A. Kusiak, Engineering Design: Products, Processes, and ...user.engineering.uiowa.edu/~coneng/lectures/Chapter_2.pdfC1 M1 Activity 2 O1 C2 M2 O2 Activity 1 I1 C1 M1 O2 Activity 1 (a)

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

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CHAPTER 2

ANALYSIS OF PROCESS MODELS AND PROBLEMS

1. INTRODUCTION 2. OBSERVATIONAL ANALYSIS OF PROCESS MODELS

2.1. Reducing Duration of Processes 2.2. Eliminating Redundant Activities 2.3. Partitioning Activities 2.4. Merging Serial Activities 2.5. Eliminating Cycles

3. COMPUTATIONAL ANALYSIS OF PROCESS MODELS 3.1. Graph Representation of Dependencies in Process Models 3.2. Types of Dependencies in Process Models 3.3. Ordering Non-cyclic Graphs 3.4. Ordering Graphs with Cycles

4. ANALYSIS OF DESIGN PROBLEMS 5. ANALYSIS OF DESIGN PROCESSES 6. ANALYSIS OF CRITICAL PATTERNS 7. SUMMARY

REFERENCES QUESTIONS PROBLEMS

1. INTRODUCTION A process model created through a modeling effort, often referred to as “as-is” model, needs to be transformed into “to-be model” in order to meet the goals of the reengineering effort. This transformation can be accomplished with various analyses of the “as-is” model. The analyses are divided into two types: • observational (manual) analysis • computational analysis. Of course, reengineering approaches other than the transformation of the “as-is” process model into “to-be” model are possible. Some favor building the final “to-be” model without representing the current process. The analysis methods presented next and the subsequent chapters indicate the difficulty in reengineering process models without supporting tools. The experience demonstrates that at some point a model has be viewed as “as-is” model and it has to be reengineered with the analysis tools. 2. OBSERVATIONAL ANALYSIS OF PROCESS MODELS Due to the qualitative nature of process models, analysis is often based on observational methods. This section describes several rules for observational analysis that enable the analyst to identify


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deficient areas in the model. The rules are applied to the IDEF3 models and, in most cases, exploit the features of this modeling methodology.

2.1. Reducing Duration of Processes The extended IDEF3 model (with resources) may be used to identify activities that can be eliminated or shortened. The hierarchical nature of IDEF allows the analyst to study the underlying process on many levels of detail (Menzel et al. 1994). As the analyst explores the decomposition, controls and mechanisms are frequently recognized as constraints for each activity. Therefore, methods of modifying controls and mechanisms must be explored. For example, human resources may be replaced by automated equipment. Similarly, the analyst should attempt to eliminate or simplify controls. Typically, a control is a piece of information that is required to perform the activity. Reducing the amount of information needed to perform the activity will often reduce the duration and simplify the process, e.g., simplifying procedures, eliminating activities.

2.2. Eliminating Redundant Activities Figure 1 illustrates the principle of eliminating redundant activities in the context of an IDEF3 model with inputs I1, I2; outputs O1, O2, mechanisms M1, M2; and controls C1, C2. Redundant activities often exist, e.g., inspection and partially automated processes. The analyst should also focus on areas of the model that are separated by physical and/or logical boundaries in the actual system. At these points, material and/or information are passed between departments, personnel, teams, and so on. The experience indicates that the likelihood of redundant activities is greater at these points.

I1

C1

M1Activity 2

O1

C2

M2

O2

Activity 1I1

C1

M1

O2

Activity 1

(a)

(b)

Figure 1. Eliminating a redundant activity: (a) “as-is” process, (b) “to-be” process

2.3. Partitioning Activities

In many cases, a single activity may be partitioned into two concurrent activities, each with a shorter duration. If this is possible, the overall duration of the process may be reduced. For example, if a random sample is to be obtained from a batch of parts for two types of testing, the testing activity may


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split into two concurrent activities, one for each test. Figure 2 illustrates the concept of partitioning an activity in an IDEF3 model.

Activity 1 Activity 2

Activity 1

Activity 2'

Activity 2"

XX

(a)

(b)

Figure 2. Result of partitioning an activity: (a) “as-is” process, (b) “to-be” process

2.4. Merging Serial Activities

Merging two or more serial activities into one activity may shorten the overall duration of the process. This approach works particularly well in systems with high volume of material and/or information handling. By combining activities, work performed in different areas may be performed in one, thus, eliminating the need for material and/or information handling. Figure 3 illustrates the principle of combining serial activities in an extended IDEF3 model. Note the difference between Figure 1 and Figure 3; in Figure 1, activity 2 and its associated inputs (I), outputs (O), controls (C), and mechanisms (M) – called ICOMs – are eliminated from the process. However, in Figure 3, activity 1' and its associated ICOMs replace activities 1 and 2.

Activity 1I1

C1

M1Activity 2

O1C2

M2

O2

Activity 1'I1

C1'

M1'

O2

(a)

(b)

Figure 3. Result of merging two activities:


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(a) “as-is” process, (b) “to-be” process

2.5. Eliminating Cycles In Section III of this Chapter, a formal method of identifying all the cycles in an IDEF model is presented. However, many of the shorter cycles in a model may become apparent during the observational analysis. In order to shorten the overall duration of the process, it is necessary to eliminate unnecessary cycles. Figure 4 illustrates the result of eliminating a cycle in an IDEF3 model. Note that activity 3 has been removed by eliminating the cycle.

Activity 1 Activity 2 X

Activity 3

X

Activity 1 Activity 2

(a)

(b)

Figure 4. Result of eliminating a cycle: (a) “as-is” process, (b) “to-be” process

3. COMPUTATIONAL ANALYSIS OF PROCESS MODELS The most frequently recognized shortcoming of process modeling may be the lack of use and/or incomplete analysis of models. In this section, algorithms for analysis of graphs extracted from IDEF models are discussed. Without a loss of generality, an IDEF3 diagram can be represented as a graph and a corresponding incidence matrix. Elements of the matrix may then be manipulated using the concepts of graph theory to identify the underlying structure of the IDEF model.

3.1. Graph Representation of Dependencies in Process Models The dependencies among activities in an IDEF3 process model are represented as a directed

graph (named process graph), where a vertex denotes an activity and an arrow denotes a dependency. For example, the arrow between and b “a → b” means that activity b is dependent on activity a. The directed graph is extracted from an IDEF model.

