Operations Research for Health Care 2 (2013) 75–85

Contents lists available at ScienceDirect

Operations Research for Health Care

journal homepage: www.elsevier.com/locate/orhc

Centralized versus distributed sterilization service:A location–allocation decision modelHouda Tlahig a,b,∗, Aida Jebali a,c, Hanen Bouchriha a, Pierre Ladet ba Laboratoire ACS, Ecole Nationale d’Ingénieurs de Tunis, Université Tunis El Manar, B.P. 37-le Belvédère-1002 Tunis, Tunisieb GIPSA-Lab, B.P 46, Rue de la Houille Blanche 38402 St Martin d’Hères, Francec Prince Sultan University, P.O. Box 53073 Riyadh 1158, Saudi Arabia

a r t i c l e i n f o

Article history:Received 12 November 2012Accepted 24 May 2013Available online 6 June 2013

Keywords:Hospital networkSterilization service configurationCentralized vs. distributedOptimizationLocation–allocation modelMILPValid cuts

a b s t r a c t

The concept of ‘‘networking’’ has become central to the reform of healthcare systems. The objective isto reduce costs while improving the quality of service. This paper deals with the problem of sterilizationservice configuration within a hospital network. Two alternatives are considered: (1) each hospital inthe network maintains its sterilization service in-house; (2) a central sterilization service ensures thisfunction for all hospitals in the network. This decision is based on a location–allocation model of thesterilization service. A Mixed Integer Linear Program (MILP) is proposed to find the optimal configurationof the sterilization service (centralized vs. distributed), the optimal location and the optimal capacity ofthe centralized sterilization service over a multi-period planning horizon. The objective is to minimizecosts related to transportation, production and resource acquisition and transfer. A solutionmethod basedon the addition of appropriate customized cuts to the originalMILP is thenproposed. The proposedmodelsare applied to 30 scenarios extracted from a real-life case study. The obtained results show that theconsidered problem can be solved to optimality for moderate size scenarios with the use of commercialMILP solvers and the addition of the proposed customized cuts to the original model. Further analysis wasconducted and pointed out how network configuration is sensitive to the number of human and materialresources available in each hospital of the network.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The last two decades have witnessed the emergence of net-works in the healthcare sector. In fact, there has been continuousgrowth in the number of hospital networks in both theUSA and Eu-rope. For example, in the USA alone, between 1980 and 1997, thenumber of hospitals organized in networks increased from 32.1%to 73.4% [1].

Hospital networking seems to represent an organizationalchoice providing interesting opportunities to cope with cost andquality issues. In [1], the author states that the pooling of avail-able resources should improve efficiency and effectiveness due tosynergies and cost savings. The need for efficient resource alloca-tion in hospitals is obvious and represents themain objective of thenetworked organization. Henceforth, restructuring the location offacilities and incrementally concentrating some services to fewerlocations becomes one of the major focuses of managerial tasks inthe hospital environment.

∗ Correspondence to: EIGSI, 26 Rue De Vaux De Foletier, 17041 LA ROCHELLE,France. Tel.: +33 546458015; fax: +33 546458010.

E-mail address: houda.tlahig@eigsi.fr (H. Tlahig).

2211-6923/$ – see front matter© 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.orhc.2013.05.001

Healthcare providers strive to minimize patient contaminationrisks and nosocomial infections. In the operating room this riskis particularly important. That is why surgical items have to befree of contamination at the time of use. This is accomplishedby subjecting them to a validated sterilization process andmaintaining the sterility up to the time of use. In France, hospitalsterilization is regulated and restricted by the guide of goodpractices [2]. Minimizing the costs and ensuring a high qualitylevel of hospital sterilization services is subscribed as one ofthe challenges of healthcare providers. These objectives couldbe reached through the optimization of the configuration ofsterilization services within a hospital network.

In this paper, we intend to investigate the opportunities ofgroupinghospital sterilization services and the economical interestof resource sharing. In order to ensure patient safety, hospitalsin developed countries are investing millions of euros in sterileinstruments; in The Netherlands, for example, the investmentin sterile equipment can be estimated to exceed 500 millioneuros [3]. In France, the cost incurred by the sterilization of 1 m3 iswidely varying from one hospital to another. While considering 21hospitals of theArc Alpin Region, the cost of sterilizing 1m3 in 2004ranged from 187 to 1174 euros [4]. It was noticeable, however,that this cost tends to be lower in hospitals with relatively

76 H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85

large demand. This sparked the interest of studying sterilizationdepartment configuration within hospital network in order toidentify and seize eventual opportunities for cost reduction whilemaintaining a high quality and traceability of sterilization process.

Within a hospital network, two major alternatives could beconsidered: (1) each hospital performs in-house its sterilizationactivities in its premises independently of the other hospitals;(2) all hospitals of the network opt for the sharing of the commonresources requested by sterilization services by grouping them inone Central Sterilization Service (CSS). The first configuration isreferred to as ‘‘distributed sterilization service’’; the second one isreferred to as ‘‘centralized sterilization service’’. The centralizationcould lead to better resource utilization and considerable costsavings through the advantage of economy of scale. However,this alternative can be considered only if the different hospitalsare located in the same region as well as the sterilization centreand incurs transportation costs. Moreover, sterilization servicecentralization increases the risk of sterile item unavailability. Thatis why it requires a high level of management to ensure thecoordination and the satisfaction of all the network actors. Reapingthe benefits of a centralized sterilization service is contingent to aneffective and efficient management at the tactical and operationallevels.

The focus of this paper is placed on finding the best choice be-tween the two alternatives concerning the configuration of thesterilization service within a hospital network: (1) a distributedsterilization service; (2) a centralized sterilization service. If thesecond alternative is chosen, the common sterilization servicelocation and sizing are also determined. The objective is tominimize the total cost of sterilization service which includes thetransportation cost, the sterilization process cost and the resourcetransfer and acquisition costs. The constraints that have been takeninto account are essentially related to resource capacities and de-mand satisfaction. This multi-site, multi-product andmulti-periodplanning horizon, location–allocation problem, is formulated as aMixed Integer Linear Program (MILP).

The remainder of this paper is organized as follows: Sec-tion 2 presents a brief literature review on hospital networklocation–allocation problems. In Section 3, the optimization prob-lem and the considered assumptions are described. Furthermore,the proposed mathematical model is presented. Section 4 detailsthe proposed solution approach. Computational experiments andresults are reported in Section 5. The last section highlights someconclusions; future extensions of this work are also discussed.

2. Literature review

Some researchers have been investigating whether mergersand networks should take the place of independent operations,focusing on hospitals’ quest for better location, optimal resourcedimensioning and improvement of the healthcare service andreduction of total health expenditure [5–8], etc. Other researchershave focused on the consequences of mergers and networking interms of benefits versus drawbacks [1,9,10], etc.

Location–allocation models have been used quite extensivelyfor quantitative analysis in health services. The common ob-jective is to minimize travel costs. Classical mathematical lo-cation–allocation models like p-median or maximum coveringlocation models have been proposed [11]. Rahman and Smith [12]reviewed a number of location–allocation studies for health ser-vice development planning and found that most of the loca-tion–allocation models and methods have been formulated eitheras p-median problems or covering problems.

Some studies have dealt with the location–allocation problemin hospital network organization and have reported a particu-lar interest in the considered configuration problem. Or and Pier-skalla [6] treated the transportation–location–allocation problem

regarding the case of regional blood banking. They suggested algo-rithms to decide howmany banks to set-up, where to locate them,how to allocate the hospitals to the banks and how to root the sup-ply operations in such a way that the transportation and systemcosts are minimized. In [7], a multi-objective heuristic approachhas been developed for determining the location and the size ofmedical departments in a hospital network. The authors aimed tominimize the patient travel cost, the total cost incurred by the lo-cation–allocation plan and the total number of unit moves neces-sary for the restructuring of the new allocation. They proposed atwo-phased solution procedure to solve the proposed mathemat-ical model. This approach sought efficient solutions by means ofmulti-objective Tabu Search in the first phase. In the second phase,they proposed clustering to allow the decision makers to explorethe solution space interactively until the ‘‘optimal’’ configurationwas found. Gunes and Yaman [8] studied the modelling change inhealthcare networks with particular reference to the implicationonpatient flows and resource allocation. They alsomodelled hospi-tal mergers at a facility planning level using a resource-based viewof hospitals. Their objective was to find the optimal resource allo-cation after a merger of two networks. They focused on the gainsin network design and flow related costs.

