jurnal GPsepty

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R jeas Research Journal in Engineering and Applied Sciences 2(2) 95-105 R jeas Emerging Academy Resources (2013) (ISSN: 2276-8467)

www.emergingresource.org

95

GOAL PROGRAMMING: - AN APPLICATION TO BUDGETARY ALLOCATION

OF AN INSTITUTION OF HIGHER LEARNING

Ekezie Dan Dan And Onuoha Desmond .O.Department of Statistics,

Imo State University, PMB 2000, Owerri Nigeria.

Corresponding Author:E kezieD an D an__________________________________________________________________________________________

In this paper we discuss the application of Goal Programming in Budgetary allocation of Institutions of higher

learning using Imo State University Owerri as a case study. This paper shall impact positively to the readingcommunity in that, it will enlighten the masses that with the use of Goal programming problems, the aims of an

organization can be achieved financially and otherwise. Data on the budget estimates of Imo State UniversityOwerri were collected from Imo State Ministry of Finance and Imo State University Bursary Department

between 2006 and 2008. Five goals in the budget estimates of the University namely; Personal cost, Overhead

cost, Capital expenditure, Revenue (internally generated) and the Total budget were considered for the study inorder of precedence (priorities). The data collected were used to formulate a goal programming problem and theformulated problem was used solved using Simplex method (using TORA software). Based on the solution

obtained, it was discovered that the optimum value of Z (Z=4.24) satisfied goal 1(the personal cost goal), goal

3(capital expenditure goal) and goal 5(the total budget goal), but failed to satisfy goal 2 and goal 4, which are

Overhead cost and Revenue goals respectively. From the findings, it was concluded that the Imo StateUniversity, Owerri should come within 4.24 billion naira to satisfy goal 2 and goal 4, which are Overhead cost

and Revenue goals respectively. It was also recommended that the University should have a minimum budget of4.24 billion naira in 2010 and should be reviewed upward annually, which should be properly and timely

monitored by active Government budget monitoring team.

Emerging Academy Resources

KEYWORDS: Goal Programming, Budgetary Allocation, Aspiration level and Goal Programming Algorithm.

__________________________________________________________________________________________INTRODUCTION

Budgetary allocation is a complex task that requires

cooperation among multiple functional units in anyinstitution. There is need to have a sound knowledge

of organizational budgeting processes in order todesign an efficient and effective budgetary allocation

model. Despite the fact that such allocation procedure

exists in the University, it is not properly structureddue to the presence of multiple conflicting objectives.

A formal decision analysis that is capable ofhandling multiple conflicting goals through the use

of priorities is the Goal Programming Model.

Goal programming (GP) is an extension of Linear

Programming (LP) which is a mathematical tool tohandle multiple, normally conflicting objectives.

For example,(i) The Vice Chancellor of any NigerianUniversity may decide to increase capital

expenditure and simultaneously reduce

revenue.

(ii) A Governor of any Nigerian State maypromise to reduce the State debt andsimultaneously offer tax relief to workers.

In such situations, it will be difficult to find a single

solution that optimizes the conflicting objectives.

Goal Programming provides a way of strivingtoward such conflicting objectives simultaneously.

According to Ignizio (1978), Goal Programming is a

tool that has been proposed as a model and approachfor analysis of problems involving multiple

conflicting objectives.

He pointed out that actual real world problems

invariably involve non-deterministic system forwhich a variety of conflicting non-commensurable

objectives exist.

The major strength of Goal Programming is its

simplicity and ease of use. This accounts for thelarger number of Goal Programming applications in

many and diverse areas such as in management ofsolid waste, accounting and financial aspect of stock

management marketing, quality control, human

resources, production, transportation and site

selection, space studies, agriculture,

telecommunication, forestry and aviation (Aouni andKettani, 2001).

Goal programming problems can be solved by widelyavailable linear programming computer packages as

either a single linear programming, or in the case of


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Research Journal in Engineering and Applied Sciences (ISSN: 2276-8467) 2(2):95-105Goal Programming: - An Application To Budgetary Allocation Of An Institution Of Higher Learning

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lexicographic variant, a series of connected linearprogramming. Goal programming can therefore

handle relatively large number of variables,constraints and objectives.

