Human Resource Allocation in a CPA Firm: A ... - Semantic Scholar

Review of Quantitative Finance and Accounting, 20: 277–290, 2003 C 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

Human Resource Allocation in a CPA Firm: A Fuzzy Set Approach WIKIL KWAK∗ Department of Accounting, College of Business Administration, University of Nebraska at Omaha, U.S.A. Tel.: (402) 554-2821 E-mail: [email protected] YONG SHI College of Information Science and Technology, University of Nebraska at Omaha, Omaha, Nebraska 68182 KOOYUL JUNG Department of Management Information Systems, Korea Advanced Institute of Science and Technology, Seoul, South Korea 207-43

Abstract. The review of existing human resource allocation models for a CPA firm shows that there are major shortcomings in the previous mathematical models. First, linear programming models cannot handle multiple objective human resource allocation problems for a CPA firm. Second, goal programming or multiple objective linear programming (MOLP) cannot deal with the organizational differentiation problems. To reduce the complexity in computing the trade-offs among multiple objectives, this paper adopts a fuzzy set approach to solve human resource allocation problems. A solution procedure is proposed to systematically identify a satisfying selection of possible staffing solutions that can reach the best compromise value for the multiple objectives and multiple constraint levels. The fuzzy solution can help the CPA firm make a realistic decision regarding its human resource allocation problems as well as the firm’s overall strategic resource management when environmental factors are uncertain. Key words: a fuzzy set, human resource allocation, accounting firm JEL Classification: M4

Introduction “Xerox Faces Criminal Inquiry Tied to Financial Restatement” (The Wall Street Journal, September 24, 2002). This is a common story we read in today’s business news. Because of earnings manipulations or misstatements from major corporations such as Xerox, the accounting profession as well as top management are under pressure to reform from their stakeholders such as investors, congress, government, union, etc. They expect auditors to do a lot better job even though auditors are now required spend more time on fraud issues. Otherwise, soon the government will control the profession. Therefore, formal planning for staff personnel in a CPA firm is very important because it enables a firm to function ∗ Corresponding

author.

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more efficiently and effectively, given the current uncertain market situation and difficulty in retaining highly qualified accountants. The current situation of declining numbers of accounting graduates and new CPA candidates requires accounting firms to rethink their human resource allocation problems. Information released in the American Institute of CPAs’ based on the 1998–1999 academic year, shows a 22-percent drop in the number of accounting graduates compared to the 1995–1996 academic year (Accounting Today, February 12–25, 2001). If this trend continues, accounting firms will be forced to do formal planning to address their human resource allocation problems. Firms will be required to perform a variety of services such as management advisory services, auditing, and tax planning for their clients with less manpower. If a firm decides to have a formal planning model, the model should incorporate economic and professional objectives of their personnel. As one of the professional objectives, 120 hours of continuing professional education for all members in public practice every three years was proposed and accepted by the council of American Institute of Certified Public Accountants (AICPA) several years ago. In addition, rotation of audit partners and staff was proposed to increase “auditor independence” by the Charted Accountants’ Joint Ethics Committee by the calendar year 1995 (Business Review Weekly, 1997). Generally, planning audit staffing in a large accounting firm is a complex task and multiple goals are necessary in today’s highly competitive environment. This paper will use audit staffing as an example of human resource allocation problem in a CPA firm. External auditors who are highly trained and experienced certified public accountants with staff members need to collect information about the client firm’s financial conditions to provide audit opinions to the public. If auditors fail to detect material misstatements of financial statements prepared by management, investors could sue auditing firms because they usually have deep pockets. Auditing is so complex and ambiguous that random or convenient assignment of auditors to clients is impossible (Summers, 1972). In addition, mixing of skill levels of auditors, dealing with qualitative appraisals, and undefined, inconstant processes in its assignment function is natural in auditing. Generally, a managing partner decides the size of audit team based on client risk and amount of work. Then managers decide training and selection of personnel based on preliminary audit planning. In this paper, audit planning is a broad term that includes personnel selection and training in a CPA firm. Linear programming focuses on a single goal—usually profit maximization or cost minimization—but this is not the situation for an accounting firm, which is a service organization with multiple goals. Besides, in today’s uncertain capital market, with higher risk of financial fraud by management, proper audit planning, as one example of a CPA firm’s resource allocation problem, helps auditors and management share the common multiple goals (Hubbard, 2000). Proper audit planning also helps match up the skills of the auditors to the areas of their expertise. Since the fuzzy set approach provides a simultaneous solution to a complex system of competing objectives, it seems to be a proper tool for an accounting firm’s human resource allocation problem in an ambiguous environment. In this paper, we will discuss an application of fuzzy set approach of this audit staff assignment problem that incorporates the multiple ambiguous objectives of a CPA firm. The same approach can be used to expand general human resource allocation problems of other functional areas such as management advisory services or tax planning of a CPA firm. Fuzzy set theory has been applied in accounting in several areas such as an audit sampling model, zero-based

