 1) + (Under Goal 1) = 125
Goal 2: 5P1 + 3P2 + 4P3
– (Over Goal 2) + (Under Goal 2) =
40
Goal 3: 5P1 + 7P2 + 8P3
– (Over Goal 3) + (Under Goal 3) =
55
and
Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.15
© The McGraw-Hill Companies, Inc., 2003
Weighted Goal Programming Spreadsheet
B
3
4
5
6
7
8
9
10
11
12
13
14
15
Goal 1 (Prof it)
Goal 2 (Employ ment)
Goal 3 (Inv estment)
Units Produced
C
D
E
Contribution per Unit Produced
Product 1
Product 2
Product 3
12
9
15
5
3
4
5
7
8
Product 1
8.33333333
McGraw-Hill/Irwin
Product 2
0
Product 3
1.66666667
F
G
Goals
H
Lev el
Achiev ed
125
>=
48.333333 =
55
<=
Goal
125
40
55
Penalty
Weights
Prof it
Employ ment
Inv estment
11.16
I
J
K
Deviations
Amount Amount
Ov er
Under
0
0
8.333333
0
0
0
Ov er
Goal
2
3
Under
Goal
5
4
L
M
Constraints
Balance
(Lev el - Ov er + Under)
125
40
55
N
O
=
=
=
Goal
125
40
55
Weighted Sum
of Dev iations
16.66666667
© The McGraw-Hill Companies, Inc., 2003
Suppose that we have an educational program
design model with decision variables x1 and x2,
where
x1 is the hours of classroom work
x2 is the hours of laboratory work
Assume the following constraint on total program
hours:
x1 + x2 < 100 (total program hours)
Two Kinds of Constraints In the goal programming
approach, there are two kinds of constraints:
1. System constraints (so-called hard
constraints) that cannot be violated.
2. Goal constraints (so-called soft constraints)
that may be violated11.17
if necessary.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Now, suppose that each hour of classroom work
involves
12 minutes of small-group experience and
19 minutes of individual problem solving
Each hour of laboratory work involves
29 minutes of small-group experience and
11 minutes of individual problem solving
The total program time is at most 6,000 minutes
(100 hr * 60 min/hr).
There are two goals: Each student should spend as
close as possible to
¼ of the maximum program time working in
small groups and
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
11.18
¹/3 of the time on problem
solving.
These conditions are:
12x + 29x ~
= 1500 (small-group experience)
1
2
19x1 + 11x2 ~
= 2000 (individual problem solving)
Where ~
= means that the left-hand side is desired to
be “as close as possible” to the right-hand side.
In order to satisfy the system constraint, at least one
of the two goals will be violated.
To implement the goal programming approach, the
small-group experience condition is rewritten as the
goal constraint:
12x1 + 29x2 + u1 – v1 = 1500 (u1 > 0, v1 > 0)
Where u1 = the amount by which total small-group
experience falls short of 1500
v1 = the amount by which total small-group
11.19
experience exceeds
1500
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Deviation Variables Variables u1 and v1 are called
deviation variables since they measure the amount
by which the value produced by the solution
deviates from the goal.
Note that by definition, we want either u1 or v1 (or
both) to be zero because it is impossible to
simultaneously exceed and fall short of 1500.
In order to make 12x1 + 29x2 as close as possible to
1500, it suffices to make the sum u1 + v1 small.
The individual problem-solving condition is written
as the goal constraint:
19x1 + 11x2 + u2 – v2 = 2000 (u2 > 0, v2 > 0)
As before, the sum of u2 +11.20
v2 should be small.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
The complete (illustrative) model is:
Min u1 + v1 + u2 + v2
s.t. x1 + x2 < 100 (total program hours)
12x1 + 29x2 + u1 – v1 = 1500 (small-group experience)
19x1 + 11x2 + u2 – v2 = 2000 (problem solving)
x1, x2 , u1, v1, u2, v2 > 0
Note: Both u1 and v1 can be 0
Now this is an ordinary LP model and can be easily
solved in Excel. The optimal decision variables will
satisfy the system constraint (total program hours).
McGraw-Hill/Irwin
11.21
© The McGraw-Hill Companies, Inc., 2003
Solver will guarantee that either u1 or v1 (or both) will
be zero, and thus these variables automatically
satisfy this desired condition.
