Goal Programming Models
for Managerial Strategic Decision
Making
Cinzia Colapinto, Raja Jayaraman and Davide La Torre
Abstract The Goal Programming (GP) model is an important Multiple Objective
Programming (MOP) technique that has been widely utilized for strategic decision
making in presence of competing and conflicting objectives. The GP model
aggregates multiple objectives and allows obtaining satisfying solutions where the
deviations between achievement and the aspirations levels of the attributes are to be
minimized. The GP model is easy to understand and to apply: it is based on
mathematical programming techniques and can be easily solved using software
packages such as LINGO, MATLAB, and AMPL. The GP describes the spectrum
of the Decision Maker’s preferences through a user-friendly and learning
decision-making process. This chapter aims to present the state-of-the-art of GP
models and highlight its applications to strategic decision making in portfolio
investments, marketing decisions and media campaign.
C. Colapinto
Department of Management, Ca’ Foscari University of Venice, Venice, Italy
e-mail: cinzia.colapinto@unive.it
C. Colapinto
Graduate School of Business, Nazarbayev University, Astana, Kazakhstan
R. Jayaraman (&)
Department of Industrial & Systems Engineering, Khalifa University
of Science & Technology, Abu Dhabi, UAE
e-mail: raja.jayaraman@kustar.ac.ae
D. La Torre
Dubai Business School, University of Dubai, Dubai, UAE
e-mail: dlatorre@ud.ac.ae
D. La Torre
Department of Economics, Management, and Quantitative Methods, University of Milan,
Milan, Italy
© Springer Nature Switzerland AG 2020
H. Dutta and J. F. Peters (eds.), Applied Mathematical Analysis: Theory,
Methods, and Applications, Studies in Systems, Decision and Control 177,
https://doi.org/10.1007/978-3-319-99918-0_16
487
488
C. Colapinto et al.
1 Introduction
Success in value investing mainly depends on the Decision Maker (DM)’s understanding of the business, however there are some basic frameworks the DM can use
to analyze any business decisions. Indeed, quantitative techniques in portfolio
selection and management can be valuable to managers, as they can better understand interrelationships among assets and the marketplace, and use this knowledge
to their advantage. Nowadays the portfolio approach is useful in a wide range of
fields. For instance, it could be applied to optimize paid search marketing, as
diversification of a search-marketing portfolio can ameliorate the advertiser’s bottom line. Portfolio Management can also be used to select a portfolio of new
product development projects: the DM manages the product pipeline and makes
decisions about the product portfolio trying to achieve different goals such as
maximization of the profitability of the portfolio, balance and support of the
strategy of the firm. Indeed, rapidly changing technologies, shorter product life
cycles, and intense global competition makes portfolio management for product
innovation crucial for a company’s survival and success. Obviously, Research and
Development (R&D) investments are often treated like financial investments in the
stock market. R&D managers strive to maximize the value of the portfolio,
leveraging return on R&D spending by appropriately designing a balanced portfolio, and a portfolio investment strategy that is aligned with the company’s overall
business strategy. Goal Programming (GP) formulations are a particular class of
optimization models, which are well-suited for portfolio construction under multiple competing objectives and investment goals.
This chapter is organized as follows. Section 2 provides a brief background of
Multiple Objective programming (MOP) and Pareto optimality. Section 3 provides
an overview of popular goal programming techniques. Section 4 presents three
different applications of GP formulations. We close the chapter with a discussion
about the advantages and possible applications and extensions of GP model in
decision making.
2 Background: Multiple Objective Programming
Multiple Objective Programming is a discipline that considers decision-making
situations involving multiple and conflicting criteria. Some examples of conflicting
criteria that have been considered in literature includes cost or price, quality, satisfaction, risk, and others. For instance, in portfolio management the DM is interested in getting high returns but at the same time reducing the risks: the stocks that
have high return typically are also associated with a high risk of losing value. In a
service industry, customer satisfaction and the cost of providing service are two
conflicting criteria that very often need to be considered simultaneously.
Goal Programming Models for Managerial Strategic Decision Making
489
Considering multiple criteria explicitly leads to more informed and better
decisions. However, typically a unique optimal solution does not exist and it is
necessary to use DM’s preferences to differentiate between available solutions.
Many important advances have been made in this field since the start of the
modern MOP discipline in the early 1960s, including new approaches, innovative
methods, hybrid techniques, and sophisticated computational algorithms.
The general formulation of a MOP model can be formulated
as follows [33]:
Given a set of p criteria f1 ; f2 ; . . .; fp optimize the vector f1 ð xÞ; f2 ð xÞ; . . .; fp ð xÞ
under the condition that x 2 DRn where D designates
the set of feasible
solutions.
Let us define a vector-valued function f ðxÞ :¼ f1 ð xÞ; f2 ð xÞ; . . .; fp ð xÞ ; according to
this, a classical MOP problem can be formulated as (assuming all objectives have to
be minimized):
Min f ðxÞ
x2D
We say that a point ^x 2 D is a global Pareto optimal solution or global Pareto
efficient solution if
f ðxÞ 2 f ð^xÞ þ ðRpþ nf0gÞc
for all x 2 D, this definition of optimal solution is based on the notion of Pareto
ordering induced by the cone. Practically speaking, a Pareto optimal solution
describes a state in which the input variables are distributed in such a way that it is
not possible to improve a single criterion without also causing at least one other
criterion to become worse off than before the change. In other words, a state is not
Pareto efficient if there exists a certain change in allocation of input variables that
may result in some criteria being in a better position with no criterion being in a
worse position than before the change. If a point x 2 D is not Pareto efficient, there
is potential for a Pareto improvement and an increase in Pareto efficiency. We refer
the readers to recent advances and various mathematical techniques of MOP in
Zopounidis and Pardalos [42].
