Systems, Man & Cybernetics - July 2015 - 30

A role assignment or an agent assignment is defined
as a tuple of an agent and a role. This tuple makes a role
attach to an agent. In formalizing role-assignment problems [40], [57], [61]-[63], [69], [72], [80], only agents and
roles are emphasized. Current agents or roles are the focus
of role assignment. Environments and groups are simplified into vectors and matrices, respectively. Furthermore,
the nonnegative integers m ^= A h express the size of
the agent set A; n ^= R h expresses the size of the role set
R; i, i 1, i 2, f expresses the indices of agents; and j, j 1, j 2, f
expresses the indices of roles.
Role range vectors are two vectors, i.e., L and U, of
the lower and upper bounds of roles in environment e of
group g. L6 j@^0 # j 1 n h expresses how many agents must
be assigned to role j, and U 6 j@ expresses the most number of agents that can be assigned to role j. L and U are
the valuable components in supplementing the E-CARGO
model. They reveal many challenges in role assignment
and the process of RBC. Without L, we would not discover
many related problems discussed in this article. L and U
are, in fact, the simplified and derived components from
the environment component in the E-CARGO model. They
show the smallest/largest numbers of agents for each role
in a group to be in the working state.
A qualification matrix Q is an m # n matrix, where
Q 6i, j@ e 60, 1@ expresses the qualification value of agent
ieN ^0 # i 1 m h for role jeN ^0 # j 1 n h. Q 6i, j@ = 0
means the lowest, and one is the highest. Note that a Q
matrix can be obtained by comparing all the qualifications, i.e., Q s, of agents with all the requirements, i.e.,
R s, of roles. Q is, in fact, a component to depict one
specific relationship between roles and agents. It significantly affects the problem of role assignment in RBC.
Because this component is an assumed one, it brings in
the significance of another problem, agent evaluation, in
RBC. A role-assignment matrix T is defined as an m # n
matrix, where T 6i, j@ e " 0, 1 ,^0 # i 1 m, 0 # j 1 n h expresses if agent i is assigned to role j. T 6i, j@ = 1 means yes
and zero means no.
The group performance v of group g is defined
as the sum of the assigned agents' qualifications, i.e.,
n-1
m-1
v = | i = 0 | j = 0 Q 6i, j@ # T 6i, j@. Role j i s work a ble
in group g if it is assigned with enough agents, i.e.,
| im=-01 T 6i, j@ $ L6 j@. T is workable if each role j is workm-1
able, i.e., 6^0 # j 1 n h | i = 0 T6i, j@$ L6 j@h. Group g is
0.71
0.29
0.69
0.0
0.97
0.58

0.6 0.0
0.67 0.44
0.92 0.92
0.0 0.53
0.51 0.77
0.64 0.24
(a)

0.22
0.76
0.6
0.0
0.65
0.0

1
0
0
0
1
0

0 0
0 0
0 0
0 1
0 0
1 0
(b)

0
1
1
0
0
0

Figure 2. (a) A qualification matrix Q. (b) An
assignment matrix T.
30

IEEE Systems, Man, & Cybernetics Magazine July 2015 	

workable if T is workable. For example, Figure 2(a) is a
qualification matrix. Figure 2(b) is an assignment matrix
that makes the group work with vector L = 62, 1, 1, 2@. The
group performance is 4.21.
From the previously given definitions, group g can
be simplified by Q, an L, and T. By scrutinizing these
three elements, we discover the related optimization and
search problems.
Problems Discovered and Solved by RBC
Research efforts have discovered and solved some significant problems related to RBC [61], [72], [80]-[82].
Definition 1 [80]
Given Q and L, GRA is to find a matrix T to
max v =

| | Q6i, j@ # T 6i, j@

m-1 n-1
i=0 j=0

subject to
T 6i, j@ ! " 0, 1 ,

| T 6i, j@ = L6 j@

m-1
i=0

| T 6i, j@ # 1
n-1
j=0

^0 # i 1 m, 0 # j 1 n h
^0 # j 1 nh

^ 0 # i 1 m h,

(1)
(2)
(3)

