05_chapter1ANINTRODUCTIONTOGOALPROGRAMMING.pdf

CHAPTER I
AN INTRODUCTION TO GOAL PROGRAMMING
1.1 INTRODUCTION
After the II world war, the, Industrial world faced a depression and to solve the
various industrial problems. Industrialist tried the models, which were successful in
solving their problems. Industrialist learnt that the techniques of OR can conveniently
apply to solve industrial problems. Then onwards, various models of OR/GP have
been developed to solve industrial problems. In fact GP models are helpful to the
managers to solve various problems; they face in their day to day work. These models
are used to minimize the cost of production, increase the productivity and use the
available resources carefully and for healthy industrial growth.
The purpose of this chapter is to describe the goal programming (GP) and
distinguish the models, methods and applications in industry. In addition, the
relationship of GP within the fields of multiple criteria decision making (MCDM) will
be discussed. Specifically, this chapter seeks to introduce and describe different type
of GP models that will be used throughout this thesis. Over the past five decades,
multi-objective mathematical programming (MOMP) has been an active area of
research in the field of industry. This chapter will also introduce some definition of GP
related.

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1.2 DEFINITIONS
Decision Maker(s): The decision maker(s) refer to the person(s), organization(s), or
stakeholder(s) to whom the decision problem under consideration belongs.
Decision Variable: A decision variable is defined as a factor over which the decision
maker has control.
Criterion: A criterion is a single measure by which the goodness of any solution to a
decision problem can be measured. There are many possible criteria arising from
different fields of application but some of the most commonly arising relate at the
highest level to
Cost
Profit
Time
Distance
Performance of a system
Company or organizational strategy
Personal preferences of the decision maker(s)
Safety considerations
A decision problem which has more than one criterion is therefore referred to
as a multi-criteria decision making (MCDM) or multi-criteria decision aid (MCDA)
problem. The space formed by the set of criteria is known as criteria space.

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Achievement Function: The function that serves to measure the achievement of the
minimization of unwanted goal deviation variables in the goal programming model.
Goal Function: A mathematical function that is to be achieved at a specified level
Goal Program: A mathematical model, consisting of linear or nonlinear functions and
continuous or discrete variables, in which all functions have been transformed into
goals.
Multiplex: Originally this referred to the multiphase simplex algorithm employed to
solve linear goal programs. More recently it defines certain specific models and
methods employed in multiple- or single-objective optimization in general.
Negative Deviation: The amount of deviation for a given goal by which it is less than
the aspiration level.
Positive Deviation: The amount of deviation for a given goal by which it exceeds the
aspiration level.
Satisfice: An old Scottish word referring to the desire, in the real world, to find a
practical solution to a given problem, rather than some utopian result for an
oversimplified model of the problem.
Constraint: A constraint is a restriction upon the decision variables that must be
satisfied in order for the solution to be implementable in practice. This is distinct from
the concept of a goal whose non-achievement does not automatically make the

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solution non-implementable. A constraint is normally a function of several decision
variables and can be equality or an inequality.
Sign Restriction: A sign restriction limits a single decision or deviational variable to
only take certain values within its range. The most common sign restriction is for the
variable to be non-negative and continuous.
Feasible Region: The set of solutions in decision space that satisfy all constraints and
sign restrictions in a goal programming form the feasible region. Any solution that
falls within the feasible region is deemed to be implementable in practice.
Trade-off: A trade-off (or tradeoff) is a situation that involves losing one quality or
aspect of something in return for gaining another quality or aspect. It often implies a
decision to be made with full comprehension of both the upside and downside of a
particular choice; the term is also used in an evolutionary context, in which case the
selection process acts as the "decision-maker".
Goal programming: GP is a branch of multiobjective optimization, which in turn is a
branch of multi-criteria decision analysis (MCDA), also known as multiple-criteria
decision making (MCDM). This is an optimization programme. It can be thought of as
an extension or generalization of linear programming to handle multiple, normally
conflicting objective measures. Each of these measures is given a goal or target value
to be achieved. Unwanted deviations from this set of target values are then minimized
in an achievement function. This can be a vector or a weighted sum dependent on the
goal programming variant used. As satisfaction of the target is deemed to satisfy the

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decision maker(s), an underlying satisficing philosophy is assumed. Goal
programming is used to perform three types of analysis:
(1) Determine the required resources to achieve a desired set of objectives.
(2) Determine the degree of attainment of the goals with the available resources.
(3) Providing the best satisfying solution under a varying amount of resources and
priorities of the goals.
1.3 DECISION ANALYSIS FOR MULTIPLE OBJECTIVES
rational, but also suggests that he is the optimizer who strives to allocate scarce
resources in the most economic manner. It is assumed that he possesses knowledge of
relevant aspects of the decision environment, a stable system of preference, and ability
to analyze alternative courses of action. However, recent developments in the theory
of the firm have raised a question as to whether such assumptions regarding economic
man can be applied to the decision maker in any realistic sense. For an individual to be
perfectly rational in decision analysis, he must be capable of attaching a definite
preference to each possible outcome of the alternative courses of action. Furthermore,
he should be able to specify the exact outcomes by employing scientific analysis.
According to broad empirical investigation, however, there is no evidence that any one
individual is capable of performing such exact analysis for a complex decision
problem.

