Page 1

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.

Page 2

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 113

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

Page 3

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.

Page 4

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:

Page 5

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 116

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.

Page 6

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 117

Minimize

+

3

+

2

+

1

++

321

dPdPdP

The model has three goal constraints due to three goals for the location of blood

banks. Since they are to be minimized, we choose all the aspiration

levels

0

321

===ρρρ

. The first goal is the minimization of the total fixed cost of

locating blood banks,

∑

=

j

−

1

+

1

=+−

n

jj

ddZf

1

0.

The second goal is the minimization of the total traveled distance from hospitals to

blood banks,

∑∑

=

i

1

=

−

2

+

2

=+−

m

n

j

ij ij

ddzd

1

0

where the Euclidean distance

()

()22

jiji ij

yyxxd

−+−=

. Here, the center of

gravity is weighted by two important factors, which are basic indicators for hospital-

blood bank relations, the annual transactions [1] and the population size of the site at

which the hospital is located. The weighted center of gravity coordinates are, for

all

nj

,..., 3 , 2 , 1

=

,

∑

=

i

∑

=

m

+

=

ijii

m

i

ijiii

j

zph

zxph

x

1

1

ε

and

ε+

=

∑

=

i

∑

=

m

ijii

m

i

ijiii

j

zyh

zyph

y

1

1

.

For the above ratios, the undetermined cases due to binary nature of

ij z ’s are avoided

by means of

0

>ε

, which is in the neighborhood of 0.

Page 7

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 118

The last goal constraint is an inequality index, to be minimized, the coefficient of

variance [19] aiming the equality and fairness distribution of sources. In the case of

blood bank location, it tries to smooth the extreme distances between hospitals and

blood banks as much as possible, which is also an important effort for health care

facility location. The goal constraint can be written as

()

0

1

33

1

2

=+−−

+

−+

=

∑

dddd

d

m

i

ij

ε

,

mj

,..., 3 , 2 , 1

=∀

,

where d is the arithmetic mean of the all

ij

d ’s so that the fairness platform is

determined for all hospitals and blood banks that we deal. Here, again

0

>ε

is

employed to ensure the absence of undetermined cases.

There are naturally some system constraints. The first is the capacity constraint

imposing that any blood bank must not exceed its annual transaction capacity, that is,

any other extra hospital must not assign to the blood bank,

∑

=

i

≤

m

j iji

czh

1

,

nj

,..., 3 , 2 , 1

=∀

.

There must be sufficient total capacity to supply the total demand, which can be stated

as

∑

=

i

∑

=

j

≤−

mn

ji

ch

11

0,

mi

,..., 3 , 2 , 1

=∀

,

nj

,..., 3 , 2 , 1

=∀

.

The following constraints stipulates that each hospital (demand node) must be

covered by exactly one of the selected blood banks, since the primary decision

variables are binary,

Page 8

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 119

∑

=

j

1

=

j

1

=

n

ijjzZ

1 and ∑

=

n

ijz

1,

mi

,..., 3 , 2 , 1

=∀

.

The number of located blood banks must not excess the maximum number of blood

bank, which may be located,

∑

=

j

≤

n

j

pZ

1

.

The following constraints impose that there must not be any empty blood bank to

which no hospital assigned,

{} { }

Z

, 0max ,...,,, max

321

=−

j mjjjj

zzzz

nj

,..., 3 , 2 , 1

=∀

.

Finally, revisiting the decision variables completes the mathematical model,

{ } 1 , 0

∈

j

Z

,

{ } 1 , 0

∈

ijz

and

0,

≥

−

k

+

kdd

,

mi

,..., 3 , 2 , 1

=∀

,

nj

,...,3 , 2 , 1

=∀

,

. 3 , 2 , 1

=∀k

Note that the foregoing model implicitly locates the selected blood banks with

respect to weighted center of gravity and assigns the hospitals to appropriate blood

banks. The mathematical model, which works without any candidate geographical

location information, is a nonlinear binary goal programming model.

It is a fact that since the developed hybrid model, which includes

6 ) 1

+

(

+

mn

decision variables, inherits from NP-hard like models [1,21], the solution of the model

is technically hard. In addition to some heuristics for the general class of location

covering models [22-23], as in a recent study [24] proposing MS Excel’s Solver,

which uses branch-and-bound methodology [25], it may be employed to reach at least

near optimal solutions. Although the nonlinearities in the model may be handled by

some appropriate transformations to reduce the CPU time, our empirical observations

Page 9

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 120

reports that MS Excel’s Solver may solve the original model in reasonable times.

Because of the multi-objectivity, the solutions are efficient at the same time for all

objectives. The solutions will be at least near optimal due to the nature of the

problem.

