Generating the Discrete Efficient Frontier to the Capital Budgeting Problem

Generating the Discrete Efficient Frontier to the Capital Budgeting ProblemAuthor(s): Meir J. Rosenblatt and Zilla Sinuany-SternReviewed work(s):Source: Operations Research, Vol. 37, No. 3 (May - Jun., 1989), pp. 384-394Published by: INFORMSStable URL: http://www.jstor.org/stable/171058 .Accessed: 23/06/2012 19:13

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GENERATING THE DISCRETE EFFICIENT FRONTIER TO THE CAPITAL BUDGETING PROBLEM

MEIR J. ROSENBLATT Technion-Israel Institute of Technology, Haifa, Israel, and Washington University, St. Louis, Missouri

ZILLA SINUANY-STERN Ben Gurion University of the Negev, Beer Sheva, Israel

(Received August 1984; revisions received July 1986; January 1988; accepted February 1988)

In this paper, we characterize the capital budgeting problem by two objective functions. One is maximizing the present value of accepted projects and the other is minimizing their risk. As we assume that the weights assigned to these objectives are unspecified, we utilize a Discrete Efficient Frontier (DEF) approach to represent all the efficient combinations. We found an optimality range for each efficient combination covering the entire possible range of weights (zero to one). Furthermore, we present different properties and characteristics of the DEF, and develop two algorithms for constructing the DEF. The first one is a simple heuristic and the second one is an optimal algorithm. We conducted experiments measuring the effectiveness of the heuristic algorithm and the effect of terminating the optimal algorithm before its completion. We have shown that the heuristic algorithm, which is the first phase of the branch-and-bound algorithm, has an average error of about 2%. Furthermore, we have shown that this average error can be reduced by applying only part of the optimal algorithm and terminating it before its actual completion.

T his paper deals with the problem of finding the optimal Discrete Efficient Frontier (DEF) for the

capital budgeting problem with two objectives. One objective is maximizing the present value of the accepted projects and the other objective is minimizing the risk associated with those projects, as measured by their variance. Researchers have used efficient frontiers in the financial literature for relating risk and return and determining efficient portfolios (see Markowitz 1952, Sharpe 1970, Brealey and Myers 1981). However, in this paper a discrete version of the problem is discussed and projects either are accepted or rejected.

Several authors have dealt with the capital budgeting problem by considering two objectives. They have used a quadratic objective function to formulate the capital budgeting problem assuming projects are dependent and covariances are known and different from zero (see Baum, Carlson and Jucker 1976, 1978, Mao 1969, and Weingartner 1966). Researchers have suggested different approaches and solution procedures for these formulations (see Lawler and Bll 1966, McBride 1981, and McBride and Yormark 1980). In this paper, we formulate the capital budgeting problem as a linear objective function, assuming that the projects are independent and, thus, their covariance equals zero. We consider this assumption quite reasonable, since, in practice, it is very difficult,

if not impossible, to obtain the values of projects' covariances; whereas, projects' variances may be derived in a similar manner to that used in PERT (i.e., a 3-point estimate for each period's cash flow). This assumption results in an objective function where a weighted average of projects' present value and variance need to be maximized.

The purpose of this paper is to develop heuristic and optimal algorithms for constructing the DEF. Also, we obtain the optimality range of each efficient combination over the entire range of weights (zero to one). In the first section, we formulate the problem and derive several properties of the DEF. In Section 2, we present the two algorithms, and we discuss some properties and stopping rules. In Section 3, we present the effectiveness of the heuristic algorithm. Finally, we discuss some conclusions and suggestions for further research.

1. Formulation and Properties

The formulation of the capital budgeting problem is given by

max Z(ae) n n (

-max ae pjXj- - -ao) E vjx,f 1 j=l ~~~~~~j= I

Subject classifications: Finance: capital budgeting. Programming, integer: branch-and-bound and multiple criteria.

Operations Research 0030-364X/89/3703-0384 $01.25 Vol. 37, No. 3, May-June 1989 384 ? 1989 Operations Research Society of America

Capital Budgeting Problem DEF / 385

subject to n

E s x, < S (2) 1= 1

xj= O or 1 (3)

where

pj = the expected present value of project j; Vj = the variance of project j; sj =the first period expected cash outflow of

project j; S = budget available for allocation to all projects

(in period 1); Xj= decision variable, equals 1 if project j is ac-

cepted and 0 otherwise; a = weight assigned to project's present value;

the corresponding weight to the variance is (1 - ae), where 0 < a c< 1;

n = number of projects considered; Z(a) = the objective function value as a function

of ae.

The following definitions are added: n

Mi= pji present value of the ith combination j=

where

[1 if project j is included in the xJ = ith combination

O otherwise

n

wi = E v1x,, variance of the ith combination j=1

n

Si= E SixJ, j=1

cash flow required for the ith combination

Z'(a) = aM - (1 -)W,

the objective function value of the ith combination.