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The primary goal of economic man as optimizer is assumed to be
maximization of profits. If this were the situation, decision analysis would not be such
a difficult task. In reality, the decision maker may have only a vague idea as to what is
the best outcome for the organization in a global sense. Furthermore, he often is
incapable of identifying the optimal choice due to either his lack of analytical ability
or the complexity of the organizational environment. There is an abundance of
evidence which suggests that the practice of decision making is affected by the
epistemological assumptions of the individual who makes the decision as given by
Schubik [1964]. Indeed, scientific methodology and rational choice are not always
directly applicable to decision analysis. The decision maker constantly is concerned
with his environment, and always relates possible decision outcomes and their
consequences to its unique conditions. This concern with the environment context of
the decision results in modifications which further remove him from the classical
concept of economic man.
Decision making, purely based on past experiences, judgment and intuition has
become rather difficult. The human mind is also not capable of perceiving in all details
more than seven parameters, on an average, at a time. The decision making is no more
an art where the decision maker can apply mental models to find solution. It is
gradually becoming more and more scientific. In scientific decision making
mathematical models are applied to find solution to organizational problems.

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Today, effective and timely decisions are crucial for successful management of
organizations. The application of quantitative technique is, therefore, becoming more
useful. These techniques were found application to industries. In the present scenario,
the decision maker has to deal with vast data, number of alternatives and different
decision situations before taking any decision. At the same time, the rapid
diversification in industries is also adding to the complexity by making organizations
multi-objective type.
The main aim of decision making is measured by the degree of organizational
objectives achieved by the decision. Therefore, the organizational objectives provide
the foundation for decision making. Decisions are also constrained by environmental
factors such as government regulations, welfare of the public and long-run effects of
the decision on environmental conditions (i.e., pollution, quality of life, use of non-
renewable resources etc). In order to determine the best course of action, therefore, a
comprehensive analysis of multiple and often conflicting organizational objectives and
environmental factors must be undertaken. Indeed, the most difficult problem in
decision analysis is the treatment of multiple conflicting objectives Schubik [1964].
The issue becomes one of value trade-offs in the complex socio-economic structure of
conflicting structure of conflicting interests. Regardless of the type of problem on
hand, it is extremely difficult to answer questions such as what should be done now,
what can be deferred, what alternatives are to be explored, and what should be the
priority structure for the objectives?

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Consequently, one of the most important and difficult aspects of any decision
problem is to achieve an equilibrium among multiple and conflicting interests and
objectives of various components of the organization. Many recent researches
concerning the future of the industrialized society have echoed the same theme. When
the society is based on enormous technological development and change, stability of
the system must be obtained by achieving a delicate balance among such multiple
objectives as industrial output, food production, pollution control, population growth,
and use of natural resources, international co-operation for economic stability, and
civil rights and equal opportunity provisions. There is obviously a need for continuous
research in the analysis of multiple conflicting objectives.
ecision maker is regarded
as one who attempts to achieve a set of objectives to the fullest possible extent in an
environment of conflicting interests, incomplete information, and limited resources as
studied by Simon [1956]. To handle multi-objective decision making a unique
-making problems. The
advantage of using goal programming over other techniques is with dealing with real-
world decision problems is that it reflects the way manages actually make decisions.
Goal programming allows decision maker to incorporate environmental,
organizational, and managerial consideration into model through goal levels and
priorities. Goal Programming, although far from a panacea, often represents a
substantial improvement in the modeling and analysis of the real life situation. The

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present state-of-the-art in the field permits the systematic analysis of a class of
(deterministic) multi-objective problems that may involve both linear or nonlinear
functions and continuous or discrete variables. Further, the general goal programming
model provides a relatively reasonable structure under which the traditional, single
objective tools (such as linear and nonlinear programming) may be viewed simply as
special cases.
Interest in goal programming has increased significantly in the recent past, as
has its actual implementation. The initial development of the concept of goal
programming was due to Charnes and Cooper, in a discussion of which appeared in
1961 although Charnes, Cooper and Ferguson claim that the idea actually originated in
1955. In essence they proposed a model and approach for dealing with certain linear
programming problems in which
constraints. Since it might well be impossible to satisfy exactly all such goals, one
attempts to minimize the sum of the absolute values of the deviation from such goals.
Goal programming now encompasses any linear, integer, zero-one, or nonlinear multi-
objective problem, for which preemptive priorities may be established, the field of
application is increasing rapidly.
The goal programming model is also formulated and entered in a similar
manner as for linear programming, the difference being that the details of all the
objective functions are entered in the desired priority. Another approach to goal
programming is to state the goals as constraints in addition to the normal constraints of

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the problem. The objective function is then to minimize the deviation from the stated
goals. The deviations represented by the objective function are given weights as
coefficients in accordance with priorities assigned to the various goals. The problem is
then solved using the linear programming model; hence sensitivity analysis is also
feasible.
Therefore, the goal programming is one of the mathematical tools, designed in
context of solving the multi-objective problems in different areas for taking the
efficient, timely and accurate decision. The various researches have been made so far
and the researchers have been continually exploring this field for more than five
decades and even today the process is on to gets a lucid picture of this tool attributing
to clearly understanding the meaning of this technique in the perspective of problem
solving relating to industry.
1.4 REVIEW OF RELATED RESEARCH
In order to solve such multi dimensional planning problems, a flexible and
practical methodology, known as goal programming, was conceived by Charnes and
Cooper [1961]. The tool was extended and enhanced by their student and, later, by
other investigators, most notably Ijiri [1965], Jaaskelanen [1969], Lee and Clayton
[1970], Ignizio [1976], Gass [1986], Romero [1991], Tamiz and Jones [1996]. Since
then many researchers have done a lot of work about extensions of goal programming
methodology such as preemptive/lexicographic linear goal programming, integer goal
programming, zero-one goal programming by Schniederjans and Hoffman, [1992],

