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Expert Systems with Applications Expert Systems with Applications 36 (2009) 1155–1163 www.elsevier.com/locate/eswa



The use of a fuzzy logic-based system in cost-volume-profit analysis 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 analytical tool cost-volume-profit, commonly called breakeven analysis (BE), is used widely in managerial decision making. In its basic form, CVP analysis 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 model is a great step forward in providing more useful information for profit planning. Nevertheless, so far a powerful approach for solving the problem is still lacking because there remains imprecision in an expert’s assessment of uncertainty factors. This paper presents a model that utilizes experts’ knowledge, employs the fuzzy set concept to handle imprecision, and then to establish a fuzzy logic-based system for managers to access and evaluate the cost-volume-profit decision making process, and finally to make the right decision. Ó 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 in which three of four variables (i.e., sales volume, variable costs, fixed costs, and selling price) are assumed to be known. One of the shortcomings of conventional CVP analysis is its inability to account for uncertainty and risk. Therefore, the restrictive assumptions of the conventional CVP model limit its usefulness to only certainty equivalent conditions that do not exit in the business environment (Charnes, Cooper, & Ijiri, 1963). Since then, probabilistic, simulation and stochastic models have been developed and received considerable attention (Adar, Barnea, & Lev, 1977; Constantinides, Ijiri, & Leitch, 1981; Hillard & Leitch, 1975; Ismail & Louderback, 1979; Jaedicke & Robicheck, 1964; Liao, 1975; Maloo, 1991; Shih, 1979). Although there are many probabilistic and stochastic models that analyze variables uncertainties, some managers choose *



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0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.11.025



not to use them because the models are either too complex or costly for small or medium size firms, and some managers without any experience of using rigorous statistical and mathematical analysis might not be able to justify the use of sophisticated probabilistic models. Therefore, managers need a practical and simplified method that could minimize these complexities and that requires minimal resources in solving breakeven problems under conditions of uncertainty. Furthermore, available stochastic and simulation models are restrictive in application because they are based on varying assumptions. Probabilistic models require the assumption of a standard distribution such as a normal distribution, which is inflexible in accommodating dynamic business conditions. Simulation techniques need the availability of probabilistic data on relevant inputs; data that are not readily available. Historical distributions do not always cast light on unfolding future events; as such, they are inadequate for handling conditions involving uncertainty (Maloo, 1991). When market fluctuations can not be predicted with certainty, managers have to make decisions under conditions of



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uncertainty. Under these conditions, decisions to make or not are often based on managers’ human intuitions, common sense and experience, rather than on the availability of clear, concise and accurate data. Fuzzy logic is used for reasoning about inherently vague concepts (Lukasiewicz, 1970), such as ‘profit is good or not’, where level of profit is open to interpretation. A firm’s projection of profit is based on relatively precise forecasts of sales and cost behavior. Differences between planned versus actual profit are attributed to fluctuations in costs, selling prices, and volume. The identification of all these intricate interrelationships is very important for managers to be successful in planning and control. Once they identify these interrelationships, managers can concentrate on strategies or products that can yield maximum profits. Thus, a technique that can provide a reliable range of estimates of costs and revenues for planning purposes, thereby minimize the differences between the planned and actual results should be seriously considered. The purpose of this research is therefore to apply the fuzzy logic to human reasoning where we specifically focus on the reasoning processes behind CPV analysis. Under uncertain circumstances, a person’s deduction and thinking process contains fuzzy factors. A human usually thinks in imprecise terms such as high and low, fact and slow, and heavy and light (Black, 1937). If such fuzziness has not been incorporated into the decision model, the real situations are not being represented correctly and thus the decision made can be erroneous. Chan and Yuan (1990) pioneered the use of fuzzy set theory in CVP decision model, but they did not use fuzzy logic approach. Their models are conceptual that requires additional definition and refinement for practical applications. A fuzzy expert system can model imprecise information by attempting to capture knowledge in a similar fashion to the way in which 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, intelligent systems that might be used to understand the CVP decision making process. The purpose of this paper is to demonstrate how fuzzy set concept can be applied to CVP decision analysis, and then adopt a fuzzy logic approach to easily analyze the interrelationships among uncertainty variables on decision making utilizing a mathematical research toolset, Matlab fuzzy logic toolboxÒ as a means of coping with uncertainty that are often present in determining profit level in a CVP model. To build a fuzzy expert system for a CVP model that is based on fuzzy logic, the researcher has captured, organized and used human expert knowledge by interviewing sales managers. 2. Methodology CVP Model and related variables The basic stochastic cost-volume-profit analyses were essentially based upon the following traditional relationship:



T ¼ SðP  V Þ  F where T is the total profit, S is sales volume in units, P is unit selling price, V is unit variable cost, and F is total fixed cost. Generally, a company can not have precise information on products’ selling prices, sales demand, variable costs, and even fixed costs (which can not remain constant in total if the activity falls outside of the relevant range). Usually, sales managers assign the values of variables based on their experience, guesses and rules-of-thumb. For instance, a sales expert may believe that the range of product price between $36 and $48 is reasonable. Consequently, fuzziness in the selling price is involved. Its membership function as well as those of other input variables and output variable are constructed with the sales expert’s assistance and are given in Tables 1 and 2, respectively. Since these variables are often associated with many uncertainties resulting in varied impacts on profit, they are assessed subjectively. For example, the exposure level of input variables is regularly expressed linguistically as low, moderate, and high, whereas the level of the output variable is classified as very low, low, moderate, high, and very high. These linguistic variables with non-crisp information are consistent with the imprecise nature. While traditional quantitative analyses do not address the issue of such imprecision, the concept of fuzzy set permits mathematical operations on this imprecise information or knowledge (AbouRizk & Sawhney, 1993).



Table 1 Membership functions of input variables Input variable



Level



Range



Selling price



Low Moderate High



$36–$41 $39–$44 $43–$48



Sales volume



Low Moderate High



160,000–180,000 170,000–190,000 180,000–200,000



Variable cost



Low Moderate High



$23–$27 $26–$30 $29–$33



Fixed cost



Low Moderate High



$180,000–$210,000 $200,000–$230,000 $220,000–$250,000



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



Level



Range



Profit



Very low Low Moderate High Very high



$0–$800 $800–$1600 $1600–$2400 $2400–$3200 $3200–$4000



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



Fuzzy set membership functions are used in this study to represent the imprecise values related to price classifications and other related variables. Here, price levels, P are represented by fuzzy sets defined by the following: P¼



3 X



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Thus, the membership value, or the degree belonging to the set, for low and moderate price is 0.33. The remaining level is assigned a membership value of 0. 2.1. Fuzzy rules and Fuzzy expert system



Pi



i¼1



where Pi is universe fuzzy subset of price levels; i is an indicator of price level, i = 1, 2, 3, representing low, moderate, and high, respectively. These price levels represent the linguistic terms characterized by fuzzy sets rather than quantity terms. Similarly the variable cost, sales volume, and fixed cost are also divided into three levels (low, moderate, and high). Total profit will have five fuzzy subsets (very low, low, moderate, high, and very high). Membership functions can also be designated graphically, in such way that they overlap to account for uncertainty on the boundaries. Based on general characterization accepted by the sales manager, the trim typed membership functions are utilized to characterize the fuzzy sets of price levels as well as other variables as shown in Figs. 1–5, respectively. The values of the membership function, for instance lp(x), on the elements (x) of its associated fuzzy sets are measures of relative degree of price. Thus, membership function for low price is as follows: n X lp1 ðxi Þ=ðxi Þ ¼ ðx; 36; 38; 41Þ Low price : P 1 ¼ i¼1



¼ 0:0=36 þ 1=38 þ 0=41 where xi is the element of fuzzy subset P1 and lp1 ðxi Þ 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