In this study, the accumulated risk of the project is measured by its (present value of the) variance. How- ever, in the portfolio selection model, where decision variables are continuous, the risk is measured by the standard deviation of the portfolio. Another approach suggested by Mao and Brewster (1969) is to use the semivariance as a measure of risk. The variance is used in this study for several reasons, among them: the simplicity in calculating the variance of the project and the fact that the Taylor expansion series of a utility function can be approximated by the expected

value and the variance. We emphasize, however, that if the standard deviation is substituted instead of the variance, different results may be expected.

Note also that in this formulation, the budgeting constraint applies only to the first year cash outflows. However, the model can be modified for the case of multiperiod budgetary constraints. Since the weight a is assumed to be unspecified, the notion of Discrete Efficient Frontier (DEF) is adopted to represent the best set of combinations of projects. The DEF contains all the combinations of projects which are feasible and are not dominated by any other combination of projects. In the following paragraphs, several properties of the DEF are presented which provide the basis for the algorithms (proof of these properties are provided in Appendix A).

Property 1. If Ma < Mb and Wa > Wb then for any weight (O -< a - 1), combination a is inferior to b; namely, Za(af) < Zb(ac) for all ae.

Property 2. If the combinations are ordered in an increasing value of M, namely M' < M'+ and if

wi- wi-I

(. - MI.- ) + (WI._ Wi.- )

wi+1 wi

(Mi+ I - Mi) + (Wi+1 - WI)

then the ith combination is dominated by either combination i - 1 or i + I for any value of a, (O -< a < 1).

Conclusion 1. Define ai as

wi - wi- I

lM Mi-M + wi wi-1 (4)

then, the ith combination is optimal if ai - ae < ai+l, where ao = O and aXN+1 = 1.

1.1. Constructing the DEF by Full Enumeration

All feasible combinations (i.e., ,q=1 SjX, S) can be constructed by simply enumerating all the possible combinations and eliminating those that do not meet budgetary constraints. Two elimination rules can be used in the process of generating the DEF. Property 1 should be used as the first elimination rule. This results in the relationship, between combinations, that if Ma < Mb then Wa < Wb. Property 2 is used as the second elimination rule, namely, if ai > a1i+1 then the ith combination is deleted. Note that if combination i is eliminated, then the values of ai- and a1i+1 need to be updated (excluding combination i). After applying the two elimination rules, the DEF set {E} is

386 / ROSENBLATT AND SINUANY-STERN

obtained and the combinations are renumbered con- secutively. If the ai's are calculated according to (4) for each efficient combination, then Conclusion 1 determines the optimality range of each combination.

Conclusion 2. If the points in the DEF are connected, then a concave piecewise linear function is created. The slopes of the DEF are increasing. Thus, if {i- 1, i, i + I I E E, then

wi - wiI W+ 1 -

- Mi-l Mi+1 - Mi

Property 3. If there are no budgetary constraints and the projects are ordered in an increasing order of vj/pj then, the ith efficient combination consists of the first i projects, and it is optimal within the range v1/(pi + vi) -_ a < vi+i/(pi+l + vi+1). Note that there are exactly n + 1 combinations in the DEF including the empty set.

The idea of presenting the decision maker with a set of solutions as a function of a, and not a unique solution, is similar to the idea of presenting the decision maker with a set of "solutions that are better than most." Baum and Carlson (1979) show that random generation of only a relatively small number of feasible solutions will produce a Better than Most (BTM) solution. Also, a linear utility function is assumed implicitly, which implies that the decision maker is "willing to accept" any level of present value and risk in the efflcient combination of the projects.

The properties and the conclusions shown in this section are the basis for the algorithms developed in the following section.

2. The Algorithms

Constructing the DEF via full enumeration may not be practical especially for a large number of projects, since 2n combinations need to be compared. There- fore, two algorithms are developed for constructing the DEF; a heuristic algorithm and an optimal algorithm. The heuristic algorithm is simple and directly utilizes the characteristics of the projects themselves (and not their combinations).

2.1. The Heuristic Procedure

1. Order the projects in an increasing order to vj/pj, let = 1, H= 0.

2. If sj : S, let i = i + 1, H = H U 1, ..., i}, repeat 2.

4. Calculate ai where ai = vi/(p, + vi), a0 = 0, and ai*+, = 1. The ith combination, which includes the first i projects, is efficient for a within the range, ai, < ? a ti+ai,

Note that if i* = n (implying that budgetary constraints are not binding) then this is the optimal set of solutions. Otherwise, the set H is the optimal DEF for 0 < a < ai*. However, for the range ai. < a <, 1 it serves only as an approximation. The theoretical base for this heuristic procedure derives from Property 3.

The idea behind the heuristic procedure is that adding a project to a given combination increases the ratio of risk (variance) to the present value. Therefore, the new combination will be efficient only if the weight given to the present value increases (i.e., a increases). The notion of ordering the items according to vi/pi has some similarity to the procedures used for solving 0-1 knapsack problems. There, items are ordered according to their "value to weight" ratio (or "bang for buck" ratio) before solution procedures are applied. Examples of these approaches are given in: Kolesar (1967), Greenberg and Hegerich (1970), Magazine, Nemhauser and Trotter (1975), Nauss (1976), Zoltners (1978), Balas and Zemel (1980), and Faaland (1981). An earlier example of this approach applied to the capital budgeting problem is in the classical paper by Lorie and Savage (1955). Also, one may view the problem similar to a parametric 0-1 knapsack problem; see Geoffrion and Nauss (1977). However, this is not quite the same because, in our case, the parameter a affects all the coefficients in the objective function and not only one.

