A business process activity model and performance measurement using a time series ARIMA intervention analysis

Expert Systems with Applications 36 (2009) 6986–6994

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Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

A business process activity model and performance measurement using a time series ARIMA intervention analysis C.Y. Lam *, W.H. Ip, C.W. Lau Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e

i n f o

Keywords: Activity model ARIMA Business process reengineering (BPR) Intervention analysis Performance analysis Reachable matrix Simulation

a b s t r a c t The degree of performance excellence that an enterprise can achieve greatly depends on the business process flow that the enterprise adopts, where the more efficient and effective the business process flow, the greater the degree of performance excellence the enterprise can achieve. Most conventional business process analyses focus on qualitative methodologies, but these lack solid measurement for supporting the business process improvement. Therefore, a quantitative methodology using an activity model that is described in this paper is proposed. This model involves the use of an adjacent matrix to empirically identify inefficient and ineffective activity looping, after which the business process flow can then be improved. With the proposed quantitative methodology, a time series intervention ARIMA model is used to measure the intervention effects and the asymptotic change in the simulation results of the business process reengineering that is based on the activity model analysis. The approach is illustrated by a case study of a purchasing process of a household appliance manufacturing enterprise that involves 20 purchasing activities. The results indicate that the changes can be explicitly quantified and the effects of BPR can be measured. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction The business process is a framework for activities to participate and interact so as to produce a product or a service and achieve the well-defined objectives of an enterprise (Hammer & Champy, 1994; Keung & Kawalek, 1997), and the structure of it greatly affects the overall performance of the enterprise. Moreover, in this ever changing global marketplace, the better the performance of the enterprise, the greater the competitiveness the enterprise can achieve. An enterprise not only faces challenges in external global competition but also faces root challenges of internal business process improvement to drive the performance towards excellence (Schorr, 1998). It is then important to evaluate and analyze the internal business process of the enterprise for the necessary partial or complete redesign of the process to improve the business performance (Morrow & Hazell, 1992). Very often, an enterprise may need to perform a dynamic analysis of their business process so as to simulate and evaluate different sets of processes that could ensure the efficiency and effectiveness of the business process flow as well as improve the overall performance of the enterprise (Alera, Borrajoa, Camachoa, & Sierra-Alonsob, 2002). Business process analysis is based on * Corresponding author. E-mail addresses: [email protected] (C.Y. Lam), [email protected] (W.H. Ip), [email protected] (C.W. Lau). 0957-4174/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2008.08.027

the business process flow or business process model that the enterprise has adopted. Business process modeling is one of the business tools that help an enterprise to achieve competitive advantages and improve business performance. The business process flow can be systemically represented and mapped by the model, and business process analysis can then be carried out. With the analyzed results, the business process can be revolutionized, redesigned or reengineered, so the enterprise can then achieve the great benefits of enhanced competitiveness (Evans, Towill, & Naim, 1995). Business process modeling and business process analysis are two inter-related research areas. In the research area of business process modeling, research interests ranges from process enactment, such as ECA Rules (Bae, Bae, Kang, & Kim, 2004) and collaboration with business process choreography (Jung, Hur, Kang, & Kim, 2004) to process monitoring, such as enterprise information portal (Hur, Bae, & Kang, 2003) and run-time environment (Kim, Kang, Kim, Bae, & Ju, 2000). Some leading approaches to business process modeling representation have been discussed in the literature, such as the family of Integrated Computer Aided Manufacturing Definition (IDEF) (Defining IDEF, 1992; Kusiak, Larson, & Wang, 1994; US Air Force, 1981), petri-nets approach (Van der Aalst & Van Hee, 1996), hyper-graph approach (Huang, 1997), entity relationship modeling (Maker et al., 1992), the role activity diagrams approach (Ould, 1995), and state-driven approach (Lee, Kim, Kang, Kim, & Lee, 2007). On the other hand, research into

