Audit scheduling with overlapping activities and sequence-dependent setup costs

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH

ELSEVIER

European Joumal of Operational Research 97 (1997) 22-33

Theory and Methodology

Audit scheduling with overlapping activities and sequence-dependent setup costs Bajis Dodin a9*, A.A. Elimam b ’ The A. Gary Anderson Craduate School of Management, University of California.Riverside, CA 92521, USA b Business Analysis and Computing Systems, College of Business, San Francisco State University,San Francisco, CA 94132, USA Received 1July 1995; accepted 1 February 1996

Abstract Audit firms are faced witb the complex job of scheduling auditors to audit tasks. The scheduling becomes more complex as the firm needs to consider real life issues in determining an optimal schedule. Among these issues are the setup times and costs emanating from changing the assignments of the auditors and the lead and lag relationships between the audit tasks. Audit scheduling with overlapping activities and sequence-dependent setup tost has not been treated in litemture. This paper presents a formulation and a solution approach for this audit scheduling problem. First, the problem is represented by an activity network with lead/lag relationships. Then the network is analyzed to determine the early and late finish times of activities. An integer linear program (ILP), which uses the early and late finish times of activities to reduce the number of decision variables, is formulated. A four-auditor two-engagement example is used to illustrate the ILP model and its solution. The results indicate that incotporating the setup tost and the overlapping of activities yields lower tost schedules leading to sizable sav.ings in the tost of audits. The proposed treatment is of merit in providing realistic schedules that can be easily implemented Keywords:

Audit scheduling; Activity network; Integer programming; Setup time; Setup tost; Overlapping

1. Introduction

The operation of an audit firm or audit department, extemal or intemal, consists of auditors with varying experiences (efficiencies), audit engagements with pre-specified arrival and due dates; and auditing requirements. Efficient management of the operation requires the determination of optimal or near optimal feasible schedules. The schedule should specify when each audit task wil1 start and finish;

* Corresponding author.

and hence when each audit engagement wil1 start and finish. The schedule also specifies the role of every auditor; which tasks the auditor wil1 process and when. This high degree of detailed scheduling is helpful in determining the utilization factors of auditors, slack periods, overtime requirements, and travel schedules and costs among others. The preceding schedule description would take into account many real life audit scheduling issues. Examining the open literature on extemal and interna1 audit staff scheduling reveals that many important factors have been considered. However, there remain other important issues that are yet to be

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relationships

B. Dodin. A.A. Elimam / European Journal of Operational Research 97 (1997) 22-33

included. For example, Chan and Dodin [4] expanded the loading model of Balachandran and Zoltners [2] to derive optimal and near optimal working schedules accommodating: the inability of the auditor to process more than one task at a time, precedence relations among the audit tasks, different arrival and due dates for the engagements, auditor’s preferente to audit a specific engagement or sub-engagement, and the achievement of a certain utilization level. Balachandran and Steuer [3] included the desire to fulfill different objectives in determining the auditing loads. Dodin and Chan [5] examined the impact of using different audit scheduling objectives, and explained how other practica1 issues can be treated. Salewski and Bartsch [12], in an attempt to consider many of the real life scheduling issues, developed a hierarchical planning model consisting of three levels. Knechl and Benson [9] developed an optimization model to determine the frequency of auditing the internal divisions. Drexl [6] used a hybrid branch and bound/dynamic programming algorithm to find a least expensive schedule under which each engagement is completed by the given planning time horizon. It is clear from the above discussion that there are important audit scheduling issues which have not been investigated. Consider first the issue of the auditor’s travel time and tost as s/he changes assignment from one engagement to another. Then consider the issue of lead/lag relationships between the audit tasks. The existing research assumes that the relationship between audit tasks is based on strict precedence. Slight reflection on many audit activities reveals that precedence relationships among activities need not be confined to strict precedence since other relations do in fact exist between audit activities. Considering the different types of precedence relationships between tasks, known as “generalized precedence relations” is more practica1 than limiting the scheduling process to the strict precedence case; see Elmaghraby and Kamburowski [8], and Moder, Phillips and Davis [lol. In this paper, we investigate the impact of the above two issues on audit scheduling. The generalized precedence relations (GPR), and the time and tost change overs for the auditors are added to the other factors, which has been published earlier, to form a more comprehensive treatment of the audit