Few studies have considered stochastic aspectswhilemodellingand solving location–allocation problems in the case of healthcaresystems. Chao [13] used a non-linear programming approach tostudy the allocation of a limited amount of service capacity todifferent service sites in such a way that the system-wide qualityis optimized. Harper [14] proposed a simulation tool for use inplanning health services when geographical considerations (bothservice and patient locations) are of prime importance.

Someotherworks have investigated the sterilization service de-partment configuration problem. In the first study, Elshafei [15]proposed a mathematical model to find the location of a set ofcentral sterilization services within a hospital network. A surveydealing with the feasibility of the sterilization service centraliza-tion in the French context has been conducted by a group of con-sultants. This study aimed at understanding the organizationalaspects of a common centralized sterilization service and findingthe best way to group hospital sterilization departments withinthe region under consideration for optimal functioning of this ser-vice [4]. This survey was based on an estimation of the steriliza-tion service cost; no optimization models have been used. A studyconducted in Switzerland [16] dealt with the development of anintegrated logistics solution to ensure optimal sterilization for ahospital network. This study stressed on the great need of healthcare decision makers to improve hospital sterilization service, asa response to the pressure on costs in the field of public health.These observations let indeed many hospitals to consider the op-tion of building a new CSS with greater capacity to serve attachedhospitals working as a network.

In [17], we addressed the problem of the centralization vs.decentralization of the sterilization service within a hospital: thecase of a Tunisian hospital where many surgical services werelocated in different wings, with each surgical service having itsown sterilization department. A two-stepped iterative approachsolution was proposed. The first step consisted of finding thebest configuration between the centralization and decentralizationof the various sterilization service departments; in the secondstep we aimed to find the optimal size for the configurationachieved in the first step. The developed approach did not takeinto account demand and cost variation over the planning horizon.In addition, sterilization service location–allocation and resourcedimensioning are addressed separately.

In [18], we proposed a model for finding the optimal choicebetween the internalization vs. externalization of the hospitalsterilization process. In the externalization case, two types of third-party providers have been considered: (1) an industrial company

H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85 77

and (2) a hospital located in the same region. The proposed modelconsidered all the sterilization process activities and took intoaccount the variation in demand and cost from one period toanother.

In both [17,18], the emphasis was placed on optimizing thecosts incurred by sterilization activity of one hospital. The decisionis made by the management of the hospital under consideration.However, it has beennoted thatmanyhospitals located in the samecountry region could have a need to optimize their sterilizationactivity. Optimizing sterilization activity separately for eachhospital overlooks the opportunities of better employment ofscarce and costly resources that could be grasped when thisoptimization integrates all the hospitals. Optimizing sterilizationactivity within a hospital network is besides encouraged by thesignificant cost disparities mentioned above. In France, such adecision is made by Regional Healthcare Agencies which missionincludes the improvement of healthcare systems efficiency withineach region of the country.

To the best of our knowledge, there have been no studies deal-ing with the choice between the centralized vs. distributed ster-ilization service in a hospital network using optimization tools.There have been papers dealing with the location–allocation prob-lem in healthcare systems but without integrating resource di-mensioning. In [19], the authors addressed the optimization of thesterilization costs through the grouping choices of medical devicesinto packages. They developed an Integer Linear Program defin-ing the items grouped in each package. They showed how group-ing choices impact process and storage costs of the sterilizationactivity. In [3], the authors developed optimization models to sup-port sterilization logistics. They defended the option of maintain-ing in-house sterilization against outsourcing sterilization tasks.Hospitals are opting for outsourcing as an attempt to achieve costsavings. However, placing the sterilization service at a distanceentails the risks of lowering sterile item availability which couldincrease costs rather reducing them. That is why the authors arerather promoting the idea of reducing in-house sterilization cost byoptimizing sterilization logistics and the composition of the nets ofsterile items.

In the present paper, we rather embrace the idea that firstwe need to optimize the sterilization service configuration withina hospital network; then adequate management tools will bedeveloped and implemented to deal with operational decisionsin order to minimize the risk of sterile item unavailability andreach the intended objectives in terms of cost reduction. Hence,we address the optimization of hospital sterilization cost throughthe configuration of the sterilization services within a hospitalnetwork. Two options are considered: (1) distributed sterilizationservice; (2) centralized sterilization service. A location–allocationmodel is proposed; in the model, resource capacity dimensioningis also integrated, in that we specify the number of resources to betransferred from the hospitals to the CSS.

3. Model formulation

3.1. Problem description

In this paper, centralization can be defined as the process bywhich hospitals within a given geographical area move towardssharing the existing sterilization resources leading to a CSS.

In our study,we consider a network composed ofN hospitals forwhich it may be beneficial to group their sterilization services intoa common CSS. Our objective is to find the optimal configurationbetween the centralized vs. distributed sterilization service andto determine the CSS location. Indeed the CSS may be located ina separate new entity or located in one of the hospitals underconsideration.

The sterilization service must ensure Reusable Medical Devices(RMD) sterility which is obtained through a high-quality regulatedclosed loop process. The most important point of consumption ofthe sterile instruments is the operating room. When a surgery isfinished, all materials will be brought to the contaminated storageof the OR, from where they are taken to the sterilization service.There, they are dismounted, disinfected, perhaps precleaned, andsubsequently put into the washing machines. After washing, theRMD are grouped into sets and packaged. There are many types ofpackaging systems such as wrapping and rigid sterile containers.The packages are put into the autoclaves where the sterilizationtakes place. Once sterilized, the packages are placed in the sterilestorage which completes the closed loop.

Many human and material resources are required to performthe sterilization activity. The human resources fully assigned tothe sterilization service department are the technicians and thesterilization nurses. The technicians are responsible for instrumentcleaning and disinfection; sterilization nurses are responsible forinstrument packaging and the control of the sterilization process.Other human resources intervene in the sterilization servicedepartment, such as a pharmacist and an administrator, but theyare assigned concurrently to other activities and responsibilitiesin the hospital. Several equipments and fixtures are used in thesterilization process: autoclaves, automatic washing machines,shelves, carts, etc. The most costly and critical material resourcesare autoclaves and automatic washing machines.

Sterilization service configuration within the hospital networkseeks to find the best location and allocation of the existing criticalhospital resources to the CSS. Henceforth, in network configura-tion, human andmaterial resources required for the CSS are deter-mined while taking into account sterilization process specificities.

In the mathematical formulation, decision variables are relatedto the choice between the centralized vs. distributed configuration,the location of the common sterilization service, the quantities ofSterilized Medical Devices (SMD) to be produced for each hospital,the number of resources to be transferred from each hospital tothe CSS and the number of vehicle to be purchased as well. If thecentralization is the chosen configuration, supply is carried outby the CSS and deliveries are achieved using available vehicles.The departure point for each vehicle is the CSS. Each vehicle hasa known capacity. The objective is to minimize sterilization costscomposed of sterilization fixed and variable costs, transportationcosts, transfer costs and storage costs.

3.2. Assumptions

In problem formulation, the following assumptions have beenconsidered:

• The potential locations of the CSS are known (existing hospitalsas well as potential new locations).

• We consider only one CSS for all the network hospitals.• The delivery costs are treated as variable costs and depend on

the distance between the hospital and the CSS location.• The delivery vehicles belong to the CSS.• The sterilization service requires a set of human and material

resources (nurses, autoclaves, etc.). Each resource has a givencapacity to produce SMD.

• The transfer costs for the material resources depend on thedistance between each hospital and the CSS. The transfer costsfor human resources are represented by the remunerationpaid to each nurse/technician whose post changes [8]. Thisremuneration depends on the distance between the hospitalwhere the nurse is currently working and the CSS.

