Distribution of Medication Considering Information ... (POM) Malawi Drug... · tain transportation...

Distribution of Medication Considering Information,Transshipment, and Clustering: Malaria in Malawi

Hoda ParvinModeling and Optimization, Amazon Co., Seattle, Washington 98109, USA, [email protected]

Shervin BeygiDigital Aviation and Analytics, Boeing Global Services, Seattle, Washington 98019, USA, [email protected]

Jonathan E. HelmDepartment of Supply Chain Management, W.P. Carey School of Business, Arizona State University, Tempe, Arizona 85287, USA,

[email protected]

Peter S. LarsonDepartment of Epidemiology, Nagasaki University, Nairobi, Kenya, [email protected]

Mark P. Van Oyen*Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA, [email protected]

M alaria is a major health concern for many developing countries. Designing strategies for efficient distribution ofmalaria medications, such as Artemesinin Combination Therapies, is a key challenge in resource constrained coun-tries. This paper develops a solution methodology that integrates strategic-level and tactical-level models to better managepharmaceutical distribution through a three-tier centralized health system, which is common to sub-Saharan African coun-tries. At the strategic level, we develop a two-stage stochastic programming approach to address the problem of demanduncertainty. In the first stage, an initial round of shipments is sent before the malaria season to each local clinic from dis-trict hospitals, which receive medications from regional warehouses. After the malaria season begins, a recourse action istriggered to avoid shortages in the form of (i) lateral transshipment or (ii) delayed shipment. The optimal solutions devel-oped by the strategic model identify small clinic clusters possessing exclusive transshipment policies. Therefore, wedecompose the problem at the tactical level, solving each clinic cluster independently using a Markov decision processapproach to determine optimal periodic transshipment policies. A case study of our proposed distribution system is per-formed for 290 facilities controlled by the Malawi Ministry of Health. Numerical analysis of Malawi’s distribution systemindicates that our proposed cluster-based decomposition method could near optimally reduce shortage incidents. More-over, such an approach is robust to challenges of developing countries such as slow paper-based inventory review, uncer-tain transportation infrastructure, the need for equitable distribution, and seasonal and correlated demand associated withmalaria transmission dynamics.

Key words: humanitarian logistics; malaria treatment distribution; stochastic programming; Markov decision processes;health care operationsHistory: Received: July 2013; Accepted: September 2017 by Sergei Savin, after 4 revisions.

1. Introduction

Due to a combination of intense poverty and environ-mental and local weather conditions, Malawi suffersfrom an exceptionally high burden of malaria. Dzin-jalamala (2009) indicates that all Malawians live atyear round risk for malaria, although incidence peaksduring the December–May rainy season. The WorldHealth Organization (WHO) (2014) estimated that atleast a third of all medical consultations are malariarelated and a recent Malaria Indicator Survey showed

that more than a third of all Malawians test positivefor recent infections at any given time (see MalawiMinistry of Health 2012). Malaria spending makes upa major portion of total expenditures on health inMalawi, crowding out spending on other conditions.Despite decades of elimination and control efforts,malaria remains one of the most common causes ofchild morbidity and mortality worldwide. Accordingto the WHO, there were nearly 207 million suspectedmalaria cases in 2012. In addition to imposing animmense burden on health and welfare, malaria is a

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Vol. 27, No. 4, April 2018, pp. 774–797 DOI 10.1111/poms.12826ISSN 1059-1478|EISSN 1937-5956|18|274|000774 © 2017 Production and Operations Management Society

http://orcid.org/0000-0002-8685-7843http://orcid.org/0000-0002-8685-7843http://orcid.org/0000-0002-8685-7843

major impediment to the economic development ofimpoverished nations (see Gallup and Sachs 2001,Malaney et al. 2004). Thus, for the past decade,malaria control and elimination have been a priorityfor international and domestic health agencies, non-governmental organizations (NGOs), and health min-istries. Malaria is a treatable disease, and promptadministration of medicines for uncomplicatedmalaria such as Artemesinin Combination Therapies(ACTs) can prevent the most severe outcomes. How-ever, stock outs of essential medications are commonin developing countries, particularly those facing dis-proportionate malaria burdens (see PMI 2014, Sudoiet al. 2012). Problems in regional supply chains havebeen noted as a major barrier to timely and efficientdistribution of malaria medications to meet localdemand (see Bateman 2013, Daniel et al. 2012, Tetteh2009).

1.1. Malawi’s Existing Health SystemMalawi’s public health system is a three-tiered net-work consisting of central warehouses and regionalhospitals in the first tier, district hospitals in the sec-ond tier, and primary health centers and local com-munity clinics in the third tier. Each tier receivessupplies from and answers to the tier above it withthe exception of the central warehouses and regionalhospitals which answer directly to the Ministry ofHealth, see Figure 1. Distribution of pharmaceuticalsbegins at the Central Medical Stores (CMS) inLilongwe, Malawi, which allocate drugs to the regio-nal hospitals and central warehouses (first tier). First

tier distribution then delivers to district hospitals,which are in turn responsible for supplying primaryhealth centers and local community clinics.In this paper, we employ stochastic programming

and Markov decision models to optimize distributionapproaches and significantly decrease treatmentshortage while limiting transportation costs.

1.2. Operational ChallengesFoster (1991) claims that: (i) proper inventory man-agement of medications in Africa can reduce costs by15%–20% and (ii) transportation of drugs and medicalaid is an especially critical factor in Africa. Accordingto Claeson and Waldman (2000), the efficacy of deliv-ering health care through such systems has been thesubject of debate for decades. Underdeveloped healthsystems that rely on centralized and hierarchical sup-ply chains with a central authority acting as primarydistributor of goods can suffer from many problems.Transportation infrastructure is generally poor, fuel

shortages complicate matters and roads are often inbad condition, especially during the rainy season,when malaria is most prevalent. Cultural issues andregional rivalries lead to inequities in access and sup-ply. This observation is based on one co-author’s onthe ground experience working with malaria inMalawi.Some research has focused on strategies that cir-

cumvent, replace, or radically decentralize publichealth systems, for example, Gallien et al. (2012);however, government sponsored distribution systemsremain the most prominent source of medications inmost developing countries, including Malawi andnearly all sub-Saharan African countries.In this paper, we explore transportation schemes