TYPES OF GOALS

There are three possible types of goals:

A lower, one-sided goal: - This goal sets a lowerlimit that we do not want to fall under (but

exceeding the limit is acceptable).

An upper, one-sided goal: - This goal sets anupper limit that we do not want to exceed (butfalling under the limit is acceptable).

A two-sided goal: - This goal sets specific targetsthat we do not want to miss on either side.

VARIANTS

Goal programming formulations ordered theunwanted deviations into a number of priority levels,

with the minimization of a deviation in a higher

priority level being of infinitely more importance

than any deviation in lower priority levels. This is

known as lexicographic or preemptive goalprogramming.

Ignizio (1976) gave an algorithm that shows how a

Lexicographic Goal Programming (LGP) can besolved as a series of linear programming model.

Lexicographic Goal Programming should be used

when there is a clear priority ordering amongst thegoals to be achieved.

BUDGETING

Revenue budgeting is an approach to the budget

decision rather than a particular budget system.However, revenue budgeting emphasizes on the

ascendancy of the revenue constraint in budgetingcalculations. Decision makers are constrained by

actual limitation on revenue raising power and/or theperception of impending limitations and fears aboutthe revenue sources.

The aims are:

To increase personnel cost (salary andallowance of staff).

To reduce overhead cost; To increase capital expenditure; To increase revenue (internally generated); To reduce the total budget.

STATEMENT OF THE PROBLEM:

The statements of the problem are as follows:

Capital and revenue were allocatedinadequately, and without order ofimportance. This inadequate allocation was

due to not using powerful quantitativeallocation models.

It is observed that allocated funds were notproperly utilized with the result that moneyallocated for the laboratories for example is

diverted into hostel maintenance. In thesame vain, other diversions also take place.

It is not a hidden fact that, the fundsallocated to tertiary institutions are often

mismanaged. Imo State University, Owerri

is not an exception. This retarded the growthof the institution.

There is no active budget monitoring teamwith the result that budgets are allowed to

operate any how. If there were active budget

monitoring teams the problems ofmismanagement and improper utilization

would be reduced.

THE OBJECTIVES OF THE STUDY

The objectives of the study are as follows:

To apply goal programming model to a real-life budgeting situation to find acompromise solution among the different

conflicting goals of the Imo StateUniversity, Owerri.

To minimize the total weights associatedwith meeting the annual budget

requirements of the institution.

SIGNIFICANCE OF THE STUDY

The insight gained from this study will: Guide and assist decision makers of the

institution in achieving the institution goalsof optimum utilization of funds in improving

the institution;

Guide the institution in budgeting; Help the institution to forecast its budget

annually; Assist in Optimization/Operational Research

students for further research.

LIMITATION OF THE STUDY

The study is limited to the budgetary allocation ofImo State University, Owerri. The budget estimates

of the University were used for the study. It is alsolimited to Goal Programming problems.

LITERATURE REVIEW

Multi-Objective and Multi-Criteria Decision

Analysis

Sundaram (1978) and Lee (1972) affirmed that the

distinguishing characteristic of goal programming isthat it allows for ordinal solution.

Ignizio (1978) states that, this approach, known as aGoal Programming, although far from a panacea;often represents a substantial improvement in the

modeling and analysis of multi objective problems.

This presents state of the art in this field now permitsthe systematic analysis of a class of multi-objective

problems that may involve either linear or non-linearfunctions in continuous or discrete variables. Further,

the goal programming model provides a relatively

reasonable structure under which the traditional,


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single objective tools may be viewed simply asspecial cases.

Sundaram (1978) argued that goal programming

combines the logic of optimization in mathematical

programming with the decision makers desire tosatisfy several goals. According to Winston (1994);

Goal Programming is the technique used to chooseamong alternative optimal solutions.

According to Winston (1994), Suppose a decisionmaker has an additive linear cost function of the

form: C(x1, x2,, xn) = c1x1 + c2x2 + + cnxn.

A decision maker with this type of cost function canoften use goal programming to determine his

decision. He observed that a cost function of theabove form defines the same trade off between each

pair of attributes xi andxj.

Chowdary and Slomp (2002) considered goal

programming as an appropriate powerful and flexible

technique for decision analysis of the troubled

modern decision maker who is burdened with

achieving multiple conflicting objectives undercomplex environmental constraints.