23

staff scheduling problem. These two issues and the difficulties associated with them are discussed in the next section. Methods of activity networks are used to overcome some of these difficulties. The basic concepts of our work and how it treats the practica1 audit scheduling issues are given in Section 2. The ILP formulation is presented in Section 3. We illustrate our approach by presenting an example case in Section 4. In the same section, we also analyze the model solution computational complexity based on the problem parameters. Finally, the summary and conclusions are given in Section 5.

2. Basic concepts and practica1 audit scheduling In this section, we introduce the basic concepts of our extensions to handle the practica1 issues faced in real life audit scheduling problems. Before proceeding with such presentation, we summarize the notations used throughout this paper: 2.1. Dejìnitions and notations 2.1.1.

Constant parameters

Aij = The set of audit tasks, belonging

to audit engagement(s) other than that of i, which can be processed after task j by auditor i. ajk = Portion of activity j to be completed before the start of activity k, where j precedes k, and O
24

B. Dodin. AA. EIimam/European

Journal of Operationnl Research 97 (1997) 22-33

j= 1,2 , . . . , J task index, J is the set and number of al1 tasks in al1 engagement.% Ji = The set of audit tasks which can be processed by auditor i. lj = The latest completion time of task j. pB = Cost of having engagement g late (tardy) for one period. sijk = The tost of having auditor i process task k after completing task j. SS,, = The time lag required for the start of task ik after the start of task j. tij = The time it takes auditor i to process audit task j-

/ZIT1

j OreJ I

:F

:

i-1

8.

Finish-to-Start

I

b. Start-to-Start

FF,.

j

Ip-/

1

171

si #c

4

c. Finish-to-Finish

ei

d. Start-to-Finish

Fig. 1. Fonns of lead/lag relations.

2.1.2. Decision variables 1 Xijh= i0

1 zijk

=

l0

if auditor i completes task j at the end of period h; otherwise. if auditor i processes task k after j and tasks j and k belong to two different audit engagements; otherwise.

Using the above parameter and variable definitions, the paper adds the following two factors to what has been published in [5], and builds on the discussion presented therein. 2.2. Overlapping relationships In al1 previous treatments of the audit staff scheduling problem it is always assumed that an activity cannot start until al1 its predecessors are completed. This is known in project management and activity networks literature as “strict precedence”. It is also known as finish-to-start (FS) precedence with a zero lag [7]. However, in many instances an activity starting time need not wait until al1 its predecessors are completed, it may start after certain number of periods from the start of its predecessors. The overlap between two dependent activities speeds up the auditing process, entourages the formation of audit teams to work on the engagements, and makes good use of the highly specialized in the audit team and audit firm. These factors lead to more audit efficiency.

The literature of project management presents four types of lead/lag relations between the tasks. In many cases, the nature of the project or technology necessitates the form of the lead/lag relationships. Other times the desire to develop efficient schedules requires benefiting from the feasibility of the lead/lag relationships wherever possible. Refs. [8] and [lO] present a comprehensive analysis of these forms. These forms are depicted in Fig. 1 for two consecutive activities j and k, where k is the successor of j. They are: 1. Finish-to-Start (FS ,): the best known form specifying the minimum number of time units that must transpire from the completion of j prior to the start of k; if FS, = 0, then we have the usual CPM/PERT. 2. Start-to-Start (SS,,): it is used often, and is equal to the minimum number of time units that must be complete on j before k can start. 3. Finish-to-Finish @Fik): it is equal to the minimum number of time units that must remain to be completed on k after the completion of j. 4. Start-to-Finish (SFi,): it is equal to the minimum number of time units that must transpire from the start of j to the completion of k. Looking at the precedence relationships between the audit tasks shows that type 2 above, Start-to-Start, as wel1 as the usual Finish-to-Start are the most applicable forms of overlapping; provided that for any two activities j and k, where j precedes k, FFj, 2 0. Consequently, in this paper we wil1 emphasize these two types of overlapping. However, if any