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business process analysis ranges from process analysis to process improvement, entailing aspects such as workflow mining (Van der Aalst & Aj, 2004), process measurement (Cardoso, 2005), and process optimization (Ha, Bae, Park, & Kang, 2006). Moreover, there are a number of analysis techniques that have been discussed in the literature, such as reachability graph (Yang & Liu, 1998), the structural analysis approach (Sadiq & Orlowska, 1999), Queueing theory (Kleinrock, 1975), self-organizing map (SOP) or dimensional output space (Díaz, Domínguez, Cuadrado, & Fuertes, 2008). The research discussed in the literature and the approaches for business process modeling and analysis all have different advantages and disadvantages. However, research into business process modeling and analysis still has room to improve, since most of the work focuses on the qualitative approach that concerns the logical correctness of the defined process instead of the performance of the defined process. Therefore, in this paper, we propose a quantitative approach using an activity model for business modeling and analysis, in which, adjacent matrixes can be applied to provide explicit performance indicators for the enterprise to identify the inefficient and ineffective activity looping, and the business process flow can then be improved. Moreover, the proposed quantitative methodologies use a time series intervention ARIMA model to measure and compare the simulation results of the business process reengineering. This is based on the process activity analysis, so that the intervention effects and the asymptotic changes can be determined. After this introduction, the rest of this paper is organized as follows: Section 2 contains the description of the basic six types of business activity interactions. The description of the activity model and analysis for business process are proposed in Section 3. Then a time series intervention ARIMA model for measuring process activity changes is discussed in Section 4. The simulation analysis and the results of time series intervention ARIMA analysis of the business process reengineering based on the proposed activity model analysis are presented in Section 5. Finally, conclusions are given in Section 6.

study of purchasing business process flow in a household appliance manufacturing enterprise, with its mapping of the business activity interactions, is illustrated in Fig. 1.

2. Types of business activity interaction

2.4. Split Interaction

There are many different types of activities in a business process, the activities may be discrete events but their interactions are continuous and co-related. Despite various routing of the activities in the business process, the ultimate aim is to achieve the objectives of the enterprise. In the routing of activities in the business process, different combinations of routing may directly affect the overall performances of the enterprise, however, there is no ‘‘standard best” routing of activities that an enterprise can follow, and enterprises rarely have a distinctive business process flow. In order to effectively model and analyze the business process, a structural approach can be used to represent the routing. Since the interaction activities in the business process are continuous and co-related within the enterprise, the relationship between activities in the business process can be represented in a Boolean adjacent matrix of A = [aij]n  n, where n is the number of activities in a business process, and aij defined as

SPI is a splitting activity interaction in a business process, in which a single previous activity is split into its several succeeding activities. The interaction of SPI in the adjacent matrix with column sum equal to one, and row sum greater than one is represented as

aij ¼



2.1. Start START is an initiation and the first interaction activity in a business process, which leads to the development of the subsequent activities. The interaction of START in the adjacent matrix with column sum equal to zero is represented as

( START ¼

otherwise:

For any business process, despite the difference in the characteristics or routing of the process flow, there are basically six types of business activities interacting between entities, i.e. Start (START), Serial Interaction (SEI), Merge Interaction (MEI), Split Interaction (SPI), Merge and Split Interaction (MSI) and End (END), these six types of activities interaction can be empirically represented according to the Boolean adjacent matrix of A = [aij]n  n. A case

) aij ¼ 0;

where i ¼ 1; 2; . . . ; n :

j¼1

2.2. Serial interaction SEI is a straightforward serial activity interaction in a business process. It directly interacts with its single previous and single succeeding activity, and the interaction of SEI in the adjacent matrix with column sum and row sum both equal to one is represented as

( SEI ¼

i:

n X

aij ¼ 1 and j :

n X

j¼1

) aij ¼ 1;

where i; j ¼ 1; 2; . . . ; n :

i¼1

2.3. Merge interaction MEI is a collection activity interaction in the business process, in which several previous activities are merged and combined, and then processed in a single succeeding activity. The interaction of MEI in the adjacent matrix with column sum greater than one, and row sum equal to one is represented as

