A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

CHAPTER 3 PROCESS DECOMPOSITION

1. INTRODUCTION 2. DECOMPOSITION OF AN ACTIVITY-PARAMETER MATRIX 2.1. Decomposition Concept 2.2. Cluster Identification Algorithm 2.3. The Branch-and-Bound Algorithm 3. INDUSTRIAL CASE STUDES 3.1.Electronics Systems Design Process 3.2. Systems Engineering Process 4. SUMMARY REFERENCES QUESTIONS PROBLEMS 1. INTRODUCTION A typical process, e.g., product development process, may involve coordination of many activities and procedures. Due to the decreasing product life cycles, it is important to reduce the time and cost of product development. It has been identified that 70-80% of the final production cost is determined during the design stage (Whitney 1988). The concept of concurrent engineering attempts to incorporate various constraints related to the product life cycle, i.e., manufacturability, quality, and reliability, in the early design stages. Concurrent design aims at the improvement of product quality, reducing the development time and cost. However, due to the interaction between various activities, the complexity of the product development process increases and makes the process difficult to manage. Decomposition is a suitable approach for simplifying many processes, including the design process. Decomposition has long been recognized as a powerful tool for analysis of large and complex systems (Courtois 1985). The advantage of decomposition is that it reduces the complexity of the process. The existing literature advocates the use of decomposition as a tool that exhibits a high degree of cohesion within modules, and low coupling between modules. The literature concerning research of decomposition in design is not very extensive. Johnson and Benson (1984) developed a two-stage decomposition strategy for design optimization. The strategy assumes that all the subproblems are independent of each other. Azarm (1987) applied decomposition to problem solving using the concept of monotonicity. Intuitive methods that consider the physics of the system as the prime factor directing the decomposition were also used. However, the quality of the results generated by an intuitive method depends heavily on designers' experience. With the growing interest in process engineering, the need for formal decomposition approaches is apparent. Some of the earlier work in the decomposition applied to engineering design include the research performed at NASA (Rogers 1989 and Sobieszczanski-Sobieski 1982, 1989) was

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

concerned with the implementation of the partitioning algorithm for identification of coupled design tasks (Steward 1981). In this chapter, a process is represented as a graph, incidence matrix, or an IDEF3 model. The objective of this chapter is to present a systematic approach for the decomposition of a process model into submodels with the minimal interdependence between them to enhance concurrency of the process. The chapter is based on the premise that in order to significantly reduce the process cycle (duration), one has to decompose the process. The decomposition approach allows one to identify processes that can be performed simultaneously or formulate strategies leading to their separation, e.g., considering alternative activities, and adding new resources. Incremental improvements of individual activities are not likely to drastically reduce the process cycle. The salient features of the decomposition approaches proposed in the chapter are as follows: • Low computational time complexity • Structured and systematic approach • Suitability for human interaction • Generality Consider the relationship between various activities and parameters (inputs and outputs) represented with an incidence matrix (Warfield 1993). An entry "*" denotes that parameter j (for the matrix in Figure 1(a), j = a, ..., h) appears in activity i (in Figure 1(a), i = 1, ..., 7). Matrices can be categorized as decomposable and non-decomposable. An incidence matrix is decomposable if its rows and columns can be grouped in such a way that the matrix separates into mutually exclusive submatrices (see the matrix in Figure 1(a)). Analogously, an incidence matrix is non-decomposable if it can not be arranged into mutually separable submatrices (see the matrices in Figure 1(b) and (c)). As the activities and parameters associated with each of the three submatrices in Figure 1(a) are independent, they can be computed simultaneously. In a non-decomposable matrix, submatrices are interdependent due to the overlapping parameters (parameter g and h in Figure 1(b)) and overlapping activities (activities 5 and 6 in Figure 1(c)). In order to be able to consider simultaneously the two clusters of activities in Figure 1(b), an additional action is required, e.g., the values of the overlapping parameters g and h would have to be provided.

