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Expert Systems with Applications 36 (2009) 1155–1163

Expert Systemswith Applications

The use of a fuzzy logic-based system in cost-volume-profitanalysis under uncertainty

Fong-Ching Yuan *

Department of Information Systems, Yuan-Ze University, 135 Yuan-Tung Road, Chung-Li 320, Taiwan, ROC

Abstract

The purpose of this paper is to present an application of fuzzy logic to cost-volume-profit (CVP) analysis. The conventional analyticaltool cost-volume-profit, commonly called breakeven analysis (BE), is used widely in managerial decision making. In its basic form, CVPanalysis examines sales prices, sales volume, variable costs and fixed costs in relation to target profit levels. This traditional CVP analysis,however, ignores the risk and uncertainty features of a firm’s operations, thus severely limits its usefulness. During the past 10 years,accountants have attempted to resolve this problem by using stochastic analysis. The use of stochastic analysis in a CVP analysis modelis a great step forward in providing more useful information for profit planning. Nevertheless, so far a powerful approach for solving theproblem is still lacking because there remains imprecision in an expert’s assessment of uncertainty factors. This paper presents a modelthat utilizes experts’ knowledge, employs the fuzzy set concept to handle imprecision, and then to establish a fuzzy logic-based system formanagers to access and evaluate the cost-volume-profit decision making process, and finally to make the right decision.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Cost-volume-profit analysis; Breakeven analysis; Fuzzy logic

1. Introduction

The traditional CVP analysis is a deterministic model inwhich three of four variables (i.e., sales volume, variablecosts, fixed costs, and selling price) are assumed to beknown. One of the shortcomings of conventional CVPanalysis is its inability to account for uncertainty and risk.Therefore, the restrictive assumptions of the conventionalCVP model limit its usefulness to only certainty equivalentconditions that do not exit in the business environment(Charnes, Cooper, & Ijiri, 1963). Since then, probabilistic,simulation and stochastic models have been developed andreceived considerable attention (Adar, Barnea, & Lev,1977; Constantinides, Ijiri, & Leitch, 1981; Hillard &Leitch, 1975; Ismail & Louderback, 1979; Jaedicke & Robi-check, 1964; Liao, 1975; Maloo, 1991; Shih, 1979).Although there are many probabilistic and stochastic modelsthat analyze variables uncertainties, some managers choose

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doi:10.1016/j.eswa.2007.11.025

* Tel.: +886 03 4638800 2606; fax: +886 03 4352077.E-mail address: imyuan@saturn.yzu.edu.tw

not to use them because the models are either too complexor costly for small or medium size firms, and some manag-ers without any experience of using rigorous statistical andmathematical analysis might not be able to justify the useof sophisticated probabilistic models. Therefore, managersneed a practical and simplified method that could minimizethese complexities and that requires minimal resources insolving breakeven problems under conditions of uncer-tainty. Furthermore, available stochastic and simulationmodels are restrictive in application because they are basedon varying assumptions. Probabilistic models require theassumption of a standard distribution such as a normal dis-tribution, which is inflexible in accommodating dynamicbusiness conditions. Simulation techniques need the avail-ability of probabilistic data on relevant inputs; data thatare not readily available. Historical distributions do notalways cast light on unfolding future events; as such, theyare inadequate for handling conditions involving uncer-tainty (Maloo, 1991).

When market fluctuations can not be predicted with cer-tainty, managers have to make decisions under conditions of

Table 1Membership functions of input variables

Input variable Level Range

Selling price Low $36–$41Moderate $39–$44High $43–$48

Sales volume Low 160,000–180,000Moderate 170,000–190,000High 180,000–200,000

Variable cost Low $23–$27Moderate $26–$30High $29–$33

Fixed cost Low $180,000–$210,000Moderate $200,000–$230,000High $220,000–$250,000

Table 2Membership functions of output variable (unit: 1000)

Output variable Level Range

Profit Very low $0–$800Low $800–$1600Moderate $1600–$2400High $2400–$3200Very high $3200–$4000