that combine both strategic and tactical level opera-tions to increase the effectiveness of ACT distributionchannels within the public, centralized supply chainof Malawi. While we study a centralized governmentsupply chain, these methods can also be applied toother problems concerning the distribution supplychains for pharmaceutical products outside of thepublic sector, such as those of the NGOs like JohnSnow Inc (see http://www.jsi.com/). At the strategiclevel, we first develop and solve a large stochasticprogram capable of optimizing ACT delivery to all290 hospitals and clinics that treat malaria in Malawi.We then use this model to investigate the impact oftransshipment and delayed shipment (where someinventory is held back at the higher echelon) on bothtransportation cost/feasibility and on ACT shortages.Applying this model to data obtained from theMalawi Ministry of Health, we find a convenientstructure in the optimal solution to the stochastic pro-gram, from which the problem can be decomposedinto small clusters of clinics with exclusive

Central Medical Storehouse

Regional Storehouses (warehouses and

regional hospitals)

District Hospitals

Local Clinics

Figure 1 Malaria Pharmaceutical Distribution Network in Malawi[Color figure can be viewed at wileyonlinelibrary.com]

Parvin, Beygi, Helm, Larson, and Van Oyen: Medication Distribution: Malaria in MalawiProduction and Operations Management 27(4), pp. 774–797, © 2017 Production and Operations Management Society 775

http://www.jsi.com/

transshipment policies. This observation allows us toimplement a tactical method for transshipment usinga tractable Markov Decision Process model, whichcould not be solved in the absence of clinic clustersdue to the curse of dimensionality. By integratingboth models, we are able to analyze unique featuresof pharmaceutical aid delivery in the developingworld, such as poor road conditions, equity, seasonal-ity of malaria, and paper-based inventory systemsrequiring periodic review. Our approach reducesshortage by 40%–60% compared to the baselinemodel.

2. Literature Review

Literature from a number of areas is relevant to thispaper, including (i) global health and humanitarianresponse literature, and (ii) transshipment and multi-echelon distribution models. In our model, weconsider a two-stage response in the distribution ofmedical supplies (as in disaster preparedness) as wellas dynamic periodic lateral (bidirectional) transship-ment decisions among clinics (the lowest echelon)based on small clusters of nearby clinics that are iden-tified by the higher level two-stage model. Our con-text is distinguished by characteristics that include: acentralized distribution system, three echelons and anetwork of almost 300 stockpoints, non-stationarydemand by month, lost unfilled demand, and hetero-geneous shipping cost parameters enabling distancesand road conditions to be incorporated in the model.

2.1. Global Health and Humanitarian OperationsEmergency response research tends to focus on broadpublic health needs that must be addressed in a rapidand targeted manner after a period of prior planning,often involving inventory prepositioning. Publishedresearch regarding disaster preparedness and emer-gency response is extensive and has been well docu-mented by several survey papers, including Altayand Green (2006), Simpson and Hancock (2009), andde la Torre et al. (2011).Particularly relevant to our methodology are emer-

gency response models that employ two-stagestochastic programming. These approaches involvean initial allocation of resources before a disaster andsubsequent transportation to affected locations after alarge emergency event (see e.g., Mete and Zabinsky2010, Salmer�on and Apte 2010).Models of disaster preparedness and emergency

response share similarities with our work; however,they typically involve rare events with unknown timingthat require a rapid response. The global health oper-ations literature generally encompasses a broader per-spective. Kraiselburd and Yadav (2013) claims thatglobal health supply chains suffer due to lack of

coordination between entities, competing and/ormyopic objectives, and poor supply chain design. Ourpaper touches on these areas specifically with respectto analysis and improvement of ongoing supply chainoperations. Recent work has begun to make the dis-tinction between ongoing and emergency operationsin global health, including Stauffer et al. (2016), thatimplements a stochastic programming model to bal-ance objectives from both perspectives. Jahre et al.(2016) also considers ongoing operations and is com-plementary to our work as the authors consider posi-tioning of global warehouses and distributionnetwork construction. Our work functions on ongoingoperations within the context of an existing network.To place our work within the public-sector supplychain context, we note that Yadav (2007) provides aframework for public-sector supply chains involving:registration, selection, procurement, distribution, anddelivery. We specifically study the area of distributionand delivery.From a funding standpoint, both Gallien et al.

(2016) and Natarajan and Swaminathan (2014) con-sider the impact of funding disbursement on theeffectiveness of prevention and treatment programs,particularly in Africa. Gallien et al. (2016) finds thateffective (frequent) monitoring of resource usage andusing cash buffers rather than regional stock bufferscan improve performance. While we do not directlystudy funding mechanisms, we do analyze the impactof different levels of funding on supply chain perfor-mance. Other mechanisms that affect the delivery ofhumanitarian operations include earmarked funding,Besiou et al. (2014), and armed conflict, Jola-Sanchezet al. (2016). In the case of Malawi, the latter has neverbeen a major issue and the former does not havemuch impact on malaria medication distribution, butnonetheless are important to consider in the broadercontext of global health operations.

2.2. Transshipment and Multi-EchelonDistribution ModelsOur work also contributes to the area of transship-ment research. Paterson et al. (2011) provides a com-prehensive survey of transshipment, identifying areaswhere additional research is particularly needed.Among multiple areas in need of development, theycite the following three: (i) using transshipment toproactively redistribute/balance the stock with multi-ple transshipment epochs, (ii) further work on largernumbers of locations (rather than the typical two orthree), and (iii) larger networks with three or moreechelons. As will be seen, our paper addresses theseareas of need through a combination of modeling, the-oretical analysis, and numerical analysis.Traditionally, the literature on transshipment has

generally addressed problems with only two retailers

Parvin, Beygi, Helm, Larson, and Van Oyen: Medication Distribution: Malaria in Malawi776 Production and Operations Management 27(4), pp. 774–797, © 2017 Production and Operations Management Society

in analytical approaches for tractability (see Patersonet al. 2011 for references). Most of the literatureassumes infinite capacity for replenishment, althoughsome papers model a finite supply or productioncapacity. In our setting, the total amount of medica-tion available is restricted, having been donated orsold to the country in large up-front lots—the methodpreferred by the ministry. Thus, replenishment costsare limited to the cost of shipment or transshipment.A multi-period, multi-location approach is taken in