The same statement holds for u2 and v2 and in
general for any pair of deviation variables.
Note that the objective function is the sum of the
deviation variables.
This choice of an objective function indicates that
there is no preference among the various deviations
from the stated goals.
McGraw-Hill/Irwin
11.22
© The McGraw-Hill Companies, Inc., 2003
For example, any of the following three decisions is
acceptable:
1. A decision that overachieves the group
experience goal by 5 minutes and hits the
problem-solving goal exactly,
2. A decision that hits the group experience goal
exactly and underachieves the problemsolving goal by 5 minutes, and
3. A decision that underachieves each goal by
2.5 minutes.
McGraw-Hill/Irwin
11.23
© The McGraw-Hill Companies, Inc., 2003
There is no preference among the following three
solutions because each of these yields the same
value (i.e., 5) for the objective function.
(1) u1 = 0
v1 = 5
u2 = 0
v2 = 0
McGraw-Hill/Irwin
(2) u1 = 0
v1 = 0
u2 = 5
v2 = 0
11.24
(3) u1 = 2.5
v1 = 0
u2 = 2.5
v2 = 0
© The McGraw-Hill Companies, Inc., 2003
Weighting the Deviation Variables Differences in
units alone could produce a preference among the
deviation variables.
One way of expressing a preference among the
various goals is to assign different coefficients
(weights) to the deviation variables in the objective
function.
In the program-planning example, one might select
Min 10u1 + 2v1 + 20u2 + v2
as the objective function. Since v2 (overachievement of problem solving) has the smallest
coefficient, the program designers would rather
have v2 positive than any of the other deviation
variables (positive v2 is penalized the least).
McGraw-Hill/Irwin
11.25
© The McGraw-Hill Companies, Inc., 2003
With this objective function it is better to be 9
minutes over the problem-solving goal than to
underachieve by 1 minute the small-groupexperience goal.
To see this, note that for any solution in which
u1 > 1, decreasing u1 by 1 and increasing v2 by 9
would yield a smaller value for the objective
function.
McGraw-Hill/Irwin
11.26
© The McGraw-Hill Companies, Inc., 2003
Goal Interval Constraints Another type of goal
constraint is called a goal interval constraint.
Such a constraint restricts the goal to a range or
interval rather than a specific numerical value.
Suppose, for example, that in the previous
illustration the designers were indifferent among
programs for which
1800 < [minutes of individual problem solving] < 2100
i.e., 1800 < 19x1 + 11x2 < 2100
In this situation the interval goal is captured with
two goal constraints:
19x1 + 11x2 – v1 < 2100 (v1 > 0)
19x1 + 11x2 + u111.27> 1800 (u1 > 0)
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
When the terms u1 and v1 are included in the
objective function, the LP code will attempt to
minimize them.
Summary of the Use of Goal Constraints Each goal
constraint consists of a left-hand side, say
gi(x1, …, xn), and a right-hand side, bi.
Goal constraints are written by using nonnegative
deviation variables ui, vi.
At optimality at least one of the pair ui, vi will always
be zero.
ui represents underachievement; vi represents
overachievement.
Whenever ui is used it is added to gi(x1, …, xn).
Whenever vi is used it is subtracted from
11.28
gi(x1, …, xn).
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Only deviation variables appear in the objective
function, and the objective is always to minimize.
The decision variables xi, i = 1, …, n do not appear in
the objective.
Four types of goals have been discussed:
1. Target. Make gi(x1, …, xn) as close as possible
as possible to bi. To do this write the goal
constraint as
gi(x1, …, xn) + ui - vi = bi
McGraw-Hill/Irwin
11.29
(ui > 0, vi > 0)
© The McGraw-Hill Companies, Inc., 2003
2. Minimize Underachievement. To do this, write
gi(x1, …, xn) + ui - vi = bi
(ui > 0, vi > 0)
and in the objective, minimize ui, the underachievement.
vi does not appear in the objective function
and it is only in this constraint, hence, the
constraint can be equivalently written as
gi(x1, …, xn) + ui > bi
(ui > 0)
If the optimal ui is positive, this constraint will
be active, for otherwise ui* could be made
smaller.