( MEI ¼

i:

n X

aij > 1 and j :

j¼1

( SPI ¼

i:

n X

n X

) aij ¼ 1;

where i; j ¼ 1; 2; . . . n :

i¼1

aij ¼ 1 and j :

n X

j¼1

) aij > 1;

where i; j ¼ 1; 2; . . . ; n :

i¼1

2.5. Merge and split interaction MSI is a combination activity interaction of merge and split in a business process, in which it merges several previous activities of a business process, then processes and splits them into several succeeding activities. The interaction of MSI in the adjacent matrix, with both column and row sum greater than one, is represented as

(

1; activity i is previous to activity j 0;

i:

n X

MSI ¼

i:

n X j¼1

aij > 1 and j :

n X

) aij > 1;

where i; j ¼ 1; 2; . . . ; n :

i¼1

2.6. End END is an ending and the final interaction activity in a business process. When the business process reaches END, then the whole business process is complete and the objectives of the enterprise

C.Y. Lam et al. / Expert Systems with Applications 36 (2009) 6986–6994

$ $

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$ MIC ROSOFT C OR PORATION

1. Request Purchase 2. Planning Purchase 3. SelectSupplier (START) (SPI) (MEI)

5. Approve Price 7. Issue Order 8. Implement Contract (SPI) (SEI) (SPI)

4. Approve Planning 6.Re-Select Supplier (SEI) (SEI)

9. Check Invoice (SEI)

13. Receive Goods (SEI)

10. Accounts Payable (MEI)

11. Account Book-In (SEI)

14. Accept Goods (MSI)

15. Reject Goods (END)

12. Financial Report (END)

MIC ROSOFT C OR PORATION

19. Stock Audit (START)

20. Adjust Balance (SEI)

16. Goods Check-In 17. Inventory Accounting 18. Inventory Report (SEI) (MEI) (END)

Fig. 1. An illustration of a purchasing business process flow with mapping of business activity interaction.

will have been achieved. The interaction of END in the adjacent matrix with the row sum equal to zero is represented as

( END ¼

j:

n X

) aij ¼ 0;

where j ¼ 1; 2; . . . ; n :

i¼1

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3. Activity model and analysis The six types of business activity interactions mentioned in Section 2 are the basic elements for the modeling of the business process as A = [aij]n  n. Between the business activity interactions of START and END, there could be more than one possible business process routing to achieve the objectives of the enterprise, i.e. there exist 1, 2, . . . , k business process routing in a single business process. The Boolean adjacent matrix of A can be further represented as A = N1 + N2+  + Nk, where N is the adjacent matrix for other k possible business process flows. Moreover, if the business activity i can flow to business activity j, then business activity j is a reachable business activity of i. As a result, a reachable matrix R (Sage, 1977) of a Boolean adjacent matrix for a business process flow can be defined as

R¼

Iþ

1 X

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

!n1 Nk

0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

¼ ðI þ AÞn1 ;

Fig. 2. An illustration of the activity cycle matrix C.

k¼1

where I = n  n identity matrix. An efficient and effective business process flow can help to improve the overall performance of the enterprise. In a business process, any activity looping is considered as inefficient and ineffective, and should be avoided. From the reachable matrix R of the business process, a sub-matrix of activity cycle matrix C that is derived from R can be used to determine the status of activity looping in the business process. The activity cycle matrix is defined as C = R \ RT, where if there are any columns or rows with more than one non-zero element and interact with more than one activity of entities in matrix C, then it is a business cycle with activity looping. Moreover, as in an efficient and effective business process, the activity cycle matrix C should be similar to an identity matrix I, such that any inefficient and ineffective business activities can also be identified from the activity cycle matrix C, if there are any entities which deviate from the identity matrix I. The activity cycle matrix C in our case study in Fig. 1 is shown in Fig. 2, where the columns and rows of activities 3–6 have more than one non-zero element and they also interact with each other as more than one activity. This clearly shows that activity looping exists in the business process of activities 3–6. This means that some business processes are possibly repeated.