The extensive surveys of goal programming by

Schniederjans (1995), Tamiz, et al (1998), and Aouniand Kettani (2001), have reflected this. Therefore,

goal programming model handles multiple goals in

multiple dimensions.

Taha (2003) also said that goal programmingtechnique is for solving multiple-objective models

and the aim is to convert the original multiple

objective into a single goal. Tipparate (2005) statedthat goal programming extended itself by

reengineering many of the prior single objectivelinear programming with multiple and /or conflicting

(traded-off) objectives.

Wikipedia (2006) described goal programming as a

branch of multiple objective programming (MOP),which in turn is a branch of multi-criteria decision

analysis (MCDA), also known as multiple criteria

decision making (MCDM). It is also thought of as a

generalization of linear programming to handlemultiple conflicting objective measures. Each of

these measures is given a goal or target value to be

achieved.

Managerial and Financial Concepts

The first publication using goal programming was

formulated by Charnes, et al (1955), applied to

constrained regression. That application appliedminimization of the deviational variables as a means

to apply least absolute value regression. The reasonfor that application was that the goal programming

formulation gave the ability to constrain results to

conform to the managerial salary requirement. Thisminimized the difference between market offers and

the competitors. This idea has developed into afruitful research area, least absolute estimation.

The initial development of the goal programming was

due to Charnes and Cooper (1961), in a discussion of

which appeared in their text, although they claim thatthe idea actually originated in 1952.

They proposed a model and technique for dealing

with certain linear programming problems in which

conflicting goals of Management were included asconstraints.

Contini (1968) proposed stochastic goal

programming (SGP) model. He considered the goalsas uncertain variables with normal distribution, that

the model maximizes the probability that theconsequence of the decision will belong to a certainregion encompassing the uncertain goal. Thus, this

model tries to generate a solution that is close to theuncertain goals.

Keown (1978), Keown and Taylor (1980), proposed

an adoption of the chance constrained for stochastic

goal programming model and transforms the modelto a non-linear program, and applied it to budget

allocation production problem.

Dade (1981) applied goal programming in revenuebudgeting. He pointed out that the concern with the

process and outcome of the budgeting has regulated

the revenue constraint to a position co-equal withvariables.

Martel and Price (1981) have developed a stochastic

goal programming model for human resource

planning.

De, et al (1982), proposed a chance constrained goalprogramming model for budgeting in the production

area. They considered the following constraint as amathematical programming model:

m

aijxj bi i = 1, 2, .. . , n, j = 1, 2, . . . , m

j=1

xj are the decision variables.

aij are the corresponding constraints and bi

are the right hand side elements.

Helm (1984), Chanchit and Terrell (1993), Nembouand Murtagh (1996), Abdelaziz and Mejri (2001),Easton and Rossin (1996) proposed a stochastic goal

programming model for employee schedulingproblem. Their model integrated the labour

requirements and scheduling decision. They pointedout that this approach allowed reaching satisfactory

result.

Lee and Olson (1985), on the other hand, proposed an

algorithm of deterministic optimization for stochasticcase. This is taken into account in 1992, and the


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problem of optimal utilization of resources andreservoir operation problem.

Chanchit and Terrell (1993), proposed a model of

stochastic goal programming for the management of

hydraulic resources. This mathematical modelproposed reflects three important characteristics of

the reservoir management problem: Multiple objectives; Stochastic inflow; Large-scale nature of the reservoir systems.

The proposal of Chanchit and Terrell (1993) enabledAbdelaziz and Mejri (2001) to propose a Chance

Constrained Goal Programming (CCGP) model todeal with the operation and the control of Multi-

purpose reservoir system. They suggested a Multi-objective approach to solve the problem consideringthe release for drinking and irrigation purposes, the

pumping cost and the salinity of the water.

Abdulaziz and Majri (2001) considered the control of

drinking water supply goal as a random variable.

Other recent applications of various formulations of

goal programming have been presented by Carrizosaand Romero (2001) and Leung (2001).

Lam and Moy (2002) reviewed the intersection of

goal programming and Discriminant Analysisbeginning with Freed and Glover (1981).