(a)

a b c d e f g h

1 * * * 2 * * * 3 * * 4 * * 5 * * 6 * * * 7 * * *

(b) 1 2 3 4 5

a b c d e f g h * * * * * *

* * * * * * * * * * * * * *

Figure 1. Three types of matrices (a) decomposable matrix

2

* * * * *

(c) 1 2 3 4 5 6

a b c d e f * * * * * * * * * * * * * * * *

* * * *

* * * *

A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

(b) non-decomposable matrix with overlapping parameters (inputs and outputs) (c) non-decomposable matrix with overlapping activities In addition to activities and parameters (inputs, outputs), decomposition can be applied for managing constraints. Constraints frequently show some degree of coupling which complicates the computational process. In this chapter, a decomposition approach is used to simplify constraint management and to reduce the computational time of constraint evaluation. 2. DECOMPOSITION OF AN ACTIVITY-PARAMETER MATRIX 2.1. Decomposition Concept To design a new product of high quality and in less time, one should explore the possibility of the decomposition of the design process into groups of activities (subprocesses). The relationship between activities is analyzed in order to detect potential activities that can be performed simultaneously. The decomposition problem is formulated as follows: Decompose an activity-parameter incidence matrix representing a process into mutually separable submatrices (groups of activities and groups of parameters) with the minimum number of overlapping activities subject to the following constraints: Constraint C1: Empty groups of activities or parameters are not allowed. Constraint C2: The number of activities in a group does exceed an upper limit, ZU (or alternatively, the number of parameters in a group is does not exceed, N). Of course, other constraints, for example, a resource cannot be used for two different activities at the same time, can be considered. The objective function is “to minimize the number of overlapping parameters.” An alternative objective function may be used, for example, “to maximize the measure of effectiveness” defined in McCormick et al. (1972) To deal with the overlapping activities, the following actions can be taken: A1. One may replace an overlapping activity with an alternative activity that involves different parameters. A2. An activity can be decomposed into subactivities and each subactivity assigned to a group of parameters with some commonality with that subactivity. A3. An overlapping activity can be removed from the matrix. A4. The duration of the overlapping activity can be shortened (e.g., by better management of resources or tools used). The model decomposition simplifies the process, explores potential concurrency among activities, and reduces the time involved in the process. The challenge is to decompose the overall process into groups of activities that are of acceptable size, can be easily solved, and for which an overall solution can be generated from the partial solutions. 2.2. Cluster Identification Algorithm

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

The cluster identification (CI) algorithm presented next decomposes a binary incidence matrix, where the mutually separable clusters exist in the matrix. The greatest advantage of the CI algorithm is its efficiency and simplicity. CI Algorithm (Kusiak and Chow 1987) Step 0. Set iteration number k = 1. Step 1. Select row i of incidence matrix [aij](k) and draw a horizontal line hi through it ([aij](k) is read: matrix [aij] at iteration k ). Step 2. For each entry of * crossed by the horizontal line hi draw a vertical line vj . Step 3. For each entry of * crossed-once by the vertical line vj draw a horizontal line hk . Step 4. Repeat steps 2 and 3 until there are no more crossed-once entries of * in [aij](k). All crossed-twice entries * in [aij](k) form row cluster RC-k and column cluster CC-k . Step 5. Transform the incidence matrix [aij](k) into [aij](k+1) by removing rows and columns corresponding to the horizontal and vertical lines drawn in steps 1 through 4. Step 6. If matrix [aij](k+1) = 0 (where 0 denotes a matrix with all empty elements), stop; otherwise set k = k + 1 and go to step 1. One can notice that the cluster identification algorithm scans each element of matrix [aij] two times. Since there are mn elements in matrix [aij], its computational time complexity is O(mn). The CI algorithm is illustrated in Example 1. Example 1 Consider the incidence matrix (1) with seven activities (rows) and eight parameters (columns).

1 2 3 4 5 6 7 8 1 2

* * *

*

3 4 5 6 7

Step 0. Step 1. Step 2.

*

*

*

*

* *

* *

* *

*

*

(1)

Set iteration number k = 1. Row 1 of matrix (1) is selected and horizontal line h1 is drawn. The result of steps 1 and 2 are presented in matrix (2). Three vertical lines v2, v3, and v5 are drawn.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

1 2 3 4 5 6 7 8 1

* *

2

*

*

h1 *

3 4 * 5

*

* *

*

*

*

* *

*

*

v2 v3

v5

6 7

*

(2)

As a result of drawing these three vertical lines, five new crossed-once entries * are created in matrix (2), i.e., entries (5, 3), (5, 5), (7, 2), (7, 3), and (7, 5). Step 3. Two horizontal lines h5 and h7 are drawn through all the crossed-once non-empty entries of matrix (2), as shown in matrix (2).