• We do not consider layoff costs. Each nurse/technician issupposed to be transferred either to the CSS or to anotherservice in his/her original hospital.

78 H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85

• At this stage, we consider that non-transferred materialresources are sold.

• We consider that if the decision to move to a CSS is made over agiven period, it will remain the same for the upcoming periodsof the planning horizon.

• The decision on the location of the sterilization service involvesall the hospitals in the network: all the considered hospitalshave to make the same decision on choosing between acentralized and a distributed organization.

• The initial inventory levels are considered equal to zero.• Inventory levels at the CSS are not considered. As soon as

sterilization has been completed, the SMDs are transferred tothe appropriate hospital.

• The demand is considered to be deterministic and is based onhistorical data on demand of the operating room activity.

3.3. Notations

The following sets and indices are used in formulating theconsidered optimization problem.

H = {H1,H2, . . . ,HN} = {set of considered hospitals}.S = {CSS1, CSS2, . . . , CSSN+M} = {set of potential locations ofthe CSS}.

We can notice here that H ⊂ S as the CSS could be located inone of the hospitals within the network.

P: number of products.T : number of planning horizon periods.R: number of material resources.W : number of human resources.N: number of hospitals in the network.M: number of potential new sites.M1,M2,M3,M4: large numbers.i: hospitals, i = 1 . . .N .j: potential locations of the CSS, j = 1 . . .N + M .p: products, p = 1 . . . P .t: periods, t = 1 . . . T .r: material resource types, r = 1 . . . R.w: human resource types, w = 1 . . .W .

We consider the following parameters:

Ctransp,i,j,t Transportation cost of one unit of product p, p =

1 . . . P , between hospital i, i = 1 . . .N , and CSS j, j =

1 . . .N + M , during period t, t = 1 . . . T (in euro/unit).CAPV Transportation capacity of one vehicle (in m3).QMr,i Number of material resources of type r, r = 1 . . . R, ini-

tially available at hospital i, i = 1 . . .N .QHw,i Number of human resources of typew, w = 1 . . .W , ini-

tially working at hospital i, i = 1 . . .N .Dp,i,t Demand of hospital i, i = 1 . . .N , for product p, p =

1 . . . P , during period t, t = 1 . . . T (in units).CVp,j,t Variable processing cost of one unit of product p, p =

1 . . . P , when it is performed in the CSS j, j = 1 . . .N+M ,during period t, t = 1 . . . T (in euro/unit).

CVHp,i,t Variable processing cost of one unit of product p, p =

1 . . . P , when it is performed in hospital i, i = 1 . . .N ,during period t, t = 1 . . . T (in euro/unit).

CSp,i,t Storage cost of product p, p = 1 . . . P , at hospital i, i =

1 . . .N , during period t, t = 1 . . . T (in euro/unit).CFj,t Fixed cost associated with the use of CSS j, j = 1 . . .N +

M , during period t, t = 1 . . . T .CFMrj,t Fixed cost associated with the use of one material re-

source of type r, r = 1 . . . R, at CSS j, j = 1 . . .N + Mduring period t, t = 1 . . . T .

CFHwj,t Fixed cost associated with the utilization of human re-source of type w, w = 1 . . .W , at CSS j, j = 1 . . .N + M ,during period t, t = 1 . . . T .

CFHi,t Fixed cost associated with the use of the sterilization de-partment of hospital i, i = 1 . . .N , during period t, t =

1 . . . T .CFHMr,i,t Fixed cost associated with the use of one material re-

source of type r, r = 1 . . . R, at the sterilization depart-ment of hospital i, i = 1 . . .N , during period t, t =

1 . . . T .CFHHw,i,t Fixed cost associated with the utilization of human re-

source of type w, w = 1 . . .W , at the sterilization de-partment of hospital i, i = 1 . . .N , during period t, t =

1 . . . T .CRr,i,j,t Transfer cost of material resource of type r, r = 1 . . . R,

from hospital i, i = 1 . . .N , to CSS j, j = 1 . . .N + M , atthe beginning of period t, t = 1 . . . T .

CTw,i,j,t Transfer cost incurred of human resource of typew, w =

1 . . .W , from hospital i, i = 1 . . .N , to CSS j, j = 1 . . .N ,at the beginning of period t, t = 1 . . . T .

CAVt Purchasing cost of a vehicle at the beginning of periodt, t = 1 . . . T .

I0p,i Initial inventory level of product p, p = 1 . . . P , at hospi-tal i, i = 1 . . .N (in units).

ISp,i,t Safety stock level of product p, p = 1 . . . P , at hospitali, i = 1 . . .N , at the end of period t, t = 1 . . . T (in unit).

CAPMr Capacity of one material resource of type r, r = 1 . . . R(in m3).

CAPHw Capacity of one human resource of type w, w = 1 . . .W(in time units).

vr,i,t Income of selling one material resource of type r, r =

1 . . . R, belonging to hospital i, i = 1 . . .N , at the begin-ning of period t, t = 1 . . . T .

Vp Volume of one unit of product p, p = 1 . . . P (in m3).δw,p Number of time units of human resource of type w, w =

1 . . .W , required to produce one unit of product p, p =

1 . . . P .

The following decision variables are used:

Zi,t = 1 If sterilization service is performed in-house, in hospitali, i = 1 . . .N , during period t, t = 1 . . . T ; = 0 other-wise.

Yi,j,t = 1 If sterilization service of hospital i, i = 1 . . .N , is per-formed in CSS j, j = 1 . . .N + M , during period t, t =

1 . . . T ; = 0 otherwise.Xp,i,j,t Number of units of product p, p = 1 . . . P , of hospital

i, i = 1 . . .N , processed in CSS j, j = 1 . . .N +M , duringperiod t, t = 1 . . . T .

XHp,i,t Number of units of product p, p = 1 . . . P , of hospitali, i = 1 . . .N , processed in-house during period t, t =

1 . . . T .Ip,i,t Inventory level of product p, p = 1 . . . P , at hospital i,

i = 1 . . .N , at the end of period t, t = 1 . . . T .TRMr,i,j,t Number of material resources of type r, r = 1 . . . R,

relocated from hospital i, i = 1 . . .N , to CSS j, j =

1 . . .N + M , at the beginning of period t, t = 1 . . . T . Incase of centralization at period k, TRMr,i=j,j,k = 0.

VRMr,i,j,t Number of material resources of type r, r = 1 . . . R, re-located from hospital i, i = 1 . . .N , to CSS j, j = 1 . . .N+

M , before or at the beginning of period t, t = 1 . . . T . Incase of centralization at period k,VRMr,i=j,j,t≥k = 0.

TRHw,i,j,t Number of human resources of type w, w = 1 . . .W ,transferred from hospital i, i = 1 . . .N , to CSS j, j =

1 . . .N + M , at the beginning of period t, t = 1 . . . T . Incase of centralization at period k, TRHw,i=j,j,k = 0.

H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85 79

VRHw,i,j,t Number of human resources of type w, w = 1 . . .W ,transferred from hospital i, i = 1 . . .N , to CSS j, j =

1 . . .N + M , before or at the beginning of period t, t =

1 . . . T . In case of centralization at period k,VRHw,i=j,j,t≥k= 0.

AVj,t Number of new vehicles required by CSS j, j = 1 . . .N +

M , at the beginning of period t, t = 1 . . . T .CAPtransj,t Transportation capacity of CSS j, j = 1 . . .N +M , dur-

ing period t, t = 1 . . . T (in m3).CAPMr,j,t Capacity of material resources of type r, r = 1 . . . R, at

the CSS j, j = 1 . . .N + M , during period t, t = 1 . . . T(in m3)

CAPHw,j,t Capacity of human resources of type w, w = 1 . . .W ,at the CSS j, j = 1 . . .N + M , during period t, t = 1 . . . T(in time units).

NMr,i,t Number of material resources of type r, r = 1 . . . R, ofhospital i, i = 1 . . .N , sold at the beginning of periodt, t = 1 . . . T .