where (1) shows that an agent can only be assigned or not,
(2) makes the group workable, and (3) means that each
agent can only be assigned to one role.
GRA was formalized and solved efficiently in [80]. It
is also applied to solving practical problems, such as the
scheduling of health-care services [77].
Definition 2 [82], [86]
A p o t e nt i a l r ole m a t r i x Mp i s d e f i ne d a s
M p: A # R ! " 0, 1 ,, w h e r e M p 6i, j@ = 1 e x p r e s s e s
i ! j.A p, and M p 6i, j@ = 0 means i g j. A p, where A p
is the potential agent set of role j [82].
Definition 3 [82], [86]
Role transfer is a process to exchange ones of T
with M p, i.e., if there are agent i, and roles j 1
and
j 2, T 6i, j 1@ = 1, M p 6i, j 1@ = 0, T 6i, j 2@ = 0,
and
M p 6i, j 2@ = 1, a role transfer occurs when T 6i, j 1@ : = 0,
M p 6i, j 1@ : = 1, T 6i, j 2@ : = 1, a n d M p [i, j 2]: = 0 (0 # i 1 m,
0 # j 1, j 2, 1 n); on the other hand, if there are role j, and
agents i 1 a nd i 2, T 6i 1, j@ = 1, M p 6i 1, j@ = 0, T [i 2, j] = 0,
and M p 6i 2, j@ = 1, a role tra nsfer also occurs when
T [i 1, j]: = 0, M p 6i 1, j@ : = 1, T 6i 2, j@ : = 1, M p 6i 2, j@ : = 0 (0 # i 1,
i 2 1 m, 0 # j 1 n) (where " : = " means "is assigned with").
Definition 4 [82], [86]
Given m, n, L, M p, and T that are not workable, the RTP
is to find a workable T l by doing role transfers.
The RTP was formalized and solved efficiently in
[82], [86], and [88]. The investigation of this problem also



Table of Contents for the Digital Edition of Systems, Man & Cybernetics - July 2015