The process graph can also be represented as an activity-activity incidence matrix (Steward 1981). A non-empty element in the incidence matrix represents a dependency between the corresponding activities (rows and columns).

For example, consider six hypothetical activities represented with a digraph and the corresponding activity-activity incidence matrix (see Figure 5). An entry aij = '*' in the incidence matrix means that activity (column) j precedes activity (row) i (or the output of j becomes an input to i) . An entry ‘+’ represents a diagonal element.


5

1 2 3 4 5 6 7 8

1 2

3 4

56 12345678

+ * * * + * + ** * + * * * + + + * +

78

Figure 5. Digraph of activities and the corresponding incidence matrix

No special structure is visible in the incidence matrix in Figure 5. The triangularization algorithm discussed later in this section transforms the matrix in Figure 5 into the matrix and the corresponding digraph shown in Figure 6.

* + * +

* * * + + *

* + * +

7 6 1 3 4 2 5 8

* 1 2

56

34

7

8*

*

+

+

6 1 3 4 2 5

7

8

Figure 6. The ordered incidence matrix and the corresponding digraph

The following can be easily observed in the matrix (or digraph) in Figure 6: (1) Precedence relationship among activities is clearly visible, (2) Coupled activities can be easily recognized. In graph theory the coupled activities are called

strongly connected components. Activities 6 and 7 are independent and they can be performed simultaneously. Activity 5 has to

be performed prior to activity 8, because activity 8 depends on activity 5. Activities 1, 2, 3, and 4 involve two cycles. For a better visual effect, the coupled activities 1, 2, 3 and 4 have been circled in the digraph in Figure 6.

3.2. Types of Dependencies in Process Models To analyze and manage a process effectively, it is important to understand the dependencies

among activities that are classified as follows: - Information dependency

If the data required by activity b is produced by activity a, then activities a and b are information-dependent. This dependency can be modified by finding alternative data sources.

- Technological or causal dependency


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This dependency is due to some technical relationship between activities. For example, before initiating the finite element analysis, a design draft has to be available. Only a major change, e.g., a technological breakthrough, in the current practice may modify this dependency.

- Common-sense or pragmatic dependency The common-sense dependency is due to some sensible way of ordering activities. For example, activities in a detailed design may begin after activities in the conceptual design have been completed. This type of dependency could be ignored at an increased risk of project success.

- Resource dependency This dependency is based on a conflict of resource requirements. An activity with this type of dependency can be expedited by allocating more resources.

- Preferential dependency This type of dependency is based on the preference of an organization.

- Functional/structure dependency If activities belong to the same function or the same structure in a product, then they are functionally/structurally dependent. The process duration cab be reduced by weakening or even ignoring some dependencies, such as

preferential dependency, common-sense dependency, and resource dependency.

3.3. Ordering Non-cyclic Graphs

Digraphs that do not include cycles can be ordered with a topological sorting algorithm. If a digraph contains a directed cycle, it is not possible to arrange its vertices in a topological order.

Most of the topological sorting algorithms have been defined for digraphs (see for example Horowitz and Sahni 1983). The algorithm presented next uses the adjacency matrix corresponding to a digraph.

The Topological Sorting Algorithm

Step 1. Set i = 1. Step 2. Draw a horizontal line through row k of the incidence matrix with only one non-empty element

(corresponding to a vertex with no predecessors). Step 3. Draw a vertical line through column k (same column number k as the row number in Step 2) of

the incidence matrix. Step 4. Label i the crossed out row k and column k of the matrix. Step 5. If each row and column of incidence matrix has been labeled, stop; otherwise set i = i + 1 and

go to Step 2.

The topological sorting algorithm is illustrated next. Example 1 Consider the incidence matrix and corresponding digraph presented in Figure 7.


7

1

2

3

4

1 2 3 4+ * * + * * + * +

1234

Figure 7. Incidence matrix and the corresponding digraph with four vertices

The four iterations of the topological sorting algorithm are shown in Figure 8 through Figure 11.

The reordered matrix is presented in Figure 12.

1 2 3 4+ * * + * * + * +

1234

1

1

1

2

3

4

1

Figure 8. First iteration of the topological sorting algorithm

1 2 3 4+ * * + * * + * +

1234

1

1

1

2

3

41

2

2

2

Figure 9. First iteration of the topological sorting algorithm

1 2 3 4+ * * + * * + * +

1234

1

1

1

2

3

41

2

2

2

3

3

3

Figure 10. Third iteration of the topological sorting algorithm


8

1 2 3 4+ * * + * * + * +

1234

1

1

1

2

3

41

2

2

2

3

3

3

4

4

4

Figure 11. Fourth iteration of the topological sorting algorithm

The final reordered matrix is shown in Figure 12. 4 2 3 1+ * + * * + * * +

4231

Figure 12. The reordered matrix corresponding to the matrix in Figure 7

3.4. Ordering Graphs with Cycles The triangularization algorithm (Kusiak et al. 1993) is more general than the topological sorting algorithm because it can be applied to the graphs with cycles. A non-empty element in the upper triangular matrix verifies the existence of a cycle in the process. The algorithm is effective in analyzing the structure of process models.

To present the algorithm, a terminology is introduced. An activity is called an origin activity (OA) if there are no other activities preceding it. The OA activities can be easily identified in the incidence matrix. If the i-th row of the incidence matrix has only one non-empty entry (a diagonal entry), then i is an origin activity (OA). At least one cycle exists in a digraph with no OAs (Kusiak et al. 1993).

The Triangularization Algorithm

The triangularization algorithm identifies cycles in an incidence matrix and triangularizes the incidence matrix concurrently. The algorithm is presented in a form that is easy to implement. Step 0. Begin [with the initial sequence of the activities (1, 2, 3, n)] Step 1. End the algorithm if all the vertices are underlined. Identify an origin activity (OA) or a destination activity (DA). Go to Step 5 if neither an OA nor a DA is found. Step 2. Apply the SORTING RULE to the activity identified in Step 1. Step 3. Underline the activity identified in Step 1. Step 4. Delete the row and column associated with the underlined activity (see Step 1) from the

incidence matrix, and go to Step 1. Step 5. Find a cycle. Step 6. Merge all the activities in the cycle into one activity. Step 7. Merge the corresponding rows and columns in the incidence matrix and go to Step Step 1. Step 8. Assign the activities and cycles in the final solution to levels according the precedence relationships.