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extended lexicographic goal programming by Romero, [2001], etc, and extensive
surveys of fields of its applications by Lee, [1972]; Schniederjans, [1995]; Tamiz et
al., [1998] such as production planning, financial planning, capital budgeting planning,
etc.
The scope of this literature review is limited to applications of goal
programming in industry. A summary of the selective literature highlighting the
specific problem type with the identified multiple objectives and the solution
methodology followed is presented. Baran et al. [2013] formulate a goal programming
model by using genetic algorithm to solve economic-environmental electric power
generation problem with interval-valued target goals. Dean and Schniederjans [1990]
applied a goal programming approach to production planning for flexible
manufacturing systems. Ghosh et al. [2005] formulate a goal program in nutrient
management for rice production in West Bengal. Golany et al. [1991] proposed a goal
programming inventory control model applied at a large chemical plant. This proposed
model yielded an efficient compromise solution and the overall levels of decision
making satisfaction with the multiple fuzzy goal values. Larbani and Aouni [2011]
presented a new approach for generating efficient solutions within the goal
programming model followed by the efficient test for the goal programming solution.
Leung and Ng [2007] presents a goal programming model for production planning of
perishable products. Mukherjee and Bera [1995] discussed the solution of a project
selection by applying goal programming technique. Sen and Nandi [2012a] applied a
goal programming approach to rubber plantation planning in Tripura. Sen and Nandi

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[2012b] formulated an optimal model by using goal programming for rubber wood
door manufacturing factory in Tripura. Sen and Nandi [2012c] reviewed goal
programming and its application in plantation management. Sinha and Sen [2011]
made an attempt to formulate a strategic planning using the goal programming
approach to maximize production quantity to make tea, profit and demand and
minimize expenditure and processing time in different machines to Tea Industry of
Barak Valley of Assam in order to flourish the tea industries. Tamiz et al. [1996]
formulate an exploration of linear and goal programming models in the downstream
oil industry.
Leung and Chan [2009] developed a preemptive goal programming model for
aggregate production planning problem with different operational constraints. Sarma
[1995] studied lexicographic goal programming to solve a product mix problem in
large steel manufacturing unit.
Ghiani et al. [2003] proposed a mixed integer linear goal programming model
for allocation of production batches to subcontractors through fuzzy set theory in an
Italian textile company which resulted to outperform the hand-made solutions put to
use by the management so far. Lee et al. [1989] formulating industrial development
policies by a zero-one goal programming approach. Nja and Udofia [2009] formulated
the mixed integer goal programming model for flour producing companies. Pati et al.
[2008] formulated mixed integer goal programming model to assist in proper
management of the paper recycling logistics system. Silva da [2013] proposed multi-

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choice mixed integer goal programming optimization for real problems in a sugar and
ethanol milling company.
Belmokaddem et al. [2009] proposed a model to minimize total production and
work force costs, carrying inventory costs and rates of changes in work force using
fuzzy goal programming approach with different importance and priorities to
aggregate production planning. Fazlollahtabar et al. [2013] formulated a fuzzy goal
programming for optimizing service industry market using virtual intelligent agent.
Mekidiche et al. [2013] applied weighted additive fuzzy goal programming approach
to aggregate production planning. Petrovic and Akoz [2008] proposed a fuzzy goal
programming model for solving the problem of loading and scheduling of a batch
processing machine. Yimmee and Phruksaphanrat [2011] proposed a fuzzy goal
programming model for aggregate production and logistics planning for increase profit
and reduce change of workforce level.
Kumar et al. [2004] applied a fuzzy mixed integer goal programming
technique for solving the vender selection problem with multiple objectives. Tsai et al.
[2008] formulated a fuzzy mixed integer multiple goal programming problem
approach with priority for channel allocation problem in steel industry. Mustafa
[1989] applied an integrated hierarchical programming approach for industrial
planning. Arthur and Lawrence [1982] proposed a multiple goal production and
logistics planning in a chemical and pharmaceutical company.
Lee and Shim [1986] established priorities for small business by interactive
goal programming on the microcomputer. An interactive sequential goal

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programming; and an aggregate production planning model and application of three
multiple objective decision methods were proposed by Masud and Hwang [1981,
1980]. Sharma et al. [2010] proposed an interactive method of goal programming
along with AHP strategy for tracking and tackling environmental risk production
planning problem that minimizes damages and wastes in dairy production system.
1.5 DESCRIPTIONS OF GOAL PROGRAMMING MODELS
The formulation of goal programming problem is similar to that of linear
programming problems. According to Charnes and Cooper [1961], goal programming
extends the linear programming formulation to accommodate mathematical
programming with multiple objectives. It was refined by Ijiri in 1965. The major
differences are an explicit consideration of goals and the various priorities associated
with the different goals.
composed of deviational variables only. In the formulation two types of variable are
used. They are decision variables and deviational variables. There are two categories
of constraints. They are structural or system constraints (strict as in traditional linear
programming) and goal constraints, which are expressions of the original functions
with target goals, set priorities and positive and negative deviational variables.
The goal programming model may be categorized in terms of how the goals
are of roughly comparable importance, goal programming is known as non-
preemptive. In cases of preemptive goals programming, the goals are assigned priority

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levels. The goals are ranked from the most important (goal 1) to the least important
(goal m) and the objective function coefficient assigned for the (deviational) variable
representing goal is Pi. Rather they are convenient way of indicating that one goal is
important than the other. These coefficients indicate that the weight of goal 1 is much
larger about the value or cost of a goal or a sub goal, but often can determine its upper
or lower limits. The decision maker can determine the priority of the desired
attainment of each goal or sub goal and rank the priorities in an ordinal sequence.
Obviously, it is not possible to achieve every goal to the extent desired. Thus, with or
without goal programming, the decision maker attaches a certain priority to the
achievement of a particular goal. The true value of goal programming, therefore, is its
contribution to the solution of decision problems involving multiple and conflicting
1.5.1 General Goal Programming Model
Charnes and Cooper [1977] presented the general goal programming model
which can be expressed mathematically as:
1
m
i i
i
(1)
Subject to the linear constraints:
Goal constraints:
1
, 1,...,
n
ij j i i i
j
a x d d b for i m