1156 F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163

uncertainty. Under these conditions, decisions to make ornot are often based on managers’ human intuitions, com-mon sense and experience, rather than on the availabilityof clear, concise and accurate data. Fuzzy logic is used forreasoning about inherently vague concepts (Lukasiewicz,1970), such as ‘profit is good or not’, where level of profitis open to interpretation. A firm’s projection of profit isbased on relatively precise forecasts of sales and cost behav-ior. Differences between planned versus actual profit areattributed to fluctuations in costs, selling prices, and vol-ume. The identification of all these intricate interrelation-ships is very important for managers to be successful inplanning and control. Once they identify these interrelation-ships, managers can concentrate on strategies or productsthat can yield maximum profits. Thus, a technique thatcan provide a reliable range of estimates of costs and reve-nues for planning purposes, thereby minimize the differencesbetween the planned and actual results should be seriouslyconsidered. The purpose of this research is therefore toapply the fuzzy logic to human reasoning where we specifi-cally focus on the reasoning processes behind CPV analysis.

Under uncertain circumstances, a person’s deductionand thinking process contains fuzzy factors. A human usu-ally thinks in imprecise terms such as high and low, factand slow, and heavy and light (Black, 1937). If such fuzz-iness has not been incorporated into the decision model,the real situations are not being represented correctly andthus the decision made can be erroneous. Chan and Yuan(1990) pioneered the use of fuzzy set theory in CVP deci-sion model, but they did not use fuzzy logic approach.Their models are conceptual that requires additional defini-tion and refinement for practical applications. A fuzzyexpert system can model imprecise information by attempt-ing to capture knowledge in a similar fashion to the way inwhich it is considered to be represented in the human mind,and therefore improves cognitive modeling of a problem(Akhter, Hobbs, & Maamar, 2005; Cox, 1994). As a result,fuzzy logic is leading to new and human-like, intelligentsystems that might be used to understand the CVP decisionmaking process. The purpose of this paper is to demon-strate how fuzzy set concept can be applied to CVP deci-sion analysis, and then adopt a fuzzy logic approach toeasily analyze the interrelationships among uncertaintyvariables on decision making utilizing a mathematicalresearch toolset, Matlab fuzzy logic toolbox� as a meansof coping with uncertainty that are often present in deter-mining profit level in a CVP model. To build a fuzzy expertsystem for a CVP model that is based on fuzzy logic, theresearcher has captured, organized and used human expertknowledge by interviewing sales managers.

2. Methodology

CVP Model and related variablesThe basic stochastic cost-volume-profit analyses were

essentially based upon the following traditional relation-ship:

T ¼ SðP � V Þ � F

where T is the total profit, S is sales volume in units, P isunit selling price, V is unit variable cost, and F is total fixedcost.

Generally, a company can not have precise informationon products’ selling prices, sales demand, variable costs,and even fixed costs (which can not remain constant intotal if the activity falls outside of the relevant range). Usu-ally, sales managers assign the values of variables based ontheir experience, guesses and rules-of-thumb. For instance,a sales expert may believe that the range of product pricebetween $36 and $48 is reasonable. Consequently, fuzzinessin the selling price is involved. Its membership function aswell as those of other input variables and output variableare constructed with the sales expert’s assistance and aregiven in Tables 1 and 2, respectively.

Since these variables are often associated with manyuncertainties resulting in varied impacts on profit, theyare assessed subjectively. For example, the exposure levelof input variables is regularly expressed linguistically aslow, moderate, and high, whereas the level of the outputvariable is classified as very low, low, moderate, high,and very high. These linguistic variables with non-crispinformation are consistent with the imprecise nature. Whiletraditional quantitative analyses do not address the issue ofsuch imprecision, the concept of fuzzy set permits mathe-matical operations on this imprecise information or knowl-edge (AbouRizk & Sawhney, 1993).

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Fuzzy set membership functions are used in this study torepresent the imprecise values related to price classifica-tions and other related variables. Here, price levels, P arerepresented by fuzzy sets defined by the following:

P ¼X3

i¼1

P i

where Pi is universe fuzzy subset of price levels; i is an indi-cator of price level, i = 1, 2, 3, representing low, moderate,and high, respectively. These price levels represent the lin-guistic terms characterized by fuzzy sets rather than quan-tity terms. Similarly the variable cost, sales volume, andfixed cost are also divided into three levels (low, moderate,and high). Total profit will have five fuzzy subsets (verylow, low, moderate, high, and very high). Membershipfunctions can also be designated graphically, in such waythat they overlap to account for uncertainty on the bound-aries. Based on general characterization accepted by thesales manager, the trim typed membership functions areutilized to characterize the fuzzy sets of price levels as wellas other variables as shown in Figs. 1–5, respectively. Thevalues of the membership function, for instance lp(x), onthe elements (x) of its associated fuzzy sets are measuresof relative degree of price.

