Discovering Blood Donor Arrival Patterns Using Data Mining: A Method to Investigate Service Quality at Blood Centers

Abstract

Blood centers without fixed appointments for collecting blood often experience nonconstant donor arrival rates, which vary due to time-of-day, day-of-week, etc. When a constant workforce size is employed in such blood centers, there is either idle personnel, or donor satisfaction is compromised due to long waiting times, or both conditions alternate over time. Consequently, a method to obtain adaptive workforce requirements might be valuable. This study utilized the Two-Step Cluster method and the Classification and Regression Trees method in succession to identify both daily and hourly donor arrival patterns at Hacettepe University Hospitals’ Blood Center. A serial queuing network model of the donation process was then employed for each of the identified donor arrival patterns. By considering and accomodating variations in the donor arrival patterns, required workforce sizes and their decomposition among process steps were predicted to achieve predetermined target values of expected waiting times and to balance workforce utilizations in the blood donation processes. Although a blood center is considered for the proposed methodology, the approach is general and applications in various operations of healthcare organizations are possible.

Appendices

Appendix 1

In this appendix, the expressions developed by Whitt [17] to find the average waiting time in a GI/G/m queuing system are presented.

λ:

Arrival rate

μ:

Service rate

m:

Number of servers

ρ:

Utilization (λ/mμ)

c ²_a :

Squared coefficient of variation of inter-arrival time

c ²_s :

Squared coefficient of variation of service time

W(λ,ρ,c ²_a ,c ²_s ,m):

Average waiting time in a GI/G/m queuing system with given arguments

W^*(ρ,m):

Average waiting time in an M/M/m queuing system with given arguments

P₀(ρ,m):

Probability of zero waiting time in an M/M/m queuing system with given arguments

Case 1: m=1

$$ W\left( {\lambda {,}\rho {,}c_a^2,c_s^2,m} \right) = \frac{{{\rho^2}\left( {c_a^2 + c_s^2} \right)}}{{2\lambda \left( {1 - \rho } \right)}}\Gamma \left( {\rho, c_a^2,c_s^2} \right) $$

where $\begin{array}{*{20}{c}} {\Gamma \left( {\rho, c_a^2,c_s^2} \right) = } \hfill & {\left\{ {\begin{array}{*{20}{c}} {\exp \left( {\frac{{ - 2\left( {1 - \rho } \right){{\left( {1 - c_a^2} \right)}^2}}}{{3\rho \left( {c_a^2 + c_s^2} \right)}}} \right)\,{\hbox{if}}\,c_a^2 \leqslant 1} \hfill \\{\exp \left( {\frac{{\left( {1 - \rho } \right)\left( {c_a^2 - 1} \right)}}{{\left( {1 + \rho } \right)\left( {c_a^2 + 10c_s^2} \right)}}} \right)\,i{\hbox{f}}\,c_a^2 > 1} \hfill \\\end{array} } \right.} \hfill \\\end{array}$

Case 2: m>1

$$ W\left( {\lambda {,}\rho {,}c_a^2,c_s^2,m} \right) = \frac{{{W^*}\left( {\rho, m} \right)\left( {c_a^2 + c_s^2} \right)}}{2}\varphi \left( {\rho, c_a^2,c_s^2,m} \right) $$

where ${W^*}\left( {\rho, m} \right) = \frac{{{{\left( {m\rho } \right)}^{m + 1}}{P_0}\left( {\rho, m} \right)}}{{\lambda m\left( {m!} \right){{\left( {1 - \rho } \right)}^2}}}$ with ${P_0}\left( {\rho, m} \right) = {\left( {\left( {\sum\limits_{k = 0}^{m - 1} {\frac{{{{\left( {m\rho } \right)}^k}}}{{k!}}} } \right) + {{\left( {m\rho } \right)}^m}\frac{1}{{m!\left( {1 - \rho } \right)}}} \right)^{ - 1}}$

$$ \varphi \left( {\rho, c_a^2,c_s^2,m} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{4\left( {c_a^2 + c_s^2} \right)}}{{4c_a^2 - 3c_s^2}}{\varphi^1}\left( {m,\rho } \right) + \frac{{c_s^2}}{{4c_a^2 - 3c_s^2}}\Theta \left( {\frac{{c_a^2 + c_s^2}}{2},m,\rho } \right)} & {} & {{\hbox{if }}c_a^2 > c_s^2} \\{} & {} & {} \\{\frac{{\left( {c_s^2 - c_a^2} \right)}}{{2\left( {c_a^2 + c_s^2} \right)}}{\varphi^3}\left( {m,\rho } \right) + \frac{{c_s^2 + 3c_a^2}}{{2\left( {c_a^2 + c_s^2} \right)}}\Theta \left( {\frac{{c_a^2 + c_s^2}}{2},m,\rho } \right)} & {} & {{\hbox{if }}c_a^2 \leqslant c_s^2} \\\end{array} } \right. $$

$$ \Theta \left( {\alpha, m,\rho } \right) = \left\{ {\begin{array}{*{20}{c}} 1 & {} & {{\hbox{if }}\alpha > {1}} \\{} & {} & {} \\{{\varphi^4}{{\left( {m,\rho } \right)}^{2\left( {1 - \alpha } \right)}}} & {} & {{\hbox{if 0}} \leqslant \alpha \leqslant {1}} \\\end{array} } \right. $$

$$ \delta \left( {m,\rho } \right) = { \min }\left\{ {0.24,\frac{{\left( {1 - \rho } \right)\left( {m - 1} \right)\left( {{{\left( {4 + 5m} \right)}^{0.5}} - 2} \right)}}{{16m\rho }}} \right\} $$

$$ {\varphi^1}\left( {m,\rho } \right) = 1 + \delta \left( {m,\rho } \right),{\varphi^2}\left( {m,\rho } \right) = 1 - 4\delta \left( {m,\rho } \right) $$

$$ {\varphi^3}\left( {m,\rho } \right) = {\varphi^2}\left( {m,\rho } \right)\exp \left( {{{ - 2\left( {1 - \rho } \right)} \mathord{\left/{\vphantom {{ - 2\left( {1 - \rho } \right)} {3\rho }}} \right.} {3\rho }}} \right),\,{\varphi^4}\left( {m,\rho } \right) = \min \left\{ {1,\frac{{{\varphi^1}\left( {m,\rho } \right) + {\varphi^3}\left( {m,\rho } \right)}}{2}} \right\} $$

Appendix 2

This appendix presents the expressions that are used to compute the parameters of the arrival and departure processes of each stage j in a serial queuing network with k stages. These expressions are special cases of the ones presented in Bitran and Tirupati [18, 19].

λ_j :

Arrival rate of stage j

μ_j :

Service rate of stage j

m_j :

Number of servers for stage j

p_j :

Non-departure probability after stage j

ρ_j :

Utilization of stage j (λ_jm_jμ_j)

c ²_a,j :

Squared coefficient of variation of inter-arrival time for stage j

c ²_s,j :

Squared coefficient of variation of service time for stage j

c ²_d,j :

Squared coefficient of variation of inter-departure time for stage j

$$ {\lambda_{j + 1}} = {p_j}{\lambda_j}{\hbox{ and }}c_{a,j + 1}^2 = c_{d,j}^2{\hbox{ for }}j > 1 $$

$$ c_{d,j}^2 = 1 + \left( {1 - \rho_j^2} \right)\left( {c_{a,j}^2 - 1} \right) + \frac{{\rho_j^2\left( {c_{s,j}^2 - 1} \right)}}{{\sqrt {{{m_j}}} }} $$