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SORTING RULE

If the activity is an origin activity (OA), move it to the furthest left position in the sequence of the activities that are not underlined. If the activity is a destination activity (DA), move it to the furthest right position in the sequence of the activities that are not underlined.

It is easy to show that by applying the SORTING RULE, the entries right to or above the block

are all empty entries. The incidence matrix corresponding to the underlined sequence of the vertices is a lower triangular matrix. If one wants to construct an upper triangular matrix, only the sorting rule has to be modified.

In the triangularization algorithm, the key to finding a cycle is to determine the successive activities. Suppose one attempts to find all successive activities of activity 1, the non-empty entries in the corresponding column 1 indicate these activities. The information from the previous cycle can be utilized in Step 5 of the algorithm. For example, if during the execution of the algorithm, a path: (9, 12, 10, 12 has been found, the cycle (12, 10, 12) is considered as one activity, say G. Finding the next cycle could start from path (9, G). In other word, it is not necessary to consider the non-empty entries (edges) that have been previously examined.

Example 2 Consider a process with 12 activities represented by the digraph and its corresponding incidence

matrix shown in Figure 13. Note that rows in the matrix in Figure 13 are the inputs to the corresponding activities and columns denote the outputs. For example, input to activity 6 (row 6) is the output of activities 2 and 12 (columns 2 and 12).

3

4

6

9

7

8

10

11

12

2

1

5

+ + * + * + * * * + * * * * + * * + * * * + * * * * + * * * * + * * * * + * * * * +

123456789

101112

1 2 3 4 5 6 7 8 9 10 11 12

Figure 13. Process graph and corresponding incidence matrix

The current (initial) sequence is (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). Since row 1 includes only one non-empty entry, the activity 1 is an OA. According to the SORTING RULE, activity 1 is underlined and it is moved to the most left position in the sequence. The sequence becomes (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12). Row 1 and column 1 are deleted from the matrix in Figure 13. In the reduced incidence matrix, activity 2 is an OA. Deleting row 2 and column 2 from the matrix in Figure 13, activity 3 becomes an OA, and later activities 11 and 7. The sequence becomes now (1, 2, 3, 11, 7, 4, 5, 6, 8, 9, 10, 12). Column 7 has only one non-empty entry in the reduced incidence matrix. The reduced incidence matrix corresponding to the activities that are not underlined is shown next.


10

4 5 6 8 9 10 12

4 5 6 8 9

10 12

+ * * * + * * + * * + * * + * * + * * * +

Since there is no single activity, which is either an OA or DA in this matrix, one looks for a cycle. Begin with any activity, e.g., activity 9 that is followed by 8 and 12. An arbitrary choice of 12 leads to the path (9, 12). From the column with 12, it is seen that 4, 6, and 10 follows activity 12. Selecting activity 10, the path becomes (9, 12, and 10). The activity 10 itself is followed by 9 and 12. Since 9 is already in the path, the cycle (9, 12, 10, 9) is generated. Also, activity 2 is in the path, thus leading to another cycle (12, 10, 12).

Next, either 9, 12, 10 or 12, 10 can be combined in a group. Combining 9, 12, 10 into one group, say G1, and treating G1 as an activity, the incidence matrix becomes:

4 5 6 8 G1+ * * * + * * + * * + * * +

4 5 6 8

G1 where G1 = (9, 10, 12).

Again, no OA or DA has been identified. Search for another cycle begins. Starting with G1 which is followed by 4, 6, and 8, a path (G1, 8) is formed. Since 5 only follow activity 8, the path becomes (G1, 8, 5). 8 follow activity 5 follow 4 and 5. The path becomes (G1, 8, 5, 4, 8) in which a cycle (8, 5, 4, 8) is included. Defining G2 = (8, 5, 4), the new matrix is:

+ * * + * * +

G2 6 G1G2

6 G1

where G1 = (9, 10, 12) and G2 = (4, 5, 8).

The column with G2 has only one non-empty entry. The newly defined activity G2 is a DA in the reduced incidence matrix above. Based on the SORTING RULE, G2 is moved to the most right position in the sequence and underlined. The corresponding sequence is (1, 2, 3, 11, 7, 6, G1, G2). The incidence matrix is:

+ * * +

6 G16

G1 where G1 = (9, 10, 12).


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Neither 6 nor G1 is an OA or a DA. There is only one cycle left which is (6, G1, 6). Redefining G1 = (6, G1) = (6, 9, 10, 12) results in the final sequence (1, 2, 3, 11, 7, G1, G2). The result of the triangulization is the incidence matrix with six blocks labeled L-1 through L-6 in the transformed incidence matrix shown in Figure 14.

+

+ +

*

*

+ *

+

*

*

+

*

*

*

*

+

**

*

*

+

*

+

*

*

*

+

**

*

*

*

+

*

*

+

123

11

69

1012458

7

1 2 3 11 6 9 10 12 4 5 8 7

L-1

L-4

L-5

L-6

L-3L- 2

**

* **

Figure 14. Transformed matrix

The group of activities represented at six different levels at the matrix above is depicted in Figure 15.

1

2 3 11 107

6 9

8

4 5

L-1 L-2 L-3 L-4Group

G1

L-5Group

G2

L-6

12

Figure 15. Ordered activities

The levels of activities are identified using the following steps: 1. Draw a square around each singular activity and a cycle. 2. Identify activities at the same level (no relationship between them). 3. Label activities at the same level. It can be seen in Figure 14 and the corresponding matrix that before commencing activities at the next level, e.g., L-6, the activities at the previous level (L-5) have to be completed.

4. ANALYSIS OF DESIGN PROBLEMS Complex designs may involve a large number of design constraints and variables. In most

situations, a design constraint exhibits some degree of coupling, which tends to complicate constraint management. To simplify the constraint management problem and reduce the computational time of constraint evaluation, the analysis of constraints is performed. By analyzing a complex design, a constraint-variable incidence matrix is frequently constructed. The goal is to detect potential groups of constraints that can be evaluated concurrently and to identify the minimum number of overlapping variables or constraints.

Example 3 Consider design of the cantilever beam in Figure 16.


