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Vol. VII, No. 2, November 2002 FUZZY ECONOMIC REVIEW

CAPITAL BUDGETING IN HEALTH ORGANIZATIONS:

APPLICATION OF THE MULTICRITERIA METHOD PROMETHEE V

A.S. Fernández Castro

1 The author wishes to thank the comments of an anonymous referee.

There are two outstanding characteristics of the capital budgeting process in health organizations: qualitative factors are considered in the decision process, and organizations are subject to severe financial constraints. The first of these characteristics suggests multicritera methods to be applied to the problem, and the second one recommends using mathematical programming techniques to select the

feasible subset of alternatives that maximizes benefits. The proposal in this paper is to use the multicriteria method PROMETHEE V, based on a fuzzy outranking relationship, which meets both requirements. The analysis of the advantages and limitations of the method is illustrated with an application to an actual case.

Keywords: capital budgeting, multicriteria, outranking, PROMETHEE, hospitals.

1. INTRODUCTION Analysis of the health sector and inspection of published experts' testimonies show that the capital budgeting process in health organizations has two outstanding characteristics: qualitative factors are considered in the decision process and organizations are subject to severe financial constraints. The first of these characteristics suggests multicritera methods to be applied to the problem, and the second one recommends using mathematical programming techniques to select the feasible subset of alternatives that maximizes benefits. The proposal in this paper is to use the multicriteria method PROMETHEE V, based on a fuzzy outranking relationship, which meets both requirements. It has the additional advantage that different versions may be specified in order to adapt the model to the constraints affecting the choice and to different ways to compare the characteristics of the alternatives.

Capital budgeting in health organizations: application of the...

The aforementioned characteristics of investment selection in health organizations will be analyzed in the next section. Then, PROMETHEE V is presented, in order to highlight its utility in our context. The proposal is exemplified with an application of the model to a published real case, which illustrates the flexibility of this method. Finally, conclusions of the analysis are summarized. 2. CAPITAL BUDGETING IN HEALTH ORGANIZATIONS Most health organizations consider qualitative factors in the capital budgeting process. Kamath and Elmer (1989) find that hospital managers have lower confidence in qualitative factors than in previous research. Nonetheless, 98% of respondents still take them into account. Moreover, 96.5% answer that qualitative factors determine more than 10% of the decisions, and 53% say that they are decisive in more than 51% of cases. Several works analyzing which factors are and must be accounted for in investment selection, such as Campbell (1994), Kleinmuntz and Kleinmuntz (1999) and Hofmann (2000), highlight the importance, among the non-economic factors, of opinions of the different groups interested in the decision process, namely, physicians, patients, regulators and so on. Other kinds of factors are also included, e. g., strategic factors or ethical considerations. These peculiarities of the health sector make extremely difficult, or rather impossible, to summarize all factors affecting project selection into their financial consequences. Among these peculiarities, the following ones deserve to be mentioned:

• There are many groups involved in the decision process (management, physicians, nurses, other employees, regulators, patients, community,...) and their interests are very diverse (interest of patients in services provided, of doctors in technology, of managers in the relationship between resources and activity, and in the conflicts among the interests of the different parts, ...). Besides, interaction among those groups is very complex, with physicians occupying a powerful position, deciding the level of the consumption of clients and strongly influencing the choice of resources.

• The final good produced is health (and, ultimately, human life), whose social valuation in monetary terms is extremely cumbersome, and which brings about important ethical considerations.

• Forecasts of future benefits from an investment are usually very imprecise, due, partly, to the technological complexity of the productive process, which makes difficult to quantify the marginal effect of a particular resource, and also to the continuous technological advance.