3. Computational Results

A hypothetical numerical example is as follows. There are at most n=3 candidate

blood banks (BB1, BB2 and BB3, whose geographical locations are currently

unknown) with the capacities of annual transactions 6,000, 4,500 and 5,000,

respectively. The fixed costs of locating them are 1.5, 1 and 1.2 (in $100.000),

respectively. There exist m=25 hospitals (H1, H2 ,.., H25) waiting for supply with

annual transaction demands ranging from 100 to 1,374. The population sizes of the

sites that the hospitals serve are ranging from 4,890 to 130,500. The scatter of the

hospitals (in 10 km) with respect to a reference frame (origin) is shown in Figure 1

(Data are not shown). The problem is the location of the blood banks within the model

(nonpreemptive) objectives and constraints.

The model is optimized via MS Excel’s classical Solver tool. (See [20,25] for

spreadsheet modeling). Since the model is highly nonlinear, the result is obtained by

solutions of different starting points. For the priority levels, we take

1

321

===

PPP

to have a nonpreemptive goal programming model. Also, we choose

001 , 0

=ε

.

Although the model has 84 (78 of 84 are binary) decision variables, the CPU time of

the model solution is 58 s on a PC with Intel (R) Core (TM) 2 CPU 5600, 1.83 GHz

and 1.00 GB RAM. According to near optimal results that the model suggest, the

three candidate blood banks should be located at the coordinates

) 33. 5 68. 5 (),(

11

=

yx

,

) 27. 2073. 5 (),(

22

=

yx

and

) 08. 12 64. 3 (),(

33

=

yx

, respectively. The near Pareto

optimal location policy is also graphed in Figure 1. BB1 should serve 9 hospitals,

BB2 is assigned 9 hospitals when BB3 is assigned 7 hospitals. All annual transaction

demands are satisfied by the blood banks, namely, the demand distributions 5,077,

4,364 and 4,961, respectively. Total fixed cost is $370,000. Total traveled distance

Page 10

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 121

between the hospitals and the respective blood banks is 892 km. The inequality index,

the coefficient of variance, is 43.7 km. The maximum distance is 72.8 km (H15-BB3)

when the minimum is 7.8 km (H2-BB2), hence the range, which is another basic

inequality index, is 65 km. As seen in Figure 1, H15 is not attached to either BB1 or

BB2. The reason may be that it is so far from the center, and it has relatively lower

demand (150) and representative population (14,700). Thus, it is inadequate to attract

the weighted center of gravity to itself. Also, not only the distance is a criterion for the

model, which takes the blood bank transaction capacities into account. From the

distance standpoint, the location strategy is also reasonable, due to, note that, the

scatter of hospitals is in range 97.5 km on the x axis when 245 km on the y axis.

BB1

BB2

BB3

0

5

10

15

y

20

25

30

0 2 4 6 8 10 12

x

H10

H21

H17

H7

H6

H24

H23

H8

H9

H16

H19

H18

H20 H11

H15

H3

H5

H4

H22

H14

H12

H1

H13

H2

H25

Figure 1. The Hospitals and Location Strategies for the Blood Banks

4. Discussion and Conclusions

A blood bank location model, which is a binary nonlinear goal programming

model, is formulated in a multiobjective frame. The objectives are minimization of

total fixed cost of locating blood banks, total traveled distance between the supply and

demand nodes and the inequality index for the distances. The model stemming from

set covering model and center of gravity method is a combination of discrete and

continuous location approaches. It is initially unknown the number of blood banks

will be located, their locations and their covering area. The solution of the model

enlightens the initial darkness.

Page 11

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 122

The recipe may be extended in such a way that the coordinates of demand

population weighted center of gravity may be restricted to a specific region within the

model formulation because of some geographical restrictions such as governmental

regulations. Also, if the parameters of distance or transaction between the blood banks

are in agenda, the model may be modified for the desired regulations. Besides, any

other objectives or constraints may be integrated to the core model such as transaction

costs may appear in the model as another objective of the goal programming model.

Moreover, from computational view, this study also shows the ease and use of MS

Excel as a spreadsheet tool for technically hard problems.

The proposed model may be conducted for not only blood bank location but also

other appropriate location issues in both general and health care case. For instance,

the model may be utilized for warehouse location or location of hospitals and fire

stations with a few modifications. As another further research, the model may be

adopted to stochastic modeling nature of location problems. The road map, which the

model offers, may be a useful tool for both industry and academia particularly in

health care management science.