B. Dodin. A.A. Elimam / European Journal of Operational

of the other two types occur, then it can be transformed to either of these two as it is possible to transform the lead/lag relation from one form to the others, or treat it on its own. The overlapping relationship between two related activities, j and k, is given by the ratio ajk. Therefore, activity k can start any time after the completion of aj,tij periods of work on activity j. Hence, SS, = ]ail
[lol. The above treatment of the lead/lag relationships allows US to calculate the early and late completion time of every activity in the network of al1 engagements which require auditing within the planning horizon. The earliest completion time, ej, is calculated using the least processing times {t,j} and accommodating the start-to-start relationships. The latest completion time, Ij, is calculated based on the fact that H is the latest allowable completion time of al1 engagements. Therefore, none of the tasks can be

Research 97 (1997) 22-33

25

critical and as H increases, the gap between ej and lj increases. The interval [ej, l,] specifies the time sub-horizon within which task j must be completed. Note that ej is the minimum of al1 possible early completion times, and 1, is the maximum of al1 possible late completion times. However, this determination does not specify when each activity wil1 start and end, or which auditor wil1 be processing it. In order to be able to determine what every auditor wil1 be auditing and when, the problem is formulated as an integer linear program (ILP). The early and late finish times are used to reduce the number of decision variables in this ILP. Solution of the ILP yields the schedule for each auditor, engagement and task.

2.3. Travel tost and time

Examination of many audit staff schedules showed that it is customary for an auditor to be involved in more than one audit engagement within a given planning period. In this case, the travel time or tost, or both, should be a factor in determining the audit schedule. If an auditor switches from a task j in engagement g to another task k in a different engagement g’, then this switch may require some travel time and expense. The time and expense depend on the location of g’ relative to g or to the home base. The travel time or expense is independent of the auditor unless the value of his/her time is included in the travel tost. The travel tost and time are known as the change-over tost and time, or setup tost and time. Scheduling with sequence-dependent setup times is very difficult. It is known to be strongly NP-hard [ll. Consequently, in this paper we wil1 assume that the travel time of the auditor between the engagements occurs during non-working hours and its monetary value is included in the travel tost. Hence the travel tost is treated as auditor dependent. Planning based on the tost of the travel time is more accurate than just the travel time. The latter does not distinguish between the ranks and efficiencies of the auditors. The travel tost, which wil1 be referred to from here on as the setup tost, captures also al1 expenses incurred as the auditor changes assignments from one engagement to another.

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B. Dodin, A.A. Elimam/ European Journal of OperationalResearch 97 (1997) 22-33

The inclusion of the setup tost requires modifying the ILP model. First, the objective function is augmented to include the total setup tost. Second, constraints are added to keep track of the instances where an auditor switches from one engagement to another. The inclusion of a sequence-dependent setup tost (or time) into the scheduling process gives rise to a traveling salesman problem (TSP) formulation; the solution of such a formulation is NP-hard [1,14]. Therefore, in this paper we capitalize on the structure of the problem to avoid the TSP formulation and the difficulties associated with its solution. The following section presents the integer linear programming formulation.

3. Integer program formulation The above alphabetical listing of notations is used in the ILP formulation. The model presented in this section extends the third ILP specified in [5], to accommodate overlapping relationships and setup costs. For the sake of completeness, we present this ILP model.

our illustrative example, given in the next section, a setup tost is incurred only if j and k belong to two different engagements. A fourth term can be added to Capture the possible reward the audit fïrm may obtain by completing an audit engagement before its due date.

3.2. Constraints The model contains live sets of constraints covering auditor’s availability and assignment to tasks, precedence among tasks and sequence-dependent setup. 1. Availability of auditors and the requirement that each auditor must work no less than H,: periods and no more than H: periods within the planning horizon H. Therefore, for each auditor i we have: ‘1 c trjxij,, 2 H;

c .jeJ,

(2)

h=r,

and

c ; jEJ,

ti,xijh I

H,+.