2.2. The Optimal Algorithm

The heuristic procedure presented in the previous section is a simple one and requires a negligible amount of computation compared with the full enumeration procedure. As usual, there is a tradeoff between achieving the optimal DEF and the compu- tational effort involved.

The intention of this section is to present an optimal algorithm for generating the DEF, based on the heuristic solution procedure. There is no claim that this algorithm is the most efficient one could have developed, although, and as will be shown, it is quite an effective one. For general properties and a discussion of optimal techniques and integer programming solutions the reader is referred to Mitten (1970), Geoffrion and Marsten (1972), and Garfinkel and Nemhauser (1972).

The algorithm is composed of two phases. In phase one the heuristic procedure is used. The DEF is


initiated with the empty set. Then, each added combination, defined as a subset of projects, is composed of the projects of the previous combination and one additional project. The projects are added in an increasing order of vj/pj. Phase one is completed when the funds required to add a new project exceed the budget limitation.

In phase two, the possibility of adding projects (not according to the order of vj/pj) is investigated. Stop- ping rules are tested. If they do not apply, then there is a search for new efficient combinations when one or more projects are deleted and other projects (one or more) replace them. Properties 1 and 2 are applied to eliminate inferior combinations.

The Algorithm

Phase One

Steps 1-3. Use the heuristic procedure's Steps 1-3. Step 4. Define Io = {1, 2, ...,i*}, O {ji* +

1,, n}, and

S= s- E- s j=1

where IJ is the last combination of projects in the heuristic

procedure, and g0 is the leftover budget. If i*= n, set D = H, go to Step 9. Otherwise (if i* < n), let G = {I?} where G = the set

of temporary efflcient combinations considered in the next phase.

Phase Two

Step 5. A test for adding projects to 10:

If minj1,o sj > S?, then projects cannot be added. Go to Step 6.

Otherwise, find Bo the set of all subsets of projects that can be added to 10 without violating the budget constraint, where

Bo={bo.:boi C5,S? go Si 1 Ko iebfoi

and Ko is the number of subsets in Bo. Add the new combinations to G. Thus,

Ko

G = G U lI? + boil. i=I

Apply Properties 1 and 2 to G and eliminate any inferior combinations.

Step 6. Set m = 1, 1= 1 m = the number of projects leaving Io.

1 = the number of projects entering from T0.

Step 6.1. Stopping Rules

Am = sum of the m minimal present values in Io (m =1, i*)

A,= sum of the 1 maximal present values in T0(l = 1, ..., n i*)

Fm = sum of the m maximal cashflows in I 0

F, = sum of the 1 minimal cashflows in To.

Step 6.1.1. If Am > A, and go + Fm > Pi,+, set 1 =1 + 1, if l > n - i*, stop. Go to Step 8; otherwise repeat 6.1. 1.

Step 6.1.2. If Am > A, and NO + Fm < Fl, i,set m m + 1, if m > i*, stop. Go to Step 8; otherwise go to Step 6.1. 1.

Step 6.1.3. If Am 6 A, and go + Fm < F,, set m= m + 1. If m > i*, stop. Go to Step 8; otherwise go to Step 6.1. 1.

If none of the stopping rules of Step 6.1 apply, continue.

Step 7. m projects leave I?. Step 7.1. Define Cm as the subsets of combinations

with m projects from 10, where

Cm = Cmi:Cmi 5I?, E p, < E PX jeC,ni jET

m sj>mlnsj}

I = 1 . .., hm where hm = number of subsets of Cm. Seti= 1.

Step 7.2. Define B,ni as the set of subsets of projects from T0 that can replace the ith combination, Cmi, of m projects from 1?, where

Bmi = bmik bmik TIO, 3? + E sj > E SJ, jE C,i jEEbmik

jE Cm i j(= bm,ik

k = 1, ..., Kmi, where Kmi = the number of subsets in Bmi. Note that the number of projects in bmik will be at least 1.

Step 7.2.1. If Bmi = 0, go to Step 7.2.3. Step 7.2.2. If Bmi $ 0, then update G.

Kmi

G=G kU tII-Cmi+bmikj.

Apply Properties 1 and 2 to G and eliminate any inferior combinations.


Step 7.2.3. If i < hm, set i = i + 1, return to Step 7.2. If m < i*, set m= m + 1, l= 1. Go to Step 6.1. Otherwise, if m = i*, stop. Go to Step 8.

Step 8. The Optimal DEF, D, is

D=HUG.

Step 9. The Optimal Solution. Order the combinations in D in an increasing order of M': the ith combination in D is optimal for a,i -

a < ai+ 1, where ai is determined according to (4), with a0 = 0, aN+ I= 1, and N is the number of combinations in D. The value of the solution is given by

Z(ae)=aoM'-(1-ae) W foraog -oe< a<ai+ l.