12

F

L

² H

B

σ

Figure 16. Cantilever beam

The equations pertaining to design of the cantilever beam are as follows (Shigley and Mischke

1986):

σ = MY

I (1)

M = FL (2)

I = BH3

12 (3)

Y = H2 (4)

∆= FL3

3EI (5)

where: σ : the maximum bending stress occurring at the section closest to the support M : bending moment L : length of the beam B : width of the beam H: height of the beam Y: dimension defined in expression (4) F: applied force ∆: beam deflection E : modulus of elasticity I : second moment of area about the neutral axis

In this example, the values of F, E, H, B, and σ are known. The unknown variables are M, I, Y,

L, and ∆. The design equations are transformed so that the known variables appear on the right-hand side. The transformed expressions (1) - (5) are as follows:

M = M(σ, Y, I) Y = Y(H) I = I(B, H) L = L(M, F) ∆= ∆(F, L, E, I) The expressions (1) - (5) are represented with an incidence matrix in which the non-empty elements of the incidence matrix represent the relationship between variables (Figure 16). For example, in order to evaluate the bending moment M, the value of σ, Y, and I have to be provided. If the design variables


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could be sequenced so that each would be able to receive all the information required, then there would be no cycles in the computation process.

σ Μ Ι

F

Mσ

E∆H

I

BL

Y

+

++

+

*+

*

*

***

+ **+

*

*

*

+

+

*

+

Y F L B H ∆ E

Figure 17. The incidence matrix corresponding to the set of transformed design equations

Applying the triangularization algorithm to the incidence matrix in Figure 17 results in the matrix shown in Figure 18.

σ

HB

∆LE

Y

MF

I

B H I Y F M E L ∆

*

+

+

*

*

+

++

++

+

+

+

*

*

*

* *

* ** *

σ

Figure 18. The ordered matrix from Figure 17

The ordered matrix in Figure 18 provides a sequence of computing expressions (1) through (5). The first four rows of the matrix in Figure 18 are interpreted as follows: given the value of B and H, the value of Y and I can be computed. The next two rows σ and F are dummy and can be ignored. To compute M, the values of Y, M, and σ are required. The value of L and ? can be computed next using the parameters indicated by the asterisks in Figure 18.

In Example 3 a variable (or parameter) – variable (parameter) matrix was built. Consider another example of designing a bandpass filter, where a constraint – variable (parameter) matrix is constructed.


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Example 4 Figure 18 shows a series m-derived bandpass filter used in many electronic devices (Turner

1975).

IN OUT

L1 C 1

C C2 3

2 3L L

Figure 19. Series m-derived bandpass filter The constraints (equations) pertaining to design of the series m-derived bandpass filter e1and e5 –

e14 are borrowed from Turner (1975). The dummy constraints e2 – e4 and e15 have incorporated to make the number of equations equal to the number of variables.

e1: f1 = f2f3 / f4 e2: f2 = f2 e3: f3 = f3 e4: f4 = f4 e5: m = x / ( ( 1-f2f3 ) / f42 ) e6: x = ( ( 1- (f22/f32) (1-f32/f42) ) 0.5

e7: y = ((( 1 - m2 ) f2f3 ) / 4f12x ) * 1 - ( f12 / f42 ) e8: z = ( ( 1 - m2 ) / 4x ) * ( 1 - ( f12 / f42 ))

e9: L1 = mR

p(f3 - f2)

e10: L2 = zR

p(f3 - f2)

e11: L3 = yR

p(f3 - f2)

e12: C1 = f3 - f2

4pf2f3mR

e13: C2 = f3 - f2

4pf2f3yR

e14: C3 = f3 - f2

4pf2f3zR

e15: R = R


15

Introducing these dummy constraints does not affect the nature of the design problem as each dummy constraint is satisfied by itself. The incidence relationship between constraints and variables is shown in Figure 20. The value of parameter p is known and it is not included in the matrix.

+ * * * + + + * * * + * * * * +* * * * * +* * * * + * * * + * * * * + * * * * + * * * * + * * * * + * * * * + * +

e1e2e3e4e5e6e7e8e9

e10e11e12e13e14e15

f1 f2 f3 f4 m x y z L1 L2 L3 C1 C2 C3 R

Figure 20. The constraint-variable incidence matrix Applying the triangularization algorithm to the constraint-variable incidence matrix in Figure 20,

the structured incidence matrix in Figure 21 is obtained.

+ + + + * * * + * * * + * * * * + * * * * * + * * * * + * * * * + * * * * + * * * * + * * * * + * * * * + * * * * +

e15 e2 e3 e4 e1 e6 e5 e7 e8 e9

e10 e11 e12 e13 e14

R f2 f3 f4 f1 x m y z L1 L2 L3 C1 C2 C3

L-1

L-2

L-3

L-4

Figure 21. The triangularized constraint - variable incidence matrix corresponding to the matrix in Figure 20

Four levels of constraints are identified: L(1) = {e15, e2, e3, e4}, L(2) = {e1, e6}, L(3) = {e5, e7,

e8}, and L(4) = {e9, e10, e11, e12, e13, e14}. It is observed that the expressions at the next level are not satisfied until the expressions at the preceding level have been satisfied. Note that that expressions at the same level are concurrently met, if all elements are diagonal elements, e.g., expressions e1 and e6 of level


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2 can be satisfied concurrently provided that all expressions at level 1 are satisfied. Expressions that can not be satisfied concurrently are coupled, e.g., the expressions e5, e7, and e8.

In Examples 4 and 5, the topological sorting algorithm rather than the triangularization algorithm could be used as no cycles exist.

5. ANALYSIS OF DESIGN PROCESSES In the previous section, the trangularization algorithm was illustrated with design problems. Next, the same algorithm will be applied to analyze process models.

Example 5 This example is taken from an industrial company. Consider the design process of an electronic

product involving hundreds of activities. The entire design process was divided into seven phases (see Figure 22). Later, it will become apparent that such an arbitrary partitioning of a process into phases is an obstacle in the minimization of the process cycle time. In some companies, however, the phases might be imposed by contractual and other requirements.

Systemrequirementsanalysis

System definition/requirementsallocation

Units definition/requirementsallocation

Preliminarydesign

Detaileddesign

Fabrication, coding,integration, evaluation

Qualitytest

1 2 3

5 6 7

4

Figure 22. The seven phase product development process

At the next decomposition level, the following activities were identified in Phase 3 of the design process: 1. Update requirements 2. Develop engineering test equipment concepts/plans 3. Develop program PPSL 4. Order/receive engineering 5. Supportability analysis 6. Update LSAR 7. Update MTBF/MTTR 8. Design/Select components (HW) 9. Provide parts technology guidance 10. Support component selection/testing 11. Produce S/W specifications (SYS/SW) 12. SRU level block diagram (HW) 13. Cost update against cost goals 14. Breadboard test/equipment design 15. Module specifications (HW) 16. Create interface specifications 17. Develop package specifications (SW) 18. Order breadboard test/equipment parts 19. Procure samples (D, E)


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20. Walkthrough/review package specifications (SYS, SW) 21. Review SRS 22. Order/review breadboard parts 23. SWE review IRS 24. Release SRS 25. Approve SRS (Procedure 2) 26. Approve SRS (Procedure 1) 27. Review SRS

28. Receive SRS approval

The activities included in the product definition/requirements allocation phase are used to illustrate the concepts presented in this section.