If ui*>0 then, since vi* must equal zero, it must
be true that gi(x1, …,11.30xn) + ui* = bi .
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
3. Minimize Overachievement. To do this, write
gi(x1, …, xn) + ui - vi = bi
(ui > 0, vi > 0)
and in the objective, minimize vi, the overachievement.
ui does not appear in the objective function,
the constraint can be equivalently written as
gi(x1, …, xn) - vi < bi
(vi > 0)
If the optimal vi is positive, this constraint will
be active. The argument is analogous to that
in item 2.
McGraw-Hill/Irwin
11.31
© The McGraw-Hill Companies, Inc., 2003
4. Goal Interval Constraint. In this instance, the
goal is to come as close as possible to
satisfying
ai < gi(x1, …, xn) < bi
In order to write this as a goal, first “stretch
out” the interval by writing
ai - ui < gi(x1, …, xn) < bi + vi
(ui > 0, vi > 0)
which is equivalent to the two constraints
gi(x1, …, xn) + ui > ai
^ +a
gi(x1, …, xn) + ui - v
i
i
^ > 0)
(ui > 0, v
i
gi(x1, …, xn) - ui > bi
^ -v +b
gi(x1, …, xn) + u
i
i
i
^ > 0, v > 0)
(u
i
i
The objective function ui + vi is minimized.
^
^
Variables ui and vi are merely surplus and
McGraw-Hill/Irwin
slack, respectively. 11.32
© The McGraw-Hill Companies, Inc., 2003
Weighted vs. Preemptive Goal Programming
•
Weighted goal programming is designed for problems where all the goals are
quite important, with only modest differences in importance that can be
measured by assigning weights to the goals.
•
Preemptive goal programming is used when there are major differences in
the importance of the goals.
– The goals are liested in the order of their importance.
– It begins by focusing solely on the most important goal.
– It next does the same for the second most important goal (as is possible without
hurting the first goal).
– It continues the the following goals (as is possible without hurting the previous
more important goals).
McGraw-Hill/Irwin
11.33
© The McGraw-Hill Companies, Inc., 2003
ABSOLUTE PRIORITIES
In some cases, managers do not wish to express
their preferences among various goals in terms of
weighted deviation variables, for the process of
assigning weights may seem too arbitrary or
subjective.
In such cases, it may be more acceptable to state
preferences in terms of absolute priorities (as
opposed to weights) to a set of goals.
This approach requires that goals be satisfied in a
specific order. Therefore, the model is solved in
stages as a sequence of models.
McGraw-Hill/Irwin
11.34
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming
•
Introduce new changing cells, Amount Over and Amount Under, that will
measure how much the current solution is over or under each goal.
•
The Amount Over and Amount Under changing cells are forced to maintain
the correct value with the following constraints:
Level Achieved – Amount Over + Amount Under = Goal
•
Start with the objective of achieving the first goal (or coming as close as
possible):
– Minimize (Amount Over/Under Goal 1)
•
Continue with the next goal, but constrain the previous goals to not get any
worse:
– Minimize (Amount Over/Under Goal 2)
subject to
Amount Over/Under Goal 1 = (amount achieved in previous step)
•
Repeat the previous step for all succeeding goals.
McGraw-Hill/Irwin
11.35
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming for Dewright
The goals in the order of importance are:
1.
2.
3.
4.
•
Start with the objective of achieving the first goal (or coming as close as
possible):
–
•
Achieve a total profit (net present value) of at least $125 million.
Avoid decreasing the employment level below 4,000 employees.
Hold the capital investment down to no more than $55 million.
Avoid increasing the employment level above 4,000 employees.