1 2 3 4 5 6 7 8 1 2

* *

*

*

h1 *

3 * * 4 * * 5 * * 6 * 7 * * * v2 v3

*

h5

*

h7

v5

(3) Step 4. Since the entries (5, 8) and (7, 8) of matrix (3) are crossed once, the vertical line v8 is drawn, as shown in matrix (4).

1 2 3 4 5 6 7 8 1 2

* * *

*

h1 *

3

* * 4 * * 5 * * 6 * 7 * * * v2 v3

v5

*

h5

*

h7

v8

(4)

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Since there are no more crossed-once nom-empty entries, all the crossed-twice entries * of matrix (4) form • Row cluster RC-1 = {1, 5, 7}, and • Column cluster CC-1 = {2, 3, 5, 8}. Step 5. Matrix (4) is transformed into matrix (5).

1 4 6 7 2 3 4 6

*

* *

*

* *

*

(5)

In the second iteration (k = 2), steps 1 through 4 are performed on matrix (5). This iteration results in incidence matrix (6).

1 4 6 7 2 3 4 6

*

* *

*

h2 *

*

h4

* v1

v6

(6)

Also, • Row cluster RC-2 = {2, 4} and • Column cluster CC-2 = {1, 6} are obtained. In the third iteration (k = 3), matrix (7) is generated 4 7 3 * * h3 6 * h6 v4 v7

(7) From this matrix RC-3 = {3, 6} and CC-3 = {4, 7} are obtained. The final decomposition result is illustrated in matrix (8).

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

CC-1

CC-2 CC-3

2 3 5 8 1 6 4 7 RC-1

1 * * * 5 * * * 7 * * * * RC-2 2 * * 4 * * RC-3 3 * * 6 *

(8) Three mutually separated subprocesses RC-1, RC-2, and RC-3 and the corresponding sets of parameters CC-1, CC-2, and CC-3 are visible in matrix (8). Example 2 illustrates the application of cluster identification algorithm to process represented with a graph. Example 2 Consider the graph representing a process involving ten activities and seventeen parameters (Figure 2). For example, x5 is the input and x15 and x13 are outputs of activity 1.

x6 9

x 12 x13

1 x5

x7

10

x8

7

x17

5

x15

x14

x9

x4

8

x10

4

6 x2

x11

x3 2

x1

x 16

3

Figure 2. A process graph The activity-parameter incidence matrix (9) corresponding to the graph in Figure 2 is shown next. Parameter x2 1 2 3 4 5 Activity 6 7 8 9 10

x4

x1

x3

*

*

x6 x5

x8 x7

x9

x10 x 12 x14 x16 x11 x13 x 15 x17

*

*

*

*

* *

* *

*

*

*

* * *

*

* *

*

(9)

* *

*

*

*

*

*

*

Matrix (9) is nonstructured and it is difficult to recognize any groupings among the activities and parameters.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Applying the cluster identification (CI) algorithm to matrix (9) results in matrix (10): Parameter G-2

G-1

G-3

x 13 x 1 x 6 x 2 x 16 x 9 x14 x12 x5 x 15 x 3 x 17 x11 x 7 x 4 x10 x 8 GA-1

Activity GA-2 GA-3

1 2 5 10 3 4 8 6 7 9

* * * * *

* * * * * * * * * * * * * * *

(10) * * * * * * * *

Three mutually exclusive groups of activities GA-1 = {1, 2, 5, 10}, GA-2 = {3, 4, 8}, GA-3 = {6, 7, 9} and three groups of parameters G-1 = {1, 3, 5, 6, 13, 15, }, G-2 = {2, 7, 9, 11, 16}, G-3 = {4, 8, 10, 12,14 }, are visible in matrix (10). Based on the clustered matrix (10), it is clear that the three groups of activities can be performed simultaneously (Figure 3).