3 Goal Programming Models: Some of the Existing
Variants
The GP model is based on mathematical programming commonly solved using
powerful mathematical programming software such as AMPL, Lindo, GAMS and
CPLEX. The GP satisfies the spectrum of the DM’s preferences where some
trade-offs can be made through a user-friendly and learning decision-making process. It is important to point out the investment decisions are actually taken by the
DM and the mathematical model is to assist and not substitute the DM.
The central idea of GP is the determination of the goal levels gi; i ¼ 1; . . .; pfor
the objective function and the minimization of any (positive or negative) deviation
490
C. Colapinto et al.
from these levels. Charnes et al. [13], and Charnes and Cooper [12] first introduced
GP and over the decades GP Models have been applied in several fields and it is
still the most popular technique within the field of MOP.
If we assume to optimize simultaneously p different conflicting criteria
f1 ; f2 ; . . .; fp , the GP model is an aggregating methodology that allows to obtain a
solution representing the best compromise that can be achieved by the DM. We
present the mathematical formulations of three popular and commonly used models
namely, Weighted, Stochastic and Fuzzy GP.
3.1
Weighted Goal Programming
The DM can show different appreciation of the positive and negative deviations
based on their relative importance in the objective. The Weighted GP (WGP) model
can express this different appreciation through corresponding weights wiþ and w
i
respectively. The mathematical formulation of the WGP, as applied to the portfolio
selection problem is as follows:
Min Z =
p
X
wiþ diþ þ w
i di
i¼1
Subject to:
fi ðxÞ diþ þ d
i ¼ gi ;
n
X
xj ¼ 1
i ¼ 1; . . .; p
j¼1
x2D
diþ ; d
i 0;
i ¼ 1; . . .; p
Among many applications we can cite Callahan [10] and Kvanli [22] illustrate a
WGP investment planning model. Blancas et al. [8] propose a synthetic sustainability indicator based on a WGP approach to support the decision making process
in the field of tourism. Jha et al. [20] include the practical aspect of segmentation
and develop a model which deals with optimal allocation of advertising budget
(through a WGP model) for multiple products which is advertised through different
media in a segmented market.
3.2
Stochastic Goal Programming
In many financial contexts, the DM has to take decisions under uncertainty. Hence
the objective functions and the corresponding goals are, in general, random variables. The Stochastic GP (SGP) model deals with the uncertainty related to the
Goal Programming Models for Managerial Strategic Decision Making
491
decision making situation as we assume that the goal values are stochastic and
follow a specific probability distribution. The general formulation of the SGP is as
follows:
Min Z =
p
X
diþ þ d
i
i¼1
Subject to:
g;
fi ðxÞ diþ þ d
i ¼
i
n
X
i ¼ 1; . . .; p
xj ¼ 1
j¼1
x2D
diþ ; d
i 0;
i ¼ 1; . . .; p
where ~gi 2 Nðli ; ri Þ. Martel and Aouni [29] develop the concept of satisfaction
function in order to incorporate explicitly the DM’s preferences. A satisfaction
function F is taking values in [0,1]. Therefore, it has a value of 1 when the DM is
totally satisfied; otherwise it is monotonically decreasing and can take values
between 0 and 1 (see Fig. 1).
We can identify three different thresholds, namely:
(a) the indifference threshold (aid): total satisfaction when the deviations are within
the interval,
(b) the nil satisfaction threshold (aio): there is no satisfaction when the deviations
reach this threshold but the solution is not rejected,
(c) the veto threshold (aiv): rejecting any solution that lead to deviations larger than
this threshold.
Considering veto thresholds leads to a partially non-compensatory model in the
sense that a bad performance on one objective cannot be compensated by a good
Fig. 1 The general form of
satisfaction function
492
C. Colapinto et al.
one on another objective. The SGP with satisfaction function [3], which incorporates explicitly the DM’s preferences, is as follows:
Max Z ¼
p
X
wiþ F
diþ þ w
F
d
i
i
i¼1
Subject to:
g;
fi ðxÞ diþ þ d
i ¼
i
n
X
i ¼ 1; . . .; p
xj ¼ 1
j¼1
x2D
0 d
i aiv ;
0
diþ
aivþ ;
i ¼ 1; . . .; p
i ¼ 1; . . .; p
Aouni et al. [3] integrate the DM’s preferences in a decision situation where the
DM wants to invest a certain amount of capital in the Tunisian stock exchange
market where stocks returns are not known with certainty.