3.4. Mathematical model

The objective is to minimize the total cost of the sterilizationservice which is composed of the delivery cost, the productioncost (both fixed and variable costs are considered), the storagecost, the purchase cost of new resources (vehicles) needed by theCSS, the cost incurred by relocating and transferring some existingresources (autoclaves, nurses, technicians, etc.) and to maximizecost savings made by selling unused material resources and bythe redeployment of human resources to other services of theconsidered hospitals.

The MILP can be modelled as follows:

MinT

t=1

Ni=1

N+Mj=1

Pp=1

Ctransp,i,j,t · Xp,i,j,t

+

Ni=1

P

p=1

CVHp,i,t · XHp,i,t +

CFHi,t

+

Rr=1

CFHMr,i,t · QMr,i +

Ww=1

CFHHw,i,t · QHw,i

· Zi,t

+

N+Mj=1

Ni=1

P

p=1

CVp,j,t · Xp,i,j,t + CFj,t ·

Yi,j,t

N

+

Rr=1

CFMr,j,t · VRMr,i,j,t +

Ww=1

CFHw,j,t · VRHw,i,j,t

+

Ni=1

Pp=1

CSp,i,t · Ip,i,t +

N+Mj=1

CAVt · AVj,t

+

Ww=1

Ni=1

N+Mj=1

CTw,i,j,t · TRHw,i,j,t

+

Rr=1

Ni=1

N+Mj=1

CRr,i,j,t · TRMr,i,j,t − vr,i,t · NMr,i,t

.

Subject to

Zi,t +

N+Mj=1

Yi,j,t = 1 ∀i = 1 . . .N, ∀t = 1 . . . T . (1)

These constraints are related to the choice of distributed vs. cen-tralized sterilization service for the considered hospital network.

Each hospital has to choose between the two options: (1) perform-ing the sterilization ‘‘in-house’’ or (2) sharing the sterilization ser-vice with the hospitals of the network:

Yi,j,t ≤ Yi,j,t+k ∀j = 1 . . .N + M, ∀i = 1 . . .N,

∀t = 1 . . . T , ∀k = 1 . . . T − t. (2)

Constraints (2) ensure that if the centralized configuration ischosen at period t , this decision should be maintained for theupcoming periods of the planning horizon:

Yi,j,t ≤ Yi′,j,t ∀j = 1 . . .N + M, ∀t = 1 . . . T ,

∀i = 1 . . .N, ∀i′ = 1 . . .N/i′ = i. (3)

Constraints (3) state that only one CSS is considered. If centraliza-tion is chosen for one hospital sterilization service then this deci-sion will be applied for the other hospitals within the network. Inthis case, all network hospitals will be assigned to only one sharedCSS:

Pp=1

Ni=1,i=j

Xp,i,j,t · Vp ≤ CAPtransj,t ∀j = 1 . . .N + M,

∀t = 1 . . . T . (4)

Constraints (4) ensure the respect of the transportation capacity:

Ni=1

Pp=1

Xp,i,j,t · Vp ≤ CAPMr,j,t ∀j = 1 . . .N + M,

∀t = 1 . . . T , ∀r = 1 . . . R (5)Ni=1

Pp=1

Xp,i,j,t · δw,p ≤ CAPHw,j,t ∀j = 1 . . .N + M,

∀t = 1 . . . T , ∀w = 1 . . .W

Pp=1

XHp,i,t · Vp ≤ QMr,i · CAPMr ∀i = 1 . . .N,

∀r = 1 . . . R, ∀t = 1 . . . T (5′)P

p=1

XHp,i,t · δw,p ≤ QHw,i · CAPHw ∀i = 1 . . .N,

∀w = 1 . . .W , ∀t = 1 . . . T .

Constraints (5) state that the quantity of DMS produced in theCSS should respect the available resource capacities. Constraints(5′) express that the quantity of DMS produced in each hospitalwhen the in-house option is chosen are limited by the availableresource capacities:

Xp,i,j,t ≤ M1 · Yi,j,t ∀i = 1 . . .N, ∀j = 1 . . .N + M,

∀p = 1 . . . P, ∀t = 1 . . . T (6)

XHp,i,t ≤ M1 · Zi,t ∀i = 1 . . .N,

∀p = 1 . . . P, ∀t = 1 . . . T . (6′)

Constraints (6) impose that production is allowed in the CSS onlyin case of centralization. Constraints (6′) are similar to constraints(6); they deal with the case of distributed configuration. M1 can beset to the biggest demand during the considered time horizon.

Ip,i,t = Ip,i,t−1 + XHp,i,t +

N+Mj=1

Xp,i,j,t − Dp,i,t

∀i = 1 . . .N, ∀t = 2 . . . T , ∀p = 1 . . . P (7)

Ip,i,1 = I0p,i + XHp,i,1 +

N+Mj=1

Xp,i,j,1 − Dp,i,1

∀i = 1 . . .N, ∀p = 1 . . . P .

80 H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85

Constraints (7) define the inventory level at each hospital of thenetwork at the end of a given period t as a function of inventorylevel at the end of period t − 1, the sterilized quantities performedduring period t (whether the sterilization is performed in-houseor in a CSS) and of the demand fulfilled during period t . Obviouslythese constraints are related to flow conservation:

Ip,i,t ≥ ISp,i,t ∀i = 1 . . .N, ∀t = 1 . . . T , ∀p = 1 . . . P. (8)

Constraints (8) imply that the stored quantities must be greaterthan or equal to the corresponding safety stock level:

CAPMr,j,t =

Ni=1

CAPMr · VRMr,i,j,t

∀j = 1 . . .N + M, ∀t = 1 . . . T , ∀r = 1 . . . R (9)

CAPHw,j,t =

Ni=1

CAPHw · VRHw,i,j,t

∀j = 1 . . .N + M, ∀t = 1 . . . T , ∀w = 1 . . .W .

Constraints (9) ensure that the capacity of the CSS is equal to thesum of the capacity of all transferred resources from hospitals ofthe considered network to the CSS:

CAPtransj,t = CAPV ·

tk=1

AVj,k ∀j = 1 . . .N + M. (10)

Constraints (10) determine the transportation capacity of the CSSas a function of the total number of purchased vehicles:

QMr,i ≥

Tt=1

N+Mj=1

(TRMr,i,j,t + NMr,i,t)

∀i = 1 . . .N, ∀r = 1 . . . R (11)

QMr,i ≤

Tt=1

N+Mj=1

(TRMr,i,j,t + NMr,i,t + M2 · (1 − Yi,j,t))

∀i = 1 . . .N, ∀r = 1 . . . R.

Constraints (11) specify that the number of material resourcesrelocated from one hospital to the CSS added to the number ofsold ones must be equal to the number of these resources initiallyavailable in the considered hospital:

VRMr,i,j,t ≤ QMr,i · Yi,j,t ∀t = 1 . . . T ,∀j = 1 . . .N + M, ∀i = 1 . . .N, ∀r = 1 . . . R (12)

VRHw,i,j,t ≤ QHw,i · Yi,j,t ∀t = 1 . . . T ,∀j = 1 . . .N + M, ∀i = 1 . . .N, ∀w = 1 . . .W .

Constraints (12) express that resources (material as well as humanresources) are transferred from hospital i to the CSS only if thecentralization option is chosen:

VRMr,i,j,t ≥ VRMr,i,j,t+k − M3 ·1 − Yi,j,t

(13)

VRMr,i,j,t ≤ VRMr,i,j,t+k ∀t = 1 . . . T ,∀i = 1 . . .N, ∀r = 1 . . . R,∀j = 1 . . .N + M, ∀k = 1 . . . T − t

VRHw,i,j,t ≥ VRHw,i,j,t+k − M4 ·1 − Yi,j,t

VRHw,i,j,t ≤ VRHw,i,j,t+k ∀t = 1 . . . T , ∀i = 1 . . .N,

∀w = 1 . . .W , ∀j = 1 . . .N + M,∀k = 1 . . . T − t.