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System constraints:
1
, 1,...,
n
ij j i
j
a x b for i m m p
with i
d , i
d , xj
where there are m goals, p system constraints and n decision variables
Z = objective function = Summation of all deviations
aij = the coefficient associated with variable j in the ith
goal
xj = the jth
decision variable
bi = the associated right hand side value
i
d = negative deviational variable from the ith
goal (underachievement)
i
d = positive deviational variable from the ith
goal (overachievement).
Both overachievement and underachievement of a goal cannot occur
simultaneously. Hence, either one or both of these variable must have a zero value;
that is,
.
Both variables apply for the non-negativity requirement as to all other linear
programming variables; that is,
.
Table 1.1 shows three basic options to achieve various goals:

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Table 1.1: Procedure for Achieving a Goal
Minimize Goal If goal is achieved
i
d Minimize the underachievement
i
d Minimize the overachievement
i i
Minimize both under- and
overachievement
The deviational variables are related to the functional algebraically as:
1 1
1
2
n n
i ij j i ij j i
j j
d a x b a x b
and
1 1
n n
i ij j i ij j i
j j
.
The GP model in (1) has an objective function, constraints (called goal
constrints) and the same nonnegative restriction on the decision variables as the LP
model. It should be mentioned that some GP researchers (see Ignizio 1985b) feel that
the term objective function is not an accurate term and the terms achievement function
or unachievement function should used in its place.

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1.5.2 Lexicographic Goal Programming Model
The initial goal programming formulations ordered the unwanted deviations
into a number of priority levels, with the minimization of a deviation in a higher
priority level being infinitely more important than any deviations in lower priority
levels. This is known as lexicographic (preemptive) or non-Archimedean goal
programming. Iserman [1982], Sherali [1982] and Ignizio [1983a] stated the
lexicographic goal programming model. Lexicographic goal programming should be
used when there exist a clear priority ordering amongst the goals to be achieved.
In preemptive goal programming, the objectives can be divided into different
priority classes. Here, it is assumed that no two goals have equal priority. The goals
are given ordinal ranking and are called preemptive priority factors. These priority
factors have the relationship P1 >>> P2 i >>> Pi+1 m where
the P1 goal is so much more important than the P2 goal and P2 goal will never be
attempted until the P1 goal is achieved to the greatest extent possible. The priority
relationship implies that multiplication by n, however large it may be, cannot make the
lower-level goal as the higher goal (that is, Pi > Pi+1).
The model can be stated as:
1
m
i i i
i
(2)

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Goal constraints:
1
, 1,...,
n
ij j i i i
j
a x d d b for i m
System constraints:
1
, 1,...,
n
ij j i
j
a x b for i m m p
with di
+
, di
-
, xj
where there are m goals, p system constaints, k priority levels and n decision variables
Pi = the preemptive priority factors of the ith
goal.
Here, the difference between equations (1) and (2) is the priority factors in the
objective function.
1.5.3 Weighted Goal Programming Model
If the decision maker is more interested in direct comparisons of the objectives
then weighted goal programming should be used. The weighting of deviational
variables at the same priority level shows the relative importance of each deviation.
Charnes and Cooper (1977) stated the weighted goal programming model as:
1
m
i i i i
i
(3)
Goal constraints:
1
, 1,...,
n
ij j i i i
j
a x d d b for i m
System constraints:
1
, 1,...,
n
ij j i
j
a x b for i m m p

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with di
+
, di
-
, xj
where i
w and i
w are non-negative constants representing the relative weight to be
assigned to the respective positive and negative deviation variables. The relative
weights may be any real number, where the greater the weight the greater the assigned
importance to minimize the respective deviation variable to which the relative weight
is attached. This model is a non-preemptive model that seeks to minimize the total
weighted deviation from all goals stated in the model.
While Ijiri (1965) had introduced the idea of combining preemptive priorities
and weighting, Charnes and Cooper (1977) suggested the goal programming model as:
1 1
i
n
m
i ik i ik i
i k
(4)
Goal constraints:
1
, 1,...,
n
ij j i i i
j
a x d d b for i m
System constraints:
1
, 1,...,
n
ij j i
j
a x b for i m m p
with di
+
, di
-
, xj
where and represent the relative weights to be assigned to each of the
i different classes within the ith category to which the non-Archimedean
transcendental value of Pi is assigned.

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1.5.4 Chebyshev Goal Programming Model
Chebyshev or fuzzy goal programming model was introduced by Flavell
[1976]. It uses to minimize the maximum unwanted deviation, rather than the sum of
deviations. For this reason Chebyshev goal programming is sometimes termed
Minmax goal programming. This utilizes the Chebyshev distancemetric, which
emphasizes justice and balance rather than ruthless optimization.
The preemptive lexicographic GP model in (2) and the non-preemptive
weighted GP model in (3) can view as the two extreme types of GP models in which
virtually all GP modeling are derived.
1.6 RELATIONSHIP OF GP TO MCDM
Multiple criteria decision making (MCDM) is a term used to describe a
subfield in operations research and management science. Zionts [1992] generally
defined MCDM as a means to solving decision problems that involve multiple
(sometimes conflicting) objectives. While that definition also applies to GP, MCDM is
a substantially broader body of methodologies of which GP is a small subset.
Furthermore, GP can provide a unifying basis for most MCDM models and methods.
With this purpose, extended lexicographic goal programming has recently been
proposed.
The various points of origin, methodology and future directions for MCDM
can be found in Starr and Zeleny [1977], Hwang et al. [1980], Rosenthal [1985],
Steuer [1983] and more in Dyer [1973], Fishburn [1974], Steuer [1986], Zionts and