The Concept of Satisfaction Functions

The concept of satisfaction functions in the goal

programming model was introduced by Martel andAouni (1990). Through the satisfactions, the decision

maker may express his/her preferences with regard to

the deviations associated with the goals set for eachobjective.

The concept of satisfaction functions was also used

for modeling the impression related to the value ofthe goals. These are expressed through the intervalsof which the upper and lower limits are set by the

decision maker.

Martel and Aouni (1998) and Aouni (1996)

concluded that the satisfactory functions have a

double role.

Rogers and Bruen (1998) explained the decision

makers sensitivity to the deviations observedbetween the set goal and the level achieved on theother hand. They took into account the imprecision or

uncertainty associated to the value of the goal.

Technology and Planning

The first engineering application of the goal

programming, due to Ignizio in 1962, was the designand placement of the antennas employed on the

second stage of the Saturn V the vehicle used to

lunch the Apollo space capsule which landed the firstmen on the moon.

Stam and Kuula (1991) applied goal programmingtechnique for selecting a flexible manufacturing

system. They, identified production volume, cost andflexibility as multiple objectives.

Shafer and Rogers (1991) used goal programmingmodel for formation of manufacturing cells.

Minimum setup time, minimum intercellularmovements, minimum investment in new equipment

and maintaining acceptable utilization levels were

identified as the multiple objectives.

Premchandra (1993) used goal programming methodfor the activity crashing in project networks. The

multiple objectives identified were as follows: Crashing time of an activity Project cost Project time

Lyu et al (1995) used a goal programming model forwhich the multiple objectives were as follow:

Demand of each boiler; Environmental requirement of the sulphur

oxide emission;

Heating value requirements; Volatile matter requirement; Inventory.

Goal programming approach has been applied for avariety of applications. Problems solved by goal

programming technique are as follows: production

planning in a ship repair company (Tabucanon andMajumder 1989) and production smoothing under

just-in-time manufacturing environment (Kim andSchnierderjans 1993).

Kalpic, et al (1995) successfully applied linear goalprogramming technique for planning objectives

which include calculating the optimum productionMix and achieving the capacity and material balance

while maximizing the contribution and minimizingthe duration of the longest resources management.

Nagarur, et al (1997) illustrated the use of zero-onegoal programming for the development of a

production planning and scheduling model in an

injection molding factory with an objective to

minimize the total cost of production, inventory andstorages.

REVIEW OF RELATED VARIANTS ANDFORMULATIONS/SOLUTIONSDykstra (1984) described that an inferior solution is

not always a problem as long as the model is correct.

It is a problem if the model does not reflect the

decision makers actual preference. In some cases,non-inferior solution may not be intrinsically superior

to an inferior solution especially in the case of naturalresources problem.

Dykstra (1984) argued that the initial run with uniqueweight can be biased, although this technique makes


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weighting more defensive than a subjectiveweighting. The goal programming solution has a high

tendency to be inferior (dominated) with respect totechnological constraints. To deal with this drawback,

Field et al (1980) presented a technique to ensure

non-inferior goal programming solutions. Thetechnique is to apply goal programming and linear

programming as a complementary package.

Winston (1994) stated that; pre-emptive goal

programming problems can be solved by anextension of the simplex known as the goal

programming simplex. To prepare a problem forsolution by the goal programming simplex, we must

compute in rows with the ith

row corresponding togoal i.

Nembou and Murtaph (1996) presented a chanceconstrained goal programming approach to solve a

hydro-thermal electricity generation planningproblem in Port Moresby, Papus New Guinea.

Their model considers explicitly the uncertainty and

seasonal variation inherent in hydro-thermal

electricity demand and supply.

Tamiz et al (1998) suggested straight restoration,

preference based restoration and interactiverestoration which employ cardinal weightings in

lexicographic goal programming to ensure non-inferior solutions.

Tamiz et al (1998) mentioned the case whereby aninefficient objective and an unbounded objective

causes the entire model inefficient and unbounded.

Hillier and Lierberman (2001) said that Goal

programming problems can be categorized accordingto the type of mathematical programming model that

it fits except for having multiple goals instead of asingle objective.

In the case of non pre-emptive goal programming, allthe goals are of roughly comparable importance but

in the case of pre-emptive goal programming, there isa hierarchy of priority levels for the goals.

