an introduction to goal programming - Shodhganga

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 nonrenewable 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 realworld 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 multiobjective 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 nonpreemptive. 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: m i

(1)

i

i 1

Subject to the linear constraints: n

aij x j

Goal constraints: j 1

di

di

bi , for i 1,..., m

16

n

System constraints:

aij x j

bi , for i

m 1,..., m

p

j 1

with d i , d i , 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 d i = negative deviational variable from the ith goal (underachievement) di = 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

di

Minimize the underachievement

di

Minimize the overachievement

i

i

If goal is achieved

Minimize

both

under-

and

overachievement

The deviational variables are related to the functional algebraically as:

1 2

di

and

n

n

aij x j bi

aij x j bi

j 1

j 1

n

n

i

ij j 1

j

i

ij

j

i

.

j 1

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: m i

i

i

i 1

Subject to the linear constraints:

(2)

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n

Goal constraints:

aij x j

di

di

bi , for i 1,..., m

j 1

n

aij x j

System constraints:

bi , for i

m 1,..., m

p

j 1

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: m i

i

i

(3)

i

i 1


Goal constraints:

aij x j

di

di

bi , for i 1,..., m

j 1

n

System constraints:

aij x j j 1

bi , for i

m 1,..., m

p

20

with di+, di-, xj where wi and wi 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: ni

m

i

ik

i

ik

(4)

i

i 1 k 1


Goal constraints:

aij x j

di

di

bi , for i 1,..., m

j 1

n

System constraints:

aij x j

bi , for i

m 1,..., m

p

j 1

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.

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Table 1.3: Citations on Weighted/Preemptive GP Methodology

Reference

What Reference Provides

Alp and Murray [1996], Arthur and Ravindran [1978], Reduced size algorithms 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] Freed and Glover [1981a, 1981b]

Used

as

discriminant

analysis Charnes and Cooper [1977], Evans and Steuer [1973], Mathematical proofs for GP Hwang et al. [1980], Larbani and Aouni [2011], Romero [2001] Schenkerman [1991]

Discussion on weighted GP

Bhargava et al. [2011], Kettani et al. [2004], Knoll and Weighted GP methodologies Engelberg [1978], Kluyver [1979], Sherali [1982], Shim and Siegel [1975], Spivey and Tamura [1970], Steuer [1979], Widhelm [1981] Arthur and Ravindran [1980], Charnes and Cooper Algorithms for both models [1961], Dauer and Krueger [1977], Ignizio [1976 book, 1982 book, 1985c], Iserman [1982], Iziri [1965 book], Lee [1972 book], Schniederjans [1994 book]

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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], General discussion of issues Clayton and Moore [1972], Gibbs [1973], Hindelang [1973], Ignizio [1978, 1983a], Rifai [1994], Ruefli [1971], Tamiz and Jones [1996, 1998]

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 branchand-bound solution method. Arthur and Ravindran [1980] developed their branch-andbound 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.

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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.

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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

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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 righthand-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

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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).

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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.

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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.

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Table 1.4: Computer Software Applications That Support GP Solution Analysis

Software

Publisher

System Features

AMPL

Boyd & Fraser Pub.

Linear GP, Integer GP, Nonlinear GP,

One Corp. Place

Duality, Sensitivity Analysis

Ferncroft Village Danvers, MA 01923 CPLEX Mixed

CPLEX Optm. Inc.


Integer Optm.

930Tahoe Blvd.


Incline Village, NV89451 GAMS

Boyd & Fraser Pub.


One Corp. Place

Sensitivity Analysis

Ferncroft Village Danvers, MA 01923 Extended LINDO

LINDO Systems

Linear GP, Integer GP, Duality, Sensitivity

1415 N. Dayton Str.

Analysis

Chicago, IL 60622 Extended GINO

HS/LP

LINDO Systems

Linear GP, Nonlinear GP, Duality,

1415 N. Dayton Str.

Sensitivity

Chicago, IL 60622

Analysis

Haverly Syst. Inc. P.


O. Box 919 Denville,

Sensitivity

NJ 07834

Analysis

39

MINOS, NPSOL and Stanford Business LSSOL


Software Inc. 2672 Bayshore Pkwy. Mtn.

Sensitivity Analysis

View. CA 94043 MPSX-MIP/370

Altium, of IBM IBM


MS 936

Duality,

Neighborhood Rd.

Sensitivity Analysis, Fuzzy GP

Kingston, NY 12401 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

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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.