(3)

h=r,

3.1. Objectioe function The objective function minimizes the total tost of: * mismatching between the auditors and audit tasks; - setups as discussed above; - delay in the completion of the audit engagements beyond their respective due dates. Therefore the objective function is to minf(r,z)

=

i

c

E

i-1

jeJ,

h=ei

C,jXijh

+

i i-

c

1 jGJi

c

2. Each audit task must be processed auditor; therefore for each audit task j

c ; is/,

g-1

(4)

1.

3. An audit task cannot start before its preceding requirements are satisfied. Consequently for every task j

SijkZijk ,$

kGAi,

E 2 ? pg(h-dg)XiM,h,

i=l

=

h=e,

,$ k’

+

Xljh

by one

(1)

10

(h-FSkj)Xikh-

bie

k

==J

foreach

kGBj,

iz.(h-‘ij)Xijh I

(5)

h-d,+l

where M, is the last task in engagement g. The second term of this objective function can be used to Capture the genera1 case of sequence-dependent setup costs between al1 pairs of tasks (j, k). In

where FS,, = tik - SS,,. Note that ej and lj are calculated according to the discussion in Section 2.2, where the lead/lag relationships and multiprocessing times of tasks are accommodated.

B. Dodin, A.A. Elimam/European Journal of OperationalResearch 97 (1997) 22-33

4. An auditor cannot process more than one task at a time. A task j is processed by auditor i, in period h if it is completed in period r where

Table 1 Auditors information Auditor

Auditor title

i

h5r
1 and

c je J,(h)

“J c xijr< 1 foreach h= 1,2, . . . . H,

(6)

5

xikr=O

foreach kEAij

r- n,,+ 1

Pa)

and c ksA,,

1 2 3 4

Partner Senior Junior No. 1 Junior No. 2

Travel tost

Minimum time

Maximum time

between engagements

H;

H?

($ /hr)

5 6 6 6

8 8 10 10

600 400 300 300

r= uj

where J,(h) is the set of tasks auditor i can process in period h. 5. Setup costs: Whenever an auditor changes audit engagements a setup tost is incurred. The setup tost depends on the auditor and on the “from” and “to” engagements. It was stated in Section 2 that including a sequence-dependent setup tost in the scheduling problem gives rise to a TSP formulation, which makes the solution of the problem NP-hard. In our instance (audit staff scheduling) we use the precedence relations to avoid the TSP formulation. As an auditor, i, completes a task in a given engagement, a setup tost is incurred only if she/he is assigned next to a task in another audit engagement. Therefore, the set Aij needs to be formulated to include the potential tasks in other engagements auditor i can choose from to process after completing task i. The sets Aij are determined a priori from the subset Ji after determining the intervals [ei, lil. Therefore, as auditor i completes task j she/he might be assigned to a task k E Aij: zijk-

Availability

ej
Therefore, let uj = max{ej, h) and vj = min{li, h + rij - l}, then for each auditor i:

27

tijk 5 1 foreach A,,,

(Tb)

where nij is the earliest completion time of task j if it is processed by auditor i. Analyzing this constraint structure indicates that the price to be paid for avoiding the TSP formulation is the big jump in the number of constraints and variables. Constraints type (7.a) and (7.b) lead to more than doubling the number of constraints if compared with the scheduling problem where the

setup tost is not sequence dependent. More discussion of this issue is presented in the next section.

