AMERICAN JOURNAL OF SOCIAL AND MANAGEMENT SCIENCES ISSN Print: 2156-1540, ISSN Online: 2151-1559, doi:10.5251/ajsms.2012.3.1.1.7

© 2012, ScienceHuβ, http://www.scihub.org/AJSMS

Comparative analysis of linear and multi-objective model application in a private hospital healthcare planning in Nigeria

Salami, A. O.

Department of Business Enterprise Management, College of Management Science, Federal University of Agriculture, Abeokuta, Nigeria

ABSTRACT

Goal Programming is designed to provide decision makers with an opportunity to satisfy diversified goals while linear programming is designed to optimize an objective function subject to several constraints. These techniques, when used properly can effectively aid decision makers in short and long range strategic planning. The objective of this study is thus to compare the solution methodologies of both the optimization (Linear) and Multi-Objective model in arriving at policy guided decision making in health care industry. Ola-Olu Hospital, a medium size private hospital in Ilorin was used as a case study for the paper. The findings of the study reveal that Goal Programming (GP) can be used to solve single as well as multiple objective decision problemss, subject to linear constraint equations. It has also been shown in this study that, in order to derive a meaningful solution, decision makers must give considerable attention to the formulation of both the LP and GP model. It is thus concluded that GP is applicable to many settings and types of decisions, and from a practical perspective, is worthy of a trial application.

Keywords: Healthcare Planning; Optimization; Multi-Objective and Modeling

INTRODUCTION

The Nigerian health care has suffered several down-falls, despite Nigerian's strategic position in Africa, the country is greatly underserved in the health care sphere. Health facilities (health centers, personnel, and medical equipments) are inadequate in this country, especially in rural areas (Onwujekwe et. al., 2010). While various reforms have been put forward by the Nigerian government to address the wide ranging issues in the health care system, they are yet to be implemented at the state and local government area levels. According to the 2009 communiqué of the Nigerian national health conference, health care system remains weak as evidenced by lack of coordination, fragmentation of services, dearth of resources, including drug and supplies, inadequate and decaying infrastructure, inequity in resource distribution, and access to care and very deplorable quality of care. The communiqué further outlined the lack of clarity of roles and responsibilities among the different levels of government to have compounded the situation.

Unarguably, problems in the health care system of any country abound to a certain extent.[Dougherty and Conway, 2008; Davis, et. al., 2005; Green-Pedersen and Wilkerson, 2006 and Moe et. al.,] Although health has the potential to attract considerable political attention, the amount of attention it actually receives varies from place to place. In their commentary of the 3T's road map to transform US health care, Denise Dougherty and Patrick H. Conway rightly stated a step by step transformation of the US health care system from 1T →2T →3T which is required to create and sustain an information-rich and patient-focused health care system that reliably delivers high-quality care (Dougherty and Conway, 2008).

Provision of timely information aimed at combating possible health menace among many other things is an important function of public health. Hence, inadequate tracking techniques in the public health sector can lead to huge health insecurity, and hence endanger national security, etc. (Moe et. al., 2007; Jones et. al., 2009)

In the complex financial environment of today’s healthcare industry, there is great need for administrators to be familiar with efficient methods of allocating scarce resources. Many healthcare facilities [HCF] have become similar to large business enterprises; they are complex organizations established to achieve certain objectives through integration and use of limited resources. One might think of an HCF as an organization whose primary production output goal is service. The inputs consist of:

(1) labour of various types (physicians, nurses, aides, technicians),

(2) fixed assets (buildings, beds, equipment) and

(3) working capital, converted to tangible variable assets (food, drugs, bedding, supplies).

The services output of an HCF can measured in terms of quantity and quality. Quantity can be measured in terms of admissions, discharges, duration of stay, number of outpatients, number of procedures, etc. Output is more difficult to measure because of many subjective elements. However, the quality component of a service is comparable with the quality component of a product; it is a function of its input quality. In other words, if an HCF has highly qualified physicians and nurses, and technologically advanced equipment, the quality of service received by its patients is likely to be high.

Various quantitative techniques have been suggested in the literature that can efficiently optimize the allocation of

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scarce resources. Linear programming [LP] is considered to be one of the most powerful quantitative techniques available. Many readers are familiar with the basic assumptions and concepts related to LP, a method successfully utilized over four decades. It deals with solving management problems of optimization of a single objective, subject to several linear constraints. Although LP is widely used in decision-making processes, it has a major limitation that restricts the users of the technique to narrowing their problems to a single objective function.

