Integrating genetic algorithms and spreadsheets: a capital budgeting application

Copyright © 2007 John Wiley & Sons, Ltd. Intell. Sys. Acc. Fin. Mgmt. 14, 87–97 (2006)DOI: 10.1002/isaf

INTEGRATING GENETIC ALGORITHMS AND SPREADSHEETS 87INTELLIGENT SYSTEMS IN ACCOUNTING, FINANCE AND MANAGEMENTIntell. Sys. Acc. Fin. Mgmt. 14, 87–97 (2006)Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/isaf.278

Copyright © 2007 John Wiley & Sons, Ltd.

INTEGRATING GENETIC ALGORITHMS ANDSPREADSHEETS: A CAPITAL BUDGETING APPLICATION

R. H. BERRYa* AND D. H. F. MANONGGAb

a Nottingham University Business School, Jubilee Campus, Wollaton Road, Nottingham NG8 1BB, UKb Satya Wacana University, Salatiga, Indonesia

SUMMARYThe role of the tax system in generating interactions between the post-tax cash flows of different projects isdiscussed. When such interactions can occur, the capital budgeting process should be based around projectcombinations rather than individual projects. Evaluation of a project combination in net present value termscan easily be done using a spreadsheet. If the number of individual projects is large, then project combinationscan be generated and an optimum combination of projects searched for using a genetic algorithm. The geneticalgorithm approach has an advantage over alternative computational approaches, such as mixed integer pro-gramming, because of the more understandable representation of the problem it allows. Copyright © 2007 JohnWiley & Sons, Ltd.

1. INTRODUCTION

The genetic algorithm (GA) approach to problem solving was introduced by Holland (1975) andis described in Goldberg (1989) and many other sources. Like many other intelligent systemapproaches, it is increasingly applied to problems in finance. However, published work oftenappears to promise little if any improvement on the performance of other, more traditionalapproaches. It is a worrying thought that much current literature appears to be aimed at demon-strating that the GA can be applied to a task without any consideration of why it might be worth-while to do so.

This paper examines the role of the GA in the traditional finance area of capital budgeting. Thistopic is a significant task for many practitioners and features in financial management, corporatefinance, and management accounting textbooks. The claim made here is not that the decision cannotbe analysed by other methods, nor that the GA identifies an optimal decision when other approachescannot. Rather, it is that the GA allows a more sensible partition of the model building task, henceoffering the potential to improve the fit between model and reality and, consequently, improving thequality of the decision-making process. Experience suggests that, in both consultancy and teachingcontexts, the GA’s use appears to generate greater client involvement in the analysis than do alter-native optimization approaches.

* Correspondence to: R. H. Berry, Nottingham University Business School, Jubilee Campus, Wollaton Road, NottinghamNG8 1BB, UK. E-mail: [email protected]

88 R. H. BERRY AND D. H. F. MANONGGA


2. FINANCIAL BACKGROUND

There is a standard approach to the problem of investment appraisal advocated in most financetextbooks. A firm is assumed to be faced with a need to evaluate and select among several possiblereal, as opposed to financial, investments. The essence of the standard approach is to calculate thenet present value (NPV) of the cash flows of each project, taking each project in turn, and toundertake all projects having a positive NPV. Among the significant practical aspects of theapproach is the need to carry out the analysis of each project on an after-tax basis. Brealey et al.(2006) offer a clear treatment of this standard approach.

The project-by-project approach, however, is potentially flawed. Research literature has recog-nized for some time that an adequate investment appraisal on an after-tax basis cannot alwaysproceed in this fashion, since the tax system can cause projects to interact. The impact of tax-inducedinteractions between projects, or tax exhaustion as the phenomenon is sometimes known, was iden-tified and analysed by Buckley (1975), Berry and Dyson (1979) and Cooper and Franks (1983).Buckley’s (1975) analysis emphasizes the interaction between a single project and the ongoingactivities of the firm that is considering whether or not to undertake the project, whereas Berry andDyson (1979) recognize that interactions also occur between projects. It must be stressed that theseinteractions are not generated by synergies or conflicts between physical aspects of the projects;projects are defined so that physical interactions between projects are eliminated. For example, thefact that locating complementary activities in close physical proximity can generate cash flowsgreater than the sum of the cash flows that would be generated if the facilities were built miles aparthas long been recognized. In these circumstances, the combination of the two complementaryprojects is treated as a single project (Haley and Schall, 1979). The source of the tax-based inter-action is the fact that tax is charged at the company level rather than the project level.