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budgeting, a materiality judgment problem, a variance investigation model, target costing, internal control evaluation model, and capital budgeting problems. Recently, Chandra and Agrawal (1998) use fuzzy logic that can translate linguistic ambiguity into operational numbers for cost management systems design. However, no previous papers try to apply a fuzzy set approach in an audit-staffing problem for a CPA firm. Zebda (1998) suggested that the basic calculus of fuzzy set theory be effectively used to the ambiguity in accounting, such as vague probability judgments, imprecise payoffs, and varying degrees of precision. Similarly, other methods of decision making under ambiguity, such as rough set approach can also be applied to the inconsistency of audit judgments (Siegel et al., 1998). Fuzzy set theory can be applied to other business problems whenever there is a need to do modeling with imprecise reasoning processes or ambiguity in human decision-making. For example, Ruefli and Sarrazin (1981) proposed a fuzzy set approach in strategic control of corporate development in ambiguous situations. Fuzzy set theory should receive more attention in the United States to simplify the modeling of complex business decision-making with ambiguity. With currently available software, the solution is practical for numerous real world problems. In the following section, prior research is discussed. The next section presents multiple objective audit staff-planning models including a fuzzy solution method. Then, a numerical example with analysis and managerial implications, and conclusions and limitations will be addressed in order.

Prior research Several papers have discussed audit-staffing problems of a CPA firm. Summers (1972) used a linear programming approach to solve the audit staff-assigning problem. He saw the limitation of linear programming, which focused on benefit maximization and assumed that the audit office was trying to maximize a mixture of monetary and non-monetary benefits. Therefore, the key factor for his model was assigning monetary equivalent benefits for several non-monetary benefits. Killough and Souders (1973) used a goal programming approach to incorporate multiple objectives. Bailey et al. (1974) advocated using the goal programming approach for the audit staff assignment problem against a linear programming approach. They solved Summer’s model using goal programming and the results showed that goal programming provided better solutions by incorporating several goals compared with solutions of linear programming. The goal programming formulation permits explicit consideration of trade-offs in changing priority of goals. Welling (1977) also used a goal programming approach to take into account human resource interactions in an audit staff allocation problem. Welling used job productivity, human resource development, and individual satisfaction as no monetary benefits in human resource valuation. Balchandran and Steur (1982) used a multiple objective linear programming (MOLP) model for the audit staff assignment problem. They thought that goal programming had limitations because weights should be assigned a priori. Gardner et al. (1990) also proposed a multi-period MOLP approach to solve audit staff planning problems. The MOLP approach does not require the decision-maker to specify weights in advance. However, if the person in charge of planning audit staff has previous experience for the