Hillier and Lierberman (2001) concluded that goal

programming and its solution procedures provide aneffective way of dealing with problems where

management wishes to strive towards several goals

simultaneously. The key is a formulation technique ofintroducing auxiliary variables that enable convertingthe problem into a linear programming format.

Hillier and Lierberman (2001) stated that to complete

the conversion of the goal programming problem to alinear programming model, we must incorporate the

definitions of the deviational variables di+

and di-

directly into the model because the simplex method

considers only the objective function and the

constraints that constitute the model.

According to Romero and Rehman (2003) BothLexicographic Goal Programming (LGP) and

Weighted Goal Programming (WGP) are best knownand widely used as goal programming variants.

However, the variants lie heavily on the great amount

of information that this goal targets, weights as wellas pre-emptive ordering of preferences. These

requirements can cause possible weakness if thedecision makers are not confident of the value of

these parameters.

In many cases, the decision makers could not specify

meaningful information on weights, or the goals areunrelated to each other so that it is not possible to

drive objectively measured preference weights. If it isthe case, the relative value of the shadow prices of

constraints from the unique weight goal programmingrun can be used to drive the real preference weights.

Diaz-Balteiro and Romero (2003) affirmed thatweighted goal programming (WGP) and

lexicographic goal programming (LGP) can be mixed

in a model; for example, weighted lexicographic and

min-max lexicographic goal programming.

Taha (2003) also contributed that the weights and the

pre-emptive methods convert the multiple goals intoa single objective function stating that these methods

do not generally produce the same solution. Neithermethod, however, is superior to the other because

each technique is designed to satisfy certain decision-

making preferences.

RESEARCH METHODOLOGY/DATA

COLLECTION

Goal Formulation

Let fi(x) be the mathematical representation of theobjectives which can be linear or nonlinear (usually

linear). Let gibe the aspiration level, three possiblegoals are

(i) fi(x) gi(ii) fi(x) gi(iii) fi(x) = gi

In regular Linear Programming, these would be hard

constraints but in Goal programming, we measure the

deviation from the goal.

Goal Programming Formulation

The general form of the goal programming model is

given byMinimize Z = (di

++ di

-),

im

Subject to

. . . (1)

n

(aij xj + di-- di

+) = bi

(i = 1,2,,m)j = 1

(j = 1,2,,n)

di+, di

-, xj 0,


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where di+

is the positive deviation variable fromoverachieving the i

thgoal;

di-

is the negative deviation variable fromunderachieving the i

thgoal and

xj is the decision variables;

aij is the decision variable coefficients;

It is pertinent to note here that; di+

, di-

are calleddeviational variables because they represent the

deviations above and below the right hand side of the

constraint i.

The deviational variable cannot be basic variablessimultaneously, because by definition they are

dependent. This implies that in any simplex iteration,at most, one of them can assume a positive value. If

the original ith

inequality is of the form and its di+

0, then the ith

goal will not be satisfied.

However, di+

and di-allow us to meet or violate the i

th

goal at will. A good compromise solution aims atminimizing the amount by which each goal is

violated.

Goal Programming Algorithms

There are two algorithms for solving goal

programming problems. These algorithms convert the

multiple goals into a single objective function.The methods are:

The weights method; The Preemptive method;

The Weights Method

In the weights method, the single objective function

is the weighted sum of the functions representing thegoals of the problem.

The weights goal programming model is of the form;

n

Minimize Z = {(wi+

+ wi-)di},

i=1

Subject to

. . . (2)m

(aijxj + di+

di-) = gi

j=1

(i = 1, 2, . . ., n)

(j = 1, 2, . . . , m)di

+, di-, xj 0,

where wi+

and wi-

are non-negative constraints and

can be real numbers representing the relative weightsassigned within a priority level to the deviational

variables.

The parameter wi+

represents positive weights thatreflect the decision makers preference regarding the

relative importance of each goal; while wi-

is the

negative weights of the decision makers preference.The determinant of the specific values of these

weights is subjective.

The Pre-Emptive Method

In this method, the decision maker must rank the

goals of the problem in order of importance. Themodel is then optimized using one goal at a time such

that the optimum value of a higher priority goal is

never degraded by a lower priority goal. This variantis known as lexicographic goal.