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antially
described in a variety of publications including Romero [1991] and Ringuest [1992].
On the conceptual level the relationship of MCDM and GP can be seen in what
Zionts [1992] calls the four subareas that make up MCDM. These four subareas that
comprise MCDM are listed in Table 1.2. According to Zionts [1992] the subarea of
multiple criteria mathematical programming refers to solving primarily deterministic,
mathematical programming problems that have multiple objectives. Linear goal
programming is one of the many methodologies that are considered a significant
contributor to this subarea of MCDM. Indeed, Zoints and Wallenius [1976] suggest
that the development of GP was a beginning point for MCDM, particularly this
subarea. How can one distinguish a GP model from the other multiple criteria
mathematical programming models? In most cases, the MCDM models in this subarea
have decision variables in their objective function, while GP models do not.
Table 1.2: MCDM Subarea and Their Related GP Topics
MS/OR Subarea Related GP topics
Multiple Criteria Mathematical Programming Linear Goal Programming
Multiple Criteria Discrete Alternatives Integer Goal Programming and Zero-One
Goal programming
Multiattribute Utility Theory Linear Goal Programming, Nonlinear GP and
Fuzzy GP
Negotiation Theory Interactive Goal Programming

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1.7 GOAL PROGRAMMING FOR MULTIPLE-OBJECTIVE DECISION
ANALYSIS
One of the most promising techniques for multiple objective decision analysis
is goal programming. Goal programming is a powerful tool which draws upon the
highly developed and tested technique of linear programming, but provides a
simultaneous solution to a complex system of competing objectives. Goal
programming can handle decision problems having a single goal with multiple sub
goals. The technique was originally introduced by Charnes and Cooper [1961], and
further developed by Jaaskelainen [1969], Lee and Bird [1970], Lee [1972] and
Ignizio [1976]. Then many researchers such as Kwak and Schniederjans [1979, 1985],
Ignizio [1987, 1989], Hallefjord and Jornsten [1988], Reaves and Hedin [1993],
Hemaida and Kwak [1994], Bryson [1995], Easton and Rossin [1996] etc., surveyed,
case study and applications of goal programming and multiple criteria decision
making (MCDM) and concentrate his views for overview of techniques for solving
multiple objective mathematical programming problems. However, the classification
of MCDM methods given by Zanakis and Gupta [1985], Steuer [1986], Romero
[1986], Tamiz and Jones [1995] etc. is usual practice to differentiate methods based on
the classifications of the problem. MCDM is an extremely important discipline that
deals with decision making problem with multiple objectives. Often goals set by
management compete for scarce resources. Furthermore, these goals may be
incommensurable.

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Thus, there is a need to establish hierarchy of importance among these
conflicting goals so that low order goals are satisfied or have reached the point beyond
which no further improvements are desirable. If the decision maker can provide an
ordinal ranking of goals in terms of their contributions or importance to the
organization and if all relationships of the model are linear, the problem can be solved
by goal programming.
In goal programming, instead of attempting to maximize or minimize the
objective criterion directly, as in linear programming, the deviations between goals
and what can be achieved within the given set of constraints are minimized. In the
simplex algorithm of linear programming such deviations are called slack variables.
These variables take on a new significance in goal programming. The deviational
variable is represented in two dimensions, both positive and negative deviations from
each sub goal or goal. Then the objective function becomes the minimization of these
deviations based on the relative importance or priority assigned to them.
The solution of any linear programming problem is based on the cardinal value
such as profit or cost. The distinguishing characteristic of goal programming is that it
allows for an ordinal solution. The decision maker may be unable to obtain
information about the value or cost of a goal or a sub goal, but often can determine its
upper or lower limits.

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1.8 GOAL PROGRAMMING SOLUTION METHODOLOGY
Goal Programming was introduced i There was computer
software (or computers) to help support the growth of this computationally dependent
methodology. The GP software required the availability of GP algorithms used to
generate the primary GP problem solutions. In addition, a collection of supporting
algorithms are also necessary to permit a post- solution analysis or secondary
consideration of the solutions obtained in the primary solution. Collectively, these
primary and secondary algorithms can be called GP solution methodologies.
The purpose of this chapter is to review all of the various types of GP solution
methodologies that have appeared in this thesis. This review includes the primary GP
algorithms and methodology used to generate linear GP, integer GP and nonlinear GP
solutions. In addition, secondary GP methodologies including duality and sensitivity
analysis used to obtain post-solution information will also be discussed.
1.8.1 Primary GP Solution Methodology
There are many different methodologies and algorithms used to generate
solutions for GP models. We will begin by categorizing them into four groups of
Linear GP (which includes all linear based GP solution methods), Integer GP (which
includes methodology used to generate all integer, mixed integer and zero-one integer
solutions), Nonlinear GP (which includes all nonlinear based GP solution methods),
and a final other group for all methodology that does not fit into the other three
groups.

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1.8.1.1 Linear Goal Programming Algorithms and Methodology
The first linear GP algorithm is actually an LP algorithm. The methodological
proof for solving LP models structured as GP problems can be found in Charnes and
Cooper [1961, pp. 210-215]. With the improvements of preemption, the generalized
inverse approach and the illustrative use of the simplex based algorithm by Iziri
[1965], as well as the publication of a software program by Lee [1972], substantially
increased linear GP research in methodological improvements. While it was assumed
the LP proof by Charnes and Cooper [1961] was sufficient to justify the mathematical
workings of GP algorithms, it is interesting to note that no mathematical proof of a
simplex-based linear GP methodology actually appeared until Evans and Steuer
[1973].
Some GP algorithms can only be used with a single type of GP model; others
have been designed to handle a wider variety of GP models. This logic has been taken
to the extreme in MULTIPLEX model and algorithm see Ignizio [1985a], which
claims to be able to work with LP, weighted GP, preemptive GP and fuzzy GP
models.
The basic algorithms used to solve the weighted GP, preemptive GP and their
combinations are available in Ignizio [1976, 1982], Iziri [1965], Lee [1972] and
Schniederjans [1984]. Other extensions of methodology can be found in Table 1.3.