Constraints (13) specify that if the centralization option is retained,the number of human and material resources required at the CSSis constant from the period of the centralization until the end ofthe considered horizon. M3 can be set to the maximum number of

available material resources and M4 can be set to the maximumnumber of available human resourcesVRMr,i,j,t+1 ≤ TRMr,i,j,t+1 + M3 ·

1 −

Yi,j,t+1 − Yi,j,t

(14)

VRMr,i,j,t+1 ≥ TRMr,i,j,t+1 ∀t = 1 . . . T − 1,∀i = 1 . . .N, ∀r = 1 . . . R, ∀j = 1 . . .N + M

VRMr,i,j,1 ≤ TRMr,i,j,1 + M3 ·1 − Yi,j,1

VRMr,i,j,1 ≥ TRMr,i,j,1 ∀i = 1 . . .N,

∀r = 1 . . . R, ∀j = 1 . . .N + M

VRHw,i,j,t+1 ≤ TRHw,i,j,t+1 + M4 ·1 −

Yi,j,t+1 − Yi,j,t

VRHw,i,j,t+1 ≥ TRHw,i,j,t+1 ∀t = 1 . . . T − 1,

∀i = 1 . . .N, ∀w = 1 . . .W , ∀j = 1 . . .N + M

VRHw,i,j,1 ≤ TRHw,i,j,1 + M4 ·1 − Yi,j,1

VRHw,i,j,1 ≥ TRHw,i,j,1 ∀i = 1 . . .N,

∀w = 1 . . .W , ∀j = 1 . . .N + M.

Constraints (14) define the number of material resources to berelocated and the number of human resources to be transferredfrom each hospital to the CSS:

TRMr,i,j,t+1 ≤ QMr,i ·Yi,j,t+1 − Yi,j,t

∀t = 1 . . . T ,

∀j = 1 . . .N + M, ∀i = 1 . . .N, ∀r = 1 . . . R (15)

TRMr,i,j,1 ≤ QMr,i · Yi,j,1 ∀j = 1 . . .N + M,∀i = 1 . . .N, ∀r = 1 . . . R

NMr,i,t+1 ≤ QMr,i ·

N+Mj=1

Yi,j,t+1 − Yi,j,t

∀t = 1 . . . T − 1,

∀i = 1 . . .N, ∀r = 1 . . . R

NMr,i,1 ≤ QMr,i ·

N+Mj=1

Yi,j,1 ∀i = 1 . . .N,

∀r = 1 . . . R

TRHw,i,j,t+1 ≤ QHw,i ·Yi,j,t+1 − Yi,j,t

∀t = 1 . . . T − 1,

∀j = 1 . . .N + M, ∀i = 1 . . .N, ∀w = 1 . . .W

TRHw,i,j,1 ≤ QHw,i · Yi,j,1 ∀j = 1 . . .N + M,∀i = 1 . . .N, ∀w = 1 . . .W .

Constraints (15) ensure that if the centralization option is chosenat period t , then the number of resources needed by the CSS shouldbe transferred once at the beginning of that period:Yi,j,t ∈ {0, 1} ∀i = 1 . . .N, ∀j = 1 . . .N + M, ∀t = 1 . . . T (16)Zi,t ∈ {0, 1} ∀i = 1 . . .N, ∀t = 1 . . . T

TRMr,i,j,t ≥ 0 ∀i = 1 . . .N, ∀j = 1 . . .N + M,∀r = 1 . . . R, ∀t = 1 . . . T (17)

TRHw,i,j,t ≥ 0 ∀i = 1 . . .N, ∀j = 1 . . .N + M,∀w = 1 . . .W , ∀t = 1 . . . T

VRMr,i,j,t ≥ 0 ∀i = 1 . . .N, ∀j = 1 . . .N + M,∀r = 1 . . . R, ∀t = 1 . . . T

VRHw,i,j,t ≥ 0 ∀i = 1 . . .N, ∀j = 1 . . .N + M,∀w = 1 . . .W , ∀t = 1 . . . T

AVj,t ≥ 0 ∀j = 1 . . .N + M, ∀t = 1 . . . TCAPMr,j,t ≥ 0 ∀j = 1 . . .N + M, ∀r = 1 . . . R, ∀t = 1 . . . TCAPHw,j,t ≥ 0 ∀j = 1 . . .N + M, ∀w = 1 . . .W , ∀t = 1 . . . TNMr,i,t ≥ 0 ∀i = 1 . . .N, ∀r = 1 . . . R, ∀t = 1 . . . TCAPtransj,t ≥ 0 ∀j = 1 . . .N + M, ∀t = 1 . . . T

Xp,i,j,t ≥ 0 ∀i = 1 . . .N, ∀j = 1 . . .N + M,∀p = 1 . . . P, ∀t = 1 . . . T (18)

XHp,i,t ≥ 0 ∀i = 1 . . .N, ∀p = 1 . . . P, ∀t = 1 . . . TIp,i,t ≥ 0 ∀i = 1 . . .N, ∀p = 1 . . . P, ∀t = 1 . . . T .

H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85 81

Constraints (16)–(18), respectively, specify binary, integer andpositive decision variables.

4. Solution method

In this paper, we are proposing to investigate the resolution ofthe proposed MILP by commercial solvers which are increasinglyknown for their power to solve these kinds of mathematicalmodels. More precisely, we propose here to implement and solveour model with the solver IBM ILOG CPLEX 12.2.

Obviously, the first solution method that we are proposing toexamine consists in the proprietary branch and bound algorithmused by the solver. Some cuts (such as Gomory fractional cutsor flow cuts, etc. [20]) are automatically generated by the solver.In this case, the obtained solution will be referred to as ‘‘defaultsettings solution’’.

Besides, the solver can be configured to aggressively generatethe cuts that could speed up the resolution. Furthermore, specificvalid cuts can be directly added to strengthen the model formu-lation. These cuts generally permit the speeding up of the resolu-tion by ‘‘cutting’’ away regions that contain no feasible solutions[21,22]. In the following, we propose to consider some valid cutsrelated to capacity dimensioning. These cuts will be referred to ascustomized cuts and are defined by Eqs. (19) and (20).Cuts based on the minimum number of material resources:

Ni=1

CAPMr · VRMr,i,j,t

≥

Mint=1...T

Ni=1

Pp=1

Dp,i,t · Vp

· Yi,j,t

∀r = 1 . . . R, ∀j = 1 . . .N + M, ∀t = 1 . . . T . (19)

Cuts based on the minimum number of human resources required:Ni=1

CAPHw · VRHw,i,j,t

≥

Mint=1...T

Ni=1

Pp=1

Dp,i,t · δw,p

· Yi,j,t

∀w = 1 . . .W , ∀j = 1 . . .N + M, ∀t = 1 . . . T . (20)

Cuts (19) calculate theminimumnumber ofmaterial resources (forinstance the minimum number of autoclaves) required to satisfythe total network demand. Cuts (20) are similar to cuts (19) asthey define the minimum number of human resources necessaryto ensure the total network demand.

First, we propose to solve theMILP using CPLEX default settingswhile enabling automatic cut generation. We note the cuts mostlyused in the first resolution. Then, we solve the MILP while usingaggressive settings for the generation of these cuts. Third, in addi-tion to aggressive settings for cut generation, we integrate in themodel the proposed customized cuts. The objective is to point outthe interest of using CPLEXparameter settings and customized cutsin order to speed up the resolution of the considered combinatorialoptimization problem. Even if most location–allocationmodels aredefined as NP-hard optimization problems [23], we propose first toexamine optimal settings solutions.