Thus, membership function for low price is as follows:

Low price : P 1 ¼Xn

i¼1

lp1ðxiÞ=ðxiÞ ¼ ðx; 36; 38; 41Þ

¼ 0:0=36þ 1=38þ 0=41

where xi is the element of fuzzy subset P1 and lp1ðxiÞ is its

corresponding membership value with respect to low price.Similarly, the rest of the price levels are defined as shown

in Fig. 1. For instance, given $40, P is represented by

P ¼ ½P 1; P 2; P 3� ¼ ½0:33=40; 0:33=40; 0=40�

Fig. 1. Membership functions o

Thus, the membership value, or the degree belonging to theset, for low and moderate price is 0.33. The remaining levelis assigned a membership value of 0.

2.1. Fuzzy rules and Fuzzy expert system

2.1.1. Fuzzy rules associated with profit levels

Before developing the fuzzy expert system, we need toestablish fuzzy rules. The total number of rules dependson the number of hedges for each fuzzy set. Hence thenumber of fuzzy rules for determining the level of totalprofit can be derived as: price (three), sales volume (three),variable cost (three), and fixed cost (three), which com-bined the results in 81 distinct fuzzy rules as shown inFig. 6. The rules describing the basis for a given profit levelwere based on the degrees of price, sales volume, variablecost, and fixed cost. A rule from Table 3 can be extractedas:

If (price = high) and (variable cost = low) and (salesvolume = high) and (fixed cost = high) then (profit = veryhigh)

2.1.2. Fuzzy expert system

In order to get a complete picture of the fuzzy expertsystem, an inference diagram is used to give a detailedexplanation of the processes involved, as shown in Fig. 7.The crisp inputs include price, sales volume, variable cost,and fixed cost to get a value for the profit level. These val-ues are converted from a numerical level to a linguisticlevel. Following that the fuzzy rules are applied and Mam-dani’s fuzzy inference method is executed, which will leadto an output (profit). After aggregating all outputs, thedefuzzification process will be executed to extract anumeric value for the profit.

f price levels ðP ¼P3

i¼1P iÞ.

Fig. 2. Membership function of variable cost levels ðV ¼P3

i¼1V iÞ.

Fig. 3. Membership function of sales volume levels ðS ¼P3

i¼1SiÞ.

Fig. 4. Membership function of fixed cost levels ðF ¼P3

i¼1F iÞ.

1158 F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163

Fig. 5. Membership function of profit levels ðT ¼P5

i¼1T iÞ.

Fig. 6. Rules associated with profit in the knowledge base.

F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163 1159

3. Analysis of profit versus other factors

Since the primary objective of management is to pro-duce maximized profits, executives should have at their dis-posal the tools that can be used to set the course of actions

and control the planned activities to reach their goal. Exec-utives could be easily misdirected by incorrect analyses ofprofit planning without adequate data. In order to fullyunderstand the contributions from various factors to theprofit level, it is required to examine the contribution from

Table 3Formation of profit rules

Ruleno.

Pricelinguisticvalue

Variable costlinguisticvalue

Sales volumelinguisticvalue

Fixed costlinguisticvalue

Profitlinguisticvalue

1 High Low High Low Veryhigh

2 High Low High Moderate Veryhigh

3 High Low High High Veryhigh

4 High Low Moderate Low Veryhigh

5 High Low Moderate Moderate Veryhigh

76 Low High Moderate Low Low77 Low High Moderate Moderate Low78 Low High Moderate High Low79 Low High Low Low Low80 Low High Low Moderate Very low81 Low High Low High Very low

1160 F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163

each factor separately. Fig. 6 shows the contribution toprofit originating from the price. The contribution fromvariable cost, sales volume, and fixed cost has been keptconstant at three levels, namely, low, moderate and high,corresponding to rule levels 1–10. A general observationis that Profit is positively related to price for any givenvalue of variable cost, sales volume, and fixed cost. Thisobservation is also plausible to the human mind. There-fore, price strategy is the most important factor in makingdecision for management.