4. Example and computational 4.1. Description

analysis

of the example

Consider the case study presented in [5], where we do not allow for overlapping of activities and setup tost is not included. We wil1 refer to this case, given in [5], as OPT-STRICT. It consists of four auditors, two audit engagements, and a planning horizon of 12 weeks. We modified this case by including travel tost and by allowing for overlapping of activities to yield the new case OPT-OVERLAP. Table 1 contains the modified auditors’ information. It consists of the auditor’s rank, available time, and travel tost between engagements. Table 2 contains the audit engagement’s information. ‘Ihe first engagement consists of 12 tasks whereas the second engagement consists of seven tasks. The tasks are numbered from 1 to 19 starting with those of the fïrst engagement. Table 3 contains the modified audit tasks’ parameters. It consists of the description of the

Table 2 Audit engagement Audit engagement

information

Description

Arrival time

Due date Cd,)

Delay tost (p,) ($/wk)

0

10 9

2000 2OQO

Cg) 1 2

A new audit A contimdng

audit

1

Audit of cash receivable and other current assets Audit of Audit of Audit of Audit of accounts Review, prepare and issue fínancial statements and audit reports

14 15 16 17 18 19

1

2 1 2 2

4

1 3

I

3 1 1

3 2

141

policy and audit quality control

_

2

13 14,15 16, 17

18

2 1 2 2

_ 13

1 3

1

3 1 1

3 2

'3j

4

f2J

2

1

1 1

1

-

1

2

1 1 _

Ilj

Duration for auditor i (t,,) in weeks

13

8 9 9 10, II

6.7

2 2 3.4, 5

1 2

Preceding tasks

by this auditor as a matter of company

liabilities tïxed and other assets revenues and assets capita1 and retained earnings

inventories

Assess, follow-up and complete interim audit

13

Global review Clean up al1 outstanding payments Draft tïnancial statement Review statements, prepare and issue audit report

9 10 11 12

’ A dash (-) indicates that the audit task will not bc processed

2 (Food processing, a continuing audit)

Audit liabilities capita1 and retained eamings

8

Planning audit engagement Complete procedures Observe inventory and validation Accounts receivable confirmation and audit procedures Cash audit and reconciliation Fixed and other assets depreciation Audit sales and other revenues and expenses

1 2 3 24

1 (Plastic manufacturing, a new audit)

Task description

5 6 1

Task i

g

’

Engagement

Table 3 Audit tasks information

0

1

_

1

0

1 0

0

1

0

0 1

CIJ

0

0

5

0

5

0

0

5 0 _

c2j

_

0 0 0 5

5

0 5

0

5 0 0

5 0

‘31

Mismatching tost for auditor i (cij) in $1000

0 0 0 5

5

0 5

0

5 0 0

5 0

“4j

29

B. Dodin. A.A. Elimam / European Journal of Operational Research 97 (1997) 22-33

Table 5 The earliest and latest completion times of the audit tasks

Table 4 Lead/lag relations Pmdecessor task, j 2 2 2 9 14 1.5 17

Successor task, k

Ratio ajk

0.33 0.33 0.33 0.50 0.50 0.50 0.50

3 4 5 10 17 17 18

Start-to-start relations SSjk a

Task j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 1 1 1 1.2 1 1

LI Note that SS, is the minimum time that must be completed on j before k can start.

task, its precedence relations, processing times and mismatching costs. Al1 precedence relationships are assumed to be of the finish-to-start type with lag zero, i.e., FS,, = 0, except as specified in Table 4. In Table 4 some activities have start-to-start relationships. 4.2. Computational

results

Earliest completion time

Latest completion time

ei 1 2 4 4 3 3 5 6 7 7 8 9 2 4 4 3 5 6 7

‘j 4 6 7 7 7 8 8 9 10 II 11 12 6 8 9 9 10 11 12

the two time analysis procedures, presented in [5] and [lol, is used to determine the earliest and latest start and completion times of each task. Table 5 lists the audit tasks and the earliest and latest completion time of each task, ei and Ij respectively.

The information given in the above tables is used

to construct the audit scheduling activity network. The network is depicted in Fig. 2 using the activityon-node mode of representation. The modification of

Ker

Task Dwation

Fig. 2. Activity network of the audit engagements.