27
Table 1.3: Citations on Weighted/Preemptive GP Methodology
Reference What Reference Provides
Alp and Murray [1996], Arthur and Ravindran [1978],
Bryson [1995], Kwak and Schniederjans [1982,
1985a], Leunge and Chan [2009], Leung and Ng
[2007], Pati et al. [2008], Schniederjans and Kwak
[1982], Sharma et al. [2010], Tamiz et al. [1996]
Reduced size algorithms
Freed and Glover [1981a, 1981b] Used as discriminant
analysis
Charnes and Cooper [1977], Evans and Steuer [1973],
Hwang et al. [1980], Larbani and Aouni [2011],
Romero [2001]
Mathematical proofs for GP
Schenkerman [1991] Discussion on weighted GP
Bhargava et al. [2011], Kettani et al. [2004], Knoll and
Engelberg [1978], Kluyver [1979], Sherali [1982],
Shim and Siegel [1975], Spivey and Tamura [1970],
Steuer [1979], Widhelm [1981]
Weighted GP methodologies
Arthur and Ravindran [1980], Charnes and Cooper
[1961], Dauer and Krueger [1977], Ignizio [1976
book, 1982 book, 1985c], Iserman [1982], Iziri [1965
book], Lee [1972 book], Schniederjans [1994 book]
Algorithms for both models

28
Lee [1983], Lee and Rho [1979b, 1985] Decomposition
methodologies
Crowder and Sposito [1987], Ignizio [1985a, 1987] Solution by dual solution
Akgul [1984], Alvord [1983], Baran et al. [2013],
Clayton and Moore [1972], Gibbs [1973], Hindelang
[1973], Ignizio [1978, 1983a], Rifai [1994], Ruefli
[1971], Tamiz and Jones [1996, 1998]
General discussion of issues
1.8.1.2 Integer Linear GP Algorithms and Methodology
In GP problem situations where decision variables are restricted to integer
values, special integer GP methodologies were developed. Most of the GP
methodologies are based on integer LP methodologies. For example, in all or mixed
integer LP problems one of the most common integer methodologies is the branch-
and-bound solution method. Arthur and Ravindran [1980] developed their branch-and-
bound integer GP algorithm on this same LP algorithm. In the case of zero-one LP
integer solutions the most commonly approach is some type of enumeration method.
Garrod and Moores [1978] developed their zero-one GP solution methodology using
the same approach. Ali et al. [2011] applied an integer goal programming approach for
finding a compromise allocation of repairable components. Tamiz et al. [1999]
analyses the extension of Pareto efficiency to integer goal programming.

29
1.8.1.3 Nonlinear GP Algorithms and Methodology
According to Saber and Ravindran [1993] there are four major approaches to
nonlinear GP:
(1) Simplex based nonlinear GP
(2) Direct search based nonlinear GP
(3) Gradient search based nonlinear GP
(4) Interactive approaches to nonlinear GP
We will discuss each of these in this section.
(1) The simplex based nonlinear GP approaches include the method of
approximation programming, which was developed by Griffith and Steward [1961]
adapted by Ignizio [1976] for GP. This methodology permits nonlinear goal
constraints to be included in a GP model.
Another simplex based approach to nonlinear GP is called separable
programming. Originally developed by Miller [1963], this approach was modified for
GP by Wynne [1978]. This methodology allows nonlinear goal constraints to be
included in the GP model by restricting the range of the decision variables into
separable functions that are assumed linear. This methodology is based on the logic of
piece-wise linear approximations.
Still another simplex based approach to nonlinear GP is quadratic goal
programming. Quadratic GP permits quadratic goal constraints and quadratic
deviation variables in the objective function. For a good review of the mechanics of

30
nonlinear GP algorithms and methodology see (Ringuest and Gulledge [1982] and
Gupta and Sharma [1989]).
(2) Direct search based nonlinear GP methods utilize some type of logical search
pattern or methods to obtain a solution that may or may not be the best satisfying
solution. The logic process is based on repeated attempts to improve a given solution
by evaluating its objective function and/or goal constraints. The basic search idea
originated with Box [1965], but was applied to GP by many others including Nanda et
al. [1988]. Hooke and Jeeves [1961] developed a single objective, continuous variable,
unconstrained optimization method that was later adopted for GP by Ignizio [1976]
and Hwang and Masud [1979].
(3) Gradient based nonlinear GP methods use calculus or partial derivatives of the
nonlinear goal constraints or the objective function to determine the direction in which
the algorithm is to search for a solution and the amount of movement necessary to
achieve that solution. While gradient based methods are generally more efficient in
obtaining a solution, they may not be appropriate for GP models whose goal
constraints or objective function is nondifferentiable.
Lee [1985a], Lee and Olson [1985], and Olson and Swenseth [1987] all
developed a version of a gradient method for GP called the chance constraint method.
The chance constraint method allows parameters to be distributed along a probability
distribution. The introduction of the probability distribution is where this methodology
obtained its probabilistic or chance name. The use of the chance constraint method
requires the assumption that the technological coefficients are normally distributed.