2.1.1. Fuzzy rules associated with profit levels Before developing the fuzzy expert system, we need to establish fuzzy rules. The total number of rules depends on the number of hedges for each fuzzy set. Hence the number of fuzzy rules for determining the level of total profit can be derived as: price (three), sales volume (three), variable cost (three), and fixed cost (three), which combined the results in 81 distinct fuzzy rules as shown in Fig. 6. The rules describing the basis for a given profit level were based on the degrees of price, sales volume, variable cost, and fixed cost. A rule from Table 3 can be extracted as: If (price = high) and (variable cost = low) and (sales volume = high) and (fixed cost = high) then (profit = very high) 2.1.2. Fuzzy expert system In order to get a complete picture of the fuzzy expert system, an inference diagram is used to give a detailed explanation 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 values are converted from a numerical level to a linguistic level. Following that the fuzzy rules are applied and Mamdani’s fuzzy inference method is executed, which will lead to an output (profit). After aggregating all outputs, the defuzzification process will be executed to extract a numeric value for the profit.



Fig. 1. Membership functions of price levels ðP ¼



P3



i¼1 P i Þ.



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



P3



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



P3



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



i¼1 V i Þ.



i¼1 S i Þ.



P3



i¼1 F i Þ.



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



Fig. 5. Membership function of profit levels ðT ¼



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P5



i¼1 T i Þ.



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



3. Analysis of profit versus other factors Since the primary objective of management is to produce maximized profits, executives should have at their disposal the tools that can be used to set the course of actions



and control the planned activities to reach their goal. Executives could be easily misdirected by incorrect analyses of profit planning without adequate data. In order to fully understand the contributions from various factors to the profit level, it is required to examine the contribution from



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Table 3 Formation of profit rules Rule no.



Price linguistic value



Variable cost linguistic value



Sales volume linguistic value



Fixed cost linguistic value



Profit linguistic value



1



High



Low



High



Low



2



High



Low



High



Moderate



3



High



Low



High



High



4



High



Low



Moderate



Low



5



High



Low



Moderate



Moderate



76 77 78 79 80 81



Low Low Low Low Low Low



High High High High High High



Moderate Moderate Moderate Low Low Low



Low Moderate High Low Moderate High



Very high Very high Very high Very high Very high Low Low Low Low Very low Very low



each factor separately. Fig. 6 shows the contribution to profit originating from the price. The contribution from variable cost, sales volume, and fixed cost has been kept constant at three levels, namely, low, moderate and high, corresponding to rule levels 1–10. A general observation is that Profit is positively related to price for any given value of variable cost, sales volume, and fixed cost. This observation is also plausible to the human mind. Therefore, price strategy is the most important factor in making decision for management. Profits are affected by the interplay of cost, sales volume, and price. Therefore, executive should have at its disposal analyses that will allow reasonably accurate prediction of the effect a change in any one of these factors would have



on the profit picture. In control, these analyses can be useful in determining whether performances were as profitable as they should have been. Therefore, certain questions must be answered when making plans for a coming period, such as ‘‘Should emphasis be placed on increasing sales volume or reducing sales prices?”, ‘‘Should sales prices be reduced in an attempt to increase sales volume, or an increase in selling prices, even though accompanied by a decrease in sales volume, result in more profits?”, or ‘‘Should reducing costs instead of increasing volume be exerted as a step toward increased profits?”, and ‘‘Should efforts be directed toward fixed or variable costs if cost reduction is the best strategy?”. In order to answer these questions, we now attempt to visualize the Profit level as a continuous function of its input parameters. Fig. 8 portrays variation of profit related to sales volume and price. The highest gradient for profit is when 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 make a decision between price and sales volume. For example, if the manager wants to set the price as $42, the sales volume must be around 190,000–200,000 units. Otherwise, if the price is set ‘high’ around $46, the sales volume could be around 160,000–200,000 units. Fig. 9 portrays variation of profit related to price and variable cost. The highest gradient for Profit is only when price is ‘high’ and variable cost is ‘low’. Fig. 10 portrays variation of Profit related to sales volumes and variable cost. It shows the highest gradient for profit is when variable cost is ‘low’ to ‘moderate’ and sales volume is ‘low’ to ‘high’. This suggests that variable cost has a significant impact on profit. Figs. 11 and 12 portray variation of profit related to fixed cost and price, fixed cost and sales volume, separately.