30

B. Dodin. A.A. Elimam/ European Journal of Operarional Research 97 (1997) 22-33

The ILP for this example consisted of 250 constraints and 310 variables. It was solved using LINDO [13]. LINDO is one of several available software packages used to solve 0-1 integer linear programs on mainframes as wel1 as personal computers. These software packages are expected to terminate with the optimal schedule. However, for some large scheduling problems the procedures may terminate before reaching an optimal schedule. For these scheduling problems we normally accept a feasible, instead of optimal feasible, schedule. In such a case, the heuristic procedures evaluated by Patterson [ 111, for the constrained-resource project scheduling problem, can be applied to the above integer program. Applying LINDO to the above integer program provided the optimal schedule depicted in Fig. 3. Analysis of the optimal schedule, Fig. 3, provides the the following information: 4.2.1. Schedule and load of auditors The schedule of auditor i is given by the Gantt chart headed by i = 1,2,3,4 in Fig. 3. For example the senior auditor is scheduled to process tasks 2 and 5 of engagement 1 then moves on to tasks 14 and 18 (2 and 6 of engagement 2) before returning to engagement 1 to process task ll. 4.2.2. Start@

and completion

times of each audit

task

For example, junior auditor 2 starts on task 17 (5 of engagement 2) in period 6 and completes it at the end of period 7. The schedule shows that the first engagement is late by two periods and engagement 2 is not. 4.2.3. Overlapping relationships Tasks 3 and 4 overlapped with task 2 for one period; also task 17 overlapped with its preceding tasks 14 and 15 by one period, and task 18 overlapped with task 17. In fact the overlapping increases the number of feasible schedules if compared with the strict precedence case. Hence, overlapping helps in reducing the value of the objective function. 4.2.4. Values of diferent objective jùnction costs These costs are: * The mismatching costs are equal to zero. * The setup costs add up to $3200. * The delay costs reached $4000.

012345678

9

10 ll,

12

Fig. 3. The optimal auditing schedule.

Therefore, the total relative tost of the schedule presented in Fig. 3 is $7200. The tost is relative since there are other tost items that are independent of the schedule, such as the salaries of the auditors or the overhead which are not included in the objective function. 4.2.5. OPT-STRICT

and OPT-OVERLAP optimal schedules The schedule for OPT-OVERLAP is presented in

Fig. 3, whereas the OPT-STRICT optimal schedule is similar to that given in [5] except both engagements are late due to the modification of some of the paramaters in Tables 2 and 3. By comparing the two cases, we observe the following: ca> Due to the inclusion of the setup tost in the objective function, the auditors moved less frequently between audit engagements in OPT-OVERLAP than in OPT-STRICT. This indicates that in the OPT-OVERLAP case the schedule tost would always be lower than that for the OPT-STRICT. For example, the partner in OPT-OVERLAP switched engagements twice, where in OPT-STRICT s/he switched four times, similarly for the senior auditor. (b) The existente of the overlapping in case of OPT-OVERLAP resulted in fewer delays, hence less delay tost. In case of OPT-OVERLAP only the first engagement is late; where in OPT-STRICT both engagements are late. The existente of overlapping in OPT-OVERLAP increased the number of feasible schedules and led to the reduction in the value of the objective function. (c) Both schedules lead to the same utilization factor of auditors, and the same total mismatching costs. This is due to the large mismatching tost if compared with the delay or setup costs. (d) The OPT-OVERLAP case required more computer time to solve. This occurred since the inclusion of the setup tost in the objective function made it more complex on one hand. It also led to increasing

B. Dodin, A.A. Elimam /European

Journal of Operatiomd Research 97 (1997) 22-33

the number of variables by 104 and the number of constraints by 152. The CPU time required to reach OPT-OVERLAP is more than ten times that of OPT-STRICT. However, the CPU time in both cases remains reasonable. In both cases it is less than two minutes on the minicomputer (VAX). The above schedule and conclusions provide the planner with information necessary for the final decisions as to how the auditing should be catried out. These conclusions by no means eliminate the role of the decision maker or the human judgment in the final scheduling decision. The methodology presented in this paper can be used to aid the decision maker in reaching more objective conclusions, and in examining the impact of various factors on his/her decisions. 4.3. Analysis of computational complexity The difficulty in implementing this scheduling methodology is the computational requirements. These requirements are a function of the model size