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company, he could assign weights and priority of goals quite accurately for a given situation. A decision maker can have some idea of trade-offs among priority of goals by doing sensitivity analysis for the priorities and parameters of goals. Thus, goal programming appears to provide a means by which a choice of goals may be better evaluated and manipulated until some optimal mix can be found. However, a goal programming or MOLP approach cannot deal with organizational differentiation or multiple decision maker situations in a real world audit staff assignment problem. Prior research shows that auditors are not likely to make significant audit plan changes unless management has an explicit incentive to misrepresent their financial statements (Glover et al., 2000). In other words, auditors need to decide the extent of audit risk of a client in preliminary audit planning and staffing. Management explanations for significant fluctuations between expected results and actual report data may affect the nature of auditors’ generated hypotheses and increase the risk of material error. Therefore, accounting firms should explore means of assisting their personnel in overcoming the influence of hypotheses from client management (Bedard et al., 1998). Based on experiments, Houston et al. (1999) found that in the presence of errors (unintentional misstatements), the audit risk model, which addresses the risks associated with issuing clean opinion on client financial statements that contain material misstatements, adequately described audit-planning decisions. However, in the presence of irregularities (intentional misstatements) it did not. Usually, auditors use the client’s earnings report to revise their beliefs about the likelihood of fraud when formulating an audit plan (Newman et al., 2001). As these audit planning studies show, audit planning and staffing requires auditors to deal with uncertainty. A fuzzy set approach will be used in this paper to deal with risk or ambiguity in an audit planning and staffing problem. Using accounting firm data, our paper will show that this approach can handle a multiple objective and multiple constraint level audit-staffing problem which is more realistic and practical in a real world situation.

Multiple objectives audit staff planning model The review of existing audit staff planning models shows that two major shortcomings in the previous mathematical models should be addressed. First, neither the linear programming approach nor the goal programming approach can provide all possible optimal trade-offs between multiple objectives under consideration. Linear programming only reflects a single objective of a corporation at a time. As a result, the linear programming approach cannot help the corporation seek to simultaneously achieve several objectives, some in conflict, in business competition. The audit staff planning models set by the goal programming or MOLP approach is an optimal compromise (i.e., trade-off) among several objectives of the corporation. However, it misses other possible optimal compromises of the objectives, which result from some linear combinations of objective weights. These compromises lead to different optimal staffing solutions for different decision situations that the corporation may face. Second, none of the past mathematical models can deal with the organizational differentiation problems. In real-world situations, when a CPA firm designs its staff planning for branch offices, the involved decision makers (executive partners or branch managers) can


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give different opinions on the same issue, such as manpower capacity and client risk factors. They need to assign more experienced senior accountants to high-risk clients. In mathematical models, these different interpretations can be represented by different “constraint levels.” Because both linear programming and goal programming presume a fixed single constraint level, they fail to mathematically describe such an organizational differentiation problem. An auditing job (planning and staffing) needs to deal with the client’s risk by nature. Therefore, the audit-planning program should be designed to respond to client risk factors because client risk factors have significant effects throughout planning, audit program effectiveness, and justification for audit tests (Wright and Bedard, 2000). More audit firms have developed decision aids to deal with uncertainty or ambiguity in audit decision making, but auditors try to ignore the aid when its prediction does not support the initial planning judgment (Boatsman et al., 1997). Previous researchers have developed decision aids based on decision rules that allow auditors to make judgments in uncertain situations, but these decision aids were not equipped to handle uncertainties caused by ambiguity and vagueness (Siegel et al., 1998). Now we need to develop a practical and realistic model to deal with uncertainty factors. Fuzzy logic is a practical method in a decision-making paradigm such as audit planning and staffing problem whenever ambiguity and vagueness are present (Omer and Andre de Korvin, 1998). To mathematically formulate the multiple objective audit-staffing models, let i be the position of audit staffs in a CPA firm under consideration. Define xi as the positions of the audit staff by the office and i = 1, . . . , t. For the coefficients of the objectives, let pi be the gross audit fees generated by the ith staff; let qi be the income made by the ith staff; let ri be the management personnel to staff men ratio to maintain audit quality. For the coefficients of the constraints, let bi be the hourly billing rates and let ci be projected profit believed by the kth manager for the firm. Here, we allow multiple constraint levels for the billing rate, which is estimated by different managers. Generally, we assume that at least two different types of personnel managers get involved in the human resource allocation problems. Then, the multiple objective audit staff-planning model is: Max Max Max

t i=1 t i=1 t

pi xi qi x i ri x i

i=1

subject to: t i=1 k j=1

bi xi ≤ (c1 γ1 + · · · + ck γk ) γ j = 1,

γ j ≥ 0,

xi ≥ 0, i = 1, . . . , t, j = 1, . . . , k.