Profits are affected by the interplay of cost, sales volume,and price. Therefore, executive should have at its disposalanalyses that will allow reasonably accurate prediction ofthe effect a change in any one of these factors would have

Fig. 7. Fuzzy ex

on the profit picture. In control, these analyses can be use-ful in determining whether performances were as profitableas they should have been. Therefore, certain questionsmust be answered when making plans for a coming period,such as ‘‘Should emphasis be placed on increasing salesvolume or reducing sales prices?”, ‘‘Should sales prices bereduced in an attempt to increase sales volume, or anincrease in selling prices, even though accompanied by adecrease in sales volume, result in more profits?”, or‘‘Should reducing costs instead of increasing volume beexerted as a step toward increased profits?”, and ‘‘Shouldefforts be directed toward fixed or variable costs if costreduction is the best strategy?”.

In order to answer these questions, we now attempt tovisualize the Profit level as a continuous function of itsinput parameters. Fig. 8 portrays variation of profit relatedto sales volume and price. The highest gradient for profit iswhen price is ‘high’ and sales volume is from ‘low’ to ‘high’,or price is ‘moderate’ and sales volume must be ‘high’.From this figure, it is very easy for the manager to makea decision between price and sales volume. For example,if the manager wants to set the price as $42, the sales vol-ume must be around 190,000–200,000 units. Otherwise, ifthe price is set ‘high’ around $46, the sales volume couldbe around 160,000–200,000 units.

Fig. 9 portrays variation of profit related to price andvariable cost. The highest gradient for Profit is only whenprice is ‘high’ and variable cost is ‘low’. Fig. 10 portraysvariation of Profit related to sales volumes and variablecost. It shows the highest gradient for profit is when vari-able cost is ‘low’ to ‘moderate’ and sales volume is ‘low’to ‘high’. This suggests that variable cost has a significantimpact on profit.

Figs. 11 and 12 portray variation of profit related tofixed cost and price, fixed cost and sales volume, separately.

pert system.

Fig. 8. Profit level related to price and sales volume.

Fig. 9. Profit level related to price and variable cost.

Fig. 10. Profit level related to sales volume and variable cost.

F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163 1161

From Fig. 11, we can see that when price is ‘low’ the profitis ‘low’, and when price is ‘high’, the profit is ‘high’,whereas fixed cost has been kept constant at three levels.The results of Fig. 11 are the same as those of Fig. 12. This

suggests that fixed cost has no impacts to profit. Fig. 13further proves this point, when variable cost is ‘low’ theProfit is ‘high’, and when variable cost is ‘high’ the profitis ‘low’, whereas fixed cost has been kept constant at three

Fig. 11. Profit level related to price and fixed cost.

Fig. 12. Profit level related to sales volume and fixed cost.

Fig. 13. Profit level related to variable cost and fixed cost.

1162 F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163

levels. Therefore, if a manager wants to reduce cost as astep toward increased profits, the efforts should be directedtoward variable cost.

4. Conclusion

Cost-volume-profit analysis is a useful managerial tool,but the estimation of profit that is often characterized by

many uncertainties has resulted in difficulties for the man-agement in decision making. Relying on point estimatesused in a CVP model for decision making can be mislead-ing if fuzziness is deemed to exist and ignored.

This study presents a model to analyze the impact ofuncertainty factors on profit. In this model, fuzzy set the-ory is applied to handle the imprecision quantitatively,the rule-based knowledge is employed, and then a fuzzy

F.-C. Yuan / Expert Systems with Applications 36 (2009) 1155–1163 1163

inference mechanism is developed based on Mamdani’sfuzzy reasoning method for assessing profits.

The proposed model is a practical approach for smalland medium size firms for the analysis of uncertainty.The illustrative examples in this paper reveal that the pro-posed model should enable managers to answer ‘‘what-if”questions without the extensive quantitative knowledgerequired in other probabilistic models, with the computa-tion time less than one minute. Since the primary objectiveof management is to produce a profit, executives can usethis tool at their disposal to set the course of actions andcontrol the planned activities in order to reach their goal.Furthermore, managers may use the proposed model as asurrogate for more complicated models in the future, suchas multi-product CVP analysis.

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