In real life, decision situations are frequently characterized by multiple, conflicting objectives. For instance, today HCF’s operates in a competitive environment, which is heavily influenced by external factors including regulators, competitors, suppliers, customers, shareholders and the public. These factors in turn make profit maximization no longer the sole objective for management to strive to achieve.

Beside profit maximization, management now has many other objectives to attain, including improved market share, improved relationships with the public, production of a quality product, minimized employee unrest, minimized pollution, and realization of a certain rate of return on investment. Many of these may conflict; and may compete for scarce resources. If the existence of multiple conflicting objectives is recognized, it follows that an ideal decision technique is one that is able to take them all into account. The objective of this study is thus to compare the solution methodologies of both the optimization (Linear) and Multi-Objective model in arriving at policy guided decision making in health care industry.

Conceptual Clarification: Optimization of a single

objective oversimplifies the pertinent objective function in some potential mathematical programming application situations. Arguments can also be made following

Simon ( 1954) that optimization is not as appropriate as statisficing. These two statements introduce the general topic of multiobjective programming. Multiobjective programming formally permits formulations where: (a) solutions are generated which are as consistent as possible with target levels of goals; ( b) solutions are identified which represent maximum utility across multiple objectives; or (c) solution sets are developed which contain all nondominated solutions. Multiple objectives can involve such considerations as leisure, decreasing marginal utility of income, risk avoidance, preferences for hired labor, and satisfaction of desirable, but not obligatory, constraints.

A discussion of this area requires some definitions. An objective is a measure that one is concerned about when making a choice among the decision variables (something to be maximized, minimized or satisfied like leisure, risk, profits, etc.). A goal implies that a particular goal target value has been chosen for an objective. We will use "multiple objective programming" to refer to any mathematical program involving more than one objective regardless of whether there are goal target levels involved.

Note, the literature contains conflicting definitions (see Blake and McCarl; Ignizio [1978,1983]; Romero [1989, 1991]). For example: ( a) goal programming has been used to refer to multiple objective problems with target levels; (b) multiobjective programming has been used to refer to only the class of problems with weighted or unweighted multiple objectives; (c) vector maximization has been used to refer to problems in which a vector of multiple objectives are to be optimized; and d) risk programming has been used to refer to multiobjective problems in which the objectives involve income and risk. Multiobjective programming involves recognition that the decision maker is responding to multiple objectives.

Generally, objectives are conflicting, so that not all objectives can simultaneously arrive at their optimal levels. An assumed utility function is used to choose appropriate solutions. Several fundamentally different utility function forms have been used in multiobjective models. These may be divided into three classes: lexicographic, multi-attribute utility and unknown utility.

The lexicographic utility function specification assumes the decision maker has a strictly ordered preemptive preference system among objectives with fixed target levels. For example, a lexicographic system could have its first priority goal as income of not less than $10,000; the second priority as leisure of no less than 20 hours a week; the third as income of no less than $12,000, etc. This formulation is typical of "goal programming models." (Charnes and Cooper (1961); Lee, (1972)). The various goals are dealt with in strict sequential order - higher goals before lower order goals. Once a goal has been dealt with (meeting or failing to meet the target level), its satisfaction remains fixed and the next lower order goal is considered. Consideration of the lower level goals does not alter the satisfaction of higher level goals and cannot damage the higher level goals with respect to target level attainment.

Multi-attribute utility approaches allow tradeoffs between objectives in the attainment of maximum utility. The most common form involves maximization of the sum of linearly weighted objectives. This type of formulation has been used by Candler and Boeljhe (1977); and Barnett, Blake and McCarl (1982). The third utility approach involves an unknown utility function assumption. Here the entire Pareto efficient (nondominated) solution set is generated so that every solution is reported wherein one of the multiple objectives is as satisfied as it possibly can be without making some other objective worse off (Geoffrion (1968)).