The design activities involved in Phase 3 are represented as the process graph shown in Figure 23. The boxes of IDEF3 model are replaced with circles and the relational links are replaced with arrows. The matrix representation of the design process is shown in Figure 23. By applying the triangularization algorithm to the matrix in Figure 24, the matrix in Figure 25 is obtained.

Six levels (groups) of activities are identified: L-1 = {1, 2, 3, C1}, L-2 = {5, C2, 13, 24, 27}, L-3 = {6, 12, 19, 26}, L-4 = {14, 15, 25}, L-5 = {C3, 18, 28}, and L-6 = {20}, where C1 = {11, 21}, C2 = {7, 4, 8, 9, 10, 22}, and C3 = {16, 17, 23. It is observed that the dependencies among activities at different levels are indicated by the elements below the block diagonal matrices. For example, activities 1 and 3 of level L-1 precede activities 5 and 8 of level L-2. Similarly, activities 11 and 21 of level L-1 precede activities 13, 24, and 27 of level L-2. One should complete the activities 1, 3, 11, and 21 as soon as possible in order to begin the downstream activities at level L-2. Another motivation for identifying the dependencies among activities at different levels, is that one may attempt to remove or redefine the dependency among groups of activities in order to increase the degree of concurrency. For example, if the dependency between activities 1 and 5 in Figure 24 would be removed, activity 5 could be performed at the first level. The concurrency among activities of the process has increased. Note that the temporal relationship between activities indicates that the activities belonging to different levels may overlap.

1

2

34

5

67

8

9

10

11

12

13

14

1516

17

1819

20

21

2223

24

25

2627

28


18

Figure 23. The process graph of Phase 3

The six levels of activities are represented as the process tree in Figure 26. A link between any two levels (groups) represents dependency between the corresponding levels. It is important to recognize the activities at the upper level that impact the lower-level activities. For example, there are some dependencies between level L-1 and level L-2 activities. In order to begin the activities in level L-2, activities 1, 3, 2, 12, 11 and 21 of level L-1 have to be completed as they impact the level L-2 activities. A manager should monitor these activities in order to avoid delay in information transfer to the next level activities.

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 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 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 * +

Figure 24. The activity-activity incidence matrix corresponding to the process graph in Figure

23

A dependency path is defined as a chain with the maximum number of activities in the process tree, i.e., the path (3-C2-12-15-C3-20) in Figure 26. A project manager should focus on monitoring activities on the dependency path. It should be noted that the result of concurrency analysis provides only guidelines for management of the design process.


19

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

L-1

L-2

L-3

L-4

L-5

L-6

C1

C2

C3

Figure 25. Thetriangularized activity-activity incidence matrix corresponding to the matrix

in Figure 24

The benefits of the triangularization analysis are as follows: • Potential groups of activities that can be performed in parallel are determined. • A stream of dependency paths is determined. One of the dependency paths from the stream is

likely to become a critical path when the process is being executed. • The duration of product development time may be reduced, due to the increased degree of

concurrency among activities.


20

L-1 L-2 L-3 L-4 L-5 L-6

2 3 1 C1

24 13 27

C2

5

12 6

14 15

18 C3 20

26 25 28

19

Dependency path

Figure 26. Ordered activities of Phase 3

The value of information contained in the ordered process in Figure 26 is tremendous. Process experts may focus on analyzing a limited number of activities in the dependency paths. By using alternative ways of providing design information, the serial dependency among activities may weaken. With the information in Figure 26 serving as another view of the process, it is easier to identify errors (e.g., missing or redundant precedences) or other discrepancies. Once the process in Figure 26 has been updated, it might be recomputed with the triangularization algorithm. No that so far we have dealt with the process structure only. Later on time will be introduced to perform temporal analysis.

Example 6 The example of IDEF3 model from an industrial company, which was deeply involved in process modeling, is shown in Figure 27. The model represents a small component of the design process at the company, and it contains 27 activities represented on a single level. Following the IDEF3 notation, inputs, controls, and mechanisms enter the left, top, and bottom of the activity box, respectively. In the preceding example, activity (output) - activity (input) analysis was performed. In this example, the input-ouput and control-output perspectives are considered. Other perspectives are certainly possible.


21

1

2

3

4

5

24

6

25

26 27

7 8

11

9 10

12 13

14

15

16

17

18 19 20

21 22

23

Figure 27. IDEF3 process model

123456789101112131415161718192021222324252627

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 77+

++

+

*

* ++

++

1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

+*

+

*

+* +

*

+*

*+

*

+

*

+

*+

+

*

*

++

*

*+

++* +

+* +

+

Figure 28. Activity (input)-activity (output) incidence matrix


22

The model in Figure 27 is used to construct input-output and control-output incidence matrices. The input-output incidence matrix is shown in Figure 28. Each non-empty element in the matrix, excluding the diagonal, corresponds to input in the IDEF3 model. Inputs that come from external sources (i.e., are not an output of an activity in the model) are not included in the matrix. Note that the matrix almost has an ideal lower triangular form prior to applying the triangularization algorithm. An control-output incidence matrix is constructed. In some cases, constructing an output-mechanism matrix might be useful. However, only a few models observed in industry contain this type of relationships. For most applications, systems utilize resources (mechanisms) that are generated by external processes or activities. For example, typical mechanisms in this process are personal computers, software, people, and so on. All of those mechanisms are generated by activities outside the scope of the model. The triangularization algorithm is performed on the input-output and control-output matrices to identify the model structure from the object flow and information flow perspectives. Figure 29 shows the triangularized matrix for the object flow perspective and Figure 30 shows the corresponding process graph. In the object flow perspective of the model, several activities do not effect the process, (i.e., 1, 2, 3, 5, 18, 23, 25, 26, 27). The object flow perspective is based on input-output relationships. By definition, inputs undergo a state transition in an activity to become outputs, i.e., value is added. Therefore, activities not influencing object flow may be considered non-value-added activities in the object flow perspective. It must be realized that these activities may still impact the process,