An alternative way to include randomness is to consider the so-called scenariobased models [2, 4, 19]. If we assume that the space of all possible events or
scenarios X ¼ fx1 ; x2 ; . . .; xN g with associated probabilities pðxs Þ ¼ ps is finite
and the objective functions and the corresponding goals are depending on the
scenario xs , the above SGP model with satisfaction function can be readily
extended to:
Max Z =
p
X
wiþ F diþ þ w
i F di
i¼1
Subject to:
fi ðx; xs Þ diþ þ d
i ¼ gi ðxs Þ;
n
X
xj ¼ 1
j¼1
x2D
0 d
i aiv ðxs Þ;
0 diþ
where xs 2 X is fixed.
aivþ ðxs Þ;
i ¼ 1; . . .; p
i ¼ 1; . . .; p
i ¼ 1; . . .; p
Goal Programming Models for Managerial Strategic Decision Making
3.3
493
Fuzzy Goal Programming
The Fuzzy GP (FGP) model was developed to deal with some decisional situations
where the DM can only give vague and imprecise goal values; in other words
aspiration levels are not known precisely. The FGP is based on the fuzzy sets theory
developed by Zadeh [37] and the concept of membership functions introduced by
Zimmerman [39]. Narasimhan [30] and Hannan’s [18] FGP formulations also use
the concept of membership functions to deal with the fuzziness of the goal values
using triangular membership functions. The general formulation of the membership
function requires two acceptability degrees (lower and upper) [41] and the functions
are assumed to be linear. Dhingra et al. [15], Rao [31] and Zimmerman [38, 40]
have developed an approximation procedure for the non-linear membership functions. The general formulation of FGP model as developed by Hannan [18] is as
follows:
Max Z ¼ k
Subject to:
fi ðxÞ
gi
diþ þ d
;
i ¼
Di
Di
n
X
xj ¼ 1
i ¼ 1; . . .; p
j¼1
x2D
k þ diþ þ d
i 1;
k; diþ ; d
i
0;
i ¼ 1; . . .; p
i ¼ 1; . . .; p
where Di is the constant of deviation of the aspiration levels gi . This constant is
pre-specified by the DM. Arenas-Parra et al. [7] have utilized FGP approach for the
portfolio selection problem.
3.4
Other GP Variants
Finally, we introduce other GP variants used to study multi-criteria problems,
highlighting their current use of combined/integrated models.
The Lexicographic or pre-emptive GP (LGP) is based on the optimization of the
objectives according to their relative importance: the most important objectives will
be at the highest levels of priority and they will be optimized first and so on. Thus
the objectives at the lowest levels of priority will have a marginal impact on the
final decision. Wang et al. [35] use a combined analytical hierarchy process - LGP
approach for supplier selection problem in supply chains.
494
C. Colapinto et al.
In the case the returns of assets are not normally distributed, higher moments
(such as skewness and kurtosis) have to be considered: the DM copes with a
trade-off between competing and conflicting objectives, i.e., maximizing expected
return and skewness, while minimizing variance and kurtosis, simultaneously. Lai
[23] used the Polynomial GP (PGP) in order to explore incorporation of investor’s
preferences in the construction of a portfolio with skewness.
The Min-Max GP model [32] seeks the minimization of the maximum deviation
from any single goal in portfolio selection: it uses essentially the same concepts as
the WGP, except instead of minimizing the sum of deviations this model seeks the
solution that minimizes the worst unwanted deviation from any single goal.
More recently, the interactive multiple GP model (IMPG)’s incorporates all the
advantages of “traditional” GP, while circumventing the unnecessary burden of
obtaining a “complete” picture of the DM’s preference pattern [17]. Indeed, an
interactive procedure progresses by seeking this information from the investor,
removing the need to make the preference structure more explicit. Lee and Shim
[26] present an interactive GP model starting on the original work by Lee et al. [24]
in strategic management. For a more extensive literature review on GP models we
refer the readers to Aouni et al. [6] and Colapinto et al. [14].
4 Applications
We present examples that illustrate different applications of GP formulations. In
details, we are going to describe three GP models applied to: (a) portfolio management, (b) strategic marketing, and (c) media planning.
4.1
A SGP Model with Satisfaction Function for Portfolio
Management
In portfolio selection problems, the Financial DM (FDM) considers simultaneously
several factors such as: return, risk, liquidity, gross book value per share, capitalization ratio, and stock market value of each firm. These objectives are usually
incommensurable and conflicting and the best portfolio requires some compromises
among various criteria by the FDM. The trade-offs are based on the FDM’s
structure of preferences.
The first bi-criteria portfolio selection model was proposed by Markowitz [28].
The main objective of the classical model is to obtain the best portfolio that may
maximize the FDM’s return while simultaneously minimize the risk of financial
losses. Given a space of events X ¼ fx1 ; x2 ; . . .; xN g with associated probabilities
pðxs Þ ¼ ps , we assume that the FDM takes his/her decisions on a stochastic linear
multi-criteria optimization model formulated as:
Goal Programming Models for Managerial Strategic Decision Making
Max Z1 ¼
Min Z2 ¼
Subject to:
Xn
x
i¼1 i
Xn
j¼1
Xn
495
lj ðxs Þxj
r ðxs Þxj
j¼1 j
¼1
x2D
where:
(a)
(b)
(c)
(d)
(e)
(f)
Z1 is the stochastic return of the portfolio
Z2 is the stochastic risk of the portfolio
xi is the proportion of the budget to be invested in security j
li is the stochastic return of security j
ri is the stochastic risk of security j
D is the set of feasible solutions.
In the above stochastic model x describes possible different scenarios. We
propose a GP model to deal with the above stochastic context, using the notion of
deterministic equivalent formulation introduced by Caballero et al. [9].
The notion of deterministic equivalent consists of replacing the initial stochastic
objective functions, which are difficult to be analyzed, with the expected value of all
objective functions. In this manner the stochastic problem is reduced to a deterministic model. Of course a lot of information is lost by reducing the stochastic
model to a deterministic one and this can lead to different definitions of the concept
of efficient solution and efficient sets that can be obtained for the same stochastic
problem. Since they deal with different characteristics of the initial problem they are
non-comparable. Caballero et al. [9] assert that the concepts of efficient solution
under certain conditions for a stochastic multi-objective programming are closely
related. Given any particular problem from the established relationships the concept
of efficiency is the one that best fits the preferences of the FDM.