Berry and Dyson (1997) discuss two particular features of tax systems that can give rise tointeractions between projects, i.e. multiple tax rates and capital allowances. They also identifywhere, across the OECD countries, relevant tax-system features exist. Multiple tax rates and systemsof capital allowances are shown to be commonplace.

The idea of tax-induced project interactions can easily be explained using the multiple tax ratefeature. Briefly, in many economies, firms with low profits pay tax at a lower rate than those thatearn high profits. In some economies, more than two tax rates exist. In the UK economy, as from1 April 2006, for example, the corporate tax function has the form shown in Figure 1.

Figure 1 shows that firms having less than L1 = £300,000 in profit pay tax at a small companies’rate of 19%, whereas firms having more than L2 = £1,500,000 in profit pay tax at a standard rateof 30%. Firms having profit between L1 and L2 pay tax at a rate that varies according to the amountof profit P. The tax payable is calculated thus:

Tax . ( , , )= − −0 3

11

4001500 000P P

(Since the aim of this paper is to comment on an approach to modelling rather than to present acase study of an investment decision, a simplified version of the UK tax system is used throughout.At this point, for example, the possibility that the firm receives franked investment income isignored, and hence the need to distinguish between profits and chargeable profits is avoided.)

Imagine a company operating at a profit level below L1, and hence paying tax at the smallcompanies’ rate. Suppose several projects are available that, when evaluated at the company’s


INTEGRATING GENETIC ALGORITHMS AND SPREADSHEETS 89

Figure 1. Multiple tax rates

current tax rate, all have positive after-tax NPVs. Further suppose that, if all such positive NPVprojects are adopted, the level of profit increases beyond the kink in the tax function. In this case,the project-by-project approach will fail to represent the true after-tax impact of the projects. Somepart of aggregate profit will be taxed at a higher rate than has been assumed in the project-by-projectanalysis. The impact of a given project on a firm’s post-tax profit can only be evaluated once it isknown which other projects will be accepted.

Another major source of project interactions is the system of capital allowances. The capitalexpenditure involved in a project can generally be treated as an expense when calculating acompany’s taxable profit. However, the way in which this expenditure is spread through the life ofa project is not generally a choice left open to a company. Rules are complicated, with allowancepatterns varying with the size of company and the type of asset acquired. For example, in the UK,since 2 July 1998, 40% of expenditure on plant and equipment by small and medium-sizedbusinesses can be claimed as a first-year allowance (FYA) against tax. The remaining 60% canbe claimed as writing-down allowances (WDAs) on a 25% reducing balance basis. (Generally,writing-down allowances are not claimed on an asset-by-asset basis. The cost of plant and equip-ment acquired, net of any FYA, is added to a ‘pool’ and WDAs are calculated on the value ofthe pool.) To demonstrate the project interaction possibility, imagine two projects, one of whichinvolves substantial capital investment, and hence makes available substantial capital allowances,but involves a significant delay before profits appear, and another which generates profit quickly.If the level of profit from activities elsewhere in the firm is low, then the allowances generatedby the first project can best be used to reduce the tax bill on the profits generated by thesecond project.

For ease of exposition later in the paper, it is worthwhile commenting here on another commonfeature of tax systems, i.e. carry forward and carry back of unused capital allowances. It is possiblethat a firm with a significant capital expenditure programme will have substantial capital allowances



available in a given year. The procedure, as has been indicated earlier, is to subtract these allowancesfrom profit before calculating tax. However, suppose the value of the allowances exceeds currentprofit. Tax systems can allow unused allowances in a given year to be carried forward to reducetaxable profits in future years and/or allow them to be carried back to previous years to generatetax rebates. Carry back is generally limited to 1 or 2 years only.

3. COMPUTATIONAL APPROACHES: MIXED INTEGER PROGRAMMING

There are three aspects to the computational problem introduced by tax interactions. The first is theneed to generate each combination of projects (2n, where n is the number of projects), the secondis to evaluate each such combination, and the third is to search for the combination with the largestNPV. Complete enumeration can be avoided. Berry and Dyson (1979, 1997) introduced mixedinteger mathematical programming models to deal with the computational aspects of the problem.The two papers differ in the complexity of the tax systems analysed. Berry and Dyson (1997)analyse a tax system that contains multiple tax rates, a system of capital allowances calculated ona reducing balance basis, an ability to carry forward unused capital allowances indefinitely to reducefuture tax bills, and an ability to carry back capital allowances for 1 year to generate a rebate oftax paid. The ability to carry back capital allowances to generate a rebate of taxes paid in previousyears is significant, since it potentially weakens project interactions.