The proposed pre-emptive model is given as

n

Minimize Z = pi (di+

+ di-),

i=1

Subject to

. . . (3)m

(aijxj + di+

di-) = gi

j=1

(i = 1, 2, . . ., n)(j = 1, 2, . . . , m)

di+, di

-, xj 0,

where pi is the preemptive factor/priority level

assigned to each relative goal in rank order (that is p1> p2 > > pn).

The weights goal programming and the preemptive or

lexicographic goal programming can be combined in

a model, for example weighted lexicographic goalprogramming. The weights and rank model according

to Kwak et al (1991) is given byn n

Minimize Z = Pi{wik+di

++ wik

-di

-}

i=1 i=1

Subject to

. . . (4)m

(aijxj + di- di

+) = gi

j=1

(i, k = 1, 2, . . . , n)(j = 1, 2, . . . , m)

di+, di

-, xj 0

where di+ and di

- are deviational variables;

xj = decision variables;

aij = decision variable coefficients;

wi+

and wi-are non-negative constraints representing

relative weights;

pi = preemptive factor/priority level assigned to each

relevant goal in rank order (that is p1 > p2 > > pn).

THE BASIC STEPS IN FORMULATING GOAL

PROGRAMMING MODEL

The basic steps in formulating a goal programmingmodel are as follows:

(i) Determine the decision variables;(ii) Specify goals including goal types (one-

way or two-way goal) and their targets;(iii) Determine the pre-emptive priorities;(iv) Determine the relative weights;(v) State the minimization objective

functions of the deviation;


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(vi) State other given requirements,example, technological constraints, non-

negativity (linear goal programming);(vii) Finally, make sure that the model can

exactly specify the decision makers

preferences.

SOURCES OF DATA COLLECTION FORANALYSIS

The data used in this research are of the secondary

type as it exists in the published budgets andunpublished budget folder.

The data for this study were collected from the ImoState Ministry of Finance, State Secretariat, Owerri

and Bursary Department of Imo State University,Owerri.

DATA ANALYSIS TECHNIQUEIn this study, the goal formulation and weights goal

programming methods were used.

The simplex method (BigM) by Tora package was

used to analyze the weighted goal programmingformulation.

PRESENTATION AND ANALYSIS OF DATA

This chapter presents and analyzes the data obtained

from the Imo State Ministry of Finance and Imo StateUniversity, Owerri, as indicated above. The five

conflicting goals in the budget estimates of the

institution over the period of three years (2006 2008), were considered.

Summary Of The Budget Estimates Over The

Three Years (2006 2008)

Table 1 gives the budget estimates summary of the

institution over the period from 2006 2008,showing the personnel cost, overhead cost, capitalexpenditure and revenue.

Table 1 Outline of the budget estimates for the three yearsAllocation in Naira/Year

ITEM (GOAL) 2006 2007 2008 TOTAL

Personnel cost 1,189,825,252 1,570,289,718 1,669,380,200 4,429,495,170

Overhead Cost 313,658,446 433,304,000 590,241,950 1,337,204,396

Capital Expenditure 863,000,000 1,019,894,886 803,306,000 2,686,200,886

Revenue (Internally Generated) 316,732,540 529,043,360 728,722,210 1,574,498,110

Total 2,683,216,238 3,552,531,964 3,791,650,360 10,027,398,562

Coded Budget Estimates over the Period of

Three Years (2006 2008)

Table 2 gives the coded budget estimates of the

personnel cost, overhead cost, capital expenditure andrevenue. The reason for coding this budget estimates

is to enable one work with small figures in theanalysis.

Table 2 Coded budget estimate for years 2006 2008

Allocation in Billion Naira/Year

ITEM (GOAL) 2006 2007 2008 TOTAL

Personnel cost 1.19 1.57 1.67 4.43

Overhead Cost 0.31 0.43 0.60 1.34

Capital Expenditure 0.86 1.02 0.8 2.68

Revenue (Internally

Generated)

0.32 0.53 0.73 1.58

Total 2.68 3.55 3.8 10.03

Assignment of Weights to the Goals

The objective function coefficient for the variablesassociated with the goal i is called the weight for the

goal i. The most important goal has the largestweight and so on.