The lexicographic multiple objective formulation is not precisely a LP problem. It has many structural characteristics in common with a LP problem; however, a conventional objective function is not defined, nor can a single LP formulation reflect imposition of the sequential ordering of the goals. Rather, an iterative procedure is needed (Lee, 1972). Essentially, the approach is to solve problems for each of the goals sequentially. When considering the ith goal solve the problem .The new variable wr gives the amount that the goal level (ΣgrjXj) is

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less than the target value (Tr), while glr is the current level of goal r. When i = 1 the problem minimizes the shortfall from the first goal target level, subject to the LP constraints. One of two solution situations will then occur. Either the optimum value of w1 (denoted w*1) equals 0,

indicating full satisfaction of the first goal, or w*1 � 0,

indicating the goal cannot be fully satisfied. Subsequently, a second problem is solved. This problem is virtually identical to the first, except w2 is minimized and a constraint is appended indicating that w1 cannot be any worse than the optimum value obtained at the end of the solution of the first stage (w1*). This requires: 1) if goal 1 was met before, then goal 1 will continue to be met (i.e., w1 must be less than or equal to zero); or 2) if goal 1 was not met, then the deviation from goal 1 will not get bigger than the minimum deviation obtained at the previous iteration. Thus, the prior objective is constrained to be no worse off than it was before. This problem, in effect, explores alternative optimums where we hold the prior objectives at their optimum values, then try to optimize the satisfaction of the subsequent objectives.

This procedure is executed for all R goals where different deviation variables are minimized at each stage and a constraint is added holding all previous deviations to maximum values prohibiting the earlier objectives from becoming worse off. Lee (1972) presents a more comprehensive discussion of the procedure while the example below gives an empirical application.

The second utility function type involves tradeoffs between various objectives. Such problems can be formulated as conventional linear programs. There have been two alternative formulations of this problem. They differ in their assumptions about target levels. The first formulation (appearing for example in Candler and Boeljhe, 1977) does not take into account target levels, maximizing the weighted sum of the quantities of each objective. This where cr is the weight which expresses the importance of the rth objective in the context of the decision maker's total utility and Nr is a normalizing factor which converts the goal values so they are valued somewhere around one.. The cr coefficients would be in utility units per percent deviation from full satisfaction at the normalizing factor for the rth objective achieved; glr is the amount of rth objective in the optimal solution and qr is the proportional satisfaction amount of rth objective relative to the normalizing factor.

The objective function maximizes multi-dimensional utility summed across all objectives. Each objective is weighted. The second equation sums the level of each objective into the variable glr. The third expresses satisfaction in terms of the normalizing factor. The fourth represents resource availability limitations, the fifth expresses nonnegativity constraints and the sixth allow the goal level to be positive or negative (note the normalizing factor must be of the appropriate sign).

This is again a linear program. The formulation is adapted from Lee (1972) and is used in Barnett, et al. (1982).

The other approach to multiobjective programming involves an unknown utility function assumption. Instead, the entire nondominated set of alternatives is generated. The formulation for this approach is exactly like the first one under the weighted tradeoff section above except that all possible weights are utilized in the problem. This particular approach has been studied extensively, (see, for example, the bibliographies in Steuer (1968); and Ignizio, 1983) but does not appear to be very empirically useful.

A CASE STUDY

Linear Programming Applied to a Healthcare Problem (Optimization Model): Ola-Olu Hospital is a medium-size

HCF, lomocated at Muritala Muhammed Way, in Ilorin, the Capital of Kwara State, Nigeria. For the sake of simplicity, assume that the HCF is specialized in performing four types of surgery: Tonsillectomies, appendectomies, hernias and cholecystectomies. The performance of these surgeries is constrained by three resources: operating room hours, recovery room bed hours, and surgical service bed-days. Assume that the director of the HCF would like to determine the optimum combination of surgical patients that maximizes the total contribution to profit, given the information in Table 1.

Where X1 =the number of tonsillectomy patients X2 = the number of appendectomy patients X3 = the number of hernia patients, and X4 = the number of cholecystectomy Let:- S1 = the idle hours of the operating room, S2 = the idle hours of recovery room, and S3 = the idle days of the surgical service beds.

Table 1: Ola-Olu Hosptial Data

TYPES OF SURVIAL PATIENTS CAPACITY

X1 X2 X3 X4

Operating room 3 4 8 6 1,100 hours

Recovery room 8 2 4 2 1,400 bed hours

Surgical services 4 6 4 2 400 bed hours

Average contribution to profit N210 N260 N280 N300

Since the above problem has only one objective function [i.e. the determination of the optimal combination of the surgical patients to maximize total average contribution], it can be solved by LP. The problem can be readily solved, giving the optimal solution shown in the final simplex tableaus in Table 2

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Table 2: Final Simplex Tableau of LP