2.3. Proofs of the Stopping Rules and Optimality

First, we present the following property.

Property 4. If vl/pj S vj+1/pj+1, (vj, pj > 0) for all j, then

I J

,- E Vj /Epj < VJ/PJ, J = 2, ... ., n. Pi j=, j=I

Using Property 4 and its proof the following conclusion is obtained.

Conclusion 3 J-1 -I J J n n E vi E pj VV E P I

VJ E V/ E Pi. j= I j=I j=1 j=I Pi 1== J=-J

Property 5. Combination R, with m projects deleted from I? and Iprojects addedfrom To, cannot be optimal if Am : A1. The proofs of Properties 4 and 5 appear in Appendix A.

Proof of Stopping Rule Step 6.1.1. From Property 5, it follows that for a given m the number of projects added, 1, should be increased (at least to 1 + 1) in order to allow Am < AI+1 . If there is enough budget to add one project (S0 + Fn > FI+,) but all the projects in To were already added (1= m - i*), then no more projects can be added. Furthermore, deleting more projects from I (m , m + 1) will increase Am which will leave the inequality Am > A,; therefore, all the possibilities for larger m cannot provide an optimal combination. Thus, the algorithm stops at this stage.

Proof of Stopping Rule Step 6.1.2. From Property 5, it follows that for a given m the number of projects added, l, should be increased at least by 1 to allow

AM < A1+,, but if there is not enough budget to allow that addition (S? + Fm < F,+ 1), then the possibility of deleting projects from I (m = m + 1) is tested to free the budget. However, if all the projects of I were deleted (m > i*), then the algorithm stops at this stage. All the possibilities of deleting m projects (m = 1, . . ., i*) were considered.

Proof of Stopping Rule Step 6.1.3. If A,,, < A1 according to Property 5 the combination R can be optimal, but if the budget is not sufficient (So + Fm < F,), then at least one project should be deleted from I (m = m + 1). However, if all the projects of I were deleted (m > i*), then the algorithm stops at this stage. All the possibilities of deleting m projects (m = 1, . . ., i*) were considered.

Proof of the Optimality of the Algorithm. There are four cases where the algorithm can terminate:

1. If the algorithm terminates at the end of phase one, namely i* = n, and all projects are accepted. Therefore, set D is the optimal set.

2, 3. Termination in stopping rules 6.1.1-6.1.3 were proven previously.

4. Termination in Step 7.2.3 means that m = i namely the possibility of deleting 1, 2, . .. , i* projects from i* and adding 1, 2, . . ., n - i* projects already have been considered, this includes the possibility of full enumeration; therefore the resulting DEF is optimal.

3. Experimentation

3.1. The Effectiveness of the Heuristic Algorithm

It is rather difficult to compare the effectiveness of the heuristic procedure with the optimal DEF because the value of a is unspecified and no single solution is available. However, by Conclusion 1 we know that the ith efficient combination is optimal for any a in the range a,i s a < a1i+, where i and i + 1 are in the efficient set. Thus, it is possible to calculate the values of Z'(a) for the overall range 0 s a s 1, where the function Z(a) is a convex piecewise linear function. The effectiveness of the heuristic approach versus the optimal approach is determined by looking at the total area of Z(a) for the range 0 s a s 1. Obviously, the higher this value is the better the solution is, since for the same a, a better Z is obtained. The resulting relative error from using the heuristic approach instead of the optimal DEF is given by the equation

AD- A, (6) DAD


where

AH = - E (ai+i - ai)(Z'(ai+ i) + Z'(ai)) 2 iEH

AD = - Y, (ai+I - ai)(Z'(ai+1) + Z'(ai)) 2 iED

and

H = the set of combinations obtained by the heuristic procedure;

D = the set of optimal combinations; e = the relative error of using the heuristic instead of

the optimal DEF.

The areas AD, for the optimal solution, and AH, for the heuristic one, are derived simply by calculating the areas of the triangles or trapezoids beneath the Z(a) function for the range 0 < a < 1. However, we emphasize that for a given a, the error of using the heuristic will be the difference between the ordinates of the connected efficient graphs. This value will gen- erally differ from e.

The objective of this subsection is to evaluate the effectiveness of the heuristic procedure via simulation. The span for each project is 4 periods. For each period, we sample 3 values from a uniform distribution with given ranges (which differ for some of the projects and over time); as in PERT, we calculated the expected value and the variance of each period for each project according to Beta distribution (based on the 3 sampled values-minimum, medium and maximum). Fi- nally, Pj and Vj are the present values of the expected cash flows and their variances over the 4 periods are discounted by r = 0.1 (see the discussion in Section 1). Experiments were performed for n = 12 and n = 18. The effect of the budget levels was studied in the following way. The budget constraints, S, were set as a proportion, f, of the average cash flow measured by

n= aj + bj

j= 2 where aj and bj represent the upper and lower bound values from which the projects' cash flow were generated (for the first year).

The errors were calculated for the cases of 12 and 18 projects and for different values of fJ For each setting, 10 examples were run and their average error is given in Table I. The values of the relative errors were calculated according to (6).