Moreover, apart from the activity looping shown in the activity cycle matrix C, the activity cycle matrix C also deviates from the identity matrix I as in activities 5, 8–18. Therefore, special attention needs to be paid to the business process routing in activities 3–6, 8–18. The business process may even need to be redesigned in order to eliminate the inefficient and ineffective activity looping and those activities which deviate from identity matrix I so as to improve the overall performance of the enterprise. Therefore, the activity model with an activity cycle matrix provides a quantitative approach for analyzing the business process of an enterprise. The activity looping, the inefficiency and ineffectiveness of business process can be determined empirically, and the major problem areas or activities in the business process can then be easily identified in order to carry out the business process reengineering or improvement so as to boost the enterprise towards performance excellence. 4. Measuring process activity changes using a time series intervention ARIMA model ARIMA stands for Auto-regressive integrated moving average process. It consists of three components of autoregression,

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integration, and a moving average. Any one of these components or a combination of them can be used for forecasting and control. A special type of ARIMA is called a time series intervention ARIMA which can be used to analyze the patterns underlying a time series based on its own historical data and previous changes. With the use of a time series ARIMA model in measuring business process activity analysis, the dynamic interruptions and changes of business process based on the activity cycle matrix can be analyzed and the behavior of the business process can be forecast and controlled accordingly (Ip, 1997). In a time series ARIMA model, the general transfer model is represented as

Yt ¼

X xj ðBÞ dj ðBÞ

Bbj X jt þ

hðBÞ at ; /ðBÞ x ðBÞ

where Y t is the output series, X jt is the input series, djJðBÞ is the transhðBÞ at defer function polynomial, bj is the delay for the jth series, /ðBÞ fines the ARIMA noise model and at is a white noise function series. Based on the general transfer model, an intervention model is then proposed by Box and Tiao, (1975) and it is represented as

Preliminary Analysis (Difference the Series to Stationary)

Yt ¼

X xj ðBÞ

Bbj Ijt þ

dj ðBÞ

hðBÞ at ; /ðBÞ

where Ijt is a binary or dummy variable (0,1) which is non-zero only during the period of intervention. To develop the time series intervention, a univariate analysis on is performed, which results in an ARIMA (p, d, q). Then ARIMA (p, d, q) can be represented as

(

rdrDS Y t ¼

hq ðBÞUQ ðBS Þ

/p ðBÞXP ðBS Þ

) at ;

where rd is the regular difference; rDS is the seasonal differencing; p is the number of AR terms; q is the number of MA terms; P is the number of seasonal AR terms and Q is the number of seasonal MA terms. The values of the parameters obtained in the general transfer model are used to estimate the parameters in the intervention model. Different types of transfer functions can be chosen to fit the response to the invention variables. Once a tentative model is identified, it is followed by the estimation of parameters and diagnostic checks of residuals. Generally, there are six procedures to model a univariate ARMIA (p, d, q) model, i.e. (1) a preliminary

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Stationary Series (Ensure the Mean, Variance, and Covariance of the Series are Invariant over Time)

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Identification (Order of Differencing)

0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

Estimation (Estimate the parameters of the tentative model)

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Diagnosis (Check for the suitability of the parameters)

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Fig. 3. A framework for ARIMA modeling approach.

Fig. 5. An illustration of the activity cycle matrix C based on the reengineering of business process.

$ $

Model Interpretation

$

1. Request Purchase 2. Book-In Request 4.Balance Purchase 5.Planning Purchase (START) (SPI) (MEI) (SEI)

6.Approve Planning (SEI)

7. Select Supplier 8. Order Examination 9. Issue Order 10. Implement Contract 11. Check Invoice (SEI) (SEI) (SEI) (SPI) (SPI)

MIC ROSOFT C OR PORATION

3. Check Store Amount (SPI)

12. AccountsPayable (MEI)

13. Goods Receive & Check-In (SPI)

14. Account & Inventory Book-In (MSI)

15. Financial & Inventory Report (END)

MIC ROSOFT C OR PORATION

17. Stock Audit (START)

18. Check Store Balance (SPI)

19. Inventory Balance Report (SEI)

20. AdjustBalance (SEI)

Fig. 4. A reengineered purchasing business process that eliminates activity looping and redesign of activities routing.