Objective row Cj 210 260 280 300 0 0 0

Objective Column

Variable column

Constraint column

X1 X2 X3 X4 S1 S2 S3

280 X3 100 -0.60 -1 1 0 0.20 0 -0.30

0 S2 900 7.8 2 0 0 -0.60 1 0.40

300 X4 50 1.3 2 0 1 -0.10 0 0.40

Zj 43,000 222 320 280 300 26 0 36

Cj – Zj row -12 -60 0 0 -26 0 -36

Solution Stub Body

Table 2 indicates that the optimal combination of the surgical patients that maximizes the total profit, given the available resources, is X3 the best way for the HCF to maximize its contribution to profit is to treat only 100 hernia patients [X3] and 50 cholecystectomy patients [X4]. This will earn the healthcare facility N43,000 and it will leave it with 900 idle hours in the recovery room [S2 = 900]. While on the surface insisting on these mixes of surgery types may appear ludicrous, two points need to be made regarding the management of patient loads:

1) It is possible to defer and/or accelerate the timing of some surgeries to accommodate, or at least approach, this type of optimal mix.

2) Where past history forecasts a non-optimal mix, it is appropriate to manage the reconfiguration of the scarce resources in such a way that it will optimally handle the mix that history predicts. Now assume that Ola-Olu Hospital seeks to achieve the following multiple objectives:

1) Achievement of at least N50,000 in profit in a specified period of time, given the available resources.

2) Minimisation of idle capacity of the available resources, i.e. operating room hours, recovery room bed-hours, and surgical services bed-days.

The solution cannot be obtained by LP since there is more than one objective, but GP is appropriate for handling the problem. The basic assumptions underlying LP are identical for GP. However, GP is capable of handling decision situations which involve multiple objectives. In fact, the solution to the previous LP problem can also be obtained through GP.

The Goal-Programming Model (Multi-Objective Model):

The basic steps used in structuring an LP model are identical to those for GP. The major difference between the two models is that the GP model does not optimize [maximize/minimize] the objective directly, as in the case of LP. Instead, it attempts to minimize the deviations between the desired goals and the realized results. Also, these goals must be prioritized in a hierarchy of importance.

To quantify this prioritization, each goal is expressed as an equation, and a deviational variable[s] is/are assigned to it. Deviational variables can be positive or negative. A positive deviation variable [d

+] represents over-achievement of the

goal. A negative deviation variable [d

-} represents under-achievement of the goal. If the desire

is not to under-achieve the goal, d should be driven to zero.

Similarly, if d is driven to zero, this means that the over-achievement of the goal will not be realized. From the foregoing discussion, it can be deduced that deviational variables are mutually exclusive. The relationship is mathematically expressed as d

+ x d = 0.

The steps needed to structure a GP model are: 1) Goals are identified and expressed as constraints. 2) Goals are analyzed to determine the correct

deviational variable[s] needed for them, d+i, d

-i, or

both 3) A hierarchy of importance among goals is

established by assigning to each of them a pre-emptive priority factor, Pj. These pre-emptive represents the highest priority, P2 the second highest, and so on. The Ps indicates a simple ordinal ordering of the goals. Once these above steps are completed, the problem can be quantified as a GP model as follows:

Minimise: m ∑Pj [d

+i + d

i1-], where j 1,2,3, …, n

i = 1 Subject to: n

∑[aijxj] - d+i + d

-i = bi, where i = 1,2,3, …, m

The following are the problem formulation and solution: Minimise: OX1 + OX2 + OX3 + OX4 + P0 [d

i-] + Od

+1 + P

-2

[d-2 + d

-3 + d

-4]

Subject to: 210X1 + 260X2 + 280X3 + 300X4 + d

i-1 + d

+1 = N50,000 ….

Target Profit [1] 3X1 + 4X2 + 8X3 + 6X4 + d

-2 = 1,100 … Operating room

hours [2] 8X1 + 2X2 + 4X3 + 2X4 + d

-3 = 1,400 … Recovery room

bed-hours [3] 4X1 + 6X2 + 4X3 + 4X4 + d

-4 = 400 … Surgical services bed-

days [4] X1, X2, X3, X4 , d

i-1, d

+1 d

-2, d

-3 and d

-4 ≥ 0

Non-negativity constraint Note that the first goal equation has two deviational variables [d

-1 and d

+1] and d

-1 appears in the objective

function with a P1 coefficient while d+

1 has a coefficient of zero. The reason for this manipulation is that, since goal one requires making at least N50,000, there is no need to put any restriction on d

+1. This is equivalent to the

maximization of profit problem in LP. Also, the insertion of d

+1 in equation [1] is required to determine if and by how

much the minimum of N50,000 profit has been exceeded. In other words, variable d

+1 will indicated the difference

between the goal profit and actual profit.