The numbers in parentheses represent the number of runs (out of 10 runs) in which the heuristic procedure resulted in the same DEF as the optimal

Table I Average Errors in Percentages for

the Different Settings 12 Projects 18 Projects

Average Optimality Average Optimality f(%) Error Range Error Range

100 0.01 (9) [0, 1] 0 (10) [0, 1] 80 1.35 (3) [0, 0.89] 0.8 (4) [0, 0.93] 60 2.8 (3) [0, 0.86] 1.6 (3) [0, 0.88] 50 4.3 (1) [0, 0.82] 3.1 (1) [0, 0.82]

Note: see the detailed results in Appendix C.

procedure. Note that when fequals 100%, it does not necessarily mean that a budget constraint was not imposed as the cash flows were randomly generated and the budgets were taken as the average of the upper and lower bounds on the cash flows. Table I presents the results of 80 examples. The average error for these 80 examples was less than 2% (e = 1.75%). Also, in about half of the examples the heuristic resulted in the same DEF as in the optimal procedure. The optimality range for the 12 and 18 projects given in Table I represents the range of values of a for which the heuristic procedure DEF is exactly the same as the optimal one. This value is an average for the 10 examples of each setting. Thus, for the 80 examples presented, the average optimality range was equal to 0.90. The following hypotheses can be made from the sets of experiments.

Hypothesis 1. For the same setting (the same number of projects and the same cash flow generator), the higher the value off, the smaller is the average error.

Hypothesis 2. For the same setting, the more projects there are, the smaller is the average error.

This is encouraging since in real life a large number of projects is usually being considered.

Finally, it is the authors' experience that the deviation of the DEF of the heuristic from the optimal DEF usually takes place for large values of a, a -o 1. Thus, for smaller values of a, the performance of the heuristic is quite satisfactory and will usually result in the optimal solution for a specific value of a (as shown by the set of experiments).

3.2. The Effects of an Early Stop in the Optimal Procedure

In this subsection, the errors incurred by terminating the optimal procedure before it is completed are tested. In essence, the effect of stopping the algorithm


after allowing only one project to leave 10 (i.e., m =

1) is evaluated. For this purpose, 60 different problems were generated, each consisting of 18 projects. Errors were again measured by comparing the total areas of Z(a), for the range of 0 6 a < 1 according to (6). In this case, A, represents the total area if the algorithm is terminated after a given step.

The results obtained in these experiments are as follows.

1) Stopping the optimal algorithm after phase 1 (the heuristic approach) resulted in an average error of 2.42%, where only in 10 out of the 60 examples the optimal DEF was derived. Also, the maximum error obtained was 13.6%.

2) Stopping the optimal algorithm after allowing only one project to be deleted from 1? (i.e., m = 1) resulted in an average error of only 0.06%! In 51 of the 60 examples solved the optimal DEE was obtained, and the maximum error in this case was only 1.5%. Also, in 5 out of the 51 examples, optimality was reached immediately after applying Step 5. These results are a considerable reduction compared to those obtained after phase 1. Furthermore, in 6 out of the 9 examples for which optimality was not reached for m = 1, the errors were less than 0.01 %. Moreover, in all of these 9 examples, optimality was reached for m = 2, namely, when only 2 projects were simulta- neously allowed to leave the set 10.

4. Conclusions and Suggestions for Further Research

In this paper, the idea of generating the optimal discrete efficient frontier (DEF) for the capital budgeting problem is discussed. The capital budgeting problem involves maximizing the present value of the accepted projects and minimizing their risk. It is assumed that a weight a is assigned to the first objective function (maximum present value) and (1 - a) to the second one (minimum variance). However, as the actual value of a is assumed to be unknown, the optimal solution is given as a function of a(0 O < a 1). Once the specific value of a is realized, or the range of values of a is assumed, the optimal combination of accepted projects is given. Thus, this paper integrates the efficient frontier approach and the traditional approach of combining objectives via linear weights.

Two algorithms are developed for generating the DEF. The first algorithm is a heuristic and the second provides an optimal DEF. The first phase of the optimal algorithm is the heuristic procedure suggested in Section 2.1, which provides an optimal solution for a, , a*. Thus, if it is known that the decision maker's

range of a is less than a *, then, in essence, there is no need to proceed with phase 2 of the algorithm. If the simulation results of Sections 3.1 and 3.2 are com- bined, then the average error of the heuristic algorithm (the first phase of the optimal algorithm) is about 2%. Thus, in general it can be concluded that the heuristic procedure is very effective and except for large values of a, (a -o 1), the DEF produced by the heuristic will be the optimal DEF.

Furthermore, simulation results show that using only part of phase 2 (m = 1) of the branch-and-bound algorithm will usually produce the optimal DEE (85% of the cases). However, even if the optimal DEF is not obtained for m = 1, then the expected error is relatively negligible (0.06% for the 60 examples considered).