Minimize (Under Goal 1)
Then, if for example goal 1 is achieved (i.e., Under Goal 1 = 0), then
–
Minimize (Under Goal 2)
subject to
(Under Goal 1) = 0
McGraw-Hill/Irwin
11.36
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Formulation for
the Dewright Co. Problem (Step 1)
Let
Pi = Number of units of product i to produce per day (i = 1, 2, 3),
Under Goal i = Amount under goal i (i = 1, 2, 3),
Over Goal i = Amount over goal i (i = 1, 2, 3),
Minimize (Under Goal 1)
subject to
Level Achieved
Goal 1: 12P1 + 9P2 + 15P3
Goal 2: 5P1 + 3P2 + 4P3
Goal 3: 5P1 + 7P2 + 8P3
Deviations
– (Over Goal 1) + (Under Goal 1) =
– (Over Goal 2) + (Under Goal 2) =
– (Over Goal 3) + (Under Goal 3) =
Goal
125
40
55
and
Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.37
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Formulation for
the Dewright Co. Problem (Step 2)
Let
Pi = Number of units of product i to produce per day (i = 1, 2, 3),
Under Goal i = Amount under goal i (i = 1, 2, 3),
Over Goal i = Amount over goal i (i = 1, 2, 3),
Minimize (Under Goal 2)
subject to
Level Achieved
Goal 1: 12P1 + 9P2 + 15P3
Goal 2: 5P1 + 3P2 + 4P3
Goal 3: 5P1 + 7P2 + 8P3
Deviations
– (Over Goal 1) + (Under Goal 1) =
– (Over Goal 2) + (Under Goal 2) =
– (Over Goal 3) + (Under Goal 3) =
Goal
125
40
55
(Under Goal 1) = (Level Achieved in Step 1)
and
Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.38
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Formulation for
the Dewright Co. Problem (Step 3)
Let
Pi = Number of units of product i to produce per day (i = 1, 2, 3),
Under Goal i = Amount under goal i (i = 1, 2, 3),
Over Goal i = Amount over goal i (i = 1, 2, 3),
Minimize (Over Goal 3)
subject to
Level Achieved
Goal 1: 12P1 + 9P2 + 15P3
Goal 2: 5P1 + 3P2 + 4P3
Goal 3: 5P1 + 7P2 + 8P3
Deviations
– (Over Goal 1) + (Under Goal 1) =
– (Over Goal 2) + (Under Goal 2) =
– (Over Goal 3) + (Under Goal 3) =
Goal
125
40
55
(Under Goal 1) = (Level Achieved in Step 1)
(Under Goal 2) = (Level Achieved in Step 2)
and
Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.39
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Formulation for
the Dewright Co. Problem (Step 4)
Let
Pi = Number of units of product i to produce per day (i = 1, 2, 3),
Under Goal i = Amount under goal i (i = 1, 2, 3),
Over Goal i = Amount over goal i (i = 1, 2, 3),
Minimize (Over Goal 2)
subject to
Level Achieved
Goal 1: 12P1 + 9P2 + 15P3
Goal 2: 5P1 + 3P2 + 4P3
Goal 3: 5P1 + 7P2 + 8P3
Deviations
– (Over Goal 1) + (Under Goal 1) =
– (Over Goal 2) + (Under Goal 2) =
– (Over Goal 3) + (Under Goal 3) =
Goal
125
40
55
(Under Goal 1) = (Level Achieved in Step 1)
(Under Goal 2) = (Level Achieved in Step 2)
(Over Goal 3) = (Level Achieved in Step 3)
and
Pi ≥ 0, Under Goal i ≥ 0, Over Goal i ≥ 0 (i = 1, 2, 3)
McGraw-Hill/Irwin
11.40
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Spreadsheet
Step 1: Minimize (Under Goal 1)
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
G
H
I
J
K
L
M
N
O
=
=
=
Goal
125
40
55
Dewright Co. Goal Programming (Preemptive Priority 1: Minimize Under Goal 1)
Goal 1 (Prof it)
Goal 2 (Employ ment)
Goal 3 (Inv estment)
Contribution per Unit Produced
Product 1
Product 2
Product 3
12
9
15
5
3
4
5
7
8
Units Produced
Product 1
3.7037
Goals
Lev el
Achiev ed
Goal
125
>=
125
40
=
40
61.481
<=
55
Deviations
Amount
Amount
Ov er
Under
0
0
0
0
6.481
0
Constraints
Balance
(Lev el - Ov er + Under)
125
40
55
Minimize (Under Goal 1)
McGraw-Hill/Irwin
Product 2
0
Product 3
5.3704
11.41
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Spreadsheet
Step 3: Minimize (Over Goal 3)
A
1
2
3
4
5
6
7
8
9
10
11
12
B
C
D
E
F
G
H
I
J
K
L
M
N
O
=
=
=
Goal
125
40
55