16. Reject Goods (END)

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analysis to check whether or not the time series is stationary; if not, then generate a stationary series by differencing or by some other form of transformation, (2) generation of a stationary series by ensuring the mean, variance, and covariance of the series are invariant over time, (3) identification of the order differencing, (4) estimation of the parameters for the tentative ARIMA model, (5) checking for the suitability of the parameters, and (6) interpretation of the formulated ARIMA model. A framework for the ARIMA modeling approach is illustrated in Fig. 3.

5. Simulation and analysis of the business process In order to verify the effect of the activity model in the case study, a simulation of the model for purchasing a business process activity is developed in which the purchasing business process activity, as shown in Fig. 1, is modified and redesigned according to the analyzed results of the activity model as described in Section 3. Then the purchasing business process is simulated, to observe the changes. A time series intervention ARIMA model is used to measure and compare the intervention effect of simulation results of the business process reengineering based on the activity model analysis so as to measure the performance.

Table 1 Assumptions for the activities in the business process simulation Assumptions

Activities

Start of business process Standard processing time

Request purchase

Process may delay

Process may on hold Double standard processing time End of business process

Accept goods, account book in, adjust balance, balance purchases, book in request, financial report, goods receive and check in, implement order, inventory accounting, inventory balance report, inventory report, receive goods, reject goods, and stock audit Check invoice, check store amount, check stock balance, goods check in, issue order, planning purchase, re-select supplier, and select supplier Approve planning, approve price, and order examination Account and inventory book in, financial and inventory report Accounts payable

5.1. Reengineering of the business process From the results in Section 3 of the purchasing business process, activity looping exists in the business process of activities 3–6; moreover, activities 5, 8–18 are the activities that need to be improved. Therefore, reengineering of the business process is strongly emphasised on the activity routing of 3–6, 8–18. After investigation of the business process, it is decided to unite activity 11 with 17, and activity 12 with 17, which means only one accounting activity for both financial and inventory activities is required. To tackle the activity looping, activity 5 and 6 are merged into activity 2, which means the approval of purchasing price is involved in the purchase planning, and selection of supplier is contained in activity 2. Moreover, activities of checking the store amount and balance are added to the business process flow to ensure the accuracy of the single accounting activity. A reengineered purchasing business process that eliminated the activity looping with the redesigning of the routing of the activities is shown in Fig. 4. According to the revised purchasing business process, an activity model can then be developed based on the relationship of the Boolean adjacent matrix A, reachable matrix R and activity cycle matrix C. The activity cycle matrix C of this case study is shown in Fig. 5, where none of the columns or rows of activity have more than one non-zero element, so that activity looping is then eliminated and does not exist in any of the activities of the business process; moreover, after the reengineering of the business process and the redesigning of the routing of the activities, the activity cycle matrix C is the same as identity matrix I. Therefore, the purchasing business process is more efficient and effective after the proposed activity model analysis has been carried out. 5.2. Simulation of the business process Simulations of the purchasing business process before and after the business process reengineering are conducted so as to verify the effect of the new activity model. The simulation data can also be observed and analyzed using a time series intervention ARIMA model to determine the intervention effects and asymptotic change of the business process reengineering that is based on the process activity model. The general assumptions for the interaction of activities in the purchasing business process are summarized in

Fig. 6. The high level mapping layer (left) and model construction layer (right) of purchasing business process before activity model analysis.

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Fig. 7. The high level mapping layer (left) and model construction layer (right) of purchasing business process after activity model analysis.