Alternative heuristics can be developed. For example, the heuristic procedure presented in Section 2.1, can be used for a < a *. Then, for a > a * the values of a, can be calculated initially assuming that no budgetary constraints are imposed. Afterwards, for any midpoint, a,' of the range (ai, ai,+j), i.e., a' =

(a,i+I + ai)/2, the optimal solution Z (a') and the values of (Mi', W,') are derived using a knapsack algorithm and include the budgetary constraints. For this purpose, an algorithm such as the direct descent approach proposed by Zoltners may be used. The combination (M', W,') represents an approximation to the range ai a < aia+ l; however, it is optimal for a'. This implies that this approach may provide a better approximation than the heuristic procedure to the optimal DEF. On the other hand, it requires additional computation.

The objective in this paper is to introduce the idea of DEF and to present simple heuristic and optimal procedures with simpler bounds and stopping rules. Therefore, the algorithms shown in this text are used. However, it may be worthwhile to test the efficiency of alternative heuristics or even, perhaps, to develop new solution procedures.

Appendix A

Proof of Property 1. Multiplying Ma < MA by a and Wa > Wb by -(I - a), and adding, the resulting inequalities yield

aMa - (1 - a) Wa < aM" - (1 - ) W"

namely, Za(a) < Z"(a).

Proof of Property 2. Define ai as in (4). If ai > a, then substituting ai from (4) and rearranging yields: aMI' - (1 -a) W'- 1 > aM' - (1 -a) W', or Z'(a)


< Z I'(a). If a > ai,I, then substituting ai, I from (4) and rearranging yields: aM's' - (1 - a) W'1 > aM' -(1- a)W'; or, Z "'(a) > Zi(a). Since ai > a?i+ , the ranges a < ai and a > ai+1 cover the whole spectrum of a(O - a s 1). Therefore, it follows that the ith combination is dominated by either combination i- 1 or i + 1 for any value of a.

Proof of Conclusion 2. Reversing the inequality sign of Property 2, and multiplying by the denominators of both sides of the condition yield

( Wi Wi- I *[Mi+

I -Mi ) + ( Wi+ I _ WA )

< (Wi+1 - Wi)[(M -Mi-M) + (Wi -Wi-)].

Note that both denominators are positive because of the conditions of Property 1. Subtracting from both sides of the above inequality the values (W -W- )( W_ ) and dividing the result by (Mi+ I- Mi)(M - MI- 1) yields inequality (5).

Proof of Property 3. It is sufficient to show that the ith combination dominates the i - 1 and the i + 1 combinations.

(a) For a 2 [vi/(pi + vi)], we will show that Z'(a) 3

Z'(aa) = E [ap -(1 -a)v1. j=1

i-l

Zi-'(a) = E [apj - (1-a)vj]. j= 1

Z' (a) - Z'.-'(a) = aP - (1 - a)vi

= a(Pi + v) -vi : 0.

(b) For a < [v,+ /(pj+1 + vi+1)], we will show that

Z'.(af) > Zi+ '(a).

i+l

Z'+'(a) = E [apj - (1 - a)vj]. j= 1

Z'(a) - Z'+1(a)

= -(api+I - (1 - a)vi+,)

= vj+I - a(Pi+, + vi+1) > 0.

Proof of Property 4. The proof is by induction. First, we prove for J = 2. As vI/pI S v2/p2, then multiplying both sides of the inequality by P1P2, we get

vIp2 < v2PI. (A.1)

Adding vIpI yields vI (pI + P2) < PI (V1 + V2) or

V1 < VI + V2 (A.2)

PI P1 +P2

Adding v2p2 to both sides of inequality (A. 1) yields P2(VI + V2) < V2(P1 + P2) or [(v1 + V2)/(P1 + P2)] <

(V2/P2).

This completes the proof for J = 2. Let's assume the property holds for J, namely

J / J j VI E IE < Pi j=i j=I Pi

and prove it for J + 1, i.e.,

J+1 1J+1

Pi j=1 j=l PJ+I

Define P =EJ= 1 pj, V= EJ I v. Given this definition, it follows that

(V/P) < (vJ/pJ) < (VJ+1/PJ+1)

namely (VIP) < (vJ+ i/P+ 1p), or

VPJ+ I1 < PVJ+ 1I* (A.4)

Adding VP to both sides yields

V(P + pJ+1) < P(V+ VJ+1)

or

V V + vJ+I +1I 1+1I p < p + = E Vj/ pj- (A.5)

Since vl/p1 < V/P by the induction assumption, it follows from (A.5) that the left inequality of (A.3) holds. Adding to both sides of inequality (A.4) vJ+ Ipi+ 1

yields

PJ+1(V + VJ+1) < VJ+1(P + pi+,)

or J+1 /J+1 V+ V1+1 V_

vi p /= I V=I + UJ+1 Pi+1I J-1 / A1P P + PJ P-+i

This proves the right-hand side of (A.3).

