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EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol 2, No. 1, 2009 (112-124)

ISSN 1307-5543 – www.ejpam.com

*Corresponding Author. Email addresses: eycetin@istanbul.edu.tr (E. Çetin),

lasinem@istanbul.edu.tr (L. Sarul). Phone: x18352 (E. Çetin), x18256 (L. Sarul)

http://www.ejpam.com 112 © 2009 EJPAM All Rights Reserved

A Blood Bank Location Model: A Multiobjective Approach

Eyüp ÇETİN*, Latife Sinem SARUL

Department of Quantitative Methods, School of Business Administration

Istanbul University, Istanbul 34320, Turkey

Office phone: + 90 212 4737070

Fax: +90 212 590 88 88

Abstract This effort derived a mathematical programming model, which is a hybrid

from set covering model of discrete location approaches and center of gravity method

of continuous location models, for location of blood banks among hospitals or clinics,

rather than blood bank layout in health care institutions. It is initially unknown the

number of blood banks will be located within capacity, their geographical locations

and their covering area. The solution of the model enlightens the initial darkness in a

multiobjective view. The objectives, which are handled via binary nonlinear goal

programming, are minimizitation of total fixed cost of location blood banks, total

traveled distance between the blood banks and hospitals and an inequality index as a

fairness mechanism for the distances. A hypothetical numerical example is solved

using MS Excel as a powerful spreadsheet tool. The recipe, which is an application of

medical operations research, may be a useful tool for health care policy makers.

Key words: Set covering model; Center of gravity method; Binary nonlinear goal

programming; Spreadsheet modeling, Medical operations research.

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1. Introduction

The location of facilities is an important issue in any application area for both

industry and academia. Any poor location decision will result in undesired

pathological situations such as increased expenses, capital costs and degraded

customer service [1]. In health care, the facility location decisions, which are strategic

not an everyday decision [2], are more critical due to that any anomaly may lead to

mortality and morbidity [1].

The availability and location of blood banks, which will serve some hospitals or

clinics, is also a strategic decision in health care delivery system. In addition to well

known importance of the subject, it is a fact that grave shortages of blood occur in

over 80% of the countries in the world, one of the reason is inadequate funding of the

local transfusion service [3], that may result from inefficient allocation of sources in

general. In investigating blood transfusion cost, one important element of variability

can be attributed to geographic location of the blood supply source [4]. Some cost-

structure analyses (eg, [5]) included distribution and delivery costs of blood as some

major variables. Besides, accessibility to a blood bank is an important component of

an organ transplant program. Transplantation requires more blood then most other

surgeries, for instance, 100 units of blood for a liver transplant patient [6]. Moreover,

blood banks may also serve as important education centers for medical staff from

hospitals and clinics [7].

It may be said that the literature is rich in general facility location and health care

location modeling. Some highlights of the literature are as follows. Hale and Moberg

[8] present a broad review of facility location and location science research. Recently,

Daskin and Schilling [9] gave the summaries of general location models. Daskin and

Dean [1] reviewed some selected location models in both general and health care

focus. Some authors studied the location analysis from stochastic standpoint. For

instance, Chen et al. [10] developed a model for stochastic facility location modeling.

Verter and Lapierre [11] developed a binary programming model for location of

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E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 114

preventive health care facilities. Flessa [12] studied a linear programming model for

allocation of health care resources in developing countries. Wu, Lin and Chen [13]

presented the optimal location model adopting the modified Delphi method, the

analytical hierarchy method and sensitivity analysis for Taiwanese hospitals. Şahin et

al. [14] present a review of health care location and also blood bank location models

and developed several location-allocation models to solve the problems of

regionalization based on an hierarchical structure. Or and Pierskalla [15] consider a

regional blood management problem where hospitals are supplied by a regional blood

bank in their region and developed a location-allocation model that minimizes the

sum of the transportation costs and the system costs. Shen et al. [16] consider a joint

location inventory problem involving regional centers nearby hospitals assigned to

them for the supply of the most perishable and most expensive blood product and

present a location-allocation model and a set covering model. Nemet and Bailey [17]

explore the relationship between distance and the utilization of health care by a group

of elderly residents in rural Vermont. Ndiaye and Alfares [18] focus on nomadic

population groups that occupy different locations according to the time of the year

and develop a binary integer programming model that is formulated to determine the

optimal number and locations of primary health units for satisfying a seasonally

varying demand.

The set covering model is a basic location structure [1] from discrete location

models when the center of gravity model is from continuous location models. The

latter is useful when the geographic position of a location is important in terms of

distribution of the services or materials. As instances, it can be used for specialty

laboratories, blood banks and ambulance services [2]. There are some crucial factors

for blood bank location, namely, transaction demands (or shipments) [2], population

size that the blood bank will serve, blood bank capacities, fixed cost of locating blood

bank [1], the distances between hospitals and blood banks and egalitarian distribution

of distances [19]. The last three elements may be taken as objectives of a blood bank

location model.

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E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 115

This study developed a mathematical programming model, which is a hybrid from

the set covering model of discrete location approaches and the center of gravity

method of continuous location models, holding three objectives; minimizing the total

fixed cost of locating blood banks, minimizing total distance between hospitals and

blood banks and minimizing an inequality index as a fairness mechanism for the

distances. The model deals with location of blood banks in a region rather than blood

bank layout in health care institutions. The objectives are transformed into a single

objective via goal programming, a kind of multiobjective programming (See [20] for

further discussion about goal programming). In this effort, the set covering model is

modified as that the classical covering parameters are taken as decision variables that

are initially unknown. More extensively, that is, it is uncertain at the beginning that

how many blood banks are located within capacity, their geographical location and

which hospitals are assigned to them. The run of the model clarifies them within the

objectives.

The paper is organized in the following way. The mathematical model, which is a

binary nonlinear goal programming model, is derived in the Methods, Computational

results via MS Excel’s Solver as a powerful spreadsheet tool regarding a hypothetical

numerical example are presented and some Discussion and conclusions are drawn.

2. The Method

The developed model is a hybrid from basic set covering model of discrete

location class and center of gravity method of continuous location models. As a

consequence, the model holds some natures and assumptions of both location models.

This model assumes that demands can be aggregated to a finite number of discrete

points (discrete location) when facilities can be located anywhere in the region

(continuous location) [1]. The model can be constructed in the following way.

We develop the model using the following notation:

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mi

,..., 3 , 2 , 1

=

: index of hospitals (demand nodes)

nj

,..., 3 , 2 , 1

=

: index of candidate blood banks

3 , 2 , 1

=

k

: index of goals

ih : total annual transaction demand by hospital i

ij

d : distance from hospital i to candidate blood bank j

p : maximum number of blood banks to locate

jf : fixed cost of locating blood bank j

j c : annual transaction capacity of blood bank j

ip : population size of the site at which hospitali is located

ix : x coordinate of hospitali with respect to a reference frame,

iy : y coordinate of hospitali with respect to a reference frame,

j x : x coordinate of the weighted center of gravity for blood bank j

j y : y coordinate of the weighted center of gravity for blood bank j

k P : priority coefficient for goalk

k

ρ : aspiration level for goalk .

+

k d : positive deviation from the aspiration level for goalk

−

k d : negative deviation from the aspiration level for goalk

We also define the following decision variables:

=

notif

j bankblood candidate locatewe if

Zj

0

1

=

not if

j bank bloodby eredbe cani hospital if

zij

0

cov1

The mathematical programming model can be formulated as follows. The

objective is to minimize the weighted sum of the appropriate deviations from the

aspiration levels as a classical effort for goal programming structure.