. Author manuscript; available in PMC: 2019 Apr 1.

Published in final edited form as: Stat Biosci. 2018 Mar 1;10(1):255–279. doi: 10.1007/s12561-018-9214-7

Table 7.

Notation

Section 2.1

G = (V,E)

Graph representation for KPD; pairs/NDDs comprise the vertex set, and VMs comprise the edge set

V^P,V^A

Set of donor-candidate pairs and NDDs respectively

j Indices representing vertices

Index representing an edge

q_i

Probability of success of pair i

p_e

Probability of success of match e

u_e

Utility of match e

S = (V_S,E_S)

Graph representation for a transplant arrangement, typically an LRS

x,y,L

Maximum length/size of cycle, chain, and LRS respectively

ℓ

Index representing length

LRS(x,y,L)

Class of LRSs in G, under the length/size constraints x,y,L

c = (V_c,E_c) = 〈i₁, …, i_ℓ〉

Graph and shorthand notation for a cycle of size ℓ

c′ = (V_c_′,E_c_′) = 〈i₀, i₁, …, i_ℓ〉

Graph and shorthand notation for a chain of length ℓ

EU_S

Expected utility of LRS S

LRS₍_i₎

Collection of LRSs in LRS(x,y,L) that involved vertex i

Y_S

Decision variable for selection of LRS S

S̄ = (V_S̄,E_S̄)

Graph representation for an observed subgraph of S

𝒞_S̄

Set of cycles and chains in observed subgraph S̄

U_c

Utility of a cycle

C_{S}^{*}

Set of potential solutions of S

C^* = (V_C^*,E_C^*)

Graph representation for a potential solution

U_C^*

Utility of a potential solution

Section 2.2

P(V̄ )

Probability that only vertices represented by V̄ succeed to transplantation

S_V̄ = (V̄,E_V̄)

Subgraph of S induced by V̄

P(Ē |V̄)

Probability that only edges represented by Ē succeed to transplantation, given vertices represented by V̄ are successful

C_{\bar{S}}^{*}

Solution for observed subgraph S̄

U_{C_{\bar{S}}^{*}}

Utility of the solution for observed subgraph S̄

EU_c,EU_c_′

Expected utility of a simple cycle or chain

Section 2.3

Number of potential solutions in an LRS

P (C_{k}^{*})

The probability that potential solution

C_{k}^{*}

will be viable for transplantation

P_k

The probability that potential solution

C_{k}^{*}

will be realized

Section 2.4

A,B

Matrix representations of subgraphs of and potential solutions of S respectively

Number of observable subgraphs

Index for observable subgraph

S_h = (V_h,E_h)

Graph representation of an observable subgraph of S

Section 2.5

Matrix representation of a set of sampled subgraphs of S

Number of sampled subgraphs

Index for subgraph samples

S_n = (V_n,E_n)

Graph representation of a sampled subgraph of S

Section 3

EU^*,

\hat{E U}

Exact expected utility and estimated expected utility for comparative experiments

Section 4

r_i

Candidate associated with pair i

D_i = {d_{i[m],m=1,…,M_i}}

Donors associated with pair i

M_i

Number of donors associated with pair i

Index for donors

E_ij

Set of edges between vertices i and j

e_imj

Match from d_im to candidate r_j

u_imj, p_imj

Utility and probability of match e_imj

q_{i m}^{d}, q_{i}^{r}

Success probabilities for donor m and candidate of pair i

G′ = (V,E′)

Reduced graph notation for KPD involving multiple donor pairs

Section 5

Probability that candidate joins simulation with second donor

Q,λ₁,λ₂,μ₁,μ₂

Intensity matrix and state transition specifications for Markov process describing availability of individuals (candidates and donors) in simulation experiment