5. Experimentation and results

In order to evaluate the proposed mathematical model,probable scenarios, for a case based on real life situations withina hospital network environment are developed and tested. Eachscenario is characterized by the number of hospitals in the network

and the demand. All scenarios are based on a real case study inFrance: a nine-hospital network located in the same region [4].The transportation time between any couple of hospitals or anyhospital and the CSS is less than 1 h. 6 hospitals are locatedin Grenoble (H1, H2, H3, H4, H5 and H6), and 3 hospitals arelocated in Chambéry (H7, H8 and H9). In addition to the ninehospitals, the CSS may be located in a new site. Five-year planninghorizon is considered (the period is the year). Two types of humanresources essential to the completion of sterilization activity areconsidered: technicians (HR1) and sterilization nurses (HR2). Twomaterial resources are taken into account: autoclaves (MR1) andautomatic washing machines (MR2). The demand of each hospital,fixed costs, the number of available resources in each hospital,material resource costs, vehicle cost were extracted from [4]. Twoproducts are considered: full-sized and half-sized standardizedreusable rigid containers. All surgical instruments are supposedto be placed in a standardized reusable rigid container for thepackaging, transportation and storage. Variable production costsare estimated based on the previous studies dealing with costanalysis of the sterilization process [24]. The work time neededto process each product is extracted from [25]. Transportationcapacity is estimated under the assumption that a maximum of3 trips are performed per one vehicle over one day. The capacityof autoclave and automatic washing machine are determined on abasis of 8 cycles a day.

Variable production cost is the same regardless of sterilizationservice configuration; it depends on the volume of the sterilizeditem. The fixed costs are mostly incurred by personnel salaries,building, furniture, equipments, maintenance and vehicles. Thecost of building is amortized over 20 years, the cost of equipmentis amortized over 10 years and the cost of a vehicle is amortizedover 3 years [4]. In the considered case study, the building andequipments of each hospital’s sterilization department are knownand will not be completely amortized over the five next years. Incase of centralization, the current in-house sterilization buildingswill be used by other hospital’s services; equipments will be eithertransferred to the CSS or sold. The cost estimation of CSS building isbased on the average daily volume of sterilized items. The used rulestates that 60 m2 surfaces are requested per m3 of daily sterilizeditems [4]. Variable transportation cost of one item depends on thedistance between the hospital and the CSS and includes the pay oftruck driver and fuel cost.

First, 21 scenarios based on real life data are developed andtested in order to investigate problem complexity and point outthe interest of using CPLEX aggressive settings for cut generationand adding customized cuts. These scenarios are obtained from 7basic scenarios: 3 scenarios consider groups of 3 hospitals, referredto as 3-G1 (H1, H2 and H3), 3-G2 (H4, H5 and H6) and 3-C (H7, H8and H9); 3 scenarios consider groups of 6 hospitals referred to as6-G12 (H1, H2, H3, H4, H5 and H6), 6-CG1 (H1, H2, H3, H7, H8 andH9) and 6-CG2 (H4, H5, H6, H7, H8 and H9), each one is includingthe hospitals of two of the previous three-hospital groups; andone scenario considers the 9 hospitals (H1, H2, H3, H4, H5, H6, H7,H8 and H9) and is referred to as 9-CG12. For each basic scenariowe examine several demand patterns: (a) constant demand;(b) increasing demand and (c) decreasing demand.

The MILP generated by these scenarios are solved with IBMILOG CPLEX 12.2 on a PC Pentium IV, 3.0 GHz. The three solutionmethods presented above are used. Default setting solution isdenoted by DSS, the solution method using aggressive settings forcut generation is denoted by ASCG and the solution method usingboth aggressive settings for cut generation and customized cuts isdenoted by ASCG + CC.

For scenarios 1–9 (considering three hospitals) the associatedMILPs comprise 886 variables, 3557 constraints and 80 customizedcuts can be added. For scenarios 10–18 (considering six-hospitals)

82 H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85

Table 1Computational results.

Scenario DP HG CPU1 CPU2 CPU3 Proposed configuration Z

3 hospitals

1 a3-G1

136 33 2 Centralization at hospital H1 from period 1 9649145.592 b 2 2 2 Decentralization 9919393.543 c 76 5 9 Centralization at hospital H1 from period 2 8909841.4694 a

3-G259 12 2 Centralization at hospital H4 from period 1 4742686.09

5 b 93 54 19 Centralization at hospital H4 from period 1 5165481.666 c 111 9 5 Centralization at hospital H4 from period 2 4434886.047 a

3-C2 2 1 Centralization at hospital H8 from period 1 5500852.59

8 b 2 2 2 Centralization at hospital H8 from period 1 6235369.409 c 2 2 2 Centralization at hospital H8 from period 1 5477991.40

6 hospitals

10 a6-G12

12500a 13340a 24 Centralization at hospital H1 from period 1 13300873.1411 b 12025a 11483a 37039a Centralization at hospital H1 from period 2 15512824.7712 c 14440a 17918a 495 Centralization at hospital H1 from period 2 12639863.6613 a

6-CG13094a 22 6 Centralization at hospital H1 from period 1 14817184.53

14 b 12017 48 51 Centralization at hospital H1 from period 1 16970642.6815 c 36 46 24 Centralization at hospital H1 from period 1 14753397.1416 a

6-CG1749 32 20 Centralization at hospital H4 from period 1 9857890.64

17 b 61 41 14 Centralization at hospital H4 from period 1 11531690.4418 c 6920 48 28 Centralization at hospital H4 from period 1 9818769.83

9 hospitals19 a

9-CG121162a 60 91 Centralization at hospital H1 from period 1 18816862.06

20 b 9037a 33009a 13358a Centralization at hospital H1 from period 1 21692793.4221 c 1501a 113 144 Centralization at hospital H1 from period 1 18735016.18

a The obtained solution after the indicated computational time is not necessarily optimal (the solver is aborted).

the associated MILPs comprise 2731 variables, 12 074 constraintsand 140 customized cuts can be added. For scenarios 19 to 21(considering 9 hospitals) the associated MILPs comprise 5566variables, 26 441 constraints and 200 customized cuts can beadded.

Table 1 presents some computational results. For each scenariothe following information is provided: the hospital group (HG), thedemand pattern (DP), the proposed configuration and the value ofthe objective function in euros (Z). In addition, for each solutionmethod, the computational time in seconds (CPU1 for DSS, CPU2for ASCG and CPU3 for ASCG+CC) are reported.

For scenarios 1–9, regardless the used solution method, anoptimal solution is found very rapidly (in few seconds) for alldemand patterns. Based on the obtained results, we can state thatthe resolution of the considered location–allocation problem is nottime consumingwhen a group of 3 hospitals is under investigation.Problem complexity appears for networks including 6 and 9hospitals as for themajority of these scenarios computing time hasexploded. However, it is clear that the use of aggressive settings forcut generation and customized cuts could reduce resolution timesignificantly and permits us to point out the optimal solution. Theuse of cuts and particularly customized cuts permits us to solvethe MILP to optimality for 90% of the aforementioned scenarios.For scenarios 10 and 12, the computational time is significantlyreduced only with the addition of the customized cuts.

Moreover, we note that for all scenarios, the three solutionmethods provide the same solution. But, when default settingssolution method is used, the obtained solution is not identified asoptimal, adversely to the other solution methods (see for instancescenario 13). This solution is often found after a short computingtime, but its optimality is not proven. When customized cuts areused, the starting lower bound is higher which permit us to provesolution optimality. This highlights the interest of developingcustomized cuts and solver aggressive settings for cut generationparticularly for problems considering large networks. Finally, wecan note that computational time could vary widely depending ondemand pattern as shown for scenarios 10, 11 and 12.

For scenarios 7, 8, 9, 11, 13, 14, 15, 16, 17, 18, 19, 20 and 21,a decentralized configuration is infeasible because some hospitalsof the network are under capacitated: some critical resourcesin at least one hospital are not sufficiently available to meetthe demand. In scenario 7, for instance, the existing human

resources in hospital H8 are not able to satisfy the demand. Withcentralization, it is not needed to hire additional resources. Inscenarios 3, 6 and 12, centralization is the optimal configurationstarting at period 2.We note that these scenarios are characterizedby a decreasing demand. Centralizing the sterilization services atperiod 2 permits better optimization in resource sharing over theupcoming periods (fewer resources are needed and transferredto the CSS). In scenario 11, even though characterized by anincreasing demand, the centralization is proposed from period 2:this indeed permits a better utilization of the transferred resourceswith regard to a centralization starting from period 1. In scenario2, with an increasing demand pattern, the optimal configurationis the decentralization instead of the centralization obtained inscenario 1, with a constant demand pattern. When the demandincreases, the utilization of available resources in each hospitalis improved; this justifies the interest for decentralization. Inaddition, decentralization permits us to avoid transportation andtransferring costs.