Robinson (1990), and it shares a number of featureswith our model, such as multiple retailers in a multi-stage optimization setting with random demand andeither backlogging or lost sales. A key feature of thisanalysis is time-stationarity of the model at each per-iod, which is not an appropriate approximation forour setting. We consider non-stationary demand dis-tributions over time; therefore, the control policiesbecome more complex, in part because the multi-period solution does not reduce to a single-periodsolution. Furthermore, the Markov decision process(MDP) approach taken by this paper would be intract-able for our 290 facilities; however, we use the opti-mal solution of our strategic stochastic program todecompose the network into small clinic clusters. Theresulting cluster-level transshipment problems aresolvable by MDP. We use the cluster transshipmentpolicies determined via the MDP to derive opera-tional insights. These include a strong characteriza-tion of optimal policies having a threshold structureand performing rebalancing above the threshold.Herer et al. (2006) extends the multi-period, multi-

location work of Robinson (1990), and it differs fromour work in ways that include maintaining a station-ary model, and allowing backordering; it alsoassumes replenishment from a central supplier inevery period. Rosales et al. (2013) provides a modelconsisting of two retailers and uses simulation tostudy the impact of model parameters (e.g., cost,lead-time, and demand uncertainty) on both a trans-shipment model and an allocation system structure—shipment from a centralized depot. This work alsoaddresses the issue of geographical demand correla-tion, which often is raised in practical settings. Asintuition would suggest, positive correlation indemand across suppliers reduces the benefits of trans-shipment. Intuition and experience with malaria andits mechanisms suggest that positive correlations canbe expected across clinics close to each other, cap-tured in both our stochastic programming and MDPmodels.Rottkemper et al. (2012) provides a mixed-integer

programming approach to minimize a shortage andoperational costs under demand uncertainty in thecontext of humanitarian operations. They use datafrom clinics in Kayanza province in Burundi to

illustrate the effectiveness of their approach. Since thesize of our problem is much larger, we construct amore scalable approach. Our paper aligns withRottkemper et al. (2012) in demonstrating that trans-shipments can significantly reduce the unsatisfieddemand at slightly increased overall cost. In addition,we argue that allowing transshipment actions canresult in higher robustness against poor road condi-tions—an inherent characteristic of distribution prob-lems in the developing world.A main modeling contribution to the transshipment

literature is the integration of both the strategic andtactical levels by combining a stochastic programmingapproach with a MDP approach. Previous work tends toconsider one or the other. This integration is facili-tated by the identification and use of the special geo-graphical clinic clustering structure resulting from theoptimal solution of the strategic model to decomposethe country-wide distribution problem into tractablesubproblems that could be solved using MDP.Other contributions stem from the unique features

of our application area: distribution of pharmaceuti-cals in the developing world. First, the situation inthis problem differs from conventional inventorymodels which tacitly assume an environment of ongo-ing production and consumption. However, in verypoor countries such as Malawi, pharmaceutical sup-plies are often donated annually in advance of thatyear’s malaria season with mid-season replenishmentbeing uncommon. This lack of ongoing and pre-dictable supply causes distribution and transship-ment to behave differently from traditional contexts.Second, we analyze equitable solutions addressingperceived fairness, which is not typically a considera-tion in traditional transshipment literature. Third, wecapture the impact of geographically and temporallycorrelated demand and seasonality of demand reflect-ing the characteristics of malaria. Fourth, we explorethe impact of transshipment frequency, which isimportant because many developing world clinics usetime consuming paper-based inventory systems andcannot engage in near-continuous review that elec-tronic monitoring systems prevalent in retail andwarehousing would allow. Fifth, we explore theimpact of poor road conditions along certain trans-portation routes that are common, particularly duringthe rainy season (and consequently the peak malariaseason), when roads can get washed out.

3. Strategic Optimization Model forMedication Distribution in aPublic-Sector Supply Chain

In this section, we develop a strategic-level opti-mization for distributing malaria medications


throughout a centralized, national distribution net-work. We first present a baseline that performs alldistribution up front and has no recourse mecha-nism to adapt to randomness in the demand. Thisbaseline is similar to the current state of distributionpolicies in many developing countries. We then pre-sent two recourse models that represent operationalinnovations that can better help the centralized pub-lic sector supply chain react to uncertainty: delayedshipment and transshipment. Each model hasbenefits and drawbacks, but both should be poten-tially implementable without significant additionalinvestment.

3.1. Baseline Model without RecourseTo demonstrate the effectiveness of incorporatingdemand uncertainty in distribution decisions, wefirst define a “baseline” model as a surrogate for thecurrent state of ACT distribution in Malawi. Notethat this baseline model already represents anoptimization with respect to the current practice.However, it is naive in the sense it makes all trans-shipment decisions before demand is realized byminimizing expected transportation costs and short-age penalties without any real time updates andrecourse actions. Specifically, the model assumeknowledge of future demand scenarios and theirprobabilities, but cannot react to any particular real-ization. Notation for all the distribution models thatfollow is given in Table 1.The main decision variable in the baseline model,

xij, corresponds to the number of malaria treatmentstransported on arc (i, j). All distribution decisionsare made at the same time based on an historical esti-mate of demand. An auxiliary variable, zsj is intro-duced to capture the shortage of malaria treatmentsin clinic j under scenario s, in which a demand of dsiis realized for clinic i. The min-cost flow formulation

introduced in Equations (1)–(7) represents the base-line model.

minX

ði;jÞ2Acijxij þ

Xs2S

Xi2C

pspizsi ð1Þ

s.t. Xj:ðm;jÞ2AR

xij � r ð2Þ

Xj:ðm;jÞ2AR

xij ¼X

j:ðj;iÞ2ADxji 8i 2 R ð3Þ

Xj:ði;jÞ2AD

xij ¼X

j:ðj;iÞ2ACxji 8i 2 D ð4Þ

Xj:ðj;iÞ2AC

xji þ zsi ¼ dsi 8i 2 C; 8s 2 S ð5Þ

xij � 0 8ði; jÞ 2 A ð6Þ

zsi � 0 8i 2 C; 8s 2 S: ð7Þ

The objective function (1) minimizes total cost,comprised of transportation costs and shortage pen-alty and (2) constrains the amount distributed to beat most the available supply (r). Constraints (3) and(4) represent the flow conservation constraints forregional warehouses to district hospitals and districthospitals to local clinics respectively. The left-hand-side of Equation (5) represents the total flow of ACTsinto local clinic i plus the shortage in that clinicunder scenario s (zsi ). The right-hand-side corre-sponds to the total demand of clinic i under scenarios (dsi ).

