Contents PROCESS DECOMPOSITION INTRODUCTION. Decomposition Approaches Desirable Properties

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1 PRCESS DECMPSTN Contents Andrew Kusiak ntelligent Systems Laboratory Seamans Center The University of owa owa City, owa - Tel: - Fax: - andrew-kusiak@uiowa.edu NTRDUCTN DECMPSTN F AN ACTVTY-PARAMETER MATRX NDUSTRAL CASE STUDES NTRDUCTN A typical process, e.g., product development process, may involve many activities and procedures Decomposition has been recognized as a powerful tool for analysis of large and complex systems Decomposition Approaches Desirable Properties Low computational time complexity Structure enhancement Enhancement of human interaction and simplification of interface with software Generality

2 (a) a b c d e f g h Three Types of Matrices (b) a b c d e f g h (a) decomposable matrix (b) non-decomposable matrix with overlapping parameters (inputs and outputs) (c) non-decomposable matrix with overlapping activities (c) a b c d e f Decomposition Problem 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 (or parameters) subject to the following constraints: Constraint C: Empty groups of activities or parameters are not allowed. Constraint C: The number of activities in a group does not exceed an upper limit, b (or alternatively, the number of parameters in a group does not exceed, d). Actions for the verlapping Activities A. ne may replace an overlapping activity with an alternative activity that involves different parameters. A. An activity can be decomposed into subactivities and each subactivity assigned to a group of parameters with some commonality with that subactivity. A. An overlapping activity can be removed from the matrix. A. The duration of the overlapping activity can be shortened (e.g.., by better management of resources or tools used). Cluster dentification Algorithm Step 0. Set iteration number k =. Step. 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. For each entry of crossed by the horizontal line hi draw a vertical line vj. Step. For each entry of crossed-once by the vertical line vj draw a horizontal line hk. Step. Repeat steps and 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. Transform the incidence matrix [aij](k) into [aij](k+) by removing rows and columns corresponding to the horizontal and vertical lines drawn in steps through. Step. f matrix [aij](k+) = 0 (where 0 denotes a matrix with all empty elements ), stop; otherwise set k = k + and go to step.

3 Example ncidence matrix Activity nput/utput Next Step Delete all doublecrossed elements

4 Resultant Matrix teration teration v v v v h h h h Final decomposition result Example (Repeated) GA- GA- GA- G/- G/- G/- ncidence matrix

5 Matrix after Steps and Step h h h h v v v v v v Step Step h h h v v v v

6 teration teration v v v v h h h h RC- RC- RC- Final decomposition result CC- CC- CC- Example x x x 0 x x x x x x x x x 0 x x x x x x x x 0 x x x x x x x x 0 x x x x x x Activity 0 x x x x x x x x Activity-input/output incidence matrix nput/utput x 0x x x x x x x x

7 The decomposed matrix nput/utput G- G- G- G- Three independent processes GA- x x x G- G- x 0 x x x x0 x x x x x Activity GA- GA- GA- 0 x x x x x x x x x x x x x0x x x x GA- GA- GA- 0 x x x x x x x x x x x GA- x x x x GA- x x x x x x 0 Step 0. Step. Branch-and-Bound Algorithm (nitialization) Consider the incidence matrix (aij) as the initial node. Set the upper bound ZU = + and lower bound ZL = -. (Branching) Select an active node (not fathomed) with the minimum value of lower bound ZL. Apply the C algorithm to the submatrix of this node. When the submatrix does not partition, apply the branching scheme. Branching scheme: R Step. Remove from the matrix a parameter corresponding to the largest number of activities, one at a time. Remove from the matrix an activity corresponding to the largest number of inputs and outputs, one at a time. (Bounding) For each new node, obtain a lower bound ZL.

8 Lower Bound Step. (Fathoming) Exclude a new node from further consideration if: Test : ZL = ZU, Test : Any of the submatrices of the matrix at this node violates constraint C or C. f ZL < ZU, store this solution as a new incumbent solution, set ZU = ZL, and reapply Tests and to the remaining unfathomed nodes. The lower bound ZL is the number of activities removed from the matrix. Upper Bound Step. (Stopping rule) Stop when there are no unfathomed nodes remaining; the current incumbent solution is final. therwise, go to Step. The upper bound ZU is calculated only after a feasible solution has been determined. Example Form subprocesses with no more than four activities in a subprocess 0 0 Activity-nput & utput ncidence Matrix and Enumeration Tree Matrix M

9 Branch & Bound Algorithm: Enumeration Tree M Remove Remove Remove Remove Remove Remove Remove Remove Remove Solution Remove Remove Remove Remove Fathom Fathom Fathom Fathom Matri x M 0 0 Solution ZL =, ZU = Z L =, Z U = Z L =, ZU = Z L = B&B Algorithm terations Activity removed 0 0 (Nondecomposable matrix) Apply the C Algorithm Activity removed 0 0 (Nondecomposable matrix) Apply the C Algorithm Activity removed 0 0 (Nondecomposable matrix) Apply the C Algorithm

10 Activities and removed Activities and removed 0 0 (Nondecomposable matrix) 0 0 (Nondecomposable matrix) Apply the C Algorithm Apply the C Algorithm Activities and removed Final Solution M Remove Remove Remove ZL =, ZU = 0 0 (Decomposable matrix) Apply the C Algorithm Remove Remove Remove Remove Remove Remove Solution Remove Remove Remove Remove Fathom Fathom Fathom Fathom 0 0 Solution ZL =, ZU = ZL =, ZU = ZL =

11 Example The final result Activity Parameter GA- { GA- { GP- GP- NDUSTRAL CASE STUDES Case Study : Process graph with annotated inputs and outputs (Phase ) Activity 0 Activity-input/output incidence matrix nput/output 0 0 0

12 Decomposed incidence matrix with overlapping inputs/outputs nput/output Activity 0 Decomposed process graph with overlapping inputs/outputs G- G- 0 G G- 0? Preceding Activity Question: How the information on the overlapping inputs/outputs can be utilized? nput Goal Succeeding Activity

13 Actions for the overlapping inputs/outputs 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 follow up activities Decomposed incidence matrix with overlapping activities Activity 0 0 nput/output Decomposed process graph with overlapping activities G- G- 0 G- 0 0 G- 0 0 G- G-? Question: How the information on the overlapping activities can be utilized?

14 Actions for the overlapping activities Case Study DEF model of the conceptual design phase,,,, ne may view the overlapping activities as a server that controls a group of activities (clients),,,,,,,,,,,, ne may replace an overlapping activity with an alternative activity that involves different inputs/outputs,,,,,,,,,, 0,,, 0,,, 0,,,,, 0,,,,,,,,, 0,,,, Decomposed activity-mechanism incidence matrix SUMMARY Process decomposition is useful in process reengineering Simplifies process management Allows to make strategic decisions