The GP model with satisfaction function is illustrated using simulated scenarios
(see Tables 1 and 2) obtained from the data of the Tunisian stock exchange market
available in Ben Abdelaziz et al. [1]. The choice of the ten stocks is supposed to be
independent.
The deterministic equivalent formulation of the above stochastic model reads as:
Max Z1 ¼
Min Z2 ¼
Subject to:
Xn
x
j¼1 j
x2D
Xn
E lj xj
E r j xj
j¼1
j¼1
Xn
¼1
496
C. Colapinto et al.
Table 1 Rate of return of stocks from stock exchange market
Events/securities
S1
S2
S3
S4
S5
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
Mean
Events/securities
−1.6407
1.1965
–1.2962
0.6120
–0.5761
–0.6970
–1.0075
0.2642
2.2613
0.6736
–0.02099
S6
0.0322
−0.6556
0.1922
0.7931
0.6148
1.3447
0.3509
2.1381
0.7589
–0.2285
0.53408
S7
–0.2907
0.9379
–0.3401
–0.1693
0.8296
0.5318
1.1727
–0.3094
0.6688
1.9818
0.50131
S8
–0.7878
0.3610
–0.2735
1.7931
0.7547
–0.0932
–2.0650
–0.1729
1.1859
0.2622
0.09645
S9
–0.6686
0.9944
1.0520
0.1135
2.6507
–0.5764
–0.2296
0.4338
0.3012
1.3112
0.53822
S10
1.1357
0.5725
1.3342
2.4851
0.1180
2.3149
–0.4660
1.0750
1.1440
2.5944
1.23078
–1.3194
–0.4361
–0.1004
0.0215
0.0941
0.0337
–0.1055
–0.4435
–0.7442
–0.3541
–0.51393
1.2922
–0.6783
–1.0707
0.6340
–0.1958
1.4385
–0.8466
0.9723
–1.6181
0.7902
0.07177
1.0438
0.2358
–0.0925
0.4969
–0.1010
0.7004
–1.5535
0.0004
0.0938
0.3037
0.11278
1.2933
–0.9456
1.7890
–1.1960
0.0107
0.3077
0.5256
0.6331
0.8684
1.0807
0.43669
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
Mean
Table 2 Risks of stocks from the stock exchange market
Events/securities
S1
S2
S3
S4
S5
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
Mean
Events/securities
1.1666
2.2283
2.1728
1.1592
6.1608
3.9946
1.6560
0.6371
2.7000
1.9834
2.3859
S6
2.4065
0.3635
2.5525
1.3802
0.4548
0.0734
1.5056
2.1861
1.1961
0.0559
1.21746
S7
3.3958
1.4174
0.0085
0.2343
0.3859
1.0999
0.2071
0.4652
0.0000
5.4431
1.26572
S8
2.7400
0.0719
0.4493
0.3811
0.4605
0.0645
0.1649
2.6861
1.8632
0.0388
0.89203
S9
6.1057
0.0381
0.1024
0.0083
4.3228
2.6128
0.7742
0.0819
1.2887
1.4534
1.67883
S10
x1
x2
x3
0.0153
0.1253
0.0004
0.0990
0.0007
0.0574
0.2566
0.9951
0.0393
0.0001
0.2327
0.1631
0.652
0.9584
0.2175
(continued)
Goal Programming Models for Managerial Strategic Decision Making
497
Table 2 (continued)
Events/securities
S6
S7
S8
S9
S10
x4
x5
x6
x7
x8
x9
x10
Mean
0.1943
2.1875
0.4980
1.9964
2.0239
0.3023
5.2098
1.25532
0.0012
0.0260
0.2152
0.3535
0.0541
0.0424
0.0000
0.08495
0.0314
1.0185
0.0076
3.1890
0.0475
0.4626
0.0006
0.60482
2.3941
0.0772
0.0082
0.0955
0.4885
0.0350
0.4227
0.39171
1.0640
1.5360
0.0296
0.2576
0.9609
0.1495
0.2212
0.60471
Table 3 Budget values
Events
Budget values
Events
Budget values
x1
x2
x3
x4
x5
Mean
1,001
1,003
1,005
997
996
x6
x7
x8
x9
x10
1000
990
1,001
995
1,002
1,010
where
(a)
(b)
(c)
(d)
(e)
(f)
Z1 is the expected return of the portfolio
Z2 is the expected risk of the portfolio
xj is the proportion of the budget to be invested in security j
E lj : the expected return of security j
E rj : the expected risk of security j
D is the set of feasible solutions and it takes into account the portfolio
diversification.
Let us suppose that the budget is a random variable whose distribution values are
listed in Table 3 according to the events’ occurrence.