Number of Accounts Payable Settlement

Table 1, where the simulation assumptions are classified according to the processing time of the activities in the business process. In the purchasing business process of this manufacturing company, three major parties are involved, i.e. Purchasing Department, Finance Department, and the Warehouse. The purchasing business process starts at the Purchasing Department with a purchase request and ends in the Finance Department with accounts payable. The high level mapping layer and the model construction layer for simulating the purchasing business process before and after business process reengineering following from the result of activity model analysis, are shown in Figs. 6 and 7, respectively. In simulating the purchasing business process, a maximum processing time of 30 time units for the whole purchasing business process is applied to the purchasing business process both before and after the business process is reengineered. The comparison of performance is based on the processing capacity of the purchasing business process, and the processing capacity is then measured in terms of the number of accounts payable settled within the 30 time units. The comparison of the simulation results of the purchasing business process before and after reengineering based on activity model analysis, is shown in Fig. 8 and Table 2, respectively. When we compare the processing capacity of the number of accounts

payable settled within 30 time units, we can see that the processing capacity is much higher in the reengineered purchasing business process flow. This means it can process more purchase orders within the same 30 time units, and thus the new business process is considered as more efficient and effective than the original flow. It also indicates that the activity model can help in the redesigning or reengineering of a business process and can support decision making. The results from the activity model also provide an explicit and easy interpretation method for identifying areas that need to be reengineered in order to improve the routing of activities. 5.3. Result of the time series ARIMA intervention model In order to determine the intervention effects of the business process reengineering, the simulation results obtained from the purchasing business process are used as the inputs for the development of the time series intervention ARIMA model. The graphical

Table 2 Comparison of simulation results Accounts payable settled

Mean

Minimum

Maximum

Before process reengineering After process reengineering

6 11

0 1

16 22

25 20 15

25

X jt : Simulation across time Yt : Numberof accounts payable settlement

20

10

15 5

10 5

0

1

3

5

7

9

11 13 15 17 19 21 23 25 27 29

Time Unit Fig. 8. Simulation results of the purchasing business process.

0 1 4 7 101316 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 Fig. 9. Dataset for the development of ARIMA intervention model.

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illustration of the simulation data across times are represented in Fig. 9. The intervention It is initiated at the 30th time unit, which is generated by the change of the business process flow based on the activity model analysis. In the development of the time series invention ARIMA model, Yt represents the number of accounts payable that are settled according to the purchase request, and Xjt represents the simulation across time units.

Table 3 Results of ADF unit root test Augmented Dickey–Fuller unit root test on null hypothesis: DPRE_INTERVENTION

Augmented Dickey–Fuller test statistic Test critical values

1% level 5% level 10% level

Augmented Dickey–Fuller test equation Variable Coefficient Standard error DPRE_INTERVENTION 1.505326 0.166162

*

R2

0.752453

Adjusted R2 S.E. of regression Sum squared resid Log likelihood

0.752453 5.106521 704.0671

84.87565

t-Statistic

Probability*

9.059382

0.0000

2.650145 1.953381 1.609798

t-Statistic

Probability

9.059382

0.0000

Mean dependent variable S.D. dependent variable Akaike info criterion Schwarz criterion Durbin– Watson statistics

In determining the order of differencing for the pre-intervention ARIMA model, a null hypothesis testing of Yt = 0 is performed by the Augmented Dickey–Fuller (ADF) unit root test. The testing rule is that if the ADF t-statistic value is greater than the ADF critical values at 1%, 5% and 10% level, the null hypothesis is accepted, otherwise, the null hypothesis is rejected. The ADF unit root test results as shown in Table 3 indicate that one differencing factor for d-value in ARIMA model is identified as its ADF t-statistic value is smaller than the test critical values at 1%, 5% and 10%. This means that the null hypothesis is rejected. Rejection also implies that there is no unit root problem and the ADF critical values are stationary at 1%, 5%, and 10% significance level. The corresponding correlogram is shown in Fig. 10. To estimate the (p, d, q) parameter in the pre-intervention time series ARIMA model, autocorrelation, partial autocorrelation, and a range of statistical measurements are observed and analyzed. A model of two AR factors, one MA factor and one differencing period is suggested, i.e. ARIMA (2, 1, 1). The estimated output of the preintervention ARIMA (2, 1, 1) model and its correlogram are shown in Table 4 and Fig. 11 respectively. Therefore, the pre-intervention ARIMA model for the purchasing business process is represented as

ð1  BÞY t ¼

0.046381

10.26351 6.133975

ð1 þ 1:5962BÞ ð1 þ 0:7292BÞð1 þ 0:6306B2 Þ

at :

The post-intervention ARIMA model, after business process reengineering based on the activity model analysis is estimated, and the model which includes the intervention transfer function, is represented as

80:0219Bt1 ð1 þ 0:5957BÞð1 þ 1:8152B2 Þ at : þ at ¼ ð1 þ 1:4455BÞ ð1 þ 0:6325BÞð1  0:0310B2 Þ

6.181554

ð1  BÞY t ¼

2.707179

Using the post-intervention ARIMA model, the asymptotic change can then be calculated as

MacKinnon (1996) one-sided p-values.