Let wi be the weight for goal i, that could range

from 2, 4, 6, the most important goal has thehighest weight and so on.

Coded Budget Estimates And The AssignedWeights To The Goals

The table below gives the coded budget estimateand the assigned weights to the goals.

Table 3 Coded budget estimates with weights

Allocation in

Billion

Naira/Year

ITEM (GOAL) 2006 2007

2008

TOTAL

Wi

Personnel cost 1.19 1.5

7

1.6

7

4.43 8

Overhead Cost 0.31 0.4

3

0.6

0

1.34 2

Capital Expenditure 0.86 1.0

2

0.8 2.68 6

Revenue (Internally

Generated)

0.32 0.5

3

0.7

3

1.58 4

Total 2.68 3.5

5

3.8 10.03 10

ASPIRATION LEVEL (Target Value) of the

Goals

The goals statements of the budget of the institution

were as follows:

Goal 1: Raise Personnel Cost (Salary andallowances of staff) by at least N3 billion per

annum;Goal 2: Reduce Overhead cost by at most N1.5

billion per annum;Goal 3: Raise capital expenditure by at least N1

billion per annum;


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Goal 4: Raise revenue (internally generated) by atleast N4.0 billion per annum;

Goal 5: Reduce the total Budget by at least N7billion per annum.

THE GOAL FORMULATION

Let x1 = Amount budgeted in the fiscal year 2006

x2 = Amount budgeted in the fiscal year 2007x3 = Amount budgeted in the fiscal year 2008

x1, x2, and x3 are the decision variables.

The goals can be stated mathematically as follows:

1.91x1 + 1.57x2 + 1.67x3 3 (Personnel costconstraint)

0.31x1 + 0.43x2 + 0.6x3 1.5 (Overhead costconstraint)

0.86x1 + 1.02x2 + 0.8x3 1 (Capital expenditureconstraint)0.32x1 + 0.53x2 +0.72x3 4.0 (Revenue constraint)

2.68x1 + 3.55x2 + 3.8x3 7 (Budget constraint)x1, x2, x3 0.

THE GOAL PROGRAMMING

FORMULATION

Let di+

= amount by which we numerically exceedthe ith goal.

di-

= amount by which we are numerically less thanthe ith goal

di+

and di-are referred to as deviational variables.

Let Z be the weighted sum associated with meeting

the annual budget requirements.Using the weighted goal programming model stated

in (2), the goal programming formulation can be

mathematically stated as follows:Min Z = 8d1

++ 2d2

-+ 6d3

++ 4d4

++ 10d5

-(objective

function) . . . (i)Subject to:

1.19x1 + 1.57x2 +1.67x3 + d1-- d1

+= 3 (ii)

0.31x1 + 0.43x2 + 0.60x3 + d2-- d2

+= 1.5 (iii)

0.86x1 + 1.02x2 + 0.8x3 + d3-- d3

+= 1 (iv)

0.32x1 + 0.53x2 + 0.72x3 + d4-- d4

+= 4 (v)

2.68x1 + 3.55x2 + 3.8x3 + d5-- d5

+= 7 (vi)

x1, x2, x3, d1+, d2

+, d3

+,d4

+, d5

+, d1

-, d2

-, d3

-, d4

-, di

- 0.

The Input Data

Table 4, gives the input data for the analysis of

budgetary allocation in Imo State University,

Owerri from 2006 2008 inclusive.Table 4

Basic variables x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 RHS

Name of variables d1+

d2+

d3+

d4+

d5+

d1-

d2-

d3-

d4-

d5-

Minimize Z 0 0 0 8 0 6 4 0 0 2 0 0 10

Constraint 1 1.19 1.57 1.67 -1 0 0 0 0 1 0 0 0 0 = 3

Constraint 2 0.31 0.43 0.60 0 -1 0 0 0 0 1 0 0 0 = 1.5

Constraint 3 0.86 1.02 0.80 0 0 -1 0 0 0 0 1 0 0 = 1

Constraint 4 0.32 0.53 0.72 0 0 0 -1 0 0 0 0 1 0 = 4

Constraint 5 2.68 3.55 3.80 0 0 0 0 -1 0 0 0 0 1 = 7

Footnote

In the above input data,Let x4 = d1

+, x5 = d2

+, x6 = d3

+, x7 = d4

+and x8 = d5

+.