On the other hand, let us suppose that the following additional financial constraints are to be satisfied:
• S1 þ S2 600
• S6 400
• S2 100
Let g1 and g2 be the two random aspiration levels for the objective functions Z1 and
Z2 whose values are listed in the following Table 4. As satisfaction function, let us
consider the following expression:
498
C. Colapinto et al.
Table 4 Aspiration levels
Events
g1
g2
x1
x2
x3
x4
x5
Mean
850.61
830.15
820.11
810.55
800.43
910.01
x6
910.04
x7
910.02
x8
910.10
x9
910.02
x10
g1 = 830.780
Fðx; aÞ ¼
Events
g1
g2
832.13
910.01
820.15
910.04
817.83
910.06
840.74
910.20
885.10
910.20
g2 = 910.07
1
1 þ a2 x 2
where a is a parameter. This function exhibits the behavior to be considered as a
satisfaction function and it is easy to verify the following properties:
(a)
(b)
(c)
(d)
(e)
(f)
Fð0Þ ¼ 1
Fð þ 1Þ ¼ 0
F 00 ðxÞ ¼ 0 , x ¼ 1=2a
0:9 FðxÞ 1 , 0 x 1=3a
0 FðxÞ 0:1 , x 3=a
0 FðxÞ 0:01 , x 3=a
Natural candidates for the indifference threshold and the dissatisfaction threshold
are, respectively, aid ¼ 1=3a and aio ¼ 3=a. Let us assume the veto threshold
aiv ¼ 2 aio ¼ 6=a. In the following model let us choose a ¼ 0:1. The GP Model
þ
with satisfaction function and weights w1þ ¼ 0:5; w
1 ¼ 0:5 w2 ¼ 0:1; w2 ¼ 0:1
can be written as:
þ
Max Z ¼ 0:5Fðd1þ Þ þ 0:5Fðd
1 Þ þ 0:1Fðd2 Þ þ 0:1Fðd2 Þ
Subject to:
0:02099 S1 þ 0:53408 S2 þ 0:50131 S3 þ 0:09645 S4 þ
0:53822 S5 þ 1:23078 S6 0:51393 S7 þ 0:07177 S8 þ
0:11278 S9 þ 0:43669 S10 d1þ þ d
1 ¼ 830:780
2:38588 S1 þ 1:21746 S2 þ 1:26572 S3 þ 0:89203 S4 þ
0:39171 S9 þ 0:60471 S10 d2þ þ d
2 ¼ 910:07
S1 þ S2 þ S3 þ S4 þ S5 þ
S6 þ S7 þ S8 þ S9 þ S10 ¼ 1000
S1 þ S2 600
S6 400
S2 100
S1; S2; S3; S4; S5;
S6; S7; S8; S9; S10 0
þ
d1þ ; d
1 ; d2 ; d2 0
Goal Programming Models for Managerial Strategic Decision Making
499
The solution provided by LINGO [27] is a portfolio with investments only in S2
(100 units), S6 (400 units), and S10 (500 units).
4.2
Strategic Marketing Decisions Using GP
with Satisfaction Function
New information technologies are facilitating more complex interactions that are
organized by networks. According to Castells [11], the network enterprise is a new
form of organization characteristic of economic activity, but gradually extending its
logic to other domains and organizations. Moreover, most firms are complex
organizations that market many different products in many different business areas,
thus a portfolio approach is well suited. The process of evaluating and implementing strategic marketing decisions is characterized by high levels of uncertainty,
potential synergies between different options, and long term consequences.
Moreover, each decision can affect all the business functions, thus contemplated
marketing actions must be evaluated and analyzed. Strategies and their implementation plan must be developed and executed at the corporate, business and
products levels. Besides Product marketing plan, a company needs a strategic
marketing plan. According to Wiersema [36], the strategic marketing perspective is
defined as having the dual task of providing a marketplace perspective on the
process of determining corporate direction, and guidelines for the development and
execution of marketing programs that assist in attaining the corporate objectives.
The application of GP models allows a more efficient analysis for decision
making in marketing. In a company there exist potential conflicts between different
areas (such as the production and marketing areas, see Taylor III and Anderson,
[34], thus GP model can deal with the complex trade-off decisions involved in this
kind of situations, which require greater coordination and integration between
different business functions.
We extend the case study presented in Lee and Nicely [25] concerning Raynebo,
Inc., a company managed by Mr. Rayne. The company is a lessor of color television
receivers in a large metropolitan area. In the same area, there are three other
competitors and Raynebo, Inc. is holding 25% of the market share. Mr. Rayne has
defined three broad goals for his company, in order of priority as follows:
1. Achieve the minimum ROI of 15% and strive for the target ROI of 20%.
2. Maintenance or improvement of market volume which is now 1250 leased sets.
3. Retention of present workforce by minimizing the turnover amongst the
workers.
The company is expected to have two new clients and delivery for each client 50
sets of leased TVs. One client will be served in March and the other in June. In
order to guarantee that the two new clients buy the products from the company, the
management decided to give 2 months free service for them.
500
C. Colapinto et al.
Moreover, the company’s main expenses can be categorized as salaries and
promotional activities. The company dedicates $12,500 yearly for promotional
activities and is targeting to increase them to $14,500. A sum of $118,004 of the
expensive goes for salaries to all of its employees and also the management is
intending on increasing the salaries by 5%.
Mr. Rayne has set several goals for the next business year. The goals and their
equations are presented below. For the modelling purposes the following are the
names and meanings of the variables:
•
•
•
•
•
•
X1
X2
X3
X4
X5
X6
denotes
denotes
denotes
denotes
denotes
denotes
to
to
to
to
to
to
number of leased sets from present investment base.
number of leased sets to result for normal growth.
number of leased sets required by new client first of March.
salaries of all employees.
promotional activities.
number of leased sets required by new client first of June.