The constant technological advance of the sector, together with the insatiability of the demand of health services, influence the other main characteristic of capital budgeting in health organizations, namely, the scarcity of available resources compared to desirable investments. In the case study that we will analyze in detail

Vol. VII, No. 2, November 2002 FUZZY ECONOMIC REVIEW

in the fourth section, Kleinmuntz and Kleinmuntz (1999), the proposal of the authors is to use the benefit/cost ratio to rank investment alternatives. This proposal is similar to the profitability index of Lorie and Savage (1955), with the peculiarity that benefits are not simple financial values, but an aggregation of the scorings obtained on the different factors evaluated. Validity of this sort of solution is limited to cases where there is just one financial constraint and projects may be undertaken fractionally. These conditions, especially the latter, are hardly met in our context. The existence of highly restrictive budgets is a possible explanation for the extension of the use of the benefit-cost ratio in hospitals. According to Kamath and Elmer (1989), the popularity of this indicator in hospitals (it reaches 35% of use) is much higher that in other sectors, in spite of the above limitations. Nevertheless, this method has the advantage of simplicity compared to most rigorous alternatives, and this may be decisive since, as Steffen and Nystrom (1988) conclude, hospital managers are less prone to use mathematical models than other managers. 3. THE MULTICRITERIA METHOD PROMETHEE V The name PROMETHEE comes from "Preference Ranking Organization METHods for Enrichment Evaluations". This family of methods was presented in Brans et al. (1984). They are based on a fuzzy outranking relationship, from which PROMETHEE I constructs a partial preorder, and PROMETHEE II develops a complete one (i.e., a preorder without incomparability relationships). PROMETHEE III associates an interval to each alternative, and establishes rankings among the intervals that are not transitive. The last version originally presented was PROMETHEE IV, developed for the analysis of continuous alternatives. Later on, Brans and Mareschal (1992) presented PROMETHEE V, a model that uses both the results of PROMETHEE II and Integer Linear Programming, which allows introduction of constraints into the problem. PROMETHEE methods evaluate the intensity of the preference between each pair of alternatives in each criterion through a fuzzy number in the 0-1 interval. This is done by means of six functions expressing the value of the preference of a on b regarding criterion j, Pj(a, b), depending on the difference between their values or "distance":

j jd = (a) - (b)f f

Some of the functions make use of an indifference threshold (q) and/or a preference threshold (p), representing, respectively, the minimum difference considered significant and the level above which it is completely significant.

Capital budgeting in health organizations: application of the...



  



   ≤

0 > d , 1

0 d , 0 = b) , (a P

Figure 1. Type I (Usual) criterion



  



   ≤

q > d , 1

q d , 0 = b) , (a P

Figure 2. Type II (U shaped) criterion

  

 



  

 



≤

p > d , 1

p d < 0 , p / d

0 d , 0

= b) , (a P

Figure 3. Type III (V shaped) criterion

P(a, b)

0 d

P(a, b)

d p

Vol. VII, No. 2, November 2002 FUZZY ECONOMIC REVIEW

  

 



  

 



≤

p > d , 1

p d < q , 21/

q d , 0

= b) , (a P

Figure 4. Type IV (level) criterion

  



  





  



  





≤

p > d , 1

p d < q , q - p

q - d

q d , 0

= b) , (a P

Figure 5. Type V (V shaped with indifference area) criterion

p < < q

0 > d ,e - 1

0 d ,0

= b) ,(a P d

σ  



 





 



 



 ≤

− 2

Figure 6. Type VI (Gaussian) criterion

1/2

0 d

P(a, b)

p q

0 d

P(a, b)

p q

P(a, b)

Capital budgeting in health organizations: application of the...

Once the preference index for each criterion has been computed, a global preference index of alternative a on b is calculated, using information about relative weights (w j ) of criteria:

⋅

∑

j j

j= 1

(a , b)w P

(a , b) =

The outranking character of an alternative is evaluated through the positive outranking flow or leaving flow, which is the summation of those indexes that

∈

Πφ ∑

b A

(a) = (a , b)

evaluate the preference of the alternative on all the other ones.