Table 6 Expressions Model elements

to compute

ILP size and its application

Expression

to calculate

size

31

expressed in terms of the number of constraints and decision variables; and the structure of the ILP. In the remainder of this section, we analyze the worst case scenario for both: the number of constraints and decision variables and we also compare it with the actual number of constraints and decision variables for the given case study. Table 6 presents the expressions used to calculate the model size in number of constraints and variables. In addition, the table shows the worst case for the model size. This table also includes the model size and the worst case scenario, for the case study. The number of constraints depends on the values of the parameters G, H, I and J as wel1 as the number of precedence relations. Analyzing constraints (2)-(7) shows that the number of constraints is independent of the due date, and linear with respect to H. Column (al of Table 6 shows the expressions used to count the number of constraints, by type. The upper bound on the number of these constraints is also given in column (b) of the table. It is clear from the table that the expressions used to count the number of constraints (2), (31, (41, (6) and

to the example Expression

for worst case

Apphcation Number

(a)

(b)

1

1 I J

(c)

to example Upper bound (d)

1. Constraints (2) (3) (4)

1 J

(5)

YZ;=,B,l

iM(M-

(6)

IH

IH

(7a)

&lA,,I

1[J*-$,T,2]

(7b)

C,I JiI 21+J+C~=,IBjl+/H+~ijlA,jl+~ilJil

II,I J,l I[J*-,X;=,T;]

iXfj/J

Cf=,Ij(fj-

(kt)

EijI AijI Xijl Ai,/ + Z:= i Ij(Ij - ei + 1)

Total

1) a

4 4 19

4 4 19

23

190

48

48

104

672

2:

98.5

IJH

206

912

I[ J’ - L$= ,T;]

104

672

IJH+I[J’-L$,T;]

310

1584

48

+fz+f(M--I)+J+t(2+H)

11. Variables

Total

e,+

1)

b

’ M is the number of rows/columns in the adjacency matrix of the corresponding activity netwerk. b This is stil1 a very tight upper bound. The exact number depends on the netwerk structure.

32

B. Dodin, A.A. Elimam/European

Journal of Operational Research 97 (1997) 22-33

(7b) depend on constant problem parameters. On the other hand, the numbers of constraints (5) and (7a) depend on the network structure of the problem. As a result, the upper bounds for these two constraint sets are different from their actual number. The expression to determine the number of constraints (7a) depends on: * The number of engagements, G. The subsets J, (or alternatively the subsets 1,). In the above example C] Ai,] = 104 and it is bounded from above by: ??

r

G

1

where Tg is the number of tasks in audit engagement g. The decision oariables consist of { x~,,,} and { zijk}. The x’s are bounded from above by H. I. J. However, the actual number of the x decision variables can be sharply reduced by the use of the intervals [e,, 1,] and the subsets J, (or 1,). In fact by using those intervals, a tighter upper bound on the number of {x,~,,} can be obtained. It is given by the expression i: j=

Ij(lj--ej+

1).

1

This expression would reduce the upper bound on the number of {xij,,} variables in the above example to 228. Clearly the number of the x variables (upper bound as wel1 as actual) is linearly dependent on H, but independent of the due dates. As H increases, the upper bound increases and the intervals [ei, Ij] for ah audit tasks j widen. The size of each interval depends on the gap between the “earliest” completion time of the engagement and its absolute completion time, H. Let TE, be the earliest completion time of engagement g, representing the duration of the longest path in engagement g, which is constrained by: - rj = min,, ,. {tij); * the engagement starts on time; and - the overlapping relationships are preserved. Then, the interval [e,, 1, ] for each critical task (on the defining path for TE,) is equal to (H - TE*). On the other hand, this interval for the non-critical tasks is greater than (H - TER).