The model has several constraints and they are presented below. The number of
television sets to be leased from the original investment base must be limited to sets
in stocks, that is
(P1) X1 d1þ ¼ 1250
In addition to that, the management of Raynebo, Inc. has set the following goals
in ordinal ranking of priorities:
(P2) Achieve a ROI of at least 15%
þ
192 X1 þ 150 X2 þ 81:5 X3 þ 44 X6 X4 X5 þ d
2 d2 ¼ 85; 996
(P3) Maintain at least 1200 leased sets
þ
X 1 þ d
3 d3 ¼ 1; 200
(P4) Retain the current level of employment in terms of wages and salaries
X4 d4þ ¼ 118; 004
(P5) Maintaining the promotional expenditure at their current level
X5 d þ 5 ¼ 12; 500
(P6) Achieve normal market growth of 2% or 25 additional leased sets above the
present level of 1250.
þ
X1 þ X2 þ d
6 d6 ¼ 1; 275
Goal Programming Models for Managerial Strategic Decision Making
501
(P7) Achieve the target ROI of 20%
þ
192 X1 þ 136 X2 þ 68:67 X3 þ 34:67 X6 X4 X5 þ d
7 d7 ¼ 11; 099
(P8) Grant a pay increase of at least 5% to employees
X4 d8þ ¼ 123; 904:2
(P9) Increase the promotional expenditure by at least $2000
X5 d9þ ¼ 12; 500
(P10) Expand volume of business by at least 100 sets for the two new clients each
50 sets.
X3 þ d
10 ¼ 50
X6 þ d
11 ¼ 50
As in the previous example, let us suppose that the satisfaction function assumes the
following form:
1
Fðx; aÞ ¼
1 þ a2 x 2
We analyze the proposed model with a 2 f1; 0:1; 0:01g. The GP model with the
satisfaction function is shown below:
þ
þ
Max Z ¼ F d2þ þ F d2þ þ F d
þ F d
þ F d5þ
2 þ F d3
3 þ F d4
þ F dþ
þ þ F d6
þ
þ F d7þ þ F d
þ F d9þ þ F d
7 þ F d8
10 þ F d11
Subject to:
X1 d1þ ¼ 1250
192 X1 þ 150 X2 þ 81:5 X3 þ 44 X6 X4 X5 þ d 2 d þ 2 ¼ 85996
þ
X1 þ d
3 d3 ¼ 1200
X4 d4þ ¼ 118004
X5d5þ ¼ 12500
X1 þ X2 d6þ þ d
6 ¼ 1275
192 X1 þ 136 X2 þ 68:67 X3 þ 34:67 X6 X4 X5 d7þ þ d
7 ¼ 110996
þ
X4d8 ¼ 123904:2
X5d9þ ¼ 12500
X3 þ d
10 ¼ 50
X6 þ Z
11 ¼ 50
þ
þ
þ
þ
þ
þ
þ
þ
þ
d1 ; d2 ; d
2 ; d3 ; d3 ; d4 ; d5 ; d þ ; d þ ; d7 ; d7 ; d8 ; d9 ; d10 ; d11 [ ¼ 0
X1 ; X2 ; X3 ; X4 ; X5 ; X6 [ ¼ 0
502
C. Colapinto et al.
Table 5 Output of the GP model with satisfaction function
Variable/parameter a
a=1
a = 0.1
a = 0.01
X1
X2
X3
X4
X5
X6
1250.002
31.12276
50.00209
123904.2
14500
50.00105
1251.257
28.30444
51.66706
123904.2
14500
50.80718
1250
29.64506
52.3378
123904.2
14500
51.17972
Table 6 Priorities Achieved by the GP Model with satisfaction function
Output
GP with satisfaction
function formulation
Lee & Nicely
formulation
Achieve a ROI of at least 15%
Maintain at least 1200 leased sets
Retain the current level of employment in terms
of wages and salaries
Maintaining the promotional expenditure at their
current level
Achieve normal market growth of 2%
Achieve the target ROI of 20%
Grant a pay increase of at least 5% to employees
Increase the promotional expenditure by at least
$2000
Expand volume of business by at least 100 sets
for the two new clients each 50 sets
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Achieved
Not Achieved
Achieved
Achieved
Using LINGO [27] with three different values for a produced the following
results.
From Table 5, it is obvious that the values are more or less close to each other
for different values of a. The results (Table 6) show that all the predefined goals
have been achieved using the proposed model with satisfaction functions.
Clearly, the company now can serve its two new clients in March and June since
the under achievement for d−10 = 0, and d−11 = 0 for a values. It also achieves the
growth of 2% in the market d−6 = 0 for all a values. Moreover, the ROI has reached
20% and in order to compare the under achievement that occurred in the GP model,
the results show that d−7 = 0.0000304 for a = 1, d−7 = 0.023 for a = 0.1, and
d−7 = 0.034 for a = 0.01 which are rounded to zero.
Goal Programming Models for Managerial Strategic Decision Making
4.3
503
Media Campaign Strategy Using GP with Satisfaction
Function
The model we are going to present is formulated as an extension of the model
developed by Fernandez et al. [16]. Recently, Kaul et al. [21] present a WGP
approach to multi-period media planning to determine optimal schedule of advertisements maximizing advertisement impressions and minimizing advertising
expenditures. Aouni et al. [5] formulate a SGP model with satisfaction function for
the optimal allocation of advertisements in different vehicles. The concept of
“Media diet” was developed to assess each individual’s exposure to content in the
media, based on the combination of media consumption and content. Media diet
data usually consist of several questions like:
– Consumption of the last issue,
– Time since last consumption,
– Number of saw/read/listened issues among the last five, for instance.
The readership matrix’ task is to give an overview of what the entire readership
population looks like and to ensure that each panel represents the readership on an
average day. In order to create the matrix, the most commonly used variables are
gender, age and reader frequency. For a representative individual, we suppose there
exists an index I(j) of how much a person sees/listens/reads a particular vehicle
j and we supposed to have n vehicles.
