Design Structure Matrices

A Design Structure Matrix (DSM) is a square matrix that maps dependencies between tasks in a project. In PRA, DSMs are derived from bipartite relationships: resources are shared across tasks, and risks are shared across resources. These shared dependencies create coupling between tasks that can propagate delays, cost overruns, or failures.

The Resource-Task Matrix

The starting point is the Resource-Task Matrix S, where rows represent resources and columns represent tasks. An entry S[i, j] = 1 means resource i is used by task j.

library(PRA)

# 4 resources x 5 tasks
S <- matrix(c(
  1, 0, 1, 0,
  1, 1, 0, 0,
  0, 1, 0, 1,
  0, 0, 1, 1,
  0, 1, 1, 0
), nrow = 4, ncol = 5)
rownames(S) <- paste0("R", 1:4)
colnames(S) <- paste0("T", 1:5)
S
#>    T1 T2 T3 T4 T5
#> R1  1  1  0  0  0
#> R2  0  1  1  0  1
#> R3  1  0  0  1  1
#> R4  0  0  1  1  0

Parent DSM

The Parent DSM P = t(S) %% S is a tasks-by-tasks matrix. The diagonal entry P[j, j] counts how many resources task j* uses. The off-diagonal entry P[j, k] counts how many resources tasks j and k share — a measure of coupling.

p <- parent_dsm(S)
print(p)
#> Resource-based 'Parent' Design Structure Matrix
#> Tasks: 5  Resources: 4
#> 
#>    T1 T2 T3 T4 T5
#> T1  2  1  0  1  1
#> T2  1  2  1  0  1
#> T3  0  1  2  1  1
#> T4  1  0  1  2  1
#> T5  1  1  1  1  2

plot(p)

Heatmap of the Parent DSM showing resource-based task coupling

Tasks that share more resources are more tightly coupled. If a resource is delayed or constrained, all tasks that depend on it are affected.

The Risk-Resource Matrix

The Risk-Resource Matrix R adds a second layer. Rows represent risks and columns represent resources. An entry R[i, j] = 1 means risk i affects resource j.

# 3 risks x 4 resources
R <- matrix(c(
  1, 0, 1,
  1, 1, 0,
  0, 1, 0,
  0, 0, 1
), nrow = 3, ncol = 4)
rownames(R) <- paste0("Risk", 1:3)
colnames(R) <- paste0("R", 1:4)
R
#>       R1 R2 R3 R4
#> Risk1  1  1  0  0
#> Risk2  0  1  1  0
#> Risk3  1  0  0  1

Grandparent DSM

The Grandparent DSM traces the dependency chain from risks through resources to tasks. The intermediate matrix T = R %% S gives a risks-by-tasks mapping, and the Grandparent DSM is G = t(T) %% T. The off-diagonal entries count how many risks are shared between each pair of tasks.

g <- grandparent_dsm(S, R)
print(g)
#> Risk-based 'Grandparent' Design Structure Matrix
#> Tasks: 5  Resources: 4  Risks: 3
#> 
#>    T1 T2 T3 T4 T5
#> T1  3  4  3  2  3
#> T2  4  6  4  2  4
#> T3  3  4  3  2  3
#> T4  2  2  2  2  2
#> T5  3  4  3  2  5

plot(g)

Heatmap of the Grandparent DSM showing risk-based task coupling

Interpreting the DSM