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Equation [2], [3] and [4] include only the under-achievement variables d

-2, d

-3 and d

-4, respectively, without

inserting the over-achievement variables. This is logical since the objective of the second goal is to minimize the idle capacity of all the scarce resources. The above problem was solved by the simplex method of GP. Table 3 presents the optimal solution [final simplex tableau].

Interpretation of the GP Solution: The solution stub [Cj –

Zj] of the simplex tableau of GP is similar to that of LP except that in GP each goal has its own Cj – Zj row, this due to the ordinal ranking of goals. The highest priority goal occupies the lowest Cj – Zj row, the next Cj – Zj row represents the second priority goal, and so on. The Cj – Zj row will show whether a goal is optimized or sub-optimized [partially satisfied].

Since the above GP problem has two prioritized goals, both should have a Cj – Zj row, which Table 3 verifies. In LP, if the desire is to minimize an objective function, the optimal solution will be reached when all the Cj – Zj values under the columns to the right of the constant column become zero, or positive [Cj – Zj >0]. The point of optimality for the lower priority goals, such as P2, will be reached when all the negative values of the Cj – Zj row for the P2 in the body section become zero or positive, provided that the negative values have corresponding zero values of the Cj – Zj row for P1. Interpretation of Table 3

The problem solution is to treat 100 hernia patients [X3] and 50 cholecystectomy patients [X4]. This patient mix will bring to the HCF N43,000 and leave it with 900 idle hours in the recovery room. This can be noted in the solution stub of Table 3 [d

-3 = 900]

Table 3: Goal Programming Solution

Objective row Cj 0 0 0 0 P1 0 P2 P2 P2

Objective column

Variable column

Constraint column

X1 X2 X3 X4 d-1 d

+1 d

-2 d

-3 d

-4

P1 d-1 7,000 -12 -60 0 0 1 -1 -26 0 -36

0 X3 100 -0.6 -1 1 0 0 0 0.20 0 -0.30

P2 d-3 900 7.8 2 0 0 0 0 -0.60 1 0.40

0 X4 50 1.3 2 0 1 0 0 -0.10 0 0.40

Cj – Zj P2 -900 -7.8 -2 0 0 0 0 1.6 0 0.60

Cj – Zj P1 -7,000 12 60 0 0 0 1 26 0 36

Solution Stub Body

Source: Solution of the Model using simplex method Algorithm

The solutions in Table 2 and 3 are similar. The reason is that in LP, the objective was to maximize profit, and the maximization of profit is a function of the optimal utilization of the scarce resources. In GP, the prioritized objectives were: first to achieve at least N50,000 in profit [this is equivalent to the maximization of profit in LP] and secondly, to minimize idle capacity of the available resources [this is equivalent to the optimum utilization of scarce resources in LP]. Thus, it is not surprising that the solutions in Tables 2 [LP] and III [GP] are similar. The solution stub of Table 3 also shows that both goals [P1 and P2] are under-achieved. The first priority goal [P1] was under-achieved by N7,000 and second [P2] was under-achieved by 900 hours. Measuring Goal Achievement: Short-term goal

achievement: Table 3 solution is short term because with limited resources their utilization could not generate the minimum desired profit. The questions now become: What resource[s] should be increased? By how much? What patient mix should be treated to attain all of the specified goals [profit and minimization of unused capacity]? Such questions involve the longer-term planning. Longer-term goal achievement: In order to optimize all of the previously mentioned goals [achieve at least N50,000 profit and minimize idle capacities of all the available resources], the previous GP model can be readjusted. The short-term equations [2]- [4] can be rearranged to make them suitable for the longer term, as follows:

Minimise: OX1 + OX2 + OX3 + OX4 + P1 [d-1] + d

+1 + P2 [d

-2]

+ 0d+

2 + P2 [d-3] + 0d

+3 + P2[d

+4] + 0d

+4

Subject to: 210X1 + 260X2 + 280X3 + 300X4 + d

-1 + d

+1 =

N50,000 [5] 3X1 + 4X2 + 8X3 + 6X4 + d

-2 + d

+2= 1,100 [6]

8X1 + 2X2 + 4X3 + 2X4 + d-3 – d

+3 = 1,400 [7]

4X1 + 6X2 + 4X3 + 4X4 + d-4 – d

+4= 400 [8]

X1, X2, X3, X4 , d-1, d

+1 d

-2, d

+2, d

-3, d

+3, d

-4, d

+4 ≥ 0

non-negativity constraint Comparing the above equations with those for the short term, the following differences can be noted:

1) In the longer-term model, an over-achievement variable [d

+] has been inserted in each capacity

constraint equation [equations [6] – [8]. The d+

represents capacity expansion. 2) The insertion of d

+2, d

+3, and the d

+4 variables in

the longer term model requires assigning them zero coefficients in the objective function. This means that there is no restriction on the expansion of these capacities as long as this expansion satisfies the predetermined objectives. The solution to the above formulation is shown in Table 4.