GA-1 x7 4

x2 x9

GA-2 x4

x 16

8

6

x 11

x 14

x8 9

7

x 12

GA-3 x 15

x5

3

5

1

x 10

x3 2

x18 x 13

10

x6

x1

Figure 3. Three independent processes (groups of activities and parameters) If an incidence matrix cannot be decomposed into mutually separable submatrices, a number of actions can be taken. One has to identify the overlapping parameters or overlapping activities and remove them from the matrix. A decomposition algorithm is presented next is a modified version of the clustering algorithm presented in Kusiak and Cheng (1990) and includes a new branching scheme. 2.3. The Branch-and-Bound Algorithm The branch-and-bound algorithm presented in this section analyzes the activity-parameter matrix and attempts to detect the potential parameters that can be performed simultaneously in order to reduce the process duration. 8

A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

The algorithm iteratively examines each unfathomed node in the enumeration tree. The lower bound ZL on the objective function is the number of parameters (tasks) removed from the incidence matrix. The upper bound ZU, equal to the number of parameters (activities) removed, is calculated only after a feasible solution has been determined. Parameters corresponding to the highest number of activities are being removed one at a time. The algorithm is presented next.

Step 0. Step 1.

Step 2. Step 3.

Step 4.

The Algorithm (Initialization) Consider the incidence matrix (aij) as the initial node. Set the upper bound ZU = +8 . (Branching) Based on the depth-first search strategy, select an active node (not fathomed) does not satisfy constraint C2. Apply the CI algorithm to the submatrix of this node. When the submatrix cannot be partitioned, use the branching scheme. Branching scheme: Remove a parameter corresponding to the largest number of activities, one at a time. (Bounding) For each new node, obtain a lower bound ZL. (Fathoming) Exclude a new node from further consideration if: Test 1: ZL ≥ ZU, Test 2: Any of the submatrices of the matrix at this node violates constraint C1 or C2. If ZL < ZU, store this solution as a new incumbent solution, set ZU = ZL, and reapply Test 1 and 2 to the remaining unfathomed nodes. (Stopping rule) Stop when there are no unfathomed nodes remaining; the current incumbent solution is final. Otherwise, go to step 1.

The branching scheme of this algorithm can be modified so that in the branch-and-bound algorithm activities rather than parameters are removed. The branch-and-bound algorithm is illustrated in Example 3 and 4. Example 3 Decompose the process model in Figure 4 into subprocesses with no more than four activities in the subprocess.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

1 1

5 17

3

5

7

9

8

9 3

11

2 11 13

12 6 8 4

10 16

7

14

2

6

4

15 10

Figure 4. Process model The process model in Figure 4 is represented with matrix M presented next. Note that in this matrix the letters I (input) and O (output) have been used rather than asterisks in some other examples. Matrix M 1 1 O 2 3 4 5 6 7 8 9 I 10 11

2

3 O

4

5 O

6

7

8

9

I I I I

O

O

10 11 12 13 14 15 16 17 O I O O I I I I

O I

I O

O I I

O O

O

I

I

I O

O

O

In computational process one row (activity) at atime of matrix M is removed. The resultant matrices are presented next. Activity 1 removed 1 2 3 4 5 6 2 3 4 5 6 7 8 9

7

8

9 I

I I I

O

O

O I

I O I

10 11 12 13 14 15 16 17 I O O I I I I

O I I

O

O O I

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

10 11 1 O

I

I O

O O

O

O

O

(Nondecomposable matrix) Activity 4 removed 1 2 3 4 5 6 O O O

1 2 3 5 6 7 8 9 I 10 11 4

I I

7

8

I O

O

O

9

10 11 12 13 14 15 16 17 O I O O I

I I O

O I I

O O

O

I

I

I O

O I

O I

I

I

(Nondecomposable matrix) Activity 9 removed 1 2 3 4 5 6 O O O

1 2 3 4 5 6 7 8 10 11 9 I

7

8

9

I I I I

O

O

10 11 12 13 14 15 16 17 O I O O I I I I

O I

I O

O I

I

O O I O

O I

O

O I

(Nondecomposable matrix)

11

A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Activities 1 and 4 removed 1 2 3 4 5 6 7 8 2 3 5 6 7 8 9 I 10 11 4 1 O

I I

I O

O

O

9

10 11 12 13 14 15 16 17 I O O I

I I O

O I I

O

O

I

I

I O

O I O

O

O I

I

I

O

O

(Nondecomposable matrix) Activities 1 and 9 removed 1 2 3 4 5 6 7 8 2 3 5 6 7 8 4 10 11 9 I 1 O