Mathematical programming deals with all aspects of the computational problem. However, theway in which the problem must be described before the mathematical programming approach canbe brought to bear is not one with which financial managers are typically comfortable. Describingthe tax system can cause a particular problem. Consider the tax rate system shown in Figure 1. Afinancial manager might write this down for a profit P and tax T as

If P ≤ L1 = 300, then T = 0.19PIf L1 ≤ P ≤ L2 = 1500, then T = 0.3P − (11/400)(1,500,000 − P)If P > 1500, then T = 0.3P

This IF . . . THEN . . . ELSE approach is relatively intuitive.To reflect the same calculation in a mathematical programme requires something a little more

complex. Identify each break point in the tax function by its coordinates (L1, T1), (L2, T2) (L3, T3),where L3 and T3 are arbitrary upper limits, and define variables zk and yk so that

P = z1L1 + z2L2 + z3L3

T = z1T1 + z2T2 + z3T3

z1 ≤ y1

z2 ≤ y1 + y2

z3 ≤ y2 + y3

y1+ y2 + y3 = 1z1 + z2 + z3 = 1

where 0 ≤ zk ≤ 1 for k = 1, 2, 3 and yk is binary for k = 1, 2, 3.The effect is the same as the set of IF . . . THEN statements, but experience suggests that this

formulation is less than intuitive as far as financial practitioners and students are concerned!



4. COMPUTATIONAL TOOLS: THE SPREADSHEET

One problem-solving tool that is familiar to most managers is the spreadsheet, and the IF . . . THENrepresentation of the tax system and other aspects of the calculation of the NPV for a given projectare easily represented using this technology. Such a spreadsheet can also evaluate the NPV of acombination of projects once a combination has been defined.

An extract from a capital budgeting spreadsheet is shown in Table I. The company representedin the spreadsheet is selecting among 14 projects. The data for these projects are stored in rows 11to 24. The planning horizon is 10 years, although only a subset of years is shown.

The data are drawn from Weingartner (1974). In this body of data, positive items represent taxableprofit and negative elements represent capital expenditures. (For simplicity, the distinction betweenprofits on which tax is usually calculated and cash flow on which investment decisions are usuallybased is ignored in this paper.) Cells C11 to C24 allow the project combination that is to beevaluated to be defined; ‘1’ indicates that a project is to be included. The content of row 25, althoughnot shown, deserves comment. This looks like a project that is permanently selected. In fact, it isthe cash flow that will be received by the company in the absence of any further investment. In thisrow, any negative elements represent losses.

Cells E50 to K50 are used to identify the treatment of unused capital allowances. Each representsa year in the planning horizon. If a cell contains ‘1’, then allowances are to be carried back in therelevant year. Rows 54 to 69 (not shown) generate the aggregate after tax cash flow from the selectedprojects. Cell D78 contains the post-tax NPV of the firm’s 10 years of cash flow, for the particularcombination of projects and tax policy selected.

If the number of projects is small, then all combinations are easily generated and evaluated.However, as the number of projects increases, so the spreadsheet must be paired with a mechanismthat generates and searches among possible project combinations. For many years, spreadsheets havehad built in optimizers that can be used for this purpose, but these have not in the past acceptedthe IF . . . THEN style of representation. Their use has required an understanding of the mathemati-cal programming style of representation and an ability to reflect it in the spreadsheet. Winston andAlbright (2001) show many examples of spreadsheet formulations of mathematical programs.