3.2. Two-Stage Stochastic FormulationThe models in this section contrast with the baselinemodel in the sense that additional demand informa-tion becomes available and recourse actions are trig-gered in the second stage. In the first stage, the MalawiMinistry of Health would decide how many ACTs tosend to each facility before the malaria season begins.In the second stage, the actual demand is realized andthe Ministry can take recourse actions to address thesupply and demand mismatch. Here we consider twopotential recourse actions: (i) transshipment and (ii)delayed shipment.In the transshipment model (section 3.3), all the

ACTs are distributed among all the facilities (tier 1, 2,and 3) in the first stage. In the second stage, transship-ment of ACTs between facilities occurs to adjustinventories in light of new demand information. Inthe delayed shipment model (section 3.4), an initialdelivery of ACTs is distributed to the clinics, but someis held back at the higher tier. During the malaria

Table 1 Distribution Model Notation

N Set of nodes consisting of the central medical storehouse (m),regional storehouses (R), district hospitals (D), and localclinics (C)

AR Subset of arcs connecting the central warehouse to regionalwarehouses

AD Subset of arcs connecting regional warehouses to districthospitals

AC Subset of arcs connecting district hospitals to local clinicsAT Subset of transshipment arcs connecting local clinics to

one anotherA Set of arcs (A ¼ AR [ AD [ AC [ AT ÞS Set of demand scenariosps Probability of scenario s where s 2 Spi Penalty of one unit of treatment shortage in clinic icij Cost of transporting one unit of treatment on arc (i, j)r Total available supply of treatmentsdsi Demand of local clinic i under scenario s where i 2 C and s 2 S


season, a better estimate of the demand is realizedand a second round of shipments is delivered.Delayed shipment is less cost effective, but from animplementation standpoint it has the political benefitof not needing to take stock from one clinic to give toanother. Transshipment, on the other hand, is morecost-effective but harder to centrally control. Note,however, that according to Kiczek et al. (2009) trans-shipment already occurs on an ad hoc basis in Malawiand in a more structured manner in neighboringZambia (Mtonga 2010).Figure 2 illustrates the timeline of events for the

two-stage stochastic models. Note that the recourseactions are not necessarily done all at once. Instead,the transshipments or delayed shipments are madethroughout the malaria season as needed. Therefore,the recourse decisions considered here are aggregate-level surrogates for the actual periodic adjustments inthe inventory level of each facility.

3.3. Two-Stage Transshipment ModelA necessary assumption for this stochastic program-ming formulation is that transshipment occurs imme-diately and instantaneously (as in a continuousreview system) in response to every stock out. Thisapproximation is acceptable for the planning stagethat the stochastic program represents. We assume aset of scenarios (S) where each scenario, s 2 S is real-ized with probability ps. Under scenario s, the realizedvalue of demand for clinic i is dsi . The first-stage prob-lem has objective:

minX

ði;jÞ2Acijxij þQ; ð8Þ

and constraints (2)–(4) and (6) from the baselinemodel. The expected recourse function, Q, is given by:

Q ¼ minXs2S

ps

Xði;jÞ2AC[AT

cijysij þ

Xi2C

pizsi

!ð9Þ

s.t.Xj:ði;jÞ2AC

ysij �X

j:ðj;iÞ2ADxij �

Xj:ði;jÞ2AC

xij 8i 2 D; 8s 2 S ð10Þ

Xj:ðj;iÞ2AT [AC

ysji �X

j:ði;jÞ2AT[ACysij þ zsi � �

Xj:ðj;iÞ2AC

xji þ dsi

8i 2 C; 8s 2 S ð11Þ

ysij � 0 8ði; jÞ 2 AT [ AC; 8s 2 S ð12Þ

zsi � 0 8i 2 C; 8s 2 S: ð13Þ

The decision variable ysij corresponds to the aggre-gate transshipment of ACTs from facility i to facility junder scenario s throughout the malaria season. Equa-tion (9) minimizes the expected cost of the secondstage—transshipment cost plus shortage penalty—where zsi represents shortage of medications in clinic iunder scenario s. Equation (10) ensure that the secondround of shipments from district hospitals to local clin-ics (

Pj:ði;jÞ2AC y

sij) do not exceed the available ACTs left

from the first stage (P

j:ðj;iÞ2AD xij �P

j:ði;jÞ2AC xij). Equa-

tion (11) capture the concept that the net transshipmentplus shortages at clinic i under scenario s (LHS) shouldexceed residual demand (demand minus initial alloca-tion of ACTs from stage 1) at clinic i under scenario s(RHS). Note that the value of first-stage decisions (xij)is known in the second stage, therefore

Pj:ðj;iÞ2AD xji is a

constraint here. Thus, we re-arrange terms in Equation(11) such that decision variables are on the left-hand-side and the known values are on the right-hand-side.

3.4. Two-Stage Delayed Shipment ModelIn the delayed shipment model, some ACTs arereserved at a higher tier for shipment after the start ofthe malaria season. The first-stage problem has anidentical formulation to the transshipment model.The expected recourse function, Q, is given by:

Stage 1: Malaria treatments are distributed to facili�es.