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Note that even though this model includes only three specific groups of objectives for presentation purposes, the modeling process easily adapts to all audit-staffing problems with multiple objectives and multiple constraint levels. We could use an integer LP program for this type of problem to enforce integer solutions, which may be easier to interpret and implement for the decision maker. The proposed model is an LP problem with three objectives and four constraint levels. In the case of billing rates, the multiple constraint level is used. In other words, each manager’s belief for the increased billing rate for the upcoming year can be different. The differences in the right hand side values are allowed in this model, which permits a group decision-making process. Solving this type of real-world problem is quite a task as market conditions or the firm environment changes. Therefore, this paper proposes a fuzzy set approach to making the computational task as practical as possible.

A fuzzy solution method Based on a compromised solution approach for multiple objective decision making problems, Shi and Liu (1993) and Liu and Shi (1994) proposed fuzzy MC linear programming. Their fuzzy approach adopts the compromised solution or the “satisficing solution” between upper and lower bounds of acceptability for objective payoffs. The following section presents the fuzzy set approach for an audit-staffing problem: Step 1. Given the proposed model in this paper, we may use any available computer software to solve the following series of LP problems. For illustration purposes, let’s assume t = 5 and k = 2: 5 (i) Max (and Min) i=1 pi xi subject to: 5

bi xi = (c1 γ1 + · · · + c5 γ2 )

i=1

2

γ j = 1, γ j ≥ 0,

j=1

and xi ≥ 0, for i = 1, . . . , 5. 5 (ii) Max (and Min) i=1 qi xi , i = 1, . . . , 5 subject to: (1) 5 (iii) Max (and Min) i=1 ri x i subject to: (1)

(1)


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Group (i) of the above LP problems has two types of optimization: one is maximization and the other is minimization. Groups (ii) and (iii) have four problems, respectively. All of them use the same constraints (1). The total number of problems is six. We define the objective value of the maximization problems by Ui as the upper bound of decision makers’ (DM) acceptability for the objective payoff; and the objective value of the minimization problems by L i as the lower bound of DMs’ acceptability for the objective payoff, i = 1, 2, 3. Step 2. Let f (x)i , i = 1, 2, 3, be the objective functions in Step 1. Then, we solve the following linear problem: Max β

(2)

subject to: β( f (x)i − L i )/(Ui − L i ), i = 1, 2, 3, β ≥ 0, 5

(3)

bi xi = (c1 γ1 + · · · + c5 γ2 )

i=1 2

γ j = 1,

γ j ≥ 0,

j=1

and xi ≥ 0,

for i = 1, . . . , 5.

The resulting values of x j are the fuzzy optimal solution and the value of β is the satisficing level of the solution for the DMs. The above fuzzy solution method has systematically reduced the complexity of the multiple objective audit staff-planning model because it transforms the multiple objective and multiple constraint LP problem into a simple LP problem. The above two solution steps can be easily implemented by employing available commercial computer software. In the next section, a numerical example of the audit staff-planning model in a CPA firm will be presented to demonstrate the implications for decision makers. A numerical example A numerical example, which is similar to Killough and Souder’s (1973), is developed to present the fuzzy set approach and its solution method. The proposed model in this paper is concerned only with a firm’s audit staffing problem, although a tax and management advisory service can be added if desired. In addition, the model to be designed is limited to the planning horizon of one year and the timing horizon can be easily extended if needed.