5. COMPUTATIONAL APPROACHES: THE GENETIC ALGORITHM

As has been said, the GA approach to problem solving was introduced by Holland (1975). A solutionto an optimization problem, e.g. Max f (x) where 0 ≤ x ≤ 31, has two aspects: a value of x and avalue of f (x). The value of x can be represented in a variety of ways. Although it might be usualto think of x as having a decimal representation, e.g. 27, a five-element vector of zeros and oneswould be an equivalent binary representation e.g. (1, 1, 0, 1, 1). The traditional GA operates on aset of such vectors to create a vector producing a high value of f (x). (Not all variants of the GArequire possible solutions to be represented in binary form, but this type of representation is sat-isfactory for the current application.) The operation of the GA is straightforward: a randomlygenerated population of binary vectors is selected, each member of the population is assigned asurvival fitness according to the value f (x) the ‘fitness function’, and the fittest members of thepopulation are allowed to ‘mate’ to produce a new population. Mating involves the crossoveroperation. This involves randomly selecting a pair of population members, randomly choosing acommon point along their length, and swapping the vectors’ elements to the right of that point. In



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this way, two new potential solutions are generated. Population generation, using the processes offitness testing, selection and crossover, is repeated many times, and typically the average fitness ofthe population increases with successive generations. Usually, a record of the best populationmember found to date is recorded; when the GA is halted, this best-population member found todate is put forward as a solution to the problem. A justification of the process is found in, forexample, Goldberg (1989). The analogy with human evolution is instructive. This is survival of thefittest. Fit population members are able to breed and pass their characteristics on to their offspring.On infrequent occasions, another operation, i.e. mutation, is allowed to affect a member of thepopulation. A population member is randomly selected, one of its elements randomly selected, andthat element is switched to zero if it is one and to one if it is zero. This occasional introductionof a potentially unusual population member ensures that very different types of solution to theproblem under examination are tried. Thus, the GA tends not to get trapped at a local optimum, andends up offering a good, though potentially not strictly optimal, solution to the problem underconsideration (Holland, 1975).

A key requirement of the basic GA approach is the consistency of the representation of thesolution, in this case a binary vector, and the set of operators, crossover and mutation, applied tothese vectors. Crossover and mutation, when applied to population vectors, should generate furtherfeasible population vectors. However, as will be seen, some complex versions of the after-tax capitalbudgeting problem violate this requirement, and the basic GA has to be supplemented.

The selection of an NPV-maximizing combination of projects is an ideal application for a GA.A solution is a vector of ‘1’s and ‘0’s, with a ‘0’ indicating that a project has not been selectedand a ‘1’ indicating that the project has been selected. The after-tax NPV of the combination servesas a measure of fitness of that particular vector. This insight has been explored in Berry and Smith(1993). A tax system similar to that dealt with in Berry and Dyson (1979), multiple tax rates, areducing balance system of capital allowances, and unlimited carry forward of unused allowances,is examined. Both the GA and the fitness function were coded in PASCAL. The GA proved capableof identifying the optimal, rather than just a good, combination of projects. (An optimal solutionhad previously been found using a mixed integer programming model.) However, PASCAL codingis as far removed from everyday management skills as the formulation of mixed integer programs.

6. INTEGRATING GENETIC ALGORITHMS AND SPREADSHEETS

Managers have generally proved comfortable with the spreadsheet style of presentation. The invest-ment appraisal spreadsheet in Table I contains nothing more than cash flow data and the representa-tion of the tax system in the form of IF . . . THEN rules. It is easy to manipulate. The impact ofadding or deleting a project by simply switching the contents of one of the driver cells from ‘1’ to‘0’, or vice versa, is easily discovered. As has been said, if only a few projects are being considered,then each combination can be identified and evaluated without further computational support.However, this exhaustive search option ceases to be practicable as the number of projects increases.What is required is a mechanism to generate, intelligently, a succession of feasible vectors so thatan NPV optimum is approached. A GA subroutine initiated from within the spreadsheet environmentcan effectively generate, evaluate and select combinations. The spreadsheet does not have to bere-expressed in another form, as would be required if a mathematical programming add-in wereto be used. The GA–spreadsheet combination effectively separates the requirement for financialunderstanding from the requirement to understand the operation of the optimization approach.



The process of initiating a GA varies from package to package, but usually requires littlemore than the identification of the cell containing the fitness function value and the driver cellsthat make up the binary vector that defines a combination of projects. In some GA packagesthat support variants of the GA approach the driver cells must be restricted to take only ‘0’ or‘1’ values.

When an optimum has been found, managers can immediately experiment with it by changingthe contents of specific driver cells. The case for the inclusion of favoured projects can be inves-tigated, as can ‘surprising’ inclusions or omissions in the optimum project combination suggestedby the GA. This is a much simpler process than imposing constraints on a mathematical program-ming formulation and re-solving.