Dewright Co. Goal Programming (Preemptive Priority 3: Minimize Over Goal 3)
Goals
Goal 1 (Prof it)
Goal 2 (Employ ment)
Goal 3 (Inv estment)
Contribution per Unit Produced
Product 1
Product 2
Product 3
12
9
15
5
3
4
5
7
8
Units Produced
Product 1
8.333
McGraw-Hill/Irwin
Product 2
0
Lev el
Achiev ed
125
48.333
55
>=
=
<=
Goal
125
40
55
Deviations
Amount
Amount
Ov er
Under
0
0
8.333333
0
0
0
Constraints
Balance
(Lev el - Ov er + Under)
125
40
55
Minimize (Ov er Goal 3)
(Under Goal 1) = 0
(Under Goal 2) = 0
Product 3
1.667
11.42
© The McGraw-Hill Companies, Inc., 2003
Preemptive Goal Programming Spreadsheet
Step 4: Minimize (Over Goal 2)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
C
D
E
F
G
H
I
J
K
L
M
N
O
=
=
=
Goal
125
40
55
Dewright Co. Goal Programming (Preemptive Priority 4: Minimize Over Goal 2)
Goals
Goal 1 (Prof it)
Goal 2 (Employ ment)
Goal 3 (Inv estment)
Contribution per Unit Produced
Product 1
Product 2
Product 3
12
9
15
5
3
4
5
7
8
Units Produced
Product 1
8.333
McGraw-Hill/Irwin
Product 2
0
Lev el
Achiev ed
125
48.333
55
>=
=
<=
Goal
125
40
55
Deviations
Amount
Amount
Ov er
Under
0
0
8.333
0
0
0
Constraints
Balance
(Lev el - Ov er + Under)
125
40
55
Minimize (Ov er Goal 2)
(Under Goal 1) = 0
(Under Goal 2) = 0
(Ov er Goal 3) = 0
Product 3
1.667
11.43
© The McGraw-Hill Companies, Inc., 2003
Example: Swenson’s Media Selection Model
J. R. Swenson is an advertising agency which has
just completed an agreement with a pharmaceutical
manufacturer to mount a radio and television
campaign to introduce a new product, Mylonal.
The total expenditures for the campaign are not to
exceed $120,000.
The client wants to reach several audiences,
however, radio and television are not equally
effective in reaching all audiences.
Therefore, the agency will estimate the impact of the
advertisements in terms of rated exposures (i.e.,
“people reached per month”) on the audiences of
interest.
McGraw-Hill/Irwin
11.44
© The McGraw-Hill Companies, Inc., 2003
The following data represent the number of
exposures per $1000 expenditure:
Total
Upper Income
TV
RADIO
14,000
1,200
6,000
1,200
The following are the campaign goals, listed in order
of absolute priority.
1. Total exposures will hopefully be at least
840,000.
2. In order to maintain effective contact with the
leading radio station, no more than $90,000
will be spent on TV 11.45
advertising.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
3. The campaign should achieve at least 168,000
upper-income exposures.
4. If all other goals are satisfied, the total number
of exposures would come as close as possible
to being maximized.
Note that if all of the $120,000 is spent on TV
advertising, then the maximum obtainable
exposures would be 1,680,000 (120*14,000).
To model the problem, the following notation will be
used:
x1 = dollars spent on TV ( in thousands)
x2 = dollars spent on radio (in thousands)
The objective function will be to maximize total
exposures and the other goals will be treated as
11.46
constraints.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Infeasible LP model
Media Selection
TV
Radio
Maximize
X1
X2
OF
Total Exposures (thousands)
14
6
840
Expenditures (th's)
15
105
Technological
coeffisients
LHS
RHS
Slack or
surplu
s
Max Expenditures (th's)
1
1
120
<=
120
1.6485E-11
Min Exposures (th's)
14
6
840
>=
840
-3.7517E-10
Max TV (th's)
1
15
<=
90
75
144
>=
168
-24
Min Upper-income Exposures (th's)
McGraw-Hill/Irwin
1.2
1.2
11.47
© The McGraw-Hill Companies, Inc., 2003
Since there are only two decision variables in this
model, the graphical approach can be used.
x2
140
The graph shows that there
are no points that satisfy all
the constraints.