Fig. 7. Fuzzy expert system.



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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.



From Fig. 11, we can see that when price is ‘low’ the profit is ‘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. 13 further proves this point, when variable cost is ‘low’ the Profit is ‘high’, and when variable cost is ‘high’ the profit is ‘low’, whereas fixed cost has been kept constant at three



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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.



levels. Therefore, if a manager wants to reduce cost as a step toward increased profits, the efforts should be directed toward 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 management in decision making. Relying on point estimates used in a CVP model for decision making can be misleading if fuzziness is deemed to exist and ignored. This study presents a model to analyze the impact of uncertainty factors on profit. In this model, fuzzy set theory 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



inference mechanism is developed based on Mamdani’s fuzzy reasoning method for assessing profits. The proposed model is a practical approach for small and medium size firms for the analysis of uncertainty. The illustrative examples in this paper reveal that the proposed model should enable managers to answer ‘‘what-if” questions without the extensive quantitative knowledge required in other probabilistic models, with the computation time less than one minute. Since the primary objective of management is to produce a profit, executives can use this tool at their disposal to set the course of actions and control the planned activities in order to reach their goal. Furthermore, managers may use the proposed model as a surrogate for more complicated models in the future, such as multi-product CVP analysis. References AbouRizk, S. M., & Sawhney, A. (1993). Subjective and interactive duration estimation. Canadian Journal of Civil Engineering, 20, 457–470. Adar, Z., Barnea, A., & Lev, B. (1977). A comprehensive cost-volumeprofit analysis under uncertainty. Accounting Review(January), 137–149. Akhter, F., Hobbs, D., & Maamar, Z. (2005). A fuzzy logic-based system for assessing the level of business-to-consumer (B2C) trust in electronic commerce. Expert Systems with Applications, 28, 623–628. Black, M. (1937). Vagueness: An exercise in logical analysis. Philosophy of Science, 4, 427–455.



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Charnes, A., Cooper, W., & Ijiri, Y. (1963). Breakeven budgeting and programming to goals. Journal of Accounting Research, 1(1), 39. Chan, Lilian. Y., & Yuan, Yufei (1990). Dealing with fuzziness in costvolume-profit analysis. Accounting and Business Research, 20(78), 83–95. Constantinides, G., Ijiri, Y., & Leitch, R. (1981). Stochastic cost-volumeprofit analysis with a linear demand function. Decision Sciences(July), 417–427. Cox, E. (1994). The fuzzy systems handbook a practitioner’s guide to building, using, and maintaining fuzzy systems. Cambridge: Academic Press. Hillard, J. E., & Leitch, R. A. (1975). Cost-volume-profit analysis under uncertainty: A long normal approach. The Accounting Review(January), 69–80. Ismail, B., & Louderback, J. (1979). Optimizing and satisfying in stochastic cost-volume-profit analysis. Decision Sciences(April), 205–217. Jaedicke, R. K., & Robicheck, A. A. (1964). Cost-volume-profit analysis under conditions of uncertainty. Accounting Review(October), 917–926. Liao, M. (1975). Model sampling: A stochastic cost-volume-profit analysis. Accounting Review(October), 780–790. Lukasiewicz, J. (1970). Philosophical remarks on many-valued systems of prepositional logic reprinted in selected works. Amsterdam: NorthHolland (pp. 153–179). Maloo, M. C. (1991). A practical approach for incorporating uncertainty in the conventional cost-volume-profit model. Akron Business and Economic Review, 22(4), 29–40. Shih, W. (1979). A general decision model for cost-volume-profit analysis under uncertainty. Accounting Review(October), 687–706.