If the tax system does not allow carry back of unused capital allowances, then a basic GAapproach is all that is needed. The operations of crossover and mutation cannot generate an invalidbinary vector. However, the possibility of carry back of unused capital allowances raises a poten-tially serious issue. There is now a taxation policy choice to be made by the company undertakingthe investment decision. If, in a given year, there are capital allowances that cannot be absorbedby available profit, then the possibilities of carry back of unused allowance, or carry forward,become available. In a given tax system there might be constraints on the way in which the policychoice can be made. For example, assume that unused capital allowances can be:

1. carried forward indefinitely, or,2. carried back up to 3 years, and only when the maximum use of the carry-back provision has

been made can any residual amount be carried forward.

The approach adopted in such a situation is to represent the policy choice by adding to the binaryvector that represents a solution to the capital budgeting problem an additional element for everyyear in the planning horizon. If the ‘0–1’ variable for a given year has a value of ‘1’, then carryback should occur before carry forward. If the variable has a value of ‘0’, then carry forward is thepolicy choice. At first sight it appears that nothing of significance has changed. The binary vectorhas j + k = n elements, where j represents project choices and k represents the tax policy for eachyear in the planning horizon. Unfortunately, although the normal operations of the GA, reproduction,crossover, and mutation will operate as before, they may generate infeasible solutions; a binaryvector may be generated that recommends carry back of allowances in a given year when there areno allowances to be carried back, or when the preceding years do not have taxable profits sufficientto absorb carried-back allowances.

In these circumstances, several alternative responses are theoretically possible:

1. Repair the offending solutions in some fashion and allow them to remain in the population ofpotential solutions.

2. Remove any infeasible solutions from the population of potential solutions.3. Allow infeasible solutions to remain in the population of potential solutions.

The third option seems, at first sight, to be the least acceptable of the three. However, one of thecosts of adopting the spreadsheet–GA combination may be a sacrifice of control over the way theGA operates. Repair and removal are not often available options. The third alternative of allowingthe infeasible binary vector to remain in the population may, in fact, be the only one that can beadopted.



When this approach has to be adopted, the infeasible binary vector must be assigned a fitnessvalue. The spreadsheet can check whether carry back is feasible when the GA recommends it. Ifthere is no profit in preceding years against which carried-back allowances can be set, then thespreadsheet can override the GA’s recommendation and carry forward unused allowances. All thatis required to achieve this is another ‘IF . . . THEN . . . ELSE . . .’ statement that checks the condi-tion that there are both allowances available to be carried back and earlier profits against which theallowances can be claimed. For the purpose of evaluation, the binary vector is corrected, but itreverts to its original form when the GA generates a new population.

7. SPREADSHEET IMPLEMENTATION OF DIFFERENT TAX REGIMES

The spreadsheet extract shown in Table I contains a typical ‘front end’ for the spreadsheet–GApartnership. It allows the facility to operate under different tax systems. By inputting values intocells J7 and J5, the options of single tax rate or multiple tax rate, and 0-, 1-, 2-, or 3-year carryback of allowances can be selected. Specific tax rates and capital allowance rates can also be inputin the relevant cells. Thus, the regular revisions to the tax system that governments introduce maynot involve significant rewriting of the spreadsheet. However, even if a more fundamental recon-struction of the tax system were to take place, nothing more is required of the financial managerthan would be the case if only a single project was being evaluated.

8. THE SYSTEM IN USE

Use of the approach in consultancy and teaching contexts generally starts with a discussion of theafter-tax cash flow generated by a single project. The idea that this project’s contribution will varyaccording to the situation of the company that takes it on is then explored. Next, two projects areexamined, individually and in combination. Then a spreadsheet is developed that allows for a greaternumber of projects. This spreadsheet is used to give managers an environment in which they canexplore the consequences of particular project choices. All a financial manager has to do to selecta combination of projects is to enter the required list of zeros and ones in cells C11 to C24. If the taxsystem involves a choice of capital allowance carry forward or carry back, then a policy must alsobe chosen by entering zeros and ones in cells E50 to K50. This step often causes initial problems.Users are advised to start with these cells set to zero and then to check visually where taxable profitoccurs over the planning horizon. Using the simple guideline “money now is worth more than moneytomorrow”, they are then advised to look where unused capital allowances could transform taxpayments into tax rebates. As they make changes they are advised to watch the impact on the overallNPV of the project combination. Inevitably there comes a point when a tax policy change has noimpact on the NPV. Users are then introduced to the model’s ability to override infeasible tax policydecisions, as this is a common explanation for an apparently ineffective tax policy change.