120
X1 = 90
X1 + X2 = 120
<
1200X1 +1200X2 = 168,000
>
<
McGraw-Hill/Irwin
120
140
11.48
x1
© The McGraw-Hill Companies, Inc., 2003
Swenson’s Goal Programming Model Note that the
first goal (total exposures will be at least 840,000), if
violated, will be underachieved.
The second goal (no more than $90,000 will be spent
on TV advertising), if violated, will be overachieved,
etc.
Employing this reasoning, the goals are restated, in
descending priority, as:
1. Minimize the underachievement of 840,000
total exposures.
Min u1 subject to the condition
14,000x1 + 6,000x2 + u1 > 840,000; u1 > 0
McGraw-Hill/Irwin
11.49
© The McGraw-Hill Companies, Inc., 2003
2. Minimize expenditures in excess of $90,000 on
TV
Min v2 subject to the condition
x1 – v2 < 90,000; v2 > 0
3. Minimize underachievement of 168,000 upperincome exposures
Min u3 subject to the condition
1,200x1 + 1,200x2 + u3 > 168,000; u3 > 0
4. Minimize underachievement of 1,680,000 total
exposures (the maximum possible)
Min u4 subject to the condition
14,000x1 + 6,000x2 + u4 > 1,680,000; u4 > 0
McGraw-Hill/Irwin
11.50
© The McGraw-Hill Companies, Inc., 2003
Note that the goals are now stated in terms of either
minimizing underachievement (i.e., min. ui) or
minimizing overachievement (i.e., min. vi).
In addition, the goals have been expressed as
inequalities. This method will facilitate a graphical
analysis.
Given that the priorities are formulated correctly, we
must now distinguish between
1. system constraints (all constraints that may
not be violated)
The only system constraint is: Total
expenditures will be no greater than
$120,000
x1 + x2 < 120
2. goal constraints 11.51
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
The model can now be expressed as:
Min P1u1 + P2v2 + P3u3 + P4u4
s.t.
x1 +
x2
<
120
14,000 x1 + 6,000x2 + u1 >
840,000
x1
- v2 <
90
1,200 x1 + 1,200x2 + u3 >
168,000
14,000 x1 + 6,000x2 + u4 > 1,680,000
(S)
(1)
(2)
(3)
(4)
x1, x2, u1, v2 , u3, u4 > 0
Note that the objective function consists only of
deviation variables and is of the Min form.
In the objective function, P1 denotes the highest
priority, and so on.
11.52
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
The previous problem statement precisely means:
1. Find the set of decision variables that
satisfies the system constraint (S) and that
also gives the Min possible value to u1
subject to constraint (1) and x1, x2, u1 > 0.
Call this set of decisions FR I (i.e., feasible
region I).
Considering only the highest goal, all of the
points in FR I are “optimal” and (again
considering only the highest goal), we are
indifferent as to which of these points are
selected.
McGraw-Hill/Irwin
11.53
© The McGraw-Hill Companies, Inc., 2003
2. Find the subset of points in FR I that gives the
Min possible value to v2, subject to constraint
(2) and v2 > 0. Call this subset FR II.
Considering only the ordinal ranking of the
two highest-priority goals, all of the points in
FR II are “optimal,” and in terms of these two
highest-priority goals, we are indifferent as to
which of these points are selected.
3. Let FR III be the subset of points in FR II that
minimize u3, subject to constraint (3) and
u3 > 0.
4. FR IV is the subset of points in FR III that
minimize u4, subject to constraint (4) and
u4 > 0. Any point in FR IV is an optimal
11.54
solution to the model.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Graphical Analysis and Spreadsheet Implementation
of the Solution Procedure Since there are only two
decision variables, we can use the graphical method
of LP.
1. Both the spreadsheet output and the
geometry reveal the the Min of u1 s.t. (S), (1),
and x1, x2, u1 > 0 is u1* = 0.
The important information is that u1 = 0 which
tells us that the first goal can be completely
attained.