123465791011121718192325262781420241516132122

1 2 3 4 6 5 7 9 0 1 2 7 8 9 3 5 6 7 8 4 0 4 5 6 3 1 2+

++

+*

*

+

*+

+* +

1 1 1 1 1 1 2 2 2 2 1 2 2 1 1 1 2 2

+

*

++

*

+* +

*

+

*

+

*

+

*

++

*

++

++

+

*

+

*

++

+

*

*

*

C-2

C-1

Figure 29. Triangularized matrix for object flow perspective


23

not in this perspective. The object flow perspective also displays a greater degree of concurrency. This suggests that performing concurrent activities simultaneously may reduce lead times. Once again, the degree of concurrency only reflects the constraints in the object flow perspective. The triangularization algorithm only identified two cycles in the process, i.e., (4, 6, 4) and (9, 10, 9). Each cycle only involves two activities, suggesting the possibility of an interaction rather than a cycle. In both cycles, the analyst should explore the possibility of combining the activities into a single activity.

001 002 003

004

005

024

006

025 026 027

007 008

011

009

010

012

013

014

015

016

017

018

019 020

021 022

023

Figure 30. Object flow perspective of a process model

Constraints in the information flow perspective reflect the use of controls or information in the process. A control is a piece of information that facilitates execution of the activity (i.e., a written procedure, oral commands, worker experience, and so on). Although an existing control is not transformed in an activity, a new control may be produced using inputs and mechanisms. Therefore, the information flow perspective illustrates the evolution of information in the system. The triangularized matrix for the information flow perspective is shown in Figure 31. The corresponding process graph is shown in Figure 32. The information flow perspective for the model reveals less concurrency of activities. The triangularization algorithm determines three independent groups of activities. Each group of activities may be performed concurrently, however, precedence constraints limit concurrency within the group. One group contains 13 activities and one cycle. Due to control-output constraints, the dependency path may only be minimized to 10 concurrent levels. Therefore, process lead-time is strongly effected by information flow along this path.


24

114162022242151718252631927456789101112132123

1 4 6 0 2 4 2 5 7 8 5 6 3 9 7 4 5 6 7 8 9 0 1 2 3 1 3+

*

+

*

+

**

+

*

++* +

* +

1 1 2 2 2 1 1 1 2 2 1 2 1 1 1 1 2 2

++

*

**

+

*

++

+

*

++

*

+

*

++

+**

+

C-1

+**

++***

++

+**

*

Figure 31. Triangularized control-output matrix

By representing an IDEF3 model as an activity (input)-activity (output), an incidence matrix allows the analyst to apply the triangularization algorithm. The algorithm identifies concurrent activities and cycles within the structure of the model. Viewing three perspectives of the model allows the user to recognize the impact of object flow, information flow, and resources on the process.


25

001

002

003

004

005

024

006

025

026

027

007

008

011

009

010

012

013

014

015

016

017

018

019

020

021

022

023 Figure 32. Triangularized control-output graph

VI. ANALYSIS OF CRITICAL PATTERNS The analysis introduced in this section identifies certain patterns of activities without considering

the time aspect. For each pattern, a critical activity is determined based on the behavior of the pattern. The patterns classified are ranked, e.g., two activities in parallel are preferred over two serial activities. The process can be improved by upgrading worse patterns to preferred ones and better management of critical activities in each pattern.

A process model may include one or more patterns that reflect the dependency structure among activities (Kusiak et al. 1995). The following patterns are defined: (a) interaction (I-pattern), (b) cycle (C-pattern), (c) serial (S-pattern), (d) branch (B-pattern), and (e) merge (M-pattern) (see Figure 33).

Only two vertices (activities), vi and vj, are involved in the I-pattern. Vertex vi is reachable from vj and vice verse. This pattern occurs when two concurrent activities must collaborate with each other in order to produce the outcome required. The information flows back and forth between the two activities.

Depending on the information dependency, two types of collaboration: horizontal collaboration and vertical collaboration are defined (Bond 1992). In the horizontal collaboration, the two collaborating activities increase the confidence level of their individual solutions and maintain the consistency of the overall solution. In the vertical collaboration, the relationship between the two activities is viewed as a server-client relationship. The server activity requests information from a client activity to make decisions and the client activity provides the information. The server activity modifies a decision according to the information provided by the client activity. This process continues until a


26

satisfactory outcome is produced. If the entry aij = aji = ‘*’, then activities i and j form an I-pattern. I-patterns are easy identifying in the triangularized matrix.

The critical activity in an I-pattern, is the server activity, which makes decisions and controls the interaction process. The client activities that provide information important to the server activity are critical, because they affect the design decision of the server activity.

The C-pattern occurs, when the level of uncertainty associated with the corresponding activities is high. The upper stream activities lead to a certain design decision, while the downstream activities provide some feedback information. This iterative process continues until the outcome produced is satisfactory. The triangularization algorithm identifies C-patterns.

+ + + + +

**i

j

i j

viv j

+ + + + +

ij

i j

**

k

kv i v j vk

Matrix representationGraph representation

+ + + + +

ij

i j

* *

k

k•••

•••v i

v j

vk

v i v j vk

+ + + + +

ij

i j

*

**

k

k

+ + + + +

ij

i j

**

k

k

••••••

vi vj

vk

(a) Interaction

(b) Cycle

(c) Serial

(d) Branch

(e) Merge

••••••vi

vj

vk

vp + + + + +

ij

i j

**

k

k*

pp

(f) Branch-and -merge

Figure 33. Patterns in the design process

In the case of two activities, it is not possible to distinguish I-pattern from C-pattern based on the

matrix representation. However, they behave differently. The behavior of the two patterns is not the same. The I-pattern mimics a collaborative behavior of two activities, while the C-pattern corresponds to a serial process with iterations. The length of the C-pattern measured, e.g., by the duration of serial


27

activities, should be reduced in order to reduce the process cycle. The larger the length of C-pattern, the longer the process cycle. One may improve the process structure by replacing a cycle that involves many activities with several short cycles (see Table 1). In some cases a C-pattern can be upgraded to an I-pattern.

Activities in the S-pattern are serial, i.e., an activity can not start until its predecessors have been completed. A critical activity is an activity with the maximum probability to block the downstream activities.