Similarly, the outranked character is evaluated by means of the negative or entering flow:

∈

Πφ ∑ _

b A

(a) = (b , a)

In PROMETHEE II a net outranking flow is calculated as the difference between the leaving and entering flows:

+ − φ φ φ

(a) = (a) - (a)

PROMETHEE V, using these results, allows to select the most interesting subset of alternatives (i.e. the subset with the highest aggregate net flow) meeting certain linear constraints that limit the choice. Denoting the alternatives as ai (i = 1, ..., m), the general formulation of the problem is:

− + ⋅

φ∑

α + −∑

∈ ∀

ii i=1

iri r r r i=1

Max ( ) · xa

Subject to:

s s = , r = 1,..., sβx

x {0, 1} i

The decision variables, x i , take the zero value if the project is not accepted, and one if it is undertaken. The constraints have been stated in a generic form, with slacks and surpluses. If a particular constraint expresses that the sum of the

Vol. VII, No. 2, November 2002 FUZZY ECONOMIC REVIEW

resources consumed by the projects cannot exceed the available quantity, i.e., if it is a "≤" constraint, there is no surplus (s r

+ = 0); if it is of "≥" type, there is no slack (s

r - = 0); and if it is an equality, there are none of them.

4. APPLICATION OF THE METHOD The case presented in Kleinmuntz and Kleinmuntz (1999) will be used as starting point to show the utility of PROMETHEE V in the capital budgeting process of health organizations, and some variants will be developed in order to illustrate the flexibility of the model. That paper presents the investment selection problem in a non-for-profit hospital of 300 beds with annual income of 150,000,000 $. The alternatives considered and the evaluation criteria are the following ones:

ALTERNATIVES

CRITERIA

Scanner

NPV (in 1,000 $)

Desktop hardware

Financial f2

Market share

EIS (Executive Information System)

Physicians relations

Mammography system

Operating efficiency

Physician answering system

Strategic

Network development

Osteoporosis centre

Patient outcome

Quality

f7 Patient satisfaction

Table 1. Alternatives and criteria

Next, the evaluations obtained by each alternative in each criterion, and the criteria weights are considered (Table 2). The aforementioned authors elaborate a scoring aggregating the marks of each alternative (after normalizing them) using the weights of the criteria. Finally, they advise ranking the alternatives according to the benefit-cost ratio obtained as quotient between the scoring and the investment (Table 3).

Capital budgeting in health organizations: application of the...

100

Criterion

Weight:

100

1467

-150

251

567

Table 2. Evaluations of the different alternatives on the different criteria

SCORING X1 X2 X3 X4 X5 X6

f1 16.7% 1.00 0.09 0.00 0.25 0.09 0.44

f2 16.7% 0.60 0.00 0.07 1.00 0.07 1.00

f3 13.3% 1.00 0.00 0.00 0.75 0.67 0.58

f4 11.7% 1.00 0.43 0.57 1.00 0.00 0.71

f5 11.7% 0.09 0.86 1.00 0.00 0.07 0.07

f6 16.7% 0.67 0.07 0.00 0.93 0.00 1.00

f7 13.3% 0.69 0.00 0.00 1.00 0.00 0.85

Global Scoring: 73.0 17.7 19.4 71.4 12.4 69.0

Ranking: 1 5 4 2 6 3

Investment (10

3 $)

1,000 357 300 200 250 275

BENEFIT/ COST: 7.3 4.9 6.5 35.7 5.0 25.1

Ranking: 3 6 4 1 5 2

Table 3. Scoring and benefit-cost ratios for the investment alternatives

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As it has been previously pointed out, for the choice based on the benefit-cost ratio to be optimal it must be possible to undertake projects fractionally, and this condition is met neither in the particular case we are studying nor in the general class of investment problems that we are analyzing. The very authors of the mentioned paper acknowledge this limitation, and exemplify it setting a 1,400,000$ budget: the optimal solution implies rejecting project X1 , despite its high benefit- cost ratio, and undertaking all the other investments. Although they recommend the different possible combinations of projects to be listed and analyzed by means of the ratios, they recognize that mathematical programming should be used when there are many alternatives. In fact, modelling and solving this problem by means of Integer Linear Programming (ILP) is so simple that the heuristic approach is no longer attractive:

Max 73.0X1+17.7X2+19.4X3+71.4X4+12.4X5+69.0X6 Subject to: 1000X1+357X2+300X3+200X4+250X5+275X6 ≤ 1400 X1 , X2 , X3 , X4 , X5 , X6 ∈ {0, 1} SOLUTION: (X1, X2, X3, X4, X5, X6) = (0, 1, 1, 1, 1, 1)

The indivisibility of the projects and its consequence, the possibility that idle funds exist, invalidate the solution provided by the profitability index, thus the relative importance of the error is lower when individual projects are small compared to available funds. But this reasoning can hardly be used to justify the use of the rough solution in the health sector, since the initial outlay of many projects is quite onerous, e.g. the 1,000,000$ scanner in our case study.