I I

I O

O

O

9

10 11 12 13 14 15 16 17 I O O I

I I O

O I

O

I

I I

I O

O I O

I

O I

O

O

I

O

O

(Nondecomposable matrix) Activities 4 and 9 removed 1 2 3 4 5 6 7 8 2 3 5 6 7 8 1 O 10 11 9 I

I I

I O

O

O

10 11 12 13 14 15 16 17 I O O I

I I O

O

9

O I

O O

O

O

I

I O

O I

O

O I

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

4

I

I

I

I

(Nondecomposable matrix)

1 5 3 11 2 6 10 8 7 9 4

5 O I

The final solution 1 3 7 12 17 9 O O O O O O I I

13 15 2

O I

8

4

O

11 10 14 16 6

O

O I

I

O I

I I O

I

I I

O I

O O I

I

O I

I

All steps of the branch-and-bound algorithm are summarized in Figure 5.

M

Remove 1

Remove 4

Remove 9

Remove 4

Remove 1

Remove 9

Remove 1

Solution Remove 9

Remove 4

Fathom

Fathom

Remove 9

Z L = 1, Z U= ∞

Remove 9

Remove 4

Z L = 2, Z U= ∞

Solution

Z L = 2, Z U= 2

Remove 4

Fathom

Fathom

Z =3 L

Figure 5. Branch-and bound computational tree Example 4 Consider matrix (11) representing a process with four activities and five parameters.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Parameter 1 1 2

Activity

2

3

4

*

*

*

* *

3

*

4

5

*

*

*

*

*

(11) Applying the CI algorithm to matrix (11) in Step 1 of the branch-and-bound algorithm, decomposition does not take place. Following the branching scheme, parameter 3 with the largest number of activities (asterisks) is removed. GP-1 1 GA-1 {

GA-2 {

4

GP-2 2

5

2

*

*

3 *

4

*

*

*

1

*

*

3

*

*

*

(12) Two groups of activities GA-1 and GA-2 and two groups of parameters GP-1 and GP-2 are visible in matrix (12) as well as the overlapping parameter 3. 3. INDUSTRIAL CASE STUDES The models of design processes may vary considerably between different companies and design domains. Kannapan and Marshek (1992) provided a detailed discussion of several design methodologies with the objective of developing a common view of the design process. Multiple disciplines participate in building process models. The transfer of information between design engineers defines precedence constraints for activities in the design process. The process decomposition concept is illustrated with two case studies presented next. 3.1. Electronics Systems Design Process Consider the graph representing Phase 3 of the design process in Figure 6 (same as the process in Figure 22 of Chapter 2) with annotated inputs and outputs.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

27

37

26

38

36

33

5

8

3

2

11

6

4

5

4

12

9 17

16

10

27 26

12

11

13 14 15 18

15

21

20 10

20

16

25

7

9

17 30 28 29

6

8

28

31

32

24

39

13

35

34

3

7

25

21

1

1

14

22

22

18

23 19

23

24

2

19

Figure 6. The process graph with annotated inputs and outputs The process in Figure 6 is represented with the activity-input/output incidence matrix in Figure 7. Input/output 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9

Activity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

O O O O I I O I I O O I I O I O I I O I O I O O O I O I O O I I O O I O O I O O I O O I I I O I

O I I O O I I O O I I

I I O I O I

O

I O O I I I O I O I O I

Figure 7. The activity-input/output incidence matrix Applying the branch-and-bound algorithm to this matrix results in matrix in Figure 6 with five overlapping inputs/outputs.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Input/output 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 2 3 1 2 3 6 7 8 9 0 1 2 3 4 5 6 7 8 9 1 2 3 4 6 7 8 9 0 2 3 4 5 6 7 8 9 4 5 0 5 1

Activity

1 5 6 7 3 4 8 9 10 19 22 2 12 14 15 16 17 18 20 23 11 13 21 24 25 26 27 28

O I O I I O

O I O O I I O I I O I O I O O I O I O O I I O O I I O I I O

I

O O

O O O I I O I

I O I

I O O I I O O

I

I I O I O I O O I O I O I I O I O I O I

Figure 8. The decomposed incidence matrix with overlapping inputs/outputs Four groups of activities are formed with some overlapping inputs or outputs. Figure 9 shows the decomposed process graph corresponding to the matrix in Figure 8.