Malaria season begins and the demand scenario is realized

Stage 2: During the malaria season, treatments are transshipped between facili�es to sa�sfy demand

Malaria SeasonPlanning Period

Stage 1: Ini�al round of treatments is distributed amongst facili�es

Malaria season begins and the demand scenario is realized

Stage 2: During the malaria season, another round of treatments is shipped to local clinics from district hospitals

Malaria SeasonPlanning Period

(a) (b)

Figure 2 Event Timelines for Two-Stage Stochastic Models


Q ¼ minXs2S

ps

Xði;jÞ2AC

cijwsij þ

Xi2C

pizsi

!ð14Þ

s.t.Xj:ði;jÞ2AC

wsij �X

j:ðj;iÞ2ADxij �

Xj:ði;jÞ2AC

xij 8i 2 D; 8s 2 S ð15Þ

Xj:ðj;iÞ2AC

wsji þ zsi � �X

j:ðj;iÞ2ACxji þ dsi 8i 2 C; 8s 2 S ð16Þ

wsij � 0 8ði; jÞ 2 AC; 8s 2 S ð17Þ

zsi � 0 8i 2 C; 8s 2 S; ð18Þ

where wsij denotes the amount of ACTs shippedfrom district hospital i to clinic j throughout themalaria season. The objective function (14) mini-mizes the expected transportation costs and shortagepenalties. Equation (15) is essentially similar to (10)from the transshipment model, allowing someinventory to be kept at the district hospital. Thismeans that the transshipment model does have asimilar capability to delayed shipment. In mostcases, however, the amount of inventory stored atthe district hospital in the transshipment model isnegligible. As we will discuss in section 3.5, undersome parameter regimes, especially when the cost oftransshipment arcs exceeds those of delayed ship-ment arcs, the transshipment model can result inoutcomes similar to those generated by the delayedshipment model. Constraints (16) capture the short-age in each clinic (zsi ) after the second round ofACTs is distributed. Constraints (17) and (18) ensurethe non-negativity of shortage.In addition to the two-stage models, we also formu-

late an analogous three stage model that provides twoopportunities for recourse during the malaria season.This model is used for comparison of reaction fre-quency, but the framework is nearly identical to thetwo stage models. For completeness, the formulationis presented in Online Appendix S1, where we alsopresent an alternate objective function that focuses onequity.

3.5. Scenario Analysis: Costs, ResourceAvailability, and UncertaintyIn this section, we present the results of computa-tional experiments based on actual locations of healthfacilities that were mapped in a country-wide surveyconducted by the Japanese International CooperativeAgency (JICA) in the year 2000. Facility demand wasestimated based on regular malaria case counts asreported by hospitals and clinics to the central Min-istry of Health, spanning the years 2003–2008. The

data are summarized in annual government HealthManagement Information System (HMIS) reports pre-pared by Republic of Malawi Ministry of Health(2009) These reports include case counts reported bymonth at each facility, whether the demand was metor not.As in all developing country contexts, there were

some missing and incomplete data at the facilitylevel. However, this nation-wide reporting programbecame fully operational in all districts in 2002 andremained so during our data-collection time-frame,Chaulagai et al. (2005). The incident counts in thedata were mostly complete, with well over 80% ofthe facilities reporting. We estimate the case countsat facilities where data was missing by consideringincidence rate for the region; see for example,Dzinjalamala (2009) and the catchment (population)that the facility serves. Multiplying the population ofthe catchment by the region’s incidence rate, weobtain approximate case counts. Catchments wereestimated using Thiessen polygons in conjunctionwith Malawian Census data. This general approachis widely used for estimating facility-level incidenceof malaria in Malawi and other parts of Africa in theepidemiology and public health literature (seeBennett et al. 2013, Chaulagai et al. 2001, 2005,Dzinjalamala 2009, Hay et al. 2010, Kazembe 2007,Kazembe et al. 2006).It was assumed that people used the closest health

facility. In reality, this may not always be true, aspatients may prefer one facility over another, or trans-portation (i.e., buses) might facilitate travel to a far-ther facility. Nonetheless, as in common practice, casecounts were assigned proportional to health facilitiesbased on the estimated population catchment and theprobability of contracting malaria associated witheach geographical region. If the necessary data can beobtained, an interesting follow-up study could com-pare the actual demand against the Thiessen polygoninterpolation mechanism to determine the accuracy ofsuch approximations. Note, determining the error inthe Thiessen polygon approach has been well-explored in the literature (e.g., Tatalovich et al. 2006,Yang et al. 2004), in which Thiessen polygons arefound to perform well relative to other methodsData were averaged over the five years to model a

typical malaria season in Malawi in the face of par-tially missing data at the clinic level. By observingclinics where the data were more complete, we notedthat, over the five years, there was some variabilityfrom year to year but little evidence of overall upwardor downward trend. This is further confirmed by theWorld Malaria Report 2014 (see WHO 2014), whichindicates no significant trend over the time period.Although there was a slight increase in malaria-related hospital admissions over the time frame, the


report states that data were insufficiently consistent toassess any trend in Malawi, consistent with observa-tions from our own data. Although we recognize thelimitations of the data available, it is still very signifi-cant that this effort is data-driven and reflects realtemporal and spatial patterns of malaria incidencegiven current research.Figure 3 shows the geographical and seasonal

shape of the ACT demand curve from our data. Thedarker color (red) in Figure 3a, indicating higherannual demand, tends to appear near populous urbanareas like Blantyre and in both urban and rural areasnear Lake Malawi where mosquitoes are more preva-lent. In our historical demand data, malaria preva-lence in each region grew over the malaria seasonproportional to the infected population. Figure 3a wasgenerated by creating a heat map (ARCGIS) from thecase counts at the various facilities at their locationsfrom our data). A heat map was used because ourdata use agreement did not allow us to show casecounts or relative sizes at individual, identifiable facil-ities. Figure 3b shows that malaria medicationdemand basically follows a six-month seasonal pat-tern which coincides with the seasonal patterns ofrainfall and thus of Anophelene mosquito prevalence.This was also obtained from our monthly case countreports from the HMIS annual reports from 2003 to2008. Furthermore, the geospatial and monthly stan-dard deviations in our data were 443.9 (CV = 0.11)and 326.56 (CV = 0.97), respectively.In the three-stage model, we divided the year into

three periods, each consisting of four months. In bothtwo- and three-stage models, the first period isAugust 1 through November 30, which is consideredthe pre-malaria season with an average demand of113,331 over the time frame. The other eight months,December 1 through July 31, are considered themalaria season. In the two-stage model, the secondstage is December 1 through July 31. In the three-stagemodel, the second stage is December 1 through March31, with an average demand of 674,702, and the third