Only when users have developed a familiarity with the spreadsheet are they introduced to the GAadd-in. Initially, no explanation is offered of the way in which GAs work. Some phrase such as ‘Thisis a method capable of generating and evaluating project combinations and tax policies.’ is used.Once the GA has been run, users are once again encouraged to explore the solution by makingsimple changes. Discussion is then initiated to explore whether any projects are featuring in theselected combination more for tax management than inherent attractiveness reasons.



Experiments are carried out to understand the impact of different crossover and mutation para-meters prior to users being given access to the GA. Different GA implementations often performdifferently! Hence, this experimentation has to be repeated for each new spreadsheet–GA combi-nation encountered. The adequacy of the system’s performance is checked against a mathematicalprogramming representation of the tax system.

9. CONCLUSIONS

The textbook approach to investment appraisal, project-by-project evaluation on an after-tax basis,is not guaranteed to generate an optimal combination of projects. Because tax is levelled at thecompany level rather than at the project level, tax can generate interactions between a project’s anda company’s current cash flows, and between the cash flows of different projects under considera-tion. Some projects that are unattractive when considered individually may feature in an optimalproject combination because they allow profits generated by other projects to be sheltered from tax.

The processes of combination generation, evaluation and selection can be supported in a varietyof ways. However, some approaches require model users to have the skills of model builders, aswell as those required for their own work. The mixed integer programming approach, for example,requires understanding of the tax system and an understanding of how characteristics of the taxsystem can be represented in a mathematical programming framework.

The use of a GA can also support combination generation, evaluation and selection. However,it separates the optimization mechanics from the way in which the fitness function is represented.This allows a separation of roles within the model-building process. Pairing the GA with aspreadsheet in a hybrid system also allows tax experts and financial managers to define the fitnessfunction in a form they are comfortable with. Experience has shown the GA approach to be capableof finding either optimal or near-optimal project combinations for a wide range of tax-system struc-tures. Paired with a spreadsheet, it has also been shown to provide an environment in which man-agers are able to explore the impact of changes in projects and tax policies.

It must be emphasized that this paper does not claim to support the general superiority of thisapproach in other problem areas. Manonnga (1996) confirms there are characteristics of this problemarea that render it particularly amenable to this approach. These are:

1. The ease with which decisions can be represented by binary variables.2. The lack of links between decisions, e.g. no requirement that particular project combinations or

tax policies should ‘go together’.3. The clearly defined nature of the optimum.4. The ease with which infeasible solutions can be recognized.5. The ease with which both feasible and infeasible solutions can be assigned a fitness value.

These characteristics are not always present in other applications contexts.

REFERENCES

Berry RH, Dyson RG. 1979. A mathematical programming approach to taxation induced interdependencies ininvestment appraisal. Journal of Business Finance and Accounting 6(4): 425–41.



Berry RH, Dyson RG. 1997. Tax induced project interactions. In Explorations in Financial Control, LapsleyI, Wilson RMS (eds). Thompson Business Press.

Berry RH, Smith G. 1993. Using a genetic algorithm to investigate taxation induced interdependencies incapital budgeting. In Artificial Neural Nets and Genetic Algorithms, Albrecht F, Reeves CR, Steel NC (eds).Springer Verlag.

Brealey RA, Myers SC, Allen F. 2006. Corporate Finance, 8th ed. McGraw Hill.Buckley A. 1975. The distorting effects of surplus advanced corporation tax. Accounting and Business Research

(Summer) 19: 168–176.Cooper I, Franks JR. 1983. The interaction of financial and investment decisions when the firm has unused tax

credits. Journal of Finance 38: 571–584.Goldberg DE. 1989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley.Haley CW, Schall LD. 1979. The Theory of Financial Decisions. McGraw Hill.Holland J. 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press.Manongga DHF. 1996. Using genetic algorithm based methods for financial analysis. PhD thesis, University

of East Anglia (unpublished).Weingartner HM. 1974. Mathematical Programming and the Analysis of Capital Budgeting Problems. Kershaw.Winston WL, Albright SC. 2001. Practical Management Science. Duxbury.

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Integrating genetic algorithms and spreadsheets: a capital budgeting application