Systems, Man & Cybernetics - July 2015 - Cover1
Systems, Man & Cybernetics - July 2015 - Cover2
Systems, Man & Cybernetics - July 2015 - 1
Systems, Man & Cybernetics - July 2015 - 2
Systems, Man & Cybernetics - July 2015 - 3
Systems, Man & Cybernetics - July 2015 - 4
Systems, Man & Cybernetics - July 2015 - 5
Systems, Man & Cybernetics - July 2015 - 6
Systems, Man & Cybernetics - July 2015 - 7
Systems, Man & Cybernetics - July 2015 - 8
Systems, Man & Cybernetics - July 2015 - 9
Systems, Man & Cybernetics - July 2015 - 10
Systems, Man & Cybernetics - July 2015 - 11
Systems, Man & Cybernetics - July 2015 - 12
Systems, Man & Cybernetics - July 2015 - 13
Systems, Man & Cybernetics - July 2015 - 14
Systems, Man & Cybernetics - July 2015 - 15
Systems, Man & Cybernetics - July 2015 - 16
Systems, Man & Cybernetics - July 2015 - 17
Systems, Man & Cybernetics - July 2015 - 18
Systems, Man & Cybernetics - July 2015 - 19
Systems, Man & Cybernetics - July 2015 - 20
Systems, Man & Cybernetics - July 2015 - 21
Systems, Man & Cybernetics - July 2015 - 22
Systems, Man & Cybernetics - July 2015 - 23
Systems, Man & Cybernetics - July 2015 - 24
Systems, Man & Cybernetics - July 2015 - 25
Systems, Man & Cybernetics - July 2015 - 26
Systems, Man & Cybernetics - July 2015 - 27
Systems, Man & Cybernetics - July 2015 - 28
Systems, Man & Cybernetics - July 2015 - 29
Systems, Man & Cybernetics - July 2015 - 30
Systems, Man & Cybernetics - July 2015 - 31
Systems, Man & Cybernetics - July 2015 - 32
Systems, Man & Cybernetics - July 2015 - 33
Systems, Man & Cybernetics - July 2015 - 34
Systems, Man & Cybernetics - July 2015 - 35
Systems, Man & Cybernetics - July 2015 - 36
Systems, Man & Cybernetics - July 2015 - 37
Systems, Man & Cybernetics - July 2015 - 38
Systems, Man & Cybernetics - July 2015 - 39
Systems, Man & Cybernetics - July 2015 - 40
Systems, Man & Cybernetics - July 2015 - 41
Systems, Man & Cybernetics - July 2015 - 42
Systems, Man & Cybernetics - July 2015 - 43
Systems, Man & Cybernetics - July 2015 - 44
Systems, Man & Cybernetics - July 2015 - 45
Systems, Man & Cybernetics - July 2015 - 46
Systems, Man & Cybernetics - July 2015 - 47
Systems, Man & Cybernetics - July 2015 - 48
Systems, Man & Cybernetics - July 2015 - 49
Systems, Man & Cybernetics - July 2015 - 50
Systems, Man & Cybernetics - July 2015 - 51
Systems, Man & Cybernetics - July 2015 - 52
Systems, Man & Cybernetics - July 2015 - 53
Systems, Man & Cybernetics - July 2015 - 54
Systems, Man & Cybernetics - July 2015 - 55
Systems, Man & Cybernetics - July 2015 - 56
Systems, Man & Cybernetics - July 2015 - 57
Systems, Man & Cybernetics - July 2015 - 58
Systems, Man & Cybernetics - July 2015 - 59
Systems, Man & Cybernetics - July 2015 - 60
Systems, Man & Cybernetics - July 2015 - 61
Systems, Man & Cybernetics - July 2015 - 62
Systems, Man & Cybernetics - July 2015 - 63
Systems, Man & Cybernetics - July 2015 - 64
Systems, Man & Cybernetics - July 2015 - 65
Systems, Man & Cybernetics - July 2015 - 66
Systems, Man & Cybernetics - July 2015 - 67
Systems, Man & Cybernetics - July 2015 - 68
Systems, Man & Cybernetics - July 2015 - 69
Systems, Man & Cybernetics - July 2015 - 70
Systems, Man & Cybernetics - July 2015 - 71
Systems, Man & Cybernetics - July 2015 - 72
Systems, Man & Cybernetics - July 2015 - Cover3
Systems, Man & Cybernetics - July 2015 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/smc_202310
https://www.nxtbook.com/nxtbooks/ieee/smc_202307
https://www.nxtbook.com/nxtbooks/ieee/smc_202304
https://www.nxtbook.com/nxtbooks/ieee/smc_202301
https://www.nxtbook.com/nxtbooks/ieee/smc_202210
https://www.nxtbook.com/nxtbooks/ieee/smc_202207
https://www.nxtbook.com/nxtbooks/ieee/smc_202204
https://www.nxtbook.com/nxtbooks/ieee/smc_202201
https://www.nxtbook.com/nxtbooks/ieee/smc_202110
https://www.nxtbook.com/nxtbooks/ieee/smc_202107
https://www.nxtbook.com/nxtbooks/ieee/smc_202104
https://www.nxtbook.com/nxtbooks/ieee/smc_202101
https://www.nxtbook.com/nxtbooks/ieee/smc_202010
https://www.nxtbook.com/nxtbooks/ieee/smc_202007
https://www.nxtbook.com/nxtbooks/ieee/smc_202004
https://www.nxtbook.com/nxtbooks/ieee/smc_202001
https://www.nxtbook.com/nxtbooks/ieee/smc_201910
https://www.nxtbook.com/nxtbooks/ieee/smc_201907
https://www.nxtbook.com/nxtbooks/ieee/smc_201904
https://www.nxtbook.com/nxtbooks/ieee/smc_201901
https://www.nxtbook.com/nxtbooks/ieee/smc_201810
https://www.nxtbook.com/nxtbooks/ieee/smc_201807
https://www.nxtbook.com/nxtbooks/ieee/smc_201804
https://www.nxtbook.com/nxtbooks/ieee/smc_201801
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1017
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0717
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0417
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0117
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1016
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0716
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0416
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0116
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_1015
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0715
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0415
https://www.nxtbook.com/nxtbooks/ieee/systems_man_cybernetics_0115
https://www.nxtbookmedia.com