31
Another gradient based method is called the partitioning gradient method.
Developed for linear GP by Arthur and Ravindran [1978] using a simplex based
approach, this methodology can be highly efficient in obtaining nonlinear GP
solutions. It works on the basis of finding smaller subproblems that lead to an optimal
solution. By solving these smaller problems and eliminating decision variables form
the model, the size of the model is reduced. A special version of the gradient based
method is called the decomposition method. The decomposition method can solve
linear GP or nonlinear GP problems. It is usually based on some version of the
Dantzig and Wolfe [1960] LP decomposition method, where large models are
decomposed into smaller sub models whose solution will be used to generate the
solution to the original larger model. Algorithms and research on the decomposition
methods for GP can be found in Ruefli [1971], Sweeney et al. [1978], Lee [1983], and
Lee and Rho [1979a, 1979b, 1985, 1986].
One special type of nonlinear GP methodology can be called stochastic goal
programming. In a stochastic GP model there are probability distributions present to
methodology for example can be used to model and solve some classes of stochastic
GP problems. For a good review of the basics see Contini [1968].
(4) Interactive approaches to nonlinear GP or interactive GP can be defines as a
collection of methodologies that are based on progressive articulation of a decision
decision maker using interactive GP will be lead to a better solution by interactively

32
comparing a given solution. This makes interactive GP a sequential search process,
but one that involves periodic feedback to the decision maker to guide the direction of
the search. The term sequential goal programming (SGP) is often used with the
interactive approach to better describe the step-wise nature of this methodology.
Interactive GP has been used for all types of GP models (i.e., linear GP, integer
GP and nonlinear GP). Any of the methods that are used to solve GP problems can be
used as an interactive, sequential GP search methodology. For an excellent review if
the mechanics of the various methods see Van and Nijkamp [1977], Spronk [1981].
1.8.1.4 Other GP Algorithms and Methodology
There are at least four other algorithm based methodologies that are
extensively represented in GP literature:
(1) Interval GP
(2) Fractional GP
(3) Duality solution
(4) Fuzzy GP
Each of these other methodologies can and often are used with linear GP,
integer GP and nonlinear GP models. They also offer unique modeling features that
have distinguish them in their right.
(1) Interval GP: Interval GP allows parameters, particularly the right-hand-side goal

33
values to be expressed on an interval basis. This method is based on interval LP,
where an upper boundary, bu and lower boundary, bl for the right-hand-side values can
be stated as:
bl ijxj u (1)
So the interval GP equivalence would be accomplished with two goal
constraints:
aijxj du
+
+ du
-
= bu (2)
aijxj dl
+
+ dl
-
= bl (3)
where the du
+
and dl
-
are both minimized in the objective function and the other
deviation variables are free to permit some compromised value for the resulting right-
hand-side value. This method can be used to deal with a variety of formulation issues
that are used to criticize GP models, such as the inappropriateness of predetermined
goals or targets see Min and Storbeck [1991]. Vitoriano and Romero [1994] proposed
the extended interval goal programming.
(2) Fractional GP: Fractional GP is a methodology used when modeling ratios. In a
variety of situations such as modeling return on investment problems, market share
problems or percentage type problems, fractional GP may be the most appropriate of
the GP methodologies. As Awerbuch et al. [1976] noted, there complexities in GP
model formulations that make simple multiplication of goal constraints an invalid
means for dealing with fractional values. For a review of some of the controversy see

34
Hannan [1977, 1981] and Soyster and Lev [1978]. Fractional GP is also an extension
of LP, called fractional LP see Martos [1964], Bitran and Novaes [1973].
(3) Duality Solution: It has been shown that GP models can be solved more
efficiently and without some computational problems by solving the dual formulation
of the GP model see Dauer and Krueger [1977], Ignizio [1985a]. This method is not
without its problems as observed by Crowder and Sposito [1987] and replied to by
Ignizio [1987]. An interesting extension of this method to sequential nonlinear GP can
be seen in El-Dash and Mohamed [1992].
(4) Fuzzy GP: Fuzzy GP is based on fuzzy set theory. Fuzzy sets are used to describe
imprecise goals. These goals are usually associated with objective functions and are
used to reflect both a weighting (with values from zero to one) and range of goal
achievement possibilities. The numerical relationship between the goal of profit and
utility in the profit occurrences. The relationship between the weighting and the profit
function can be linear or nonlinear. Most importantly, this methodology allows the
decision maker who cannot precisely define goals to at least express them using a
weighting structure that is not limited. This makes fuzzy programming an idea
approach when utility function type goals are to be used in the GP model.
Fazlollahtabar et al. [2013] proposed a fuzzy goal programming model for optimizing
service industry market by using virtual intelligent agent. Kumar et al. [2004]
approached a fuzzy goal programming for vendor selection problem in a supply chain.
Mekidiche et al. [2013] approached a weighted additive fuzzy goal programming to

35
aggregate production planning. Yimmee and Phruksaphanrat [2011] proposed fuzzy
goal programming for aggregate production and scientists.
1.8.2 Secondary GP Solution Methodology
Two extensions of LP are duality and sensitivity analysis. These extensions
exist in GP as well but with some unique characteristics.
1.8.2.1 Duality in GP
In LP models we seek to determine the marginal contribution (also called the
dual decision variable) of each of the right-hand-side values in terms of the single
objective function units (Fand and Puthenpura [1993], pp. 56-72). The same basic
simplex process is used in GP duality to derive the marginal contribution of each
right-hand-side values. A variety of GP concepsts and methodologies on duality can
be found in Markowski and Ignizio [1983a, 1983b], Ogryczak [1986, 1988b] and
Martinez-Legaz [1988]. An exception that makes GP duality different is that its
interpretation of the resulting marginal contribution is somewhat different from LP.
The marginal contributions of right-hand-side values or goals in GP models take on a
multidimensional characteristic see Ignizio [1984b]. The interpretation of the marginal
contribution in GP models has to be in terms of all of other goals in the model. That is,
the marginal contribution of one goal in terms of all other goals. An excellent
discussion of the mechanics and interpretations can be found in Ignizio ([1982],
Chapter 18).

