Outline. Graph Representation of Dependencies. Dependency (Design) Structure Matrix

Transcription

1 Dependency (Design) Structure Matrix Outline Andrew Kusiak Seamans Center Iowa City, Iowa - Tel: - ax: - andrew-kusiak@uiowa.edu DSM definition DSM and innovation Topological sorting Traingularization algorithm Summary Graph Representation of Dependencies Digraph of activities and the corresponding incidence matrix Output Input The ordered DSM INPUT OUTPUT

2 Case : Organizing Non-cyclic Graphs The Topological Sorting Algorithm Incidence matrix and the corresponding digraph with four vertices Step. Set i =. Step. Draw a horizontal line through unlabeled row k of the incidence matrix with only one non-empty element (corresponding to a vertex with no predecessors). Step. Draw a vertical line through column k (same column number k as the row number in Step ) of the incidence matrix. Step. abel i the cross out row k and column k of the matrix. Step. If each row and column of incidence matrix has been labeled, stop; otherwise set i = i and go to Step. Input Output irst iteration of the topological sorting algorithm Third iteration of the topological sorting algorithm Second iteration of the topological sorting algorithm ourth iteration of the topological sorting algorithm

3 The reorganized matrix The topological sorting algorithm organizes a matrix and the corresponding process model. It determines a feasible path of getting form activity to. What will the Topological Sorting Algorithm do if there is a cycle in the process graph? Use the Triangularization Algorithm!

4 Case : Organizing Graphs with Cycles The Triangularization Algorithm Step 0. egin [with the initial sequence of the activities (,,,..., m)] Step. End the algorithm if all the vertices are underlined. Identify an activity which is an origin activity (OA) or a destination activity (DA). Go to Step if neither an OA nor a DA is found. Step. Apply the SORTING RUE to the activity identified in Step. Step. Underline the activity identified in Step. Step. Delete the row and column associated with the underlined activity (see Step ) from the incidence matrix, and go to Step. Step. ind a cycle. Step. Merge all the activities in the cycle into one activity. Step. Update the corresponding rows and columns in the incidence matrix and go to Step. Step. Assign the activities and cycles in the final solutions to levels according to the earliest start time. SORTING RUE If the activity is an origin activity (OA), move it to the most left position in the sequence of activities that are not underlined. If the activity is a destination activity (DA), move it to the most right position in the sequence of activities that are not underlined. Process graph and corresponding incidence matrix 0 0 0

5 Step 0. Initial sequence {,,,,,,,,, 0,, } Step. is OA as it has no predecessors, i.e., an empty row) 0 Iteration Step. Sorting rule produces {,,,,,,,,, 0,, } Step. Cross out row and column 0 Iteration Step. Activity is OA as it has no predecessors, i.e., one non- empty element) Step. Sorting rule produces {,,,,,,,,, 0,, } Step. Cross out row and column After iterations Selected position The current sequence {,,,,,,,,,, 0, } Step. Identify a cycle O and D positions possible The reduced incidence matrix

6 0 0 Cycle G = {(, ), (,0), (0, )} is determined or short G = (,, 0) 0 0 Cycle G Collapsing Cycle G = (,, 0) No OAs and DAs G G Another Interpretation of Cycle G Collapsing Cycle G = (,, 0) has to maintain relationship with the remaining activities G 0 G rows and columns replaced with one row and one column 0 Yet, Another Interpretation of Cycle G Collapsing Retain all relationships between the body of the matrix and the cycle-activities through the activity representing the cycle.

7 G G Cycle G Collapsing G = (,, ) G G Previous sequence {,,,,,,,,,, 0, } G = (,, 0) G = (,, ) G G G G G G G G G G The corresponding sequence is (,,,,,, G, G) G G Redefine new G = (, G) = (,,, 0) The final sequence is (,,,,, G, G) Project on activity 0 Original matrix 0 Oi Original i graph 0 0 Organized matrix Ordered activities Cycle Cycle Identifying evels of Activities Step. Draw a frame around each cycle. Step. Draw a frame around activities at the same level (no relationship between them). Step. rame the remaining activities. Step. abel activities at the same level.

8 0 0 abeling Steps Step Step 0 0 Step DESIGN PROEM Parameter-Parameter Matrix Cantilever beam Step The equations = MY I M= I= where: Y= = EI : the maximum bending stress occurring at the section closest to the support M : bending moment : length of the beam : width of the beam : hight of the beam Y : dimension defined above : applied force : deflection E : modulus of elasticity I : secondmomentof areaaboutthe neutral axis The transformed expressions are as follows: M = M (, Y, I) Y = Y() I = I(, ) = (M, ) = (,, E, I) Y E M Y I E

9 Solution Interpretation: I Y M E To determine beam deflection the following values are needed: I,, E, and. In case I was not known, the value of and would be needed. I Y M E I Y M E I Y M E I Y M E DSM Applications Process analysis Product modularity Product complexity reduction Innovation