Let x9 = d1-, x10 = d2

-, x11 = d3

-, x12 = d4

-and x13 = d5

-.

The Tora software was applied on Table 4 to obtain

the table below.

Interpretation Of The Solution

The application of the simplex method (Big M Method) by Tora package, gives the optimum

solution as follows:

Z = 4.24, x1 = 0, x2 = 0, x3 = 1.84d1

+= 0.08, d2

+= 0, d3

+= 0.47, d4

+= 0, d5

+= 0, d1

-= 0,

d2- = 0.39, d3

- = 0, d4- = 2.66 and d5

- = 0.

Since the value of Z is not equal to zero, the solutionsatisfies goal 1, goal 3, and goal 5, but fails to satisfygoal 2 which is the Overhead cost and goal 4 which is

the Revenue goal. Predominantly, for d2-

= 0.39, itmeans that overhead cost level (target) of 1.5 billion

naira has a shortfall of 0.39 billion naira in theOverhead cost; which indicates that the actual

overhead cost should be 1.89 billion naira, and for d4-

= 2.66, it means that the Revenue goal level (target)

of 4 billion naira exceeded the Revenue goal by 2.66

billion naira; which indicates that the actual revenueshould be 1.34 billion naira.

On the other hand, the budget goal of at least 7 billion

naira is not violated as d5- = 0.

DISCUSSION OF FINDINGS

The study examined the problem of conflicting

goals in budgeting situation in Imo State University,

Owerri.

Each of the inequalities in the goal formulationrepresents a goal that the institution aspires to

satisfy. However, compromised solutions among the

conflicting goals were found, and the approach tofinding a compromise solution is to convert each

inequality into flexible goal in which the

corresponding constraints may be violated.

The simplex method (BigM Method) was used tosolve the weighted goal programming model

formulated, and the optimal solution obtained byTORA package is Z = 4.24, x3 = 1.84, d1

+= 0.08, d3

+

= 0.47, d2-

= 0.39, d4-

= 2.66, all the remainingvariables are zero.

Since the optimum is not equal to zero, this

indicates that at least one of the goals is not

satisfied. Z is the weighted sum associated withmeeting the annual budget requirements.


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103

All but goal 2 and 4 were fully satisfied, which is

the overhead cost and the revenue goalsrespectively. Specifically, d2

-= 0.39; shows that

Overhead cost level (target) of 1.5 billion naira has a

shortfall of 0.39 billion naira and d4- = 2.66, indicates

that the revenue goal level (target) of 4 billion

exceeded the revenue goal by 2.66 billion. On thecontrary, the budget goal of at least 7 billion target is

not violated as d5-= 0.

The Z value of 4.24 billion indicates that if goal 1, 3

and 5 were satisfied, the institution should comewithin 4.24 billion naira budget in order to meet goal

2 and 4, which are Overhead cost and Revenue goalsrespectively.

Hence, the minimum budget of the University shouldbe 4.24 billion naira in the next fiscal year 2010 and

should be reviewed upward annually.

CONCLUSION

This study, examined the budgeting system of Imo

State University using goal programming model.

The results demonstrated that all the goalsformulated were met, except the overhead cost and

revenue target.

The Universitys minimum budget should be 4.24billion naira to meet goal 2 and 4 which are the

overhead cost and the revenue goals. Optimistically,

it can be said that the University has not performedbelow expectation, the institution should continue

with their budget allocation formula with increasedadaptation to new scientific techniques.

RECOMMENDATIONBased on the findings of this study, the following

recommendations are made:(1)The personnel cost should be raised by at least

3 billion. The overhead cost should beincreased to at most 0.39 billion.

(2)The capital expenditure should be increased byat least 1 billion.

(3)The minimum budget of the University shouldbe 4.24 billion per annum and should be

reviewed upward annually.

(4)Budgeting (capital and resources allocation)should be based on new technology (preferable

goal programming model).

(5)The budget should be properly managed andutilized.

(6)Timely and active budget monitoring teamshould monitor the budget of the institution.

(7)In all, the minimum budget of 4.24 billionshould cover all the University needs.

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