stage is April 1 through July 31 with an averagedemand of 1,319,287.To generate clinic-level demand scenarios, we used

the estimated clinic-level demand (based on the his-torical prevalence data and the aforementioned inter-polation method using Thiessen polygons) as thebaseline. To make the results more robust to a rangeof possible events, 10 scenarios (details will follow)were then generated based on the expert opinion fromone of our co-authors who has performed extensivefield-work in Malawi regarding malaria. The scenar-ios developed herein were designed and confirmedbased on his personal experiences over several yearsin Malawi. Our data include demand from 290 facili-ties including 3 regional warehouses, 21 district hos-pitals, and 266 local clinics.For each clinic, we generated 10 scenarios to popu-

late demand parameters for the second (dsi ) and thirdstage (d0si ), respectively. We start with five main sce-narios, assuming each of those scenarios are equallylikely to happen. The first three key scenarios weregenerated by perturbing the original demand fromour historical data (Di). To add robustness, two otherscenarios were generated using a uniform distribu-tion such that the mean of the uniform distributionequals the average observed demand. For each ofthose five main scenarios, we generated a less likelyvariation to capture the potential for rare extremeevents. In total, we assumed each main scenario has aprobability of 0.19 and each scenario extension has aprobability of 0.01. In designing these scenarios, wealso capture correlated demand across the differentstages as malaria is a transmittable disease whosespread depends on the number of infected persons.That is, high initial demand is more likely to translateinto high demand in future stages. Details of thesescenarios are described in Table 2.Note that a more sophisticated demand forecasting

model could use the historical data on malaria casesas a baseline in conjunction with demographic censusinformation to detect key drivers of malaria case load,

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400,000

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Figure 3 Plot of Demand for Malaria Medication Based on Geography (total yearly demand) and Seasonality (by month) [Color figure can beviewed at wileyonlinelibrary.com]

Notes: Darker color (red) (a) indicates higher demand. Dots are facility locations.


and consequently, medication demand. Suchadvanced models could better characterize spatialand temporal variation in medication demand. Suchadvanced approaches, although feasible, are beyondthe scope of the current research.To capture the transportation cost (cij), we calcu-

lated the distance between each facility pair in kilo-meters. For the purposes of this research, we usedEuclidean distance. A road map for Malawi was avail-able, and more accurate measures of road distancebased on a map could have been produced, but it wasfound that the quality of the map varied by geo-graphic area. It was found that Euclidean distancesaccurately reflect road-based measures regionally inMalawi and other research has confirmed that thismeasure is satisfactory compared to more sophisti-cated methods (see Nesbitt et al. 2014). To account forcartographic shortfalls on published maps andthereby maintain consistency over the region of inter-est, we use straight line distance.Road quality in Malawi varies widely and the sys-

tem is mostly underdeveloped so the unit transporta-tion cost can vary by route, although specific data onroad quality are not available. Thus, we initially usethe average transportation cost per kilometer inMalawi reported by Lall et al. (2009) which is about 4cents (or 228.4 kwacha, the Malawian currency), butvary the costs to account for road conditions in oursensitivity analyses.In the following sections, we consider key features

in analyzing a public-sector supply chain in the devel-oping world. We begin by performing a sensitivityanalysis on the shortage penalty, transshipment costs,and supply availability. All three factors have beenshown to be of significant concern in the literature.We then conclude with a novel analysis of road condi-tions, which are known to degrade significantly dur-ing the rainy season when malaria is most prevalent.

3.5.1. Sensitivity Analysis on the Value ofShortage Penalty. Estimating the shortage penalty

for malaria medications is non-trivial. Factors such asloss of income and productivity (for patients and rela-tives) during the course of infection, and health careexpenditures should be taken into account in order toobtain a correct estimate. It should be noted that giventhe type of parasite, the symptoms and their severityvary dramatically. Some people may have alreadydeveloped immunity while for others (especially chil-dren) the disease can be deadly. Furthermore, malariacan have a higher indirect impact on children by ham-pering their physical and intellectual growth.Accounting for all these factors and monetizing theirimpact is key to determining the actual value of theshortage penalty and is beyond the scope of thispaper. Due to a lack of reliable data regarding healthcare expenditures, we performed a sensitivity analy-sis on the value of the shortage penalty. As a baseline,we begin with Malawi’s national income per capita,reported to be $810 by the World Health Organization(WHO) (2014). According to UNICEF (2004), malariacan slow the economic growth in sub-Saharan Africaby 1.3% annually. Based on these statistics, one canestimate the economic impact of malaria in Malawi atthe individual level to be about $10.5 in lost economicgrowth, which is considered a lower bound because itcaptures only the loss of economic growth. In this sec-tion, we consider shortage penalty values between$10 and $100. Based on our computational results,even a low number, $20, is high enough to triggereffective distribution of medications. Hence, we usethe $20 shortage penalty for future illustrative exam-ples and computations (e.g., section 4.5). For the fol-lowing experiments, we set the available supply ofACT to 1.5 million units as this was also a middlerange for the estimated annual supply.Figure 4 demonstrates the inverse relationship

between shortage penalty and transshipment volume,which is consistent with results reported byRottkemper et al. (2012). The three-stage delayedshipment model tends to be more effective at address-ing shortage, although at a higher transportation cost.

Table 2 Ten Demand Scenarios Were Generated Based on the Observed Historical Demand for Each Clinic (Di)

No Name Description Probability Demand in Stage 2 (dsi ) Demand in Stage 3(d0si )

1 LOW1 Low total demand, variation 1 0.19 14Di38Di

2 LOW2 Low total demand, variation 2 0.01 14Di18Di

3 MED1 Medium total demand, variation 1 0.19 12Di12Di

4 MED2 Medium total demand, variation 2 0.01 12Di38Di

5 HIGH1 High total demand, variation 1 0.19 34Di56Di

6 HIGH2 High total demand, variation 2 0.01 34Di Di

7 CONS1 Uniform total demand, variation 1 0.19 U ½0:9 D2 ; 1:1 D2� 2di8 CONS2 Uniform total demand, variation 2 0.01 U ½0:9 D2 ; 1:1 D2 110 di9 VAR1 Uniform demand variation 3 0.19 U ½0:2 D2 ; 1:8 D2 34 di10 VAR2 Uniform demand variation 4 0.01 U ½0:2 D2 ; 1:8 D2� di