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The CPA firm audit staff is assumed to consist of five levels. At the highest level is Partner A and at the second level is Partner B. In descending hierarchical order, Manager, Senior Staff Auditor, and Junior Staff Auditor. Besides these five levels, the personnel partner, who is responsible for manpower planning, will be added. In this paper, let us assume that the personnel partner is trying to develop the most effective audit staff workload allocation plan for the next year. Given projected billings and the present number of staff members at each level, he needs to know whether the firm should increase or decrease the number of its auditor employees at each level. Based on past experience, the personnel partner decides the objectives to be achieved. The first objective is to increase gross audit fees by 5 percent over the past year. To achieve this objective, the personnel partner has to have sub-goals of increasing hourly billing rate per classification by 2 percent and increasing chargeable hours by 4 percent whichever is possible. The second objective is to maintain a ratio of at least one management personnel (partners and managers) to every five staff (senior and junior staff auditors). This is a required condition to maintain the quality of the audit work and to retain the company’s good reputation. The third objective is to maintain net income of $5,500,000. Tables 1, 2, and 3 represent the relevant information. Table 1 contains the information concerning audit personnel, their working hours, the billing rate per hour, and the gross audit Table 1. Audit personnel, working hours, billing rates, and fees

Position

# Employ

Working Hr/ Person/Yr

Total Hr/ Position

Charge Hr/ Person/Yr

Non-charge/Yr

Billing Rate/Hr

Partner A Partner B Manager Senior Junior

2 5 12 40 52

2,500 2,250 2,300 2,250 2,000

5,000 11,250 27,600 90,000 104,000

2,000 2,000 2,100 2,000 1,700

500 250 200 250 300

$80 $70 $60 $50 $40

Gross Audit Fees Earned for the Past Year: Chargeable hours/Position/Year × Billing Rate/Hour = $10,068,000 Table 2. Projected information for the next year based on goals set by the firm 1. Chargeable hours (an increase of 4%) Partner A Partner B Manager Senior Junior

4,000 × 104% = 4,160 Hrs 10,000 × 104% = 10,400 25,200 × 104% = 26,208 80,000 × 104% = 83,200 88,400 × 104% = 91,936

2. Billing rates/hour (an increase of 2%) Partner A Partner B Manager Senior Junior

$80 × 102% = $81.60/Hr $70 × 102% = $71.40 $60 × 102% = $61.20 $50 × 102% = $51.00 $40 × 102% = $40.80


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Table 3. Projected revenues and expenses 1. Gross audit fees Partner A 4,160 Hrs × $81.60/Hr = $339,456 Partner B 10,400 Hrs × $71.40/Hr = $742,560 Manager 6,208 Hrs × $61.20/Hr = $1,603,929 Senior 83,200 Hrs × $51.00/Hr = $4,243,200 Junior 91,936 Hrs × $40.80/Hr = $3,750,989 Total

$10,680,134

2. Expenses Partner A ($50,000 * X 1 ) Partner B ($45,000 * X 2 ) Manager ($30,000 * X 3 ) Senior ($25,000 * X 4 ) Junior ($20,000 * X 5 ) Where X i represents number of auditors employed

Salaries

3. Other expenses Estimated Fixed Costs: (e.g., rents, insurance, depreciation expenses for office equipment, . . . etc.) Total

$2,750,000

fees earned for the past year. Table 2 contains projected information on chargeable hours, nonchargeable hours, and billing rates for the upcoming year. Table 3 presents projected revenues and expenses predicted on these same goals. Analysis Formulation of the Model variables is as follows: X1 X2 X3 X4 X5 X6 X7 X8 X9 X 10

= Number of audit partners A required = Number of audit partners B required = Number of audit managers required = Number of audit senior staffs required = Number of audit junior staffs required = New hourly billing rate for partners A = New hourly billing rate for partners B = New hourly billing rate for managers = New hourly billing rate for senior staffs = New hourly billing rate for junior staffs

Objectives and constraints for the numerical example are as follows: First, the objective of a 5% increase in gross audit fees may be expressed as: 4,160X 6 + 10,400X 7 + 26,208X 8 + 83,200X 9 + 91,936X 10 ≤ $10,680,134