The B-pattern exists when the downstream activities await information produced at an upstream activity. The critical activity in B-pattern is the diverge activity (vertex vi in Figure 33(d)). After completion of the diverge activity, the activities in the first level are performed in parallel.

The M-pattern occurs when an activity has more than one predecessor directly connected to it. The activity, called converge activity (vertex vk in Figure 33(e)), can begin only after all preceding activities have been completed. The activities that produce the output (e.g., information or components) important to the converge activity are critical. More than one critical activity may exist in this pattern. After receiving the information required, a converge activity may begin without all predecessors completed (e.g., ignoring the common sense or resource dependencies). In an M-pattern, the activity ahead of the converge activity and with the maximum probability (to take the longest time) becomes a critical activity.

If all successors of an activity have only one predecessor in the B-pattern (i.e., the diverge activity, vertex vi in Figure 33), then the pattern identified is called a “perfect” B-pattern. In practice, however, some successors may have more than one predecessor. The type of B-pattern with the successors having more than one predecessor is called a BM-pattern, because it combines the properties of the B-pattern and M-pattern.

In general, an I-pattern tends to include smaller number of activities than a C-pattern. However, a poor communication among activities in the I-pattern may deteriorate its performance (Kim et al. 1992). Also, a B-pattern, M-pattern, and BM-pattern are preferable over an S-pattern, because some of the activities in the B-pattern, M-pattern, and BM-pattern are performed in parallel. The process model may be improved by replacing one pattern with a preferable one. Numerous strategies aiming at the improvement of process models are summarized in Table 1.

Table 1. Strategies to reduce the process cycle Pattern Critical activity Example actions Interaction

• The server activity • The client activity that represents important support work.

• Better control of client activities. • Better communication between a server and clients. • Set up a meeting before an activity begins.

Serial

• Activity prone to reduction of its duration • Activity with the maximum impact on the design (e.g., maximum duration)

• Perform some activities in parallel (or I-pattern) • Pass earlier information to the downstream activities (overlap some activities)


28

Cycle

• Activity prone to reduction of its duration • Activity with the maximum impact on the design (e.g., maximum duration)

• Pass earlier information to the downstream activities (overlap some activities) • Eliminate the cycle (set up a meeting before an activity begins • Transform a long cycle into several shorter cycles

Branch • Diverge activity • Complete the diverge activity as early as possible

Merge

• Activities with the information produced important to the converge activity • Activity with the maximum duration

• Focus on the critical activity to reduce its duration

Branch- and-merge

• Diverge activity • Activities with the information produced important to the converge activity • Activity with the maximum duration

• Complete the diverge activity as early as possible • Focus on the critical activity to reduce its duration

VII. SUMMARY In this chapter, the observational and computational analyses of IDEF models were addressed. Several rules for observational model analysis were presented and related to the IDEF0 and IDEF3 models. Observational analysis is often followed a computational analysis of IDEF models. In some cases there might be no need for computational analysis, e.g., when the process model has a serial structure. IDEF models without cycles can be analyzed with the topological sorting algorithm. The triangularization algorithm applies to analysis of models with cycles. The two algorithms require a partial information from an IDEF model equivalent to a process graph. This information was represented as an incidence matrix. Ways of improving process models by identifying critical patterns and replacing them with more suitable ones were discussed. None of the analysis discussed in this chapter required duration of activities. The ultimate goal of the analyses introduced was to improve the structure of process models. Numerous illustrative examples were presented. REFERENCES 1. Blum, B. I. (1992). Software Engineering A Holistic View, Oxford University Press, New York. 2. Bowers, D. S. (1988). From Data to Database, Chapman and Hall, London. 3. Busby, J. S. and G. M. Williams (1993). "The value and limitation of using process models to

describe the manufacturing organization," International Journal of Production Research, 31, 2179-2194.

4. Colquhoun, G. J., R. W. Baines and R. Crossley (1993). "A state of the art review of IDEF0," International Journal of Computer Integrated Manufacturing, 6, 252-264.

5. Demarco, T. (1979). Structured Analysis and System Specification, Prentice-Hall, Englewood Cliffs, N.J.


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6. Gane, T. and C. Sarson (1982). Structured Systems Analysis, McDonnell Douglas, St. Louis, MO.

7. Hatley, D. J. and I. A. Pirbhai (1987). Strategies for Real-Time System Specification, Dorset House, London, UK.

8. Horowitz, E. and Sahni, S. (1983), Fundamentals of Data Structure, Computer Science Press, Rockville, Md., 312-313.

9. Kim, J.S., Ritzman, L.P., Benton, W.C., and D.L. Snyder (1992). “Linking product planning and process design decisions,” Decision Sciences, 23, 44-60.

10. Kusiak, A. and K. Park (1990). "Concurrent engineering: decomposition and scheduling of design activities," International Journal of Production Research, 28, 1883-1900.

11. Kusiak, A., Larson, T. N., and J. Wang (1994). "Reengineering of design and manufacturing processes," Computers and Industrial Engineering , 26, 521-536.

12. Kusiak, A. and J. Wang (1993a). "Qualitative analysis of the design process," Proceedings of the Winter Annual ASME Meeting, New Orleans, LA., ASME Press, New York, DE-66, 21-32.

13. Kusiak, A. and J. Wang (1993b). "Decomposition in concurrent design," in Concurrent Engineering: Automation, Tools, and Techniques, edited by A. Kusiak, John Wiley, New York, pp. 481-508. 14. Kusiak, A. and J. Wang (1993c). "Efficient organizing of design activities," International Journal of Production Research, 31, 753-769. 15. Kusiak, A., J. Wang, D.W. He, and C.X. Feng (1995) “A Structured Approach for Analysis of Design Processes,” IEEE Transactions on Components, Packaging, and Manufacturing Technology - Part A, 18, 664-673. 16. Kusiak, A., J. Zhu and J. Wang (1993). "Algorithms for simplification of the design process,"

Proceedings of the 1993 NSF Design and Manufacturing Systems Conference, Society of Manufacturing Engineers, Dearborn, MI, 1107-1111.

17. Mayer, R. J., T. P. Cullinane, P. S. deWitte, W. B. Knappenberger, B. Perakath and M. S. Wells (1992). Information Integration for Concurrent Engineering (IICE) IDEF3 Process Description Capture Method Report, Armstrong Laboratory, Wright-Patterson AFB, Ohio 45433, AL-TR-1992-0057.