The number of the z decision variables is equal to from above by expression (8). However, the number of the z’s can also be sharply reduced by using the subsets Ij. In the above example, the number of the z variables is equal to 104, compared with an upper bound of 672. The upper bound on the z variables is independent of H and of the intervals [ei, /,]. The ILP of the above example consists of 310 variables (compared to an upper bound of 1584) and 250 constraints (compared to an upper bound of 985). It is clear, from the above analysis, that for large audit scheduling problems with several engagements the number of constraints can be in thousands, and the number of decision variables can be in tens of thousands. Therefore, the use of exact solution procedures for determining optimal feasible schedules could become restrictive. In such cases, the heuristics presented in [ 111 are more practica1 and attractive to use. CIAijJ which is bounded

5. Summary and conclusions This paper deals with the problem of audit staff scheduling. It extends past contributions to handle the overlapping relationships between audit tasks and the setup times and costs emanating from having auditors involved in more than one audit engagement. For the overlapping of activities, the nature of overlapping in audit scheduling is determined and modeled. It is also shown that the use of overlapping in audit scheduling allows the derivation of more feasible schedules at lower costs. The schedule-dependent setup time and tost is known in the literature to be a strong NP-hard problem. However, in this paper we transfer the setup time into tost, and capitalize on the structure of the problem to avoid the NP-hardness. These two extensions would yield lower-tost audit schedules compared to those obtained using any of the preceding audit scheduling approaches. The audit scheduling problem with the addition of the above two extensions is formulated first as an activity network. Then the time analysis of the network, accommodating the multi-processing times of the audit task, and the lead/lag relations between the

B. Dodin. A.A. Elimam / European Journal of Operational

is used to develop an efficient integer programming model. The model is solved using the available software to yield an optimal schedule through the use of exact solution methods. For large audit scheduling problems, where “large” here means that the size of the integer program is beyond the capacity of the available integer programming computer codes, the use of heuristics becomes necessary. tasks,

33

Research 97 (1997) 22-33

European Journal

of Operational

Research

of project networks by job assignment”, Management Science 37/12 (1991) 1590-1602. [71 Elmaghraby, SE., Actiuity Networks: Project Planning and Control by Network Mode& Wiley, New York, 1977. Bl Elmaghraby, S., and Kamburowski, J., “The analysis of activity networks under generalized precedence relations (GPR)“, Management Science 38/9 (1992) 1245-1263. [91 Knechl, W.R., and Benson, H.P., “An optimization approach for scheduling intemal audits of divisions”. Decision Sciences 22/2

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Van Nostrand Reinhold, New York, 1983. 1111 Patterson, J.H., Exact and Heuristic Solution Procedures for the Constrained-Resource Project Scheduling Problem, Volumes 1-4, Monograph, College of Business Administration, University of Missouri, Columbia, 1984. [121 Salewski, F., and Bartsch, T., “A comparison of genetic and greedy randomized algorithms for medium-to-short-term audit-staff scheduling”, Working Paper No. 356, Department of Production and Logistics, Institute for Management, Christian-Albrechts University at Kiel, Germany, 1994. [131 Shrage, L., Linear, Integer and Quadratic Programming with LINDO, The Scientitìc Avenue, Palo Alto, CA, 1986. in flow shop scheduling with LI41 Simons, J.V., “Heuristics sequence dependent setup times”, OMEGA International Journal of Management Science 20/2 (1992) 215-225. agement

Wiley New York, 1974. B.V., and Zoltners, A.A., “An interactive Dl Balachandran, The Acadult-staff scheduling decision support system”, counting Reuiew LVI (1981) 801-812. K.R., and Steuer, R.E., “An interactive model [31 Balachandran, for the CPA firm audit staff planning problem with multiple objectives”, The Accounting Reuiew LVII (1982) 125-140. [41 Chan, H., and Dodin, B. “A decision support system for audit staff scheduling with precedence consttaints and due dates”, The Accounfing Reuiew LX]/4 (1986) 726-735. bl Dodin, B., and Chan, H. “Application of production scheduling methods to extemal and intemal audit scheduling”,

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