16

O

A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

27

G-4

37

26

38

36

G-1

G-2 8

3

33

5

5

2

11

6

4

6

8

15

21

27

12 10

13 14 15 18

9 16

26

12

11

10

20

16

25

20

17

17 30 28 29

4 7

9

28

31

32

24

39

13

35

34

3

7

25

21

1

1

14

22

22

23 19

24

23 G-3

18

2

19

Figure 9. The decomposed process graph with overlapping inputs/outputs Three actions can be taken to enhance the degree of concurrency among activities: • Alternative inputs/outputs could be considered for the activities with overlapping inputs/outputs • Some inputs may be assumed or ignored • The overlapping inputs/outputs can be used as goals to the preceding activities and inputs to the succeeding activities, e.g., parameter 31 in Figure 9 could become a goal of activity 11, and at the same time input of activity 17. This way the groups G-4 and G-3 containing activities 11 and 17 would decouple. Similarly, applying the decomposition algorithm to the matrix in Figure 7 results in the decomposed activity-input/output matrix with overlapping activities shown in Figure 10 (Figure 11 shows the corresponding decomposed process graph). Six groups are formed with some overlapping activities. To deal with the overlapping activities, the following actions can be taken: • One may view the overlapping activities as a server that controls a group of activities (clients). • One may replace an overlapping activity with an alternative activity that involves different inputs/outputs.

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

Input/output 1 1 1 1 1 1 1 1 1 1 3 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 2 2 1 2 3 4 5 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 8 9 0 1 2 3 4 5 6 7 8 9 6 7 8 6 7 1 5 6 7 9 10 19 22 2 12 14 15 18 11 Activity 13 17 20 21 24 25 26 27 28 3 4 23 8 16

O I O I I O O I O I

O I O O I I O O I I O I O I O I O O I I O I O I O O I O O I I O O I I O I O I I O I O I O I O O I O I O

I O I O I O I O

I O I

I O I

I O

Figure 10. The decomposed incidence matrix with overlapping activities

27

G-4

37

26

38

36

G-1

33

5

G-5 5 8

3

2

11

6

4

6

8

12 10

9 17

16

19

10

26

14

22

22

23 19

2

20

27

12

11

13 14 15 18

15

21

20

29

16

25

7

9

17 30 28

4

28

31

32

24

39

13

35

34

3

7

25

21

1

1

23

G-6

24

18 G-3

G-2

Figure 11. The decomposed process graph with overlapping activities

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

3.2. Systems Engineering Process The model shown in Figure 12 is based on systems engineering process at an industrial company (Kusiak et al. 1994). The model has been simplified to include 16 activities on a single level of abstraction. The model includes 20 mechanisms (numbered 1, ..., 20) that are required to perform the activities at the conceptual design phase. 1

1, 2, 3, 4, 5

2

4

6

1, 2, 3, 6

2, 7, 15

2, 6, 16, 15, 17, 18

3

5

7

7, 2, 6, 3

1, 13, 12, 3

8

2, 6, 3

7, 13, 3, 12, 1

10

9 16 13, 12, 3

11

14, 1, 20 14, 8, 9, 10, 11

8, 9, 10, 11

12

13

14

15

13, 3

2, 6, 16, 18, 3

19, 3, 1, 2, 20, 6

7, 2, 6, 3

Figure 12. IDEF3 model of the conceptual design phase The decomposed activity - mechanism incidence matrix of the model in Figure 12 is shown in Figure 13. 4 5 8 9 10 11 14 20 19 7 12 13 16 17 18 2 3 1 6 15 1 10 11 16 14 3 4 7 15 5 9 12 6 13 2 8

* *

* * * * * * * * * * * * * * * * * * * * * * * * * * *

* * * * * * * * * * * * * * * * * * * * * * *

* * * * * * * * * * * * * *

Figure 13. Decomposed activity-mechanism incidence matrix The process decomposes into four groups of activities and mechanisms with five overlapping mechanisms (1, 2, 3, 6, and 15). This result can be used for various purposes, e.g., physical arrangement of resources, procurement of resources, and so on.