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It is desirable to provide net income of $5,500,000 in the upcoming year for the growth and enhancement of the firm’s partners. Therefore, the second objective can be expressed as: 4,160X 6 + 10,400X 7 + 26,208X 8 + 83,200X 9 + 91,936X 10 − 50,000X 1 − 45,000X 2 − 30,000X 3 − 25,000X 4 − 20,000X 5 ≤ $8,250,000 The personnel partner believes that it is desirable to maintain a ratio of at least one management personnel (partners and managers) to every five staff persons (senior and junior audit staff). This is a desirable condition to maintain the quality of audit work. After all the field work is done at least one of the partners should review that the job is done properly. All the audit work from the planning to issuing the audit report should be supervised properly. This third objective can be expressed as follows: −5(X 1 + X 2 + X 3 ) + (X 4 + X 5 ) ≤ 0 The following are constraints: (i) Personnel requirement The constraints for the number of audit personnel required (from Tables 1 and 2) can be expressed as: 2,500X 1 ≤ 5,160 2,250X 2 ≤ 11,650 2,300X 3 ≤ 28,608 2,250X 4 ≤ 93,200 2,000X 5 ≤ 107,536 Here the nonchargeable hours are maintained at present levels to provide necessary time for formal training, the upgrading of services, and obtaining the new clients. (ii) Billing rates The multiple constraints for the new hourly billing rates can be expressed as: X 6 ≤ (81.6γ1 + 85.0γ2 ) X 7 ≤ (71.4γ1 + 75.0γ2 ) X 8 ≤ (61.2γ1 + 65.0γ2 ) X 9 ≤ (51.0γ1 + 55.0γ2 ) X 10 ≤ (40.8γ1 + 45.0γ2 ) Here the new hourly billing rates have two different levels. The first right hand side value is the rate estimated by the personnel partner and the second right hand side value is the rate

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estimated by the managing partner. The fuzzy set approach we used in this paper allows multiple constraint levels as well as multiple objectives. However, for illustration purpose, we only allow multiple billing rates. (iii) Minimization of overtime Job pressure, frequent traveling, and overtime can explain high turnover of top quality certified public accountants. The personnel partner decides to minimize overtime to retain experienced auditors. This constraint can be expressed as: 500X 1 ≤ 1,000 250X 2 ≤ 1,250 200X 3 ≤ 2,400 250X 4 ≤ 10,000 (iv) Constraints on senior and junior audit staffs It is desired to keep at least 90 senior and junior staff auditors and one managing partner to increase sales and maintain the quality of audit work. The personnel partner wants to maintain at least current partner B, manager, and senior staff levels. This constraint can be expressed as: X 4 + X 5 = 90,

X 1 ≥ 1,

X 2 ≥ 5,

X 3 ≥ 10, and

X 4 ≥ 40

A fuzzy solution method Step 1. The LINDO (Schrage, 1991) computer software was used to solve three maximization problems and three minimization problems for three objectives, respectively. The results are: U1 = 0.000000001109, U2 = 8902440, U3 = 5.000; and L 1 = 0, L 2 = −2760360, L 3 = −13.768. Step 2. Let β be the satisficing level of the fuzzy optimal solution. Then, we have the fuzzy solution problem as shown in the Appendix. Solving this problem, we obtain: β = 0.99, which implies 99% of objectives are satisfied with current constraints. The satisfaction rate is high for this numerical example because the problem is simple and well defined. The current solution shows that 1.965 Partner As, 5 Partner Bs, 12 Managers, 40 Senior Staff Auditors, and 50 Junior Staff Auditors are required for the next year. Here, 1.965 Partner As mean less than 2 Partner As. If the firm has partially retiring partners, they can hire part-time partners. However, if this is not possible, they have to use Integer LP to set inter solutions. This solution indicates that the firm needs to hire less management personnel and less junior staff auditors for the upcoming year. Finally, γ1 = 0 and γ2 = 1. This result implies that the managing partner’s opinion dominates the personnel partner’s opinion. If we force γ1 ≥ 0.3, the satisfaction rate (β) is 97% and

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Partner A is 1.889. Other values are the same. Here γ1 ≥ 0.3 means that the managing partner’s opinion still dominates the personnel partner’s opinion. If γ1 ≥ 0.5, the satisfaction rate is decreasing to 95.7% and Partner A is 1.839. This case implies that the personnel partner’s opinion is stronger than the managing partner’s opinion. However, the solutions are pretty similar in this numerical example because the constraint level is narrowly defined.