For each fixed vehicle, the corresponding element in the matrix is a synthetic
number in [0, 1] which represents a proxy of the above readership matrix and it can
be interpreted as the averaged probability that an individual will be exposed to an
advertisement placed in vehicle j. If xj advertising insertions are purchased in
vehicle j, the average expected number of exposures per individual is I(j)xj. In this
model the DM has two objectives:
1. The efficacy of a schedule x ¼ ðx1 ; x2 ; . . .; xn Þ is the number of exposures per
n
P
I ð jÞxj .
individual in the population, and this is obtained by
2. The total cost of schedule x ¼ ðx1 ; x2 ; . . .; xn Þ is
n
P
j¼1
The multi-criteria problem can be formulated as
j¼1
cðjÞxj .
504
C. Colapinto et al.
Table 7 Media consumption
in Italy
Vehicle
Media consumptions
TV
Radio
Newspaper
94.79
67.28
52.95
Max
Min
n
X
j¼1
n
X
I ð jÞxj
cðjÞxj
j¼1
Subject to
x2D
where the set D is general enough for covering a large set of restrictions found in
problems of media campaign. Given two goals g1 (2,456) and g2 (904,000), we
propose the following GP model with satisfaction function:
Max Z ¼
2
X
wiþ Fðdiþ Þ þ w
i Fðdi Þ
i¼1
Subject to
n
X
þ
I ð jÞxj þ d
1 d1 ¼ g1
j¼1
n
X
þ
cð jÞxj þ d
2 d2 ¼ g 2
j¼1
x2D
0 diþ aivþ
0 diþ aivþ
ði ¼ 1; 2Þ
ði ¼ 1; 2Þ
Let us consider the following illustrative example based on real data from the
Italian media market. Table 7 shows the real media diet in Italy according to Census
data.
Table 8 Prices’ list for an
advertising slot
Vehicle
Prices’ list for an advertising slot
TV
Radio
Newspaper
73,600
11,200
54,801
Goal Programming Models for Managerial Strategic Decision Making
505
Table 8 shows average of official prices’ list for an advertising slot and provides
the realizations of the random variable for different vehicles (TV, radio,
newspaper).
As in the previous example, let us assume the satisfaction function F to take the
expression
Fðx; aÞ ¼
1
:
1 þ a2 x 2
LINGO [27] provides the following optimal solution (x1, x2, x3) = (1, 35, 8) that is
interpreted as maximize the advertising opportunities through radio, followed by
newspaper and TV media.
5 Conclusions
In the real world, strategic management decisions often imply to harmonize different needs and interests or to balance conflicting criteria. An important way to
model such problems is through the use of a goal programming approach, which
can combine the optimization with the DM desire to satisfy several goals simultaneously. The learning offered by GP models helps generating scenarios where the
DM can interact and make changes to the model parameters to enhance the
decision-making process. Indeed, the GP approach allows for a better modelling of
real managerial situations: it is rare that criteria are to be minimized, rather the DM
need to achieve certain objectives to satisfy all stakeholders’ perspectives. For
instance in green supply chain management the DM aims at keeping the level of
pollution below a certain sustainable threshold rather than purely minimizing. Is it
realistic to reach a society with a zero level pollution? Similarly it is not possible to
reset the production costs to zero aiming at delivering a differentiate product. Again
in supply chain management, the simultaneous profit maximization and the minimization of inventory costs, or rejected rate or environmental impacts requires a
multi-objective approach. This chapter drives the reader through the main features
of the GP approach, and presents some relevant applications in strategic decision
making.
References
1. Abdelaziz, F.B., Aouni, B., El Fayedh, R.: Multi-objective stochastic programming for
portfolio selection. Eur. J. Oper. Res. 177, 1811–1823 (2007)
2. Aouni, A., Abdelaziz, F.B., La Torre, D.: The stochastic goal programming model: theory and
applications. J. Multicriteria Decis. Anal. 19(5–6), 185–200 (2012)
3. Aouni, B., Ben Abdelaziz, F., Martel, J.M.: Decision-maker’s preferences modeling in the
stochastic goal programming. Eur. J. Oper. Res. 162, 610–618 (2005)
506
C. Colapinto et al.
4. Aouni, A., Colapinto, C., La Torre, D.: Solving stochastic multi-objective programming in
multi-attribute portfolio selection through the goal programming model. J. Financ. Decis.
Mak. 6(2), 17–30 (2010)
5. Aouni, B., Colapinto, C., La Torre, D.: Stochastic goal programming model and satisfaction
functions for media selection and planning problem. Int. J. Multicriteria Decis. Mak. 2, 391–407
(2012)
6. Aouni, B., Colapinto, C., La Torre, D.: Financial portfolio management through the goal
programming model: current state-of-the-art. Eur. J. Oper. Res. 234(2), 536–545 (2014)
7. Arenas-Parra, M., Bilbao-Terol, A., Rodriguez Uria, M.V.: A fuzzy goal programming
approach to portfolio selection. Eur. J. Oper. Res. 133, 287–297 (2001)
8. Blancas, F.J., Caballero, R., Gonzalez, M., Lozano-Oyola, M., Perez, F.: Goal programming
synthetic indicators: an application for sustainable tourism in Andalusian coastal counties.
Ecol. Econ. 69, 2158–2172 (2010)
9. Caballero, R., Cerda, E., Munoz, M.M., Rey, L., Stancu-Minasian I.M.: Efficient solution
concepts and their relations in stochastic multiobjective programming. J. Optim. Theory Appl.