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4. SUMMARY In this chapter, the decomposition concept was applied to processes. It allows determining activities, inputs, outputs, and controls of a process model that need to be address in order to identify concurrent activities and bottlenecks. Decomposition is a key tool for restructuring processes. The need for the necessary data is modest. The decomposition method discussed in this chapter and the methods discussed in Chapter 2 provide a useful tool for analysis of process models. Each of them provides a unique view of the process. REFERENCES 1. Azarm, S. (1987), “Optimal Design Using A Two-Level Monotonicity-Based Decomposition Method,” Proceedings of the 1987 ASME Design Automation Conference, Boston, Mass., 41-48. 2. Courtois, P. J. (1985), “On Time and Space Decomposition of Complex Structures,” Communication of ACM, 2, 590-603. 3. Johnson, R. C. and Benson, R. C. (1984), “A Basic Two-Stage Decomposition Strategy for Design Optimization,” ASME Journal of Mechanisms, Transmissions, and Automation in Design, 106, 380-386. 4. Kannapan, S. M. and Marshek, K. M. (1992), "Engineering design methodologies: a new perspective," in Intelligent Design and Manufacturing, Ed. Kusiak, A. John Wiley & Sons, Inc., New York, NY, 3-38. 5. Kusiak, A. and Cheng, C. H. (1990), “A Branch-and-Bound Algorithm for Solving the Group Technology Problem,” Annals of Operations Research, 26, 415-431. 6. Kusiak, A. and Chow, W. S. (1987), “Efficient Solving of the Group Technology Problem,” Journal of Manufacturing Systems, 6, pp. 117-124. 7. Kusiak, A., Larson, T. N., and Wang, J. (1994), "Reengineering design and manufacturing processes," Computers and Industrial Engineering, 26, pp. 521-536. 8. McCormick, W. T., Schweitzer, P. J., and White, T. W. (1972), “Problem Decomposition and Data Reorganization by Clustering Technique,” Operations Research, 20, 992-1009. 9. Rogers, J. L. (1989), “DeMAID: A Design Manager’s Aided for Intelligent Decomposition User’s Guide,” NASA TM-101575, Langley Research Center, Hampton, Virginia. 10. Sobieszczanski-Sobieski, J. (1982), “A Linear Decomposition Method for Large Optimization Problems - Blueprint for Development,” NASA TM-83248, Langley Research Center, Hampton, Virginia. 11. Sobieszczanski-Sobieski, J. (1989), “Multidisciplinary Optimization for Engineering Systems: Achievements and Potential,” NASA TM-101566, Langley Research Center, Hampton, Virginia. 12. Steward, D. V. (1981), Systems Analysis and Management: Structure, Strategy, and Design, Petrocelli Books, New York. 13. Warfield, J. N. (1973), “Binary Matrices in System Modeling,” IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-3, No. 5, pp. 441-449. 14. Whitney, D. E. (1988), “Manufacturing by Design,” Harvard Business Review, 83-91.

QUESTIONS

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A. Kusiak, Engineering Design: Products, Processes, and Systems, Academic Press, San Diego, CA, 1999.

1. What are the salient features of the process solved by the cluster identification algorithm? 2. What are the main steps of the cluster identification algorithm? 3. What is the basic difference between the matrix constructed for the triangularization algorithm of Chapter 2 and the branch-and-bound algorithm considered in this chapter? 4. What is the basic difference between the solution provided by the triangularization algorithm (Chapter 2) and the solution generated by the branch-and-bound algorithm? 5. What actions can be taken to eliminate the overlapping inputs/outputs? 6. What actions can be taken to eliminate the overlapping activities? PROBLEMS 1. For Problem 2 from Chapter 2, construct an activity-input/output matrix and solve it with a suitable decomposition algorithm. Compare the result with the result provided by the triangularization algorithm. 2.

For Problem 3 from Chapter 2, construct an activity-input/output matrix and activity – control matrix. Solve the two problems a suitable decomposition algorithm.

3. For Problem 4 from Chapter 2 construct a formula – parameter/variable and solve it with the branch-and-bound algorithm. 4. (a) (b) (c) (d) (e)

For the IDEF3 process model in Figure A1: number (label) the inputs, outputs, and controls, set up an activity - input, output, and control incidence matrix, set up an activity - input and output incidence matrix, apply the branch-and-bound algorithm to the matrices generated in (b) and (c), discuss the results. 11

01 02

07

12

08

03 04

05

09

06 Figure A1. IDEF3 model of an industrial process

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