As the shortage penalty increases, however, weobserve that the gap between the three-stage delayedshipment model and the three-stage transshipmentmodel shrinks. Also note that under high shortagepenalty values, the two-stage delayed shipmentmodel results in higher shortage than the two-stagetransshipment model. This occurs because once theactual demand is realized, the delayed shipmentmodel can only send additional shipments of medica-tions from the district hospitals to the local clinics toaddress shortage. The transshipment model, on theother hand, has a broader base of facilities from whichto satisfy demand. In some sense, this confirms theresults of Rosales et al. (2013) that transshipmentmodels outperform generalized allocation mecha-nisms under most parameters.Managerial Insights. The delayed shipment model

incurs lower shortages than transshipment in mostcases. At first, this may seem counterintuitive becausethere is more flexibility in the transshipment model.However, this flexibility actually causes the first stageto distribute all the inventory out to the clinics to saveon transportation cost instead of prepositioning alarge stock at the district hospitals. This actuallydecreases the precision with which inventory is posi-tioned across the country, making it more likely thatsufficient inventory is not nearby the point of need.Transshipments will not be executed when the dis-tance renders the transportation cost prohibitive. Inthe delayed shipment model, on the other hand, thedistrict hospitals tend to be centrally located withmany clinics around them. Since there is a larger stockstored at these hospitals initially (by design), therewill be more incentive to take the recourse shippingaction in stages two and three due to sufficient inven-tory and proximity. This also explains why shippingcost is higher for delayed shipment, because ratherthan shipping direct, much of the product must fol-low first a route from the main dispensary to the dis-trict hospital and then a second route from thehospital to the clinics. Hence, the key insight is that ifthe government has sufficient transportation budget

and cares more about avoiding shortage, then delayedshipment may be a better structure.

3.5.2. Transshipment Cost Sensitivity Analysis. Inthis section, we analyze the sensitivity of the trans-shipment model to the transshipment cost and com-pare it with the delayed shipment model. Inparticular we explore the cases where (i) transship-ment is cheaper and (ii) more expensive than ship-ment along the main channels from the regional anddistrict hospitals. It may be possible that shipmentsare cheaper because smaller and more frequent ship-ments may be transported with smaller and cheapertransportation methods, such as a motorcycle or smallvehicle that can more easily pass difficult terrain orroads that are damaged by heavy rains. In these cases,a large truck may have difficulty navigating certainroutes and therefore be more costly relative to trans-shipment. On the other hand, it may also be possiblethat frequent transshipment loses economies of scale,making clinic-to-clinic shipping costs more expensive.Thus we analyze how the models react in both cases.To do so, we modify the cost of clinic-to-clinic trans-shipment to be X% of the standard cost of shipment,where X ranges from 0% to 150%. Costs for the otherroutes remain unchanged.In Figure 5a and b, the dashed lines represent the

expected transportation cost and expected shortagevolume respectively for the delayed shipment mod-els. These are constant across all scenarios becausedelayed shipment does not use the clinic-to-clinicroutes. The solid lines represent the transportationand expected shortage costs for the transshipmentmodel. When transshipment is not expensive, all theinventory is initially allocated to one clinic in the clus-ter, which then transships to the other clinics due tothe lower cost of transshipment relative to the cost ofthe initial distribution.Transportation cost initially increases as transship-

ment becomes more expensive; however at 60%, thetransportation cost begins to decrease while the short-age penalty increases more sharply. This inflection

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Figure 4 Sensitivity Analysis on Potential Values of Shortage Penalty (total available supply = 1.5 million units)


point occurs because at this level, clinic-to-clinictransportation becomes expensive enough that themodel will stop transshipping along certain routesaltogether, preferring some shortages rather thanincurring high shipping costs, e.g., shipping acrossthe country to fulfill a small amount of demand. Asobserved in section 3.2, the similarity between Equa-tions (10) and (15) enables the transshipment modelto store some inventory at district hospitals and delayshipments if necessary. When transshipment becomesprohibitively expensive, clinic-to-clinic shipments areavoided entirely and the transshipment modelrestricts itself only to the cheaper routes used in thedelayed-shipment model; then both costs approachthose of delayed shipment.While the shape of the curve is driven by the partic-

ular network structure as well as the shortage penaltyand original transportation cost values, changingthese values would likely shift the inflection pointwhile maintaining the overall shape. A key insight isthat, once transshipment reaches a scenario-specificcost threshold (e.g., 50%–60% in Figure 5), the trans-portation cost will remain relatively stable, as theoptimization becomes more conservative as to howfar one would be willing to ship medications to satisfyunmet demand in a different region. Essentially,transshipment eliminates routes from considerationdue to high cost thus becoming less effective in

satisfying all demand. This serves to localize thetransshipment mechanism around increasingly proxi-mate geographical clusters. As an extension of thisline of reasoning, the higher the penalty cost forunmet demand, the longer the model resists localiz-ing transshipment efforts in favor of moreregional/national transshipment. Thus, depending onthe strategic goals and constraints of the distributor,the optimal shipping network may be more localizedor more national.

3.5.3. Sensitivity Analysis on Supply Avail-ability. Supply availability is a major challenge indistributing malaria medications in holoendemicareas. As reported by Natarajan and Swaminathan(2014) the process of procuring humanitarian suppliescan be subject to delays and uncertainty. To betterassess the effectiveness of our proposed stochasticmodels, we compare their results for a range of possi-ble supply values, between 500,000 and 2,000,000units while fixing the shortage penalty at $100. Fig-ure 6b, shows that the stochastic models can betterutilize the additional supply of medications toaddress shortage compared to the baseline model.Among the stochastic models, three-stage modelstend to be better at utilizing additional supply thantwo-stage models. This insight is similar to the keytakeaway from section 3.5.1; specifically, the three-

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Figure 6 Sensitivity Analysis on Supply Availability (Shortage Penalty = $100)


stage model better utilizes the supply of ACTsthrough targeted repositioning in stages two andthree. Obviously as more medications are available,more medications will be transported, and while thetotal cost of shortage decreases, transportation costwill increase. As mentioned earlier, the analysis inthis section is performed by fixing the per unit short-age penalty (p) at $100. Based on the discussion in sec-tion 3.5.1, a higher value of p creates more incentivefor the model to transship more items and reduce theoverall shortage cost. As observed in Figure 6b, for afixed value of p, as the total supply volume increases,transportation cost almost reaches a plateau. We canexpect that increasing p will shift the plateau to theright, while reducing p will shift the plateau to theleft.