110(1), 53–74
10. Callahan. J.: An introduction to financial planning through goal programming. Cost Manag. 3,
7–12 (1973)
11. Castells, M.: The Rise of the Network Society, The Information Age: Economy, Society and
Culture, Vol. I. Cambridge, MA; Oxford, UK (1996)
12. Charnes, A., Cooper, W.W.: Management Models and Industrial Applications of Linear
Programming. Wiley, New York (1961)
13. Charnes, A., Cooper, W.W., Ferguson, R.O.: Optimal estimation of executive compensation
by linear programming. Manage. Sci. 2, 138–151 (1955)
14. Colapinto, C., Jayaraman, R., Marsiglio, S.: Multi-criteria decision analysis with goal
programming in engineering, management and social sciences: a state-of-the art review. Ann.
Oper. Res. 251, 7–40 (2017)
15. Dhingra, A.K., Rao, S.S., Kumar, V.: Nonlinear membership functions in multiobjective
fuzzy optimization of mechanical and structural systems. AIAA J. 30, 251–260 (1992)
16. Fernandez, P.J., Lauretto, M.de Souza, Pereira, C.A. de Bragança, Stern, J.M.: A new media
optimizer based on the mean-variance model. Pesqui. Oper. 27(3), 427–456 (2007)
17. Hallerbach, W., Ning, H., Soppe, A., Spronk, J.: A framework for managing a portfolio of
socially responsible investments. Eur. J. Oper. Res. 153, 517–529 (2004)
18. Hannan, E.L.: On fuzzy goal programming. Decis. Sci. 12, 522–531 (1981)
19. Jayaraman, R., Colapinto, C., Liuzzi, D., La Torre, D.: Planning sustainable development
through a scenario-based stochastic goal programming model. Oper. Res. Int. J. 17(3), 789–
805 (2017)
20. Jha, P.C., Aggarwal, R., Gupta, A.: Optimal media planning formulti-products in segmented
market. Appl. Math. Comput. 217, 6802–6818 (2011)
21. Kaul, A., Aggarwal, S., Krishnamoorthy, M., Jha, P.C.: Multi-period media planning for
multi-products incorporating segment specific and mass media. Ann. Oper. Res. 1–43 (2018)
22. Kvanli, A.H.: Financial planning using goal programming. Omega 8(2), 207–218 (1980)
23. Lai, T.Y.: Portfolio selection with skewness: a multi-objective approach. Rev. Quant. Financ.
Account. 293–305 (1991)
24. Lee, S.M., Justis, R., Franz, L.: Goal programming for decision making in closely held
businesses. Am. J. Small Bus. 3(4), 31–41 (1979)
25. Lee, S.M., Nicely, R.E.: Goal programming for marketing decisions: a case study. J. Mark.
38, 24–32 (1974)
26. Lee, S.M., Shim, J.P.: Interactive goal programming on the microcomputer to establish
priorities for small business. J. Oper. Res. Soc. 37(6), 571–577 (1986)
27. LINGO, Release 12, Lindo System Inc., Chicago, 2010
28. Markowitz, H.: Portfolio selection. J. Financ. 7, 77–91 (1952)
29. Martel, J.M., Aouni, B.: Incorporating the decision maker’s preferences in the goal
programming model. J. Oper. Res. Soc. 41, 1121–1132 (1990)
Goal Programming Models for Managerial Strategic Decision Making
507
30. Narasimhan, R.: Goal programming in a fuzzy environment. Decis. Sci. 11, 325–336 (1980)
31. Rao, S.S.: Multi-objective optimization of fuzzy structural systems. Int. J. Numer. Meth. Eng.
24, 1157–1171 (1987)
32. Romero, C.: Handbook of Critical Issues in Goal Programming. Pergamon Press, Oxford
(1991)
33. Sawaragi, Y., Nakayama, H., Tanino, T.: Mathematics in Science and Engineering: Vol. 176.
Theory of Multiobjective Optimization. Academic Press, Orlando (1985)
34. Taylor III, B.W., Anderson, P.F.: Goal programming approach to marketing/ production
planning. Ind. Mark. Manage. 8(2), 136–144 (1979)
35. Wang, G., Huang, S.H., Dismukes, J.P.: Product-driven supply chain selection using
integrated multi-criteria decision-making methodology. Int. J. Prod. Econ. 91(1), 1–15 (2004)
36. Wiersema, F.D.: Strategic marketing: linking marketing and corporate planning. Eur. J. Mark.
17(6), 46–53 (1983)
37. Zadeh, L.A.: Fuzzy sets. Inf. Control. 8, 338–353 (1965)
38. Zimmerman, H.-J.: Fuzzy programming and linear programming with several objectives
functions. Fuzzy Sets Syst. 1, 45–55 (1978)
39. Zimmerman, H.-J.: Using fuzzy sets in operations research. Fuzzy Sets Syst. 13, 201–216
(1983)
40. Zimmerman, H.-J.: Modelling flexibility, vagueness and uncertainty in operations research.
Investig. Oper. 1, 7–34 (1988)
41. Zimmerman, H.-J.: Decision making in ill-structured environments and with multiple criteria.
In: Bana and Costa, C.A., (eds.) Readings in Multiple Criteria Decision Aid, pp. 119–151.
Springer, Heidelberg (1990)
42. Zopounidis, C., Pardalos, P. M. (eds.): Handbook of Multicriteria Analysis, vol. 103. Springer
Science & Business Media, Berlin (2010)