18. Menzel, C. P., R. J. Mayer and D. E. Edwards (1994). "IDEF3 process descriptions and their semantics," in Intelligent Systems in Design and Manufacturing, Edited by C. Dagli and A. Kusiak, ASME Press, New York, 171-212.

19. Mitchell, F. H. (1991). CIM Systems, Prentice Hall, London. 20. Page-Jones, M. (1980). The Practical Guide to Structured Systems Design, Yourdon Press,

New York. 21. Pressman, R. S. (1992). Software Engineering A Practitioner's Approach, McGraw-Hill, New

York. 22. Ross, D. T. (1977). "Structured Analysis (SA): A language for communicating ideas," IEEE

Transactions on Software Engineering, SE-3, 16-24. 23. Ross, D. T. (1985). "Applications and extensions of SADT," Computer, April, 25-34. 24. Ross, D. T., J. W. Brackett, R. R. Bravoco, and K. E. Schoman (1980). Architects Manual

ICAM Definition Method- IDEF0, I-CAM, DR-80-ATPC. 25. Ross, D. T. and K. E. Schoman (1977). "Structured analysis for requirements definition," IEEE

Transactions on Software Engineering, SE-3, 6-15.


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26. Shunk, D., B. Sullivan and J. Cahill (1986). "Making the most of IDEF modeling - The triple-diagonal concept," CIM Review, 3, 12-17.

27. U. S. Air Force (1981). Integrated Computer Aided Manufacturing (ICAM) Architecture Part II, Volume IV-Functional Modeling Manual (IDEF0), Air Force Materials Laboratory, Wright-Patterson AFB, Ohio 45433, AFWAL-tr-81-4023.

28. Ward, P. T. and S. J. Mellor (1985). Structured Development for Real-Time Systems, Yourdon Press, New York.

QUESTIONS 1. What is the result of observational analysis of a process model? 2. When computational analysis of a process model is useful? 3. What are the main differences between the “reengineered” process model and “as-is”

process model? 4. What types of dependencies are encountered in process modes? 5. What contains more information: an IDEF3 process model or a corresponding input-output

graph? 6. What other relationships besides input-output dependencies can be represented with

graphs? 7. What is a cycle in a graph? 8. What is a strongly connected component in a graph? 9. What are the main steps of the triangularization algorithm? 10. Does the triangularization algorithm solve a design problem? 11. What does the object flow perspective represent? 12. What does the control-output perspective represent? PROBLEMS 1. Your role as a process analyst is to reduce cycle time of the process in Figure A1. The duration of each activity is as follows: • Release PCB1 – 2 minutes • Release PCB2 – 4 minutes • Test PCB1 – 25 minutes • Test PCB2 – 30 minutes • Analysis of Test 1 – 7 minutes • Analysis of Test 2 – 9 minutes Draw a Gantt chart of the process activities when: (a) Logical junctions a, b, and c are X (exclusive OR) junctions (b) Logical junctions a and b are X (exclusive OR) junctions, and junction c is a synchronous

AND junction. (c) Explain the impact of controls on the Gantt schedule.


31

Release PCB 2 for testing

TestPCB 1

Analysis ofTest 2 results

Analysis ofTest 1 results

aRelease PCB 1 for testing b c Test

PCB 2

Figure A1. IDEF3 model of a testing process 2. For the IDEF3 process model in Figure A2:

(1) Set-up an input-output incidence matrix (2) Apply the triangularization algorithm to the matrix created in (1) and arrange

activities and cycles into levels. Assume that the duration of each activity is 2 days. Knowing that a process model represents the space of all possible decision paths:

(3) Draw a Gantt chart of the shortest path through the model.

04

08

09

10

03

07

06

05

O X

X

X

01

02

Figure A2. IDEF3 model of an industrial process

3. Consider the design process model in Figure A3: (a) Determine cycles with the triangularization algorithm, (a) Arrange the design activities in a time sequence, (b) If you were allowed to delete one activity from the model, which one would you delete and

why?


32

1

6

2 3

7 8 9 10

4

5

11

Figure A3. The IDEF3 model of a manufacturing process

4. For the process model in Figure A4:

1 32

4 6

5

9

7

8

10

11X

Figure A4. IDEF3 process model

(a) Organize the input-output relationships of with the triangularization algorithm. (b) How many levels of activities shave you identified? (c) How many cycles have you identified? 4. Consider the torsion bar spring in Figure A5.

TD

L

Figure A5. Torsion bar spring

The equations pertaining to design of the spring are as follows:


33

f1: Twist angle? θ = 32TLpGD4

f2: Stiffness K = Tq

f3: Stress rate S = tq

f4: Volume V = pD2L

4

f5: Stress τ = 16TpD3

n

f6: Polar moment of inertia J = pD4

32

where: θ : Angular twist of the bar G : Shear module of the bar ? : Shear strength of the bar material L : Length of the bar K : Torsion stiffness of the bar V : Volume J : Polar moment of inertia n : Safety factor of the bar D : Diameter of the bar S : Stress rate in the bar T : Torque on the bar

Given the value of parameters (inputs) G, T, and n: (a) Determine the minimum number of missing parameters, if any, to design the spring

(determine the value of D and L). (b) Using the triangularization algorithm, find the best sequence of computing equations (f1) –

(f6) in order to design the spring. 6. Consider design of the helical compression spring in Figure A6.

x1

x2

F1

F2

l 1

lf

l2

Figure A6. Helical compression spring

The constraints pertaining to design of the helical compression spring are as follows:

f1 : F1x1

= F2x2


34

f2 : F1x1

= Gd4

8D3N

f3 : σ = 8DF2pd3

f4 : s = x2 - x1

f5 : V = p2d2DN

4

where:

• F1 = applied initial force, N • F2 = applied final force, N • x1 = initial deflection, m • x2 = final deflection, m • D = mean coil diameter • G = modulus of rigidity, N/m2 • σ = stress in compressed condition, N/m2 • V = volume of spring material, m2 • N = number of active coils (turns) of wire • d = diameter of spring wire, m • s = stroke, m

Determine the sequence of computing equations f1 – f5 that minimizes the number of iterations.

Documents

A. Kusiak, Engineering Design: Products, Processes, and ...user.engineering.uiowa.edu/~coneng/lectures/Chapter_2.pdfC1 M1 Activity 2 O1 C2 M2 O2 Activity 1 I1 C1 M1 O2 Activity 1 (a)