36
Other studies have extended duality analysis in GP. An iterative algorithm with
its dual formulation was discussed by Dauer and Krueger [1977]. Ben-Tal and
Teboulle [1986] added an even greater degree of complexity to the use of duality with
a stochastic, nonlinear GP model. As previously mentioned, dual formulations for GP
models have also been shown to enhance computational efficiency for solving GP
problems when compared to other standard algorithms see Dauer and Krueger [1977],
Ignizio [1985].
1.8.2.2 Sensitivity Analysis in GP
According to Ignizio ([1982], Chapter 19) there are seven types of changes that
can be implemented as a part of sensitivity analysis in GP:
1. Changes in the weighting at a priority level
2. Changes in the weighting of deviation variables within a priority level
3. Changes in the right-hand-side values
4. Changes in technologies coefficients
5. Changes in the number of goals
6. Changes in the number of decision variables
7. Reordering preemptive priorities
Most of these have been illustrated by application see in Lee [1972], Ignizio
[1982], Schniederjans [1984], all provide basic methodologies for undertaking these
seven types of sensitivity analysis in GP models.

37
1.8.3 Computer Software Supporting GP Solution Analysis
Decision Analysis, the
computer coding for a FORTRAN program presented the first published source of
software for all the various types of weighted and preemptive linear GP models. Other
ode that
came later helped to broaden software capabilities to included LP algorithms as a part
of a package of software. Other specialized computer codes whose ability to deal with
a smaller subset of GP problem soving have been developed over the years.
Unfortunately, most such codes do not end up in journal publications and even
their applications are not always reported until years after they appear in the literature.
The Lee [1993] used the AB:QM, version 3.1, microcomputer or PC which was more
adequate to do solve the small problems. The AB:QM software can handle a 50 goal
constraint by 50 decision variable or (50 row × 50 column) GP model. It does not
handle integer GP problems or nonlinear GP models, unless those models can be
converted into the linear GP equivalent. AB:QM also does not provide duality or
sensitivity analysis for GP models. Excel solver are most commonly used in GP
models.
Out of total of 112 identified developers of LP or some type of mathematical
programming software application, only 15 developers actually claimed that GP type
models could be processed by their software. There are list of some advanced
computer software of largest size are given here.

38
Table 1.4: Computer Software Applications That Support GP Solution Analysis
Software Publisher System Features
AMPL Boyd & Fraser Pub.
One Corp. Place
Ferncroft Village
Danvers, MA 01923
Linear GP, Integer GP, Nonlinear GP,
Duality, Sensitivity Analysis
CPLEX Mixed
Integer Optm.
CPLEX Optm. Inc.
930Tahoe Blvd.
Incline Village,
NV89451
GAMS Boyd & Fraser Pub.
One Corp. Place
Ferncroft Village
Danvers, MA 01923
Sensitivity Analysis
Extended LINDO LINDO Systems
1415 N. Dayton Str.
Chicago, IL 60622
Linear GP, Integer GP, Duality, Sensitivity
Analysis
Extended GINO LINDO Systems
1415 N. Dayton Str.
Chicago, IL 60622
Linear GP, Nonlinear GP, Duality,
Sensitivity
Analysis
HS/LP Haverly Syst. Inc. P.
O. Box 919 Denville,
NJ 07834
Sensitivity
Analysis

39
MINOS, NPSOL and
LSSOL
Stanford Business
Software Inc. 2672
Bayshore Pkwy. Mtn.
View. CA 94043
Sensitivity
Analysis
MPSX-MIP/370 Altium, of IBM IBM
MS 936
Neighborhood Rd.
Kingston, NY 12401
Duality,
Sensitivity Analysis, Fuzzy GP
Solvers Frontline Systs. Inc.
P. O. Box 4288
Incline Village, NY
89450
XPRESS-MP Resource Optim.,
Inc. 531 S. Gay Str.
Ste 1212 Knoxville,
TN 37010-1520
1.9 NEW DEVELOPMENTS IN GOAL PROGRAMMING
Goal Programming (GP) is a powerful and flexible technique that can be
applied to a variety of decision problems involving multiple objectives. It should,
however, be pointed out that GP is by no means a panacea for contemporary decision
problems. The fact is that GP is applicable only under certain specified assumptions

40
and conditions. Most GP applications have thus far been limited to well-defined
deterministic problems. Furthermore, the primary analysis has been limited to the
identification of an optimal solution that optimizes goal attainment to the extent
possible within specified constraints. In order to develop goal programming as a
universal technique for modern decision analysis many refinements and further
research are necessary.
After the study of the literature, we have been able to identify a number of
models and solution techniques in goal programming that have been developed and
used in problem solving. These various models and techniques of goal programming
are identified on page 42.
1.10 CONCLUSION
Virtually all models developed for managerial decision analysis have neglected
the unique organizational environment, bureaucratic decision process, and multiple
conflicting natures of organizational objectives. In reality, however, these are
important factors that greatly influence the decision process. In this study, the goal
programming approach is discussed as a tool for the optimization of multiple
objectives while permitting an explicit consideration of the existing decision
environment.
In the nearly half century since its development, goal programming has
-objective

41
optimization field. This is due to a combination of simplicity of the form and
practicality of approach.
This chapter conceptually described the relationship of goal programming (GP)
within the subject area of multi criteria decision making (MCDM). Furthermore, we
have discussed a variety of GP methodology. Included algorithms and methodology
designed to obtain a basic or primary solution for a problem. These primary types of
methods included linear GP, integer GP and nonlinear GP. Each of these types of
methodologies was subdivided into various other existing methodologies. This chapter
also discussed secondary GP methodologies including duality and sensitivity analysis.
List of computer software supporting GP solution are also included in this thesis.

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