On the other hand, once we assume the need to use a Mathematical Programming model, it is to some degree incongruous applying it on the bases of such a rough comparison among alternatives as a simple scoring. Comparisons carried out by means of the multicriteria method PROMETHEE II are undoubtedly richer:

• It makes modelling of decisor's preferences more flexible, by means of the six different types of preference functions.

• It allows grading the outranking relationships, making use, if convenient, of the concepts of preference and indifference thresholds. This gradation is especially attractive when comparisons are based on qualitative factors.

• Both the outranking and outranked character (superiority and inferiority compared to other alternatives) of each option are analyzed separately.

PROMETHEE V conjugates these virtues with the use of the LP, which allows considering not only the financial constraints, but also other type of limitations that influence the decision process, such as dependence or sustituibility between alternatives. As a first approach of PROMETHEE V to the case considered, it is applied with Type I criteria, because of their simplicity and because they do not

Capital budgeting in health organizations: application of the...

102

need more information than the reference model. The following matrixes express the superiority of the alternative of the line on that of the column:

f1 (type I) [16.7%] F2 (type I) [16.7%] f3 (type I) [13.3%] f4 (type I) [11.7%]

X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6

X1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 1 1

X2 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

X3 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0

X4 0 1 1 0 1 0 1 1 1 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1

X5 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0

X6 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 0 1 0

X1 0 0 0 1 1 1 0 1 1 0 1 0 0 1 1 0 1 0

X2 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0

X3 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

X4 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 1 1

X5 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

X6 0 0 0 1 0 0 1 1 1 1 1 0 1 1 1 0 1 0

f5 (type I) [11.7%] f6 (type I) [16.7%] f7 (type I) [13.3%]

Table 4. Distance between alternatives using Type I criteria

From these valuations and the weights of the approaches (in brackets in the previous chart) the leaving and entering flows and the net outranking flow are computed (Table 5). The LP model is constructed, using these results, pursuing maximization of the total net outranking flow, subject to the financial constraint:

Maximize 2.55X1 -2.27X2 -2.30X3+2.28X4 –2.42X5+2.15X6 Subject to: 1000X1+357X2+300X3+200X4+250X5+275X6 ≤ 1400 X1 , X2 , X3 , X4 , X5 , X6 ∈ {0, 1} SOLUTION: (X1 , X2 , X3 , X4 , X5 , X6 ) = (1, 0, 0, 1, 0, 0)

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103

+ (x)

0.00

0.88

0.42

1.00

0.53

3.72

0.12

0.00

0.33

0.12

0.40

0.12

1.08

0.12

0.40

0.00

0.12

0.23

0.12

0.98

0.47

0.88

0.00

0.88

0.38

3.50

0.00

0.30

0.12

0.00

0.13

0.85

0.47

0.88

0.45

0.75

0.00

3.43

- (x)

1.17

3.35

3.28

1.22

3.27

1.28

Φ (x):

2.55

-2.27

-2.30

2.28

-2.42

2.15

Ranking:

Table 5. PROMETHEE II ranking using Type I criteria

The new optimal solution is very far from the one provided by the first model. It shows a clear difference between the two models: while in the scoring-based model all the alternatives receive a positive valuation, in PROMETHEE V negative valuations are given to the alternatives that are outranked by others to a higher extent that outrank them. Therefore, the valuation obtained in PROMETHEE II is a statement about the acceptance or rejection of a project. Consequently, project X1 is preferred to the subset of projects X2 , X3 , X5 and X6 , since all the latter projects except X6 receive negative valuations. The previous version of the model does not make use of all the possible advantages of this approach. To illustrate them, a version with type V criteria is applied, using the indifference and preference thresholds shown in tables 6 and 7. The main change in the ranking obtained is that the first project goes down from the first to the third position. The reason is that its superiority on other alternatives, mainly regarding criteria f3 and f4 , is now graded as partial or relative.