References

[1] M.S. Daskin, L.K. Dean, Location of health care facilities. In: Sainfort F,

Brandeau M, Pierskalla W, editors. Handbook of OR/MS in health care: A Handbook

of Methods and Applications. USA: Kluwer, 2004, p. 43-76

[2] Y.A. Ozcan, Quantitative Methods in Health Care Management: Techniques and

Applications. San Francisco CA: Jossey-Bass/Wiley; 2005.

[3] M.J. Thomas, The management of overseas emergencies. Travel Med. Infec. Dis.

5, 2: 113-116 (2007).

[4] J.M. Forbes, M.D. Anderson, G.F. Anderson, G.C. Bleecker, E.C. Rossi, G.S.

Moss, Blood transfusion costs: a multicenter study. Transfusion. 31, 4: 318-23 (1991)

[5] R.A Fouty, V.E. Haggen, J.D. Sattler, Problems, personnel, and proficiency of

small hospital laboratories. Public Health Rep. 89, 5: 408-417 (1974).

Page 12

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 123

[6] P.A Lindsey, E.A. McGlynn, Pew Memorial Trust policy synthesis: 5. State

coverage for organ transplantation: a framework for decision making. Health Serv

Res. 22, 6: 881-922 (1988).

[7] M. Amin, D. Fergusson, K. Wilson, A. Tinmouth, A, Aziz, D. Coyle, P. Hebert,

The societal unit cost of allogenic red blood cells and red blood cell transfusion in

Canada, Transfusion. 44, 10: 1479-86 (2004).

[8] T.S. Hale, C.R. Moberg, Location Science Research: A Review, Annals of

Operations Research. 123, 1-4: 21-35 (2003).

[9] J. Current, M. Daskin, D. Schilling, Discrete network location models. In: Drezner

Z, Hamacher, editors. HW Facility Location: Applications and Theory. Berlin:

Springer, 2002.

[10] G. Chen, M.S. Daskin, Z. Shen, S. Uryasev, A New Model For Stochastic

Facility Location Modeling. Research Report 2005-8, ISE Dept., University of

Florida, 2005.

[11] V. Verter, S.D. Lapierre, Location of preventive health care facilities. Annals of

Operations Research. 110, 1-4: 123-132 (2002).

[12] S. Flessa, Priorities and allocation of health care resources in developing

countries: a case-study from the Mtwara region, Tanzania, Eur. J. Oper. Res. 150, 1:

67-80 (2003).

[13] C.R. Wu, C.T. Lin, H.C. Chen, Optimal selection of location for Taiwanese

hospitals to ensure a competitive advantage by using the analytic hierarchy process

and sensitivity analysis. Build. Environ. 42, 3: 1431-1444 (2007).

[14] G. Şahin, H. Süral, S. Meral, Locational Analysis For Regionalization Of

Turkish Red Crescent Blood Services, Computers & Operations Research. 34, 3: 692-

704 (2007).

[15] I. Or, W.P. Pierskalla, A Transportation Location-Allocation Model for Regional

Blood Banking, AIIE Transactions. 11, 2: 86-95 (1979).

[16] Z.M. Shen, C. Coullard, MS. Daskin, A Joint Location-Inventory Model,

Transportation Science. 37, 1: 40-55 (2003).

[17] G.F. Nemet, A.J. Bailey, Distance and Health Care Utilization among the Rural

Elderly, Social Science & Medicine. 50, 9: 1197-1208 (2000).

Page 13

E. Çetin, L. Sarul / Eur. J. Pure Appl. Math, 2 (2009) 124

[18] M. Ndiaye, H. Alfares, Modeling Health Care Facility Location for Moving

Population Groups, Computers& Operations Resesarch. 35, 7: 2154-2161 (2008).

[19] H.A. Eiselt, G. Laporte, Objectives in location problems. In: Drezner Z, editor.

Facility Location: A survey of applications and methods. Berlin: Springer-Verlag,

1995.

[20] W.L. Winston, Operations Research: Application and Algorithms, fourth ed.

Belmont:, Brooks/Cole, 2004.

[21] J. Brimberg, P. Hansen, N. Mladenovic, Decomposition strategies for large-scale

continuous location-allocation problems. IMA J Manag. Math. 17, 4: 307-316 (2006).

[22] J. Zhou, B. Liu, New stochastic models for capacitated location-allocation

problem. Computers and Industrial Engineering. 45, 1: 111-125 (2003).

[23] A.T. Ernst, , M. Krishnamoorthy, Solution algorithms for the capacitated single

allocation hub location problem. Annals of Operations Research. 86, 0: 141-159

(1999).

[24] E. Çetin S. Tolun, A weapon-target assignment approach to media allocation.

Appl. Math. Comput. 175, 2: 1266-1275 (2006).

[25] W.L. Winston, S.C. Albright, Practical Management Science, second edition.

Pacific Grove: Brooks/Cole, 2001