Table 2 presents some details concerning the optimal config-uration proposed for scenario 10. The number of the existing re-sources available in each hospital of the network is presented. Foreach hospital, the following information is provided: the numberof existing resources (# Exist.), the number of the resources to betransferred to the CSS when the centralization is the retained con-figuration (# Trans.).

For scenario 10 (as it has been shown previously in Table 1), thecentralization of the sterilization service represents the optimalconfiguration from the first period of the planning horizon. TheCSS is located in hospital H1. Indeed the demand of hospital H1is much larger than the demand of the other hospitals. Locatingthe CSS in hospital H1 incurs the smallest fixed cost andminimizestransportation costs as well as transferring costs. We can remarkthat the resource gain is of 22% for MR1, 71% for MR2, 64% for HR1and 53% for HR2. A cost saving of 12% is also achieved with regardto the current situationwhere all hospitals perform sterilization in-house. The human resources that are not assigned to the CSS willbe redeployed in other hospital services. The material resourceswhich are not transferred to the CSS will be sold.

In the following, further analysis of the considered problemis conducted. 9 scenarios are developed in order to examine theimpact of some parameters on the network design. In this analysis,

H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85 83

Table 2Resource gain achieved by the optimal configuration based on scenario 10.

Hospital MR1 MR2 HR1 HR2# Exist. # Trans. # Exist. # Trans. #Exist. #Trans. # Exist. #Trans.

H1 10 10 2 2 23 15 20 20H2 1 0 1 0 3 0 5 0H3 1 1 1 0 3 0 3 0H4 3 3 1 0 5 0 6 0H5 1 0 1 0 4 0 5 0H6 2 0 1 0 4 0 4 0

Table 3Obtained results for scenarios 22–30.

DP Proposed configuration Z

Scenario 22 a Centralization at hospital H1 from period 2 18507142.04Scenario 23 b Centralization at hospital H1 from period 4 20089742.80Scenario 24 c Centralization at hospital H1 from period 2 17289528.90Scenario 25 a Centralization at hospital H1 from period 1 18816862.08Scenario 26 b Decentralization 20358111.58Scenario 27 c Centralization at hospital H1 from period 2 17606328.45Scenario 28 a Centralization at hospital H1 from period 1 18658862.08Scenario 29 b Centralization at hospital H1 from period 4 20903142.81Scenario 30 c Centralization at hospital H1 from period 2 17454328.90

the 9 hospitals are considered. The generated MILPs are solvedusing both aggressive settings for cut generation and customizedcuts. The obtained results for the 9 scenarios are reported inTable 3.

Hospitals H7, H8 and H9 are short of human resources tomeet their future internal demand. That is why the proposedconfiguration for all networks including these hospitals was thecentralization. Subsequently, we propose first to examine theimpact of hiring additional human resources on the networkconfiguration. In scenarios 22, 23 and 24, the number of HR1 andHR2 inH7,H8 andH9 is modified in such away that each hospital isable to perform internally its sterilization activity for the first year(we suppose that one HR2 is hired in H7 and H9; and one H1 andone HR2 are hired in H8).

In case of hiring the mentioned human resources, the central-ization starting at period 2 becomes the proposed configuration forscenarios 22 and 24. In scenario 22, the produced quantity over thefirst periodmust cover both the demand and the safety stock. Post-poning the centralization to period 2 allows the transfer of fewerresources to the CSS and guarantees a better utilization of these re-sources over periods 2, 3, 4 and 5. For scenario 23, the centraliza-tion starting at period 4 is the proposed configuration. This can beexplained again by the unavailability of human resources requiredto cover the demand of periods 4 and 5.

Furthermore, the number of human resources was progres-sively increased in order to point out configuration change. Wenoted that with an increase of 12% in the number of human re-sources, the centralization (at hospital 1 from period 1) becomesthe optimal configuration. This number of resources is used in sce-narios 25, 26 and 27.

In scenario 26, the decentralization can be explained by theincrease of resource utilization notably over the last periods ofthe planning horizon. Opting for the centralization in scenario 26means that some resources are transferred to the CSS becausethey are only needed to satisfy the demand of period 5; theseresources are not necessarily well utilized over the other periodsof the planning horizon.

We consider scenario 22 while progressively decreasing thenumber of human resources in H1. With a decrease rangingbetween 2% and 21%, the decentralization is the optimal solution.With a decrease larger than 21% (27% decrease in the number ofHR1 and 15% decrease in the number of HR2), H1 will not be ableto satisfy the demand in-house and the centralization starting atperiod 1 becomes the optimal configuration.

The number of human resources available in each hospital hasan important impact on network configuration. Obviously, a num-ber of human resources exceeding the requirements of sterilizationactivity in each hospital favour centralization. Meanwhile a lackof resources in any hospital of the network can be solved by cen-tralization. But, in this case, centralization is not necessarily betterthan opting for recruiting the requested resources as we can noticewhile comparing scenarios 19 and 22.

Similarly to human resources, we propose to examine the im-pact of the number ofmaterial resources onnetwork configuration.While maintaining the same number of human resources consid-ered in scenario 22, the number ofmaterial resources was progres-sively increased. We noted that with an increase of 64% of the totalnumber of MR1 and 30% of MR2, the centralization (at hospital 1from period 1) becomes the optimal configuration. This number ofresources is used in scenarios 28, 29 and 30.

In scenario 22, the average utilization rate of MR1 is 75% andthe average utilization rate of MR2 is 27%. The centralizationbecomes the optimal solution from period 1 when these ratesreach respectively, 47% and 21% (scenario 28). Postponing thecentralization to period 2 and 3 in scenario 29 and 30 is to ensurebetter utilization of the transferred/shared resources.

We can conclude that network configuration is sensitive tothe number of material resources available in each hospital ofthe network. The centralization is favoured by an increase ofresources’ number, adversely to the decentralization. Obviously,decentralization will not be the proposed configuration if thedecrease generates a lack in resources so that one hospital will notbe able to satisfy its demand internally.

While considering scenario 22, we conducted a sensitivityanalysis study on some cost parameters. Firstly, we varied thetransportation cost. The proposed configuration changed onlywhen variable transportation costs were multiplied by 5.6:decentralization becomes the optimal configuration. However,any decrease in the transportation cost favours the centralizationat hospital H1 from period 2. The network configuration is notsensitive to transportation cost. Secondly, we varied the fixedcost of the CSS. This cost is primarily composed of the costof the building and its maintenance. With an increase of 5%,decentralization becomes the optimal configuration. Nonetheless,a decrease of CSS fixed cost does not have any effect on thenetwork configuration: centralization at hospital H1 from period2 remains the proposed solution. The network configuration is

84 H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85

very sensitive to CSS fixed cost. The estimation of this costmust be made with accuracy. An error of 5% could mislead thedecision. Besides, we varied the fixed cost of in-house sterilization.Similarly to CSS fixed cost, this cost is mainly composed of thecost incurred by the building and its maintenance. From a decreaseof 8%, decentralization becomes the optimal configuration. Aftermultiplying the in-house sterilization fixed cost by a factor of 2.6,the centralization at hospitalH1 fromperiod 1becomes the optimalconfiguration.We can state that network configuration is sensitiveto a decrease in in-house sterilization fixed cost. However, if theconfiguration decision involves hospitals which have operated thesterilization activity for some time, then the incurred in-housesterilization fixed cost should be known with some accuracy.Emphasis should be rather placed in CSS fixed cost estimation byreferring to experts in the field.