Capital budgeting in health organizations: application of the...

104

f1 (typeV:q=50,p=300) f2 (t. V: q=5, p=25) f3 (type V q=5, p=25) f4 (type V q=5, p=25)

X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6 X1 X2 X3 X4 X5 X6

X1 0 1 1 1 1 1 0 1 1 0 1 0 0 1 1 0.5 0.75 1 0 0.75 0.5 0 1 0.25

X2 0 0 0.4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.5 0

X3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.75 0

X4 0 0.8 1 0 0.8 0 1 1 1 0 1 0 0 1 1 0 0 0.25 0 0.75 0.5 0 1 0.25

X5 0 0 0.4 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0

X6 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0.25 0 0 1 0

X1 0 0 0 0.05 0 0 0 1 1 0 1 0 0 1 1 0 1 0

X2 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

X3 1 0.25 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

X4 0 0 0 0 0 0 0.75 1 1 0 1 0 0.75 1 1 0 1 0.25

X5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

X6 0 0 0 0 0 0 1 1 1 0 1 0 0.25 1 1 0 1 0

f5 (type V: q=5, p=25) f6 (t. V: q=5, p=25) f7 (type V: q=5, p=25)

Table 6. Distance between alternatives using Type V criteria

X1 X2 X3 X4 X5 X6 Φ +

(x)

X1 0.00 0.85 0.83 0.24 0.85 0.33 3.10

X2 0.12 0.00 0.07 0.12 0.18 0.12 0.59

X3 0.12 0.03 0.00 0.12 0.20 0.12 0.58

X4 0.39 0.82 0.83 0.00 0.72 0.10 2.85

X5 0.00 0.13 0.20 0.00 0.00 0.00 0.33

X6 0.37 0.80 0.77 0.17 0.75 0.00 2.85

Φ -

(x) 0.99 2.63 2.68 0.64 2.70 0.66

Φ (x): 2.11 -2.04 -2.10 2.21 -2.36 2.19

Ranking: 3 4 5 1 6 2

Table 7. PROMETHEE II ranking using Type V criteria

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Next, the three models presented are applied for each of the three budgets originally considered:

DECISION MODEL

PROMETHEE V

Type V BUDGET SCORING & ILP

Type I Basic Model X2+X3 ≥1 X4+X6 ≤1

X2+X3 ≥1

X4+X6 ≤1

1,400,000$ X2 , X3 , X4 , X5 , X6 X1 , X4 X4 , X6 X2, X4 , X6 X1 , X4 X2 , X4

1,600,000$ X1 , X4 , X6 X1 , X4 , X6 X1 , X4 , X6

1,800,000$ X1 , X3 , X4 , X6 X1 , X4 , X6 X1 , X4 , X6

Table 8. Choices of the different decision models with three different budgets

These results highlight the rejection of alternatives with negative valuations in PROMETHEE methods: only projects with positive valuations (X1 , X4 and X6) are selected, even if there are idle funds, unless additional constraints reflect that projects with negative net outranking flows are specially interesting or necessary. This is illustrated with the constraints added to the model with Type V criteria, which show several examples of information that may be included by means of linear expressions. For instance, it may be considered necessary to invest either in PCs or in the EIS, because of a strategic focus towards information systems, or because of the convenience of satisfying people involved on those activities, or for any other reason. The first constraint in the previous chart reflects this situation, and the new solution includes X2 . The second constraint would preclude choosing two projects that would benefit mainly mature women, the mammography system and the osteoporosis centre. The fact that the financial constraint becomes non-binding when the budget suffices to undertake all projects with positive net outranking flows may be regarded undesirable by decision-makers, since net outranking flows reflect just relative valuations. There are at least two ways to overcome this drawback.

May be the most obvious solution would be to add a constant to all net flows in order to make them positive, so the objective function would become:

φ + ∑   m

ii i=1

Max ( ) k · xa

Capital budgeting in health organizations: application of the...