Interpretation of the Solution

The solution stub of Table 4 shows that, in order to fully optimize all the predetermining goals in the longer term, the following strategy should be adopted:

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1. Increase the surgical service bed-days by 300. This is noted in the d

+4 row in the solution stub,

where this variable has a value of 300 in the constant column.

2. Treat only tonsillectomy and hernia patients. The best combination of these patients will be 88 hernia patients [X3] and 130 tonsillectomy patient [X1]

The above solution optimizes both priority goals. The Cj – Zj rows for P1 and P2 in the solution stub verify this point. The cells at the intersections of both Cj – Zj, P1 and Cj – Zj, P2 with the constant column both have zero values. This means there is no under-achievement value for them. Table 5 presents a comparison of the short and longer term solutions.

Table 4: GP Final Simplex Tableau – Long Range

Objective row Cj 0 0 0 0 P1 0 P2 0 P2 0 P2 0

Objective Column

Variable column

Constraint column

X1 X2 X3 X4 d-1 d

+1 d

-2 d

+2 d

-3 d

+3 d

-4 d

+4

0 d+

4 7300 0 -5 0 -3 0 0 0 0 0.5 -0.5 -1 1

0 X3 88 0 0.5 1 1.8 0 0 0.20 -0.2 -0.1 0.1 0 0

0 d+

1 2,230 0 -120 0 106.2

-1 -1 26.9 -26.9 16.2 16.2 0 0

0 X1 130 1 0 0 -0.2 0 0 -0.10 0.1 0.2 -0.2 0 0

Cj – Zj P2 15 0 0 0 0 -0 1 0 0 1 0 0 0

Cj – Zj P1 0 0 0 0 0 1 0 0 0 0 0 0 0

Solution Stub Body

Source: Solution of the Model using simplex method Algorithm Table 5: Comparison of the Short and Longer term Solutions of GP

Short-term Solution Longer-term Solution

Goal[s] optimized : none Goals optimized: All

Patient mix solution: 100 hernia patients Patient mix solution: 130 tonsillectomy

[X3] and 50 cholecystectomy patients [X4] Patients [X1] and 88 hernia patients [X3]

Source: Results in table 3 and 4

CONCLUSION

The foregoing demonstrates the advantages of GP over LP. Tables 2 and 3 shows that GP can be used to solve single as well as multiple linear objectives subject to linear constraint equations. It has also been shown that, in order to derive a meaningful solution, decision makers must give considerable attention to the formulation of a GP model. They must be precise and accurate in determining over-achievement and under-achievement variables needed for every goal, for lack of accuracy can yield a long-term solution when the intent is a short-term solution.

But let the reader beware. GP is not a panacea for a multitude of problems seeking solutions. Optimal solutions may not achieve primary goals. Some of the underlying assumptions may be found to be relatively unrealistic; for example, many relationships are really curvilinear. Solutions require implementations – which may be impractical or difficult. Decisions still

have to be couched in the best of managerial thinking – about values: personal, institutional, and environmental. And solutions do not specify their intrinsic relationships to hosts of variables not considered, and the potential consequences of decision implementation, which may be both functional and dysfunctional.

The process of building or designing a model such as goal programming model is not always straightforward. Although, such model will invoke certain simplifications and assumptions that make the problem we are dealing with analogous to some other situation for which there is a ready-made model and solution. Oladipo and Salami, [2004]. The goal programming model used in this write-up is applicable to all types of organization. However, this study has been designed to help managers in the Healthcare sector think through the practice of goal programming technique as a tool of optimizing profit and other major organizational objectives. Although, attention of the researchers was focused on Ola-Olu

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hospital, Ilorin, the study can also be extended to other manufacturing and/or service organizations.

From the foregoing, it is recommended that decision makers or policy makers should exploit the unified approach, model and philosophy provided by the multi-objective mathematical programming model so as to better solve their administrative and other similar problems.

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