The number of vehicles requested to ensure transportationbetween hospitals and the CSS is based on the annual producedquantities and the capacity of one vehicle. The latter wasdetermined under the assumption that a vehicle ensures 3 trips aday. Obviously, vehicle dimensioning, at this level, does not takeinto account short-term variation of the demand. The questionhere is what if the number of vehicles acquired is not enoughto cover the demand on a daily basis? To answer this question,we propose to assess the impact of vehicle capacity decreaseon network configuration. A decrease in one vehicle capacitycould be seen as lesser trips per day but more hospitals insertedin the route. With a decrease of 18% in vehicle capacity thedecentralization becomes the proposed solution for scenario 22.For a decrease ranging from 2% to 17%, centralization in hospitalH1from period 2 remains the proposed configuration but 6 vehiclesare required instead of 5. It means that the need for one morevehicle will not affect the quality of the decision at this level. Itis clear, however, that operating and managing the CSS will bringadditional difficulties and challenges to handle the demand ofthe different hospitals of the network on a daily basis. Integratedproduction and vehicle routing scheduling approach should bedeveloped and adequately implemented to enable conductingoperating theatre activities safely and smoothly. This supposesappropriate use of Information Technology (such as RFID, [3])to ensure timely information about operating theatre schedules,hospital demand and inventory levels.

The proposed location–allocation model serves as a decisionmaking tool for the configuration of hospital sterilization servicewithin a hospital network. Basically, it is exploited while consid-ering one CSS and two alternatives, centralization vs. decentral-ization. However, it can be used more widely (while designing aproper experimentation) in the perspective to design the networkconfiguration while considering, for example, several CSSs andseveral options of centralization vs. decentralization. Besides, theproposed model can be used as a what-if analysis tool to identifyunder which conditions the centralization vs. decentralization isthe best configuration. For example, if hospitals H7, H8 and H9 hirethe human resources indicated above and the demand has a con-stant pattern, centralization starting at period 2 becomes the pro-posed configuration by the model.

6. Conclusion

In this paper, we proposed an optimization model for theconfiguration of hospital sterilization service within a hospitalnetwork. The model examines two alternatives: the centralizationof the service in a common CSS or the decentralization, meaningthat each hospital keeps in-house sterilization service. Theproblem is formulated as a location–allocation model and permitsus to determine the location and size of the CSS.

The proposed model is solved by commercial solver IBMILOG CPLEX 12.2. Three solution methods are investigated: (1) aresolution based on default parameters of CPLEX; (2) a resolutionbased on default parameters of CPLEX while enabling aggressivesettings for cut generation; (3) a resolution using aggressivesettings for cut generation and customized cuts. In order toevaluate the proposedmathematicalmodel, 21 scenarios, for a casebased on real life situationswithin a hospital network environmentare tested. The experimentation of the proposed model highlightsthe interest of the centralization of the sterilization service withinhospital network through resource sharing and optimization.

Moreover, the obtained results show that the proposed MILPcan be solved to optimality for moderate size scenarios (like thoseused in our study) with the use of commercial MILP solvers and theaddition of appropriate customized cuts to the original model.

Further analysis was conducted and pointed out how networkconfiguration is sensitive to the number of human and materialresources available in each hospital of the network as well as tosterilization fixed costs.

The proposed model can also be used in the perspective todesign the network while considering the case of a mixed configu-rationwhere someof the network hospitalsmove to the centraliza-tion and the others keep their own sterilization services. For that,an appropriate experimentation scheme must be designed.

In further work, the case of a partial centralization where someproducts are performed in the hospital and the others are sent tothe centralized sterilization service looks to be an interesting al-ternative to study. In the former case, static vs. dynamic alloca-tion resource strategies can be evaluated and compared. Anotherprospect of this study is to generalize the obtained results and toestablish rules for the choice centralized vs. distributed steriliza-tion service depending on other criteria like order-to-delivery timeand service quality level.

References

[1] F. Lega, Strategies for multi-hospital networks: a framework, Health ServicesManagement Research 18 (2005) 86–99.

[2] Standard AFNOR FD S98–135, Stérilisation des dispositifs médicaux-Guidepour la maîtrise des traitements appliqués aux dispositifs médicaux réutilis-ables, ISSN: 0335-3931, 2005.

[3] J.V. Klundert, P. Muls, M. Schadd, Optimizing sterilization logistics in hospitals,Health Care Management Science 11 (2008) 23–33.

[4] iQualis santé & ARES santé, ARHRA Réflexion sur l’opportunité et la faisabilitéd’une mutualisation des moyens de stérilisation des établissements de l’ArcAlpin, 2005.

[5] O. Aptel, H. Pourjalali, Improving activities and decreasing costs of logistics inhospitals: a comparison for US and French hospitals, The International Journalof Accounting 36 (2001) 65–90.

[6] I. Or, W.P. Pierskalla, A transportation location–allocation model for regionalblood banking, AIIE Transactions 11 (2) (1979) 86–95.

[7] C. Stummer, K. Doener, A. Focke, K. Heidenberger, Determining location andsize of medical departments in a Hospital Network: a multi-objective decisionsupport approach, Health Care Management Science 7 (2004) 63–71.

[8] E.D. Gunes, H. Yaman, Modelling change in a health system: implication onpatient flows and resource allocations, Clinical and Investigative Medicine 28(6) (2005) 331–333.

[9] F. Pasin, M.H. Jobin, J.F. Cordeau, Application d’une approche de simulationpour analyser le partage de ressources entre des organisations du secteur dela santé, Cahier de recherche n◦ 01–02, HEC Montréal (2001).

[10] A. Calvo, H. Marks, Location of health care facilities: an analytical approach,Socio-Economic Planning Sciences 7 (1973) 407–422.

[11] S.C.K. Chu, L. Chu, A modelling framework for hospital location and serviceallocation, International Transactions in Operational Research 7 (2000)359–368.

[12] S. Rahman, D. Smith, Use of location–allocation models in health servicedevelopment planning in developing nations, European Journal of OperationalResearch 123 (2000) 437–452.

[13] X. Chao, L. Liu, S. Zheng, Resource allocation in multisite service system withintersite customer flow, Management Science 49 (2003) 1739–1752.

[14] P.R. Harper, A.K. Shahani, J.E. Gallagher, C. Bowie, Planning health serviceswith explicit geographical considerations: a stochastic location–allocationapproach, The International Journal of Management Science 33 (2005)141–152.

[15] A.N. Elshafei, An approach to locational analysis, Operational ResearchQuarterly 26 (1975) 167–181.

H. Tlahig et al. / Operations Research for Health Care 2 (2013) 75–85 85

[16] S. Georges, Le contrôle (logistique) dans le processus de stérilisationhospitalière, Forum 3 (2007) 7–11.

[17] H. Tlahig, A. Jebali, H. Bouchriha, A two-phased approach for the centralizationvs. decentralization of the hospital sterilization department, European Journalof Industrial Engineering 3 (2) (2009) 227–246.

[18] H. Tlahig, H. Bouchriha, A. Jebali, P. Ladet, A mathematical model for theinternalization/ externalization decision of the hospital sterilization process,in: Post-Conference Proceedings of the 33rd International Conference onOperational Research Applied to Health Services, -ORAHS’07, 2008.

[19] F. Reymondon, B. Pellet, E. Marcon, Optimization of hospital sterilizationcosts proposing new grouping choices of medical devices into packages,International Journal of Production Economics 112 (2008) 326–335.

[20] IBM ILOG Cplex V12.1 Parameters Reference Manual, International BusinessMachines Corporation 1987, 2009.

[21] G. Cornuéjols, Valid inequalities formixed integer linear programs,Mathemat-ical Programming (2006).

[22] M. Paquet, A. Martel, B. Montreuil, A manufacturing network design modelbased on processor and worker capabilitie, International Journal of ComputerIntegrated Manufacturing 17 (12) (2007) 117–125.

[23] X. Li, A. Gar-On Yeh, Integration of genetic algorithm in GIS: a facility locationmodelling, Journal of the Eastern Asia Society for Transportation Studies 6(2005) 2908–2920.

[24] H. Tlahig, H. Bouchriha, A. Jebali, P. Ladet, N. Taggiasco, Etude del’externalisation du secteur de stérilisation hospitalière: une analyse parles coûts, Gestions Hospitalières 483 (2009) 105–110.

[25] S. Bauler, C. Combe, M. Piallat, C. Laurencin, H. Hida, Proposal of a costingmethod for the provision of sterilization in a public hospital, AnnalesPharmaceutiques Françaises 69 (11) (2011) 209–213.