Proof of Property 5. Since MR = M'- A, + A, it follows that MR < M'?. In order to prove that combination R cannot be optimal, we will show that there are two other combinations R and 1? that dominate it, where R is a combination with m projects deleted from 10. Since MR = M- Am + H1 and A, 3 H1 it follows that MR < MR < M'?. According to Prop- erty 4 and Conclusion 3, Qm/Am < Q1/A1, where Qm is the sum of vj for the m projects deleted and Q, is the sum of vj for the I projects added. Rearranging the above inequality and subtracting A, Q- from both sides yields AI(Qi - Q,) < Q,(A,n - Al) or Q/Aj > (Q. - Ql)/(Am - A1). Since QO/M, = (WR - WI)/(MR - MR) and ( W' - WR)/


(M - MR) = (Q,i -_QI)/(Am- A), it follows that (WR WK)/(MR _ Mk) > (W' WR)/(MIO - MR). Thus, according to Property 2, combination R is dominated either by R or by 10.

Appendix B. Example

Consider the following case for 18 projects. The projects are ordered in an increasing order of the ratio of

vj/pj. The cash flow requirements, the present values, the variances and the ratios of variance/present value for the various projects are provided in Table II. The budget is set at the value of S = 1086.

Phase 1

Steps 1-4. Applying phase 1 of the algorithm, it is found that i* = 9, since 10

E s1 = 1093 > 1086. j= I

Therefore, j = 11, 2, 3, 4, 5, 6, 7, 8, 9}, 0 = { 10, 1 1, 12, 13, 14, 15, 16, 17, 181 and

9

30= 1086- E sj= 111. j= I

Thus, phase 1 results in ten efficient combinations (including the 0 combination). These are the first 10 combinations that appear in Table IV, where the 10th combination is an approximation for the interval 0.72 s a s 1 as obtained by the heuristic procedure.

Table II Data for the 18 Projects

J . . ._..Si Pi e v1/D

1 125 165 70 0.424 2 77 123 123 1.000 3 111 123 124 1.008 4 121 176 182 1.034 5 128 81 99 1.222 6 77 145 217 1.496 7 103 156 235 1.506 8 109 119 216 1.815 9 124 86 166 1.930

10 118 63 135 2.143 1 1 78 142 371 2.613 12 108 93 251 2.699 13 124 125 355 2.840 14 134 101 325 3.218 15 149 -31 160 5.161 16 127 57 306 5.368 17 116 47 415 8.830 18 132 7 418 59.714

Table III New Combinations When Project 9 Leaves Po

Value of ai in Iteration

Combination M' W 1 2 3

Io 1174 1432 I?Ut111 1316 1803 0.723 IUll, 121 - {9} 1323 1888 0.924 I0U{ II, 141 - 191 1331 1962 0.902 0.914 I0U{1l, 131 - 191 1355 1992 0.556 0.829

Phase 2

Step 5. Test for adding projects to Io. There are two projects that may be added to Io without violating budgetary constraints.

Bo = {(l1), (12)1.

Adding new combinations to Io and applying Prop- erty 1 does not eliminate any combination. However, when Property 2 is applied, one out of the two new combinations is deleted due to the inferiority of project 12. The new combination Io + I 1 11 = 1, . . . 9, 11 is added to the set of the current efficient combinations, G.

Step 6. For m = 1 the stopping rules of Step 6.1 do not hold since

Al = min p = 81 < maxp,= 142 =A. je10j =-T

Step 7. The possibility of exchanging projects from I0 with projects from T? is explored. If project 9 leaves then, candidates for entering are

B1,9 ( 1, 12), (11, 13), (11, 14)1.

Applying Property 2 to G results in adding the combination I? + Il 1, 13} - 191 to the efficient set G, as shown in Table III.

It is seen that out of the three potential new combinations only one remains, due to Property 2. In the first iteration the combination of exchanging project 9 by I11 + 12} is deleted. Then, after modifying the value of a, for 10 + I 1l, 14} - 19}, it is shown that this combination is also inferior. Similarly, the algorithm proceeds with B1 i, for i < 9. For illustration, if m = 2 and / = 1 we calculate A2= 81 + 86 167,Al = 142, S0 + Fm = 111 + 125 + 128, and F1 = 78; namely, A2 > Al and go + Fm > F1; this corresponds to 6.1.2, therefore we continue with I = 2, etc. The optimal DEF is given in Table IV with the optimality ranges (a).


Table IV The Optimal DEF and the Relevant Ranges of a

Combination Efficient Number Combination Mi a

1 0 0 0 0.30 2 1 165 70 0.50 3 1, 2 288 193 0.50 4 1,2,3 411 317 0.51 5 1, 2, 3, 4 587 499 0.55 6 1,2, 3,4, 5 668 598 0.60 7 1, 2, 3, 4, 5, 6 813 815 0.60 8 1, 2, 3, 4, 5, 6, 7 969 1050 0.64 9 1,2,3,4,5,6,7,8 1088 1266 0.66

10 1, 2, 3, 4, 5, 6, 7, 8, 9 1174 1432 0.72 11 1,2,3,4,5,6,7,8,9, 11 1316 1803 0.83 12 1,2,3,4,5,6,7,8, 11, 13 1355 1992 0.92 13 1, 2, 3, 4, 6, 7, 8, 11, 13, 14 1375 2218 1.00