Alternative optima for the current model are
provided by all values of (x1, x2) that satisfy
the conditions
x1 + x2 < 120
FR I 14,000x1 + 6,000x2 > 840,000
11.55
x1, x2 > 0
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Media Selection - GP model
TV
Radio
X1
X2
Total Exposures (thousands)
Expenditures (th's)
120
0
Minimiz
e
U1
V2
U3
U4
OF
1
1
1
1
54
0
30
24
0
Technological coeffisients
Max Expenditures (th's)
1
1
Min Exposures (th's)
14
6
Max TV (th's)
1
1
-1
Min Upper-income Exposures
(th's)
1.2
1.2
Exposures Target (th's)
14
6
McGraw-Hill/Irwin
LHS
1
1
11.56
RHS
Slack
or
surplus
120
<
=
120
0
1680
>
=
840
840
90
<
=
90
0
168
>
=
168
0
1680
>
=
1680
0
© The McGraw-Hill Companies, Inc., 2003
Media Selection -Step 1
TV
Radio
X1
X2
Penalty
Expenditures (th's)
Minimize
U1
V2
1
60
0
0
0
Technological coeffisients
1
U3
Max Expenditures (th's)
1
1
Min Exposures (th's)
14
6
1
0
U4
OF
1
0
0
LHS
RHS
Slack or
surplus
60
<=
120
60
840
>=
840
2.121E-10
2
3
4
Ra
nk
McGraw-Hill/Irwin
11.57
© The McGraw-Hill Companies, Inc., 2003
At any such point, the goal is attained (u1* = 0) so
that, in terms of only the first goal, these decisions
are equally preferable.
Thus FR I is the shaded area ABC.
u1 = 0
McGraw-Hill/Irwin
11.58
© The McGraw-Hill Companies, Inc., 2003
Media Selection -Step 2
TV
Radio
X1
X2
Minimize
U1
Penalty
Expenditures (th's)
V2
U3
1
60
0
0
0
Technological coeffisients
Max Expenditures (th's)
1
1
1
Min Exposures (th's)
14
6
2
Max TV (th's)
1
1
-1
0
U4
OF
1
0
0
LHS
RHS
Slack or
surplus
60
<=
120
60
840
>=
840
0
60
<=
90
30
0
=
0
0
3
4
Value of U1 found in step 1
1
Ra
nk
McGraw-Hill/Irwin
11.59
© The McGraw-Hill Companies, Inc., 2003
We see that: Min v2 such that x in FR I, goal (2) and
v2 > 0 is v2* = 0. x1, x2 > 0 Thus, FR II is defined by
x1 + x2 < 120
14,000x1 + 6,000x2 > 840,000
FR II
x1 < 90
x1, x2 > 0
The shaded area ABDE is a
subset of FR I and as
expected, the size of the
feasible region is smaller.
v1 = 0
u1 = 0
McGraw-Hill/Irwin
11.60
© The McGraw-Hill Companies, Inc., 2003
Media Selection - Step 3
TV
Radio
X1
X2
Minimize
U1
V2
Penalty
Expenditures (th's)
15
105
0
U4
OF
1
1
24
24
0
RHS
Slack or
surplus
<=
120
0
840
>=
840
0
15
<=
90
75
168
>=
168
0
0
=
0
0
0
=
0
0
Technological coeffisients
LHS
120
Max Expenditures (th's)
1
1
1
Min Exposures (th's)
14
6
2
Max TV (th's)
1
3
Min Upper-income Exposures (th's)
1.2
0
U3
1
-1
1.2
1
4
Value of U1 found in step 1
RaValue of V2 found in step 2
nk
McGraw-Hill/Irwin
1
1
11.61
© The McGraw-Hill Companies, Inc., 2003
FR III is the line segment BD.
In this case u3* = 24,000. Although the first
two goals were completely attained (since
u1* = v2* = 0), the third goal cannot be
completely attained because u3* > 0.
FR III
McGraw-Hill/Irwin
x1 + x2 < 120
14,000x1 + 6,000x2 > 840,000
x1 < 90
1,200x1 + 1,200x2 > 168,000 – 24,000 = 144,000
11.62
© The McGraw-Hill Companies, Inc., 2003
Media Selection - Step 4
TV
Radio
X1
X2
Minimize
U1
V2
U3
Penalty
Expenditures (th's)
90
30
0
0
24
U4
OF
1
240
240
Technological coeffisients
Max Expenditures (th's)
1
1
1
Min Exposures (th's)
14
6
2
Max TV (th's)
1
3
Min Upper-income Exposures (th's)
1.2
1.2
4
Exposures Target (th's)
14
6
Value of U1 found in step 1
Value of V2 found in step 2
Ra Value of U3 found in step 3
nk
McGraw-Hill/Irwin
LHS
1
-1
1
1
1
1
1
11.63
RHS
Slack or
surplus
120
<=
120
0
1440
>=
840
600
90
<=
90
0
168
>=
168
0
1680
>=
1680
0
0
=
0
0
0
=
0
0
24
=
24
0
© The McGraw-Hill Companies, Inc., 2003
Recall that the fourth goal is to minimize
underachievement of the maximum possible number
of exposures, which is 1,680,000.