Fig. 10. Correlogram of Yt with one differencing factor.

Table 4 The estimation output of the pre-intervention ARIMA (2, 1, 1) model Dependent variable: DPRE_INTERVENTION Variable

Coefficient

Standard error

C AR(1) AR(2) MA(1)

0.050299 0.729200 0.630619 1.596220

0.017011 0.168098 0.169236 0.299386

Inverted AR Roots

R2 Adjusted R2 S.E. of regression Sum squared resid Log likelihood Durbin–Watson statistics .36–.71i

0.885795 0.870898 2.166064 107.9122 57.01533 1.889007 .36 + .71i

t-Statistic 2.956823 4.337954 3.726267 5.331651 Mean dependent variable S.D. dependent variable Akaike info criterion Schwarz criterion F-statistic Probability (F-statistic) Inverted MA Roots

Probability 0.0071 0.0002 0.0011 0.0000 0.005705 6.028445 4.519654 4.711630 59.46386 0.000000 1.60

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Fig. 11. The correlogram of the pre-intervention ARIMA (2, 1, 1) model.

Table 5 The estimation output of the post-intervention ARIMA (2, 0, 1) model Dependent variable: DPRE_INTERVENTION Variable

Coefficient

Standard error

t-Statistic

Probability

C AR(1) AR(2) MA(1) MA(2)

10.32041 0.632512 0.031021 0.595742 1.815214

– – – – –

– – – – –

– – – – –

Inverted AR roots

R2 Adjusted R2 S.E. of regression Sum squared resid Log likelihood Durbin–Watson statistics .05

0.806690 0.773071 2.805877 181.0778 65.86439 2.781319 .68

Mean dependent variable S.D. dependent variable Akaike info criterion Schwarz criterion F-statistic Probability (F-statistic) Inverted MA roots

10.42350 5.890108 5.061742 5.299636 23.99493 0.000000 1.68

Fig. 12. The correlogram of the post-interveniton ARIMA (2, 0, 2) model.

80:0219 ¼ 33 units: ð1 þ 1:4455Þ As can be seen from the time series intervention ARIMA model, the ARIMA model has been changed from pre-intervention ARIMA (2, 1, 1) model to post-intervention ARIMA (2, 0, 2) model. The estimated output of the post-intervention ARIMA (2, 0, 2) model and its correlogram are shown in Table 5 and Fig. 12, respectively. Moreover, from the analysis it can be observed that the intervention is in a kind of gradual permanent intervention pattern. This shows that the business process reengineering based on the activity model analysis enhances the efficiency and effectiveness of the business process by an initial increase of 80 units and is followed by an alternative increase and decrease to a permanent level. The intervention effect can also be calculated as an asymptotic change of 33 units, and the lag time effect is estimated to be one period before the reengineered business process settles down to a steady state.

6. Conclusion As the business marketplace becomes more globalized, enterprises are more eager to increase their competitiveness. Research on business process modeling and analysis then become more important for driving the business process towards excellence and for gaining a competitive advantage. In this paper, a quantitative approach of activity model analysis is proposed, which involves business modeling and analysis based on the Boolean adjacent matrix A, reachable matrix R and activity cycle matrix C. The inefficient and ineffective activity looping in the business process can then be explicitly identified and tackled, and the business performance can be improved accordingly. With the proposed activity model for business analysis, a time series intervention ARIMA model is then used to provide a precise estimation of the form and magnitude of the intervention effects of the business process reengineering, based on the proposed activity model analysis.

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