Table V Computation Results of the Experimentsa

No. of Projects

12 18

f e e (%) (%) a (%) a

50 2.5 0.89 8.1 0.72 4.3 0.75 1.7 0.77 2.2 0.73 4.6 0.80 3.3 0.85 2.3 0.82 0.9 0.92 3.2 0.83 2.2 0.78 1.1 0.86

13.2 0.73 3.4 0.75 5.9 0.78 0.8 0.87 8.2 0.8 6.0 0.78

60 0.2 0.87 4.6 0.85 2.6 0.75 2.3 0.78 1.5 0.84 0.25 0.86 6.7 0.72 3.7 0.79 2.1 0.82 1.6 0.84

13.6 0.72 2.3 0.85 0.9 0.91 1.6 0.84

80 0.001 0.91 1.5 0.81 2.1 0.79 0.7 0.88 2.6 0.86 2.1 0.91 0.7 0.89 0.34 0.84 0.9 0.92 0.95 0.88 0.6 0.84 2.4 0.99 5.8 0.84 0.8 0.86

100 0.1 0.97 a These are results only for cases where there was an error

(47 out of 80); for the others e = 0, a = 1.

Appendix C

The computation results of the experiments are given in Table V.

Acknowledgment

We would like to thank the anonymous referee for the thorough and constructive comments on an earlier version of this paper.

References

BALAS, E., AND E. ZEMEL. 1980. Algorithm for Large Zero-One Knapsack Problems. Opns. Res. 28, 1130-1154.

BAUM, S., AND R. C. CARLSON. 1979. On Solutions That are Better Than Most. Omega 7, 249-255.

BAUM, S., R. C. CARLSON AND J. V. JUCKER. 1976. Applying the Portfolio Selection Model to the Cap- ital Budgeting Problem. Technical Report 76-3, Department of Industrial Engineering, Stanford University, Stanford, Calif.

BAUM, S., R. C. CARLSON AND J. V. JUCKER. 1978. Some Problems in Applying the Continuous Portfolio Selection Model to the Discrete Capital Budgeting Problem. J. Finan. Quant. Anal. 333-344.

BREALEY, R., AND S. MYERS. 1981. Principles of Corpo- rate Finance. McGraw-Hill, New York.

FAALAND, B. H. 1981. The Multiperiod Knapsack Prob- lem. Opns. Res. 29, 612-616.


GARFINKEL, R. S., AND G. L. NEMHAUSER. 1972. Integer Programming. John Wiley & Sons, New York.

GEOFFRION, A. M., AND R. E. MARSTEN, 1972. Integer Programming Algorithms: A Framework and State of the Art Survey. Mgmt. Sci. 18, 465-491.

GEOFFRION, A. M., AND R. NAUSS. 1977. Parametric and Postoptimality Analysis in Integer Linear Program- ming. Mgmt. Sci. 23, 453-466.

GREENBERG, H., AND R. L. HEGERICH. 1970. A Branch Search Algorithm for the Knapsack Problem. Mgmt. Sci. 16, 327-333.

KOLESAR, P. J. 1967. A Branch and Bound Algorithm for the Knapsack Problem. Mgmt. Sci. 13, 723-735.

LAWLER, E. L., AND M. D. BELL. 1966. A Method for Solving Discrete Optimization Problems. Opns. Res. 14, 1098-1112.

LORIE, J. H., AND L. J. SAVAGE. 1955. Three Problems in Capital Rationing. J. Business 6, 229-239.

MAGAZINE, M. J., G. L. NEMHAUSER AND L. E. TROTTER, Jr. 1975. When the Greedy Solution Solves a Class of Knapsack Problems. Opns. Res. 23, 207-217.

MAO, J. C. 1969. Quantitative Analysis of Financial Decisions. Macmillan, New York.

MAO, J. C., AND J. F. BREWSTER. 1969. An E - S, Model of Capital Budgeting. Engrg. Econ. 15, 108-122.

MARKOWITZ, H. M. 1952. Portfolio Selection. J. of Finan. 7, 77-91.

MCBRIDE, R. D. 1981. Finding the Integer Efficient Frontier for Quadratic Capital Budgeting Problems. J. Finan. Quant. Anal. 16, 247-253.

MCBRIDE, R. D., AND J. S. YORMARK. 1980. Finding All Solutions for a Class of Parametric Quadratic Integer Programming Problems. Mgmt. Sci. 26, 784-795.

MITTEN, L. G., 1970. Branch and Bound Methods: Gen- eral Formulation and Properties. Opns. Res. 18, 24-34.

NAUSS, R. M. 1976. An Efficient Algorithm for the 0-1 Knapsack Problem. Mgmt. Sci. 23, 27-31.

SHARPE, W. F. 1970. Portfolio Theory and Capital Mar- kets, McGraw-Hill, New York.

WEINGARTNER, H. M. 1966. Criteria for Programming Investment Project Selection. J. Indust. Econ. 65-76.

ZOLTNERS, A. A. 1978. A Direct Descent Binary Knap- sack Algorithm. J. Assoc. Comput. Mach. 25, 304-3 11.

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Generating the Discrete Efficient Frontier to the Capital Budgeting Problem