Managerial implications The fuzzy set approach for the audit staff planning problems as we discussed in this paper has some managerial implications. First, the model integrates multiple objectives. In this model, multiple constraint levels are also allowed to incorporate each DM’s preference of new hourly billing rates. Second, the fuzzy set approach facilitates each decision maker’s participation to avoid suboptimization of overall firm goals. Third, the fuzzy set approach is more straightforward and easy to solve using currently available computer software. Fourth, the fuzzy set approach is flexible enough to easily add more objectives or constraints according to the changing environment. Fifth, the fuzzy set approach allows a group decision-making process, which is pretty common in today’s business environment, by allowing multiple right hand side values. Finally, the fuzzy set approach allows the sensitivity analysis of the opinions of decision makers as we have illustrated in the solutions of the numerical example. Overall, CPA firms can use the fuzzy set approach model for the audit staff planning problems such as ours as their formal planning tool to function more effectively and efficiently, to provide better service to clients and the general public, and to meet the challenge of the future in this dynamic capital market.

Conclusions A fuzzy set approach for an audit staff-planning model has been developed in this paper. This model can provide a comprehensive scenario about possible optimal audit personnel allocation depending on multiple criteria and multiple constraint levels. In addition, this fuzzy set approach can better handle real-world problems with uncertainty. The fuzzy set model and its solution methods provide better solutions in audit staff planning problems than the previous linear programming or goal programming models by receiving input from all decision makers. However, the current model only shows single period model. Another limitation of this paper is not using the numerical example of a real firm’s data. We can extend the fuzzy set approach to other real-world problems in business. The framework of this model can be applied to other areas, such as financial planning, portfolio determination, inventory management, resource allocation, and audit sampling objectives for a CPA firm if the decision variables and formulation are expressed appropriately.


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Appendix Maximize β Subject to: 4,160X 6 + 10,400X 7 + 26,208X 8 + 83,200X 9 + 91,936X 10 − 1109340

β≥0 4,160X 6 + 10,400X 7 + 26,208X 8 + 83,200X 9 + 91,936X 10 − 50,000X 1 − 45,000X 2 − 30,000X 3 − 25,000X 4 − 20,000X 5 − 11662800 β ≥ 2760360 −5(X 1 + X 2 + X 3 ) + (X 4 + X 5 ) − 18.768 β ≤ 13.7680 2,500X 1 ≤ 5,160 2,250X 2 ≤ 11,650 2,300X 3 ≤ 28,608 2,250X 4 ≤ 93,200 2,000X 5 ≤ 107,536 X 6 − 81.6γ1 − 85γ2 X 7 − 71.4γ1 − 68γ2 X 8 − 61.2γ1 − 65γ2 X 9 − 51.0γ1 − 55γ2 X 10 − 40.8γ1 − 45γ2 500X 1 250X 2 200X 3 250X 4

≤0 ≤0 ≤0 ≤0 ≤0

≤ 1,000 ≤ 1,250 ≤ 2,400 ≤ 10,000

X4 + X5 X1 X2 X3 X4

≥ 90 ≥1 ≥5 ≥ 10 ≥ 40

γ1 + γ2 = 1 and β ≤ 1.

References Accounting Today 15(3), 8, (2001). Bailey, A. D., W. J. Boe and T. Schnack, “The Audit Staff Assignment Problem: A Comment.” The Accounting Review LIX(3), 572–574, (1974).

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