Thus, we wish to minimize the
underachievement u4 where
14,000x1 + 6,000x2 + u4 > 1,680,000
Since u4 = 240,000, we
achieve 1,680,000 240,000 = 1,440,000
exposures.
McGraw-Hill/Irwin
11.64
The unique optimum
is x1* = 90, x2* = 30
(i.e., spend $90,000
on TV ads & $30,000
on radio ads).
© The McGraw-Hill Companies, Inc., 2003
COMBINING WEIGHTS AND
ABSOLUTE PRIORITIES
In reviewing the results of the absolute priority
study, the older members of the Mylonal market
begins to take on importance.
The exposures per $1000 of advertising are:
EXPOSURE GROUP
50 and over
TV
RADIO
3,000
8,000
Note that radio and TV exposures are not equally
effective in generating exposures in this segment of
the population.
McGraw-Hill/Irwin
11.65
© The McGraw-Hill Companies, Inc., 2003
If there were no other considerations, then we would
like as many 50-and-over exposures as possible.
Since radio yields such exposures at a higher rate
than TV (8000 > 3000), the maximum possible
number of 50-and-over exposures would be
achieved by allocating all of the $120,000 available
to radio.
Thus, the maximum number of 50-and-over
exposures is 120 x 8000 = 960,000.
Once the first three goals are satisfied, we would
like to come as close as possible to minimizing
underachievement.
To resolve this conflict of goals, use a weighted sum
of the deviation variables as the objective in the final
11.66
phase of the absolute priorities
approach.
McGraw-Hill/Irwin
© The McGraw-Hill Companies, Inc., 2003
Media Selection - Weighted
Step 4
TV
Radio
X1
X2
Minimize
U1
V2
U3
Penalty
Expenditures (th's)
15
105
Wei
ght
0
0
24
U4
U5
OF
1
3
1065
840
75
Technological coeffisients
Max Expenditures (th's)
1
1
1
Min Exposures (th's)
14
6
2
Max TV (th's)
1
3
Min Upper-income Exposures (th's)
1.2
1.2
4
Exposures Target (th's)
14
6
4
Exposures > 50 years (th's)
3
8
Value of U1 found in step 1
Value of V2 found in step 2
Ra Value of U3 found in step 3
nk
McGraw-Hill/Irwin
LHS
1
-1
1
1
1
1
1
1
11.67
RHS
Slack
or
surplu
s
120
<=
120
0
840
>=
840
0
15
<=
90
75
168
>=
168
0
1680
>=
1680
0
960
>=
960
0
0
=
0
0
0
=
0
0
24
=
24
0
© The McGraw-Hill Companies, Inc., 2003
Note that the new objective function has moved the
optimal solution from one end of FR III to the other.
This optimal solution is as close as possible to the
more heavily weighted goal.
Sensitivity analysis
on the weights in the
objective function
could be used to see
when the solution
changes from point
B to point D.
McGraw-Hill/Irwin
11.68
© The McGraw-Hill Companies, Inc., 2003
Multi-Objective Decision Making
•
Many problems have multiple objectives:
– Planning the national budget
• save social security, reduce debt, cut taxes, build national defense
– Admitting students to college
• high SAT or GMAT, high GPA, diversity
– Planning an advertising campaign
• budget, reach, expenses, target groups
– Choosing taxation levels
• raise money, minimize tax burden on low-income, minimize flight of business
– Planning an investment portfolio
• maximize expected earnings, minimize risk
•
Techniques
– Preemptive goal programming
– Weighted goal programming
McGraw-Hill/Irwin
11.69
© The McGraw-Hill Companies, Inc., 2003