106

Despite its intuitiveness, this change is not neutral, but favours solutions consisting on a higher number of projects, as the following expression of the former function shows:

+φ∑ ∑ m m

Ki ii i=1 i=1

Max ( ) · x xa

Therefore, to guarantee that the transformation will not change the solution, investment should be the same for all projects, so the number of projects would be determined exclusively by the budget, and there should be no further constraints. Obviously, these conditions are far from representing the common scenario of our problem. A less controversial mechanism to avoid rejection of projects when there are idle funds is to include a variable expressing that surplus. An upper limit may be set for the surplus of money, transforming the model into a Mixed-Integer Linear Programming one; or a "soft" constraint may be added by means of Goal Programming, including into the objective function a penalty for the low level of expenditure. 5. CONCLUSIONS Qualitative factors are usually taken into account in the capital budgeting process of health organizations. Some of the reasons are the special nature of the good produced and the complexity of the productive process, which involve groups with varied interests and is subject to a permanent technological progress. Besides, these organizations are subject to strongly limitative financial constraints, which can explain the popularity of the benefit-cost ratio for project selection. From the analysis of the characteristics of the multicriteria method PROMETHEE V and of its application in this context, it is concluded that it offers the following contributions:

• It allows modelling the comparisons of alternatives by means of six different preference functions, and makes possible grading the superiority relationships.

• It provides an indicator of the degree of preference of an alternative on others that summarizes the results of two separated analyses of its superiority and its inferiority. The sign of this index expresses the acceptance or rejection of each investment.

• Contrary to other heuristic approaches, PROMETHEE V guarantees the maximum overall satisfaction attainable with the available budget.

• The ILP permits adding a variety of other constraints to the model, such as those expressing the need of a project or those reflecting incompatibility or dependence between alternatives.

The meaning of the sign of the PROMETHEE II index, as a statement about individual acceptance or rejection of a project, may be a controversial characteristic

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of the model when there is a surplus of money after undertaking all projects with positive net outranking flows. Anyway, several ways to overcome this problem, transforming the constraints and/or the objective function, have been proposed. Finally, it is interesting to note that PROMETHEE V may be considered an alternative to approaches such as cost-effectiveness, cost-utility and cost-benefit analyses. Nonetheless, the multicriteria nature of PROMETHEE suggests that it may be not a competitor, but a complement of those techniques. In fact, the need to include a variety of indicators in the analysis is a consequence of the practical limitations of the family of tools that evaluate all dimensions of so complex problems. Therefore, a model that handles multiple criteria in investments selection may be the vehicle to incorporate the partial results of those other approaches to the decision making process. REFERENCES BRANS, J.P.; MARESCHAL, B.; VINCKE, P. (1984). "PROMETHEE: A new family of outranking methods", in

Brans, J.P. (Ed.) Operational Research'84, Elsevier Science Publishers B.V., North-Holland, p. 477- 490.

BRANS, J.P.; MARESCHAL, B. (1992). "PROMETHEE V: MCDM problems with segmentation constraints". INFOR, 30, p. 85-96.

CAMPBELL, C. (1994). "Hospital plant and equipment replacement decisions: A survey of hospital financial managers". Hospital & Health Services Administration, 39 (4), p. 538-556.

HOFMANN, P.B. (2000). "Allocating limited capital resources". Healthcare Executive, March/April, p. 53- 54.

KAMATH, R.R.; ELMER, J. (1989). "Capital investment decisions in hospitals: Survey results". Health Care Management Review, 14(2), p. 45-56.

KLEINMUNTZ, K.E.; KLEINMUNTZ, D.N. (1999). "A strategic approach to allocating capital in healthcare organizations". Healthcare Financial Management, April, p. 52-58.

LORIE, J.H.; SAVAGE, L.J. (1955). "Three problems in rationing capital". The Journal of Business, 28 (4), p. 229-239.

STEFFEN, T.M.; NYSTROM, P.C. (1988). "Problem solving by hospital managers". Health Care Management Review, 13(4), p. 25-31.

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