3.5.4. Road Condition Analysis. As mentioned insection 1.2, poor road conditions can make certaintransportation routes difficult and therefore morecostly, requiring, for instance, special vehicles ordelayed travel during especially poor weather peri-ods. To analyze this feature of supply distribution inthe developing world, we design a scenario in which aproportion of the roads in our supply network is mademore costly to travel due to poor road conditions.Since we are not aware of any data on the actual roadconditions of the thousands of potential supply routes,we test the model’s sensitivity to poor road conditionsby varying the proportion of total routes that are con-sidered to be in poor condition from 10% up to 50%,with the poor routes being selected at random with anadditional cost of shipping along the given route alsogenerated randomly. In the scenario where 10% ofroads are considered in poor condition, we assigned aBernoulli indicator to each road where the road is con-sidered in poor condition with p = 0.1 and standardcondition with 1 � p = 0.9. If the road was found tobe poor, we multiplied the transportation cost by1 + U(0, 0.5), where U(�) is a uniform random vari-able. We modify the cost per km rather than adding arandom quantity to each route because when travelingon a poor road for a longer distance, the cost shouldincrease more than when traveling on a poor road for

a shorter distance. We then generated five outcomesamples for the entire set of routes. For the 20% sce-nario, we started with the same bad roads as the 10%scenario and then modified the remaining roads thathad not been touched in the previous scenario using aBernoulli probability that guaranteed that 20% of thetotal routes would be modified (in this case p = 0.111).This yields a coherent comparison between the differ-ent scenarios. The rest of the scenarios (30%–50%)were generated in the same manner.Figure 7 shows the results for the different percent-

ages of roads in poor condition in terms of transporta-tion cost (Figure 7a) and the expected shortage(Figure 7b). The X’s represent the solution of thetransshipment model for the five different randomscenarios we generated at each percentage of poorroads. The dashed line represents the average of thefive scenarios for transshipment. Likewise, the plussymbol and solid line represent the correspondingoutcomes for the delayed shipment model.First, note that the transportation cost remains rela-

tively stable as the percentage of bad roads increases.This is because the transportation cost is high enoughthat it becomes more beneficial to keep medicationslocally rather than ship across routes with poor roadconditions for a small reduction in shortages. Delayedshipment costs trend downward because there arefewer viable options when a key route becomesaffected by poor road conditions so the model choosesto accept more shortages. The transshipment model,on the other hand, has more flexibility because thereare many more options when clinic-to-clinic routesare added. When one route becomes more expensive,the model is able to find other viable routes to trans-ship product. The transshipment model’s transporta-tion cost demonstrates a slight upward trend as thetransshipment model seeks alternative routes thatallow for more movement of medications at a slightlyhigher price.The cost of storing more medications locally can be

measured in terms of increased shortages. As seen inFigure 7b, the slope of the increase in shortage issteeper for the delayed shipment model than thetransshipment model, which implies that the

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transshipment model is better at meeting demand asroad conditions worsen.A key takeaway from this analysis is that poor road

conditions lead to an increase in shortages and lessmovement of product around the network. However,the flexibility of the transshipment model to choosealternate routes enables more demand to be satisfiedrelative to delayed shipment, albeit at increased trans-portation cost due to using more expensive alternateroutes.

3.6. A New Distribution Structure: EstablishingClinic ClustersOne of the key insights gained from the computa-tional experiments on the strategic-level stochasticprogram is the appearance of what we call clinic clus-ters. That is, the transshipment stochastic modelsgroup clinics together into clusters such that trans-shipment often occurs within clusters only, and veryrarely between different clusters. In our transshipmentmodel experiments, for example, only 15%–25% ofclinics send ACTs to other clinics in the recoursestage. These sender clinics transship their excessinventory to between 2 and 5 proximal receiver clinicsin the vicinity of the sender clinic. Figure 8 illustratesfive representative clinic clusters in the northern areaof Malawi. This idea of clusters can be used to decom-pose the nationwide problem into tractable cluster-level problems that can be solved independently atthe tactical/operational level. This is the key to inte-grating the strategic models with the operationalmodels that we develop in section 4.This decomposition has further benefits. The geo-

graphic proximity of clinics within a cluster increasesthe likelihood that the clinics would be willing towork together toward a transshipment program; thispartially mitigates challenges associated with regionalrivalries mentioned in section 1.2. Proximity alsomakes transshipment more feasible by motorcycle orsmall vehicle that are less likely to get stuck due topoor road conditions. Finally, communication—a

major cause of breakdown of supply—is much easierfor tightly clustered clinics.

4. A Tactical/Operational Model forTransshipment in Clusters

For the strategic planning models of section 3.2 andsection A.1 in Online Appendix S1, it was necessarilyassumed that recourse occurred under continuousreview. In reality, transshipments would likely occurperiodically at regularly scheduled intervals afterstock reviews at each facility. In this section, we use aMDP to develop a mechanism to operationalize thetransshipment concept at a cluster level based on thestrategic plan developed in the previous sections.First, we formulate a MDP model to analyze the

dynamics of a periodic review system for the clinicclusters. Second, we analyze the structure of the opti-mal policy under a reasonable demand assumptionfor clinics that are within close geographic proximity.We are able to show that balancing the load evenlyacross the clinic cluster is optimal, but re-balancingoccurs only in a cluster-level “sweet spot” wherethere is not too much or too little inventory in thecluster as a whole. Third, we parameterize the modelwith historical demand data (as described at thebeginning of section 3.5) and solve the MDP numeri-cally to illustrate the behavior of the model andexplore unique features of distributing pharmaceuti-cals in the developing countries.To identify clinic clusters, we solve the strategic-

level transshipment model to optimality and calculatethe optimal values of transshipment between two clin-ics under all scenarios, i.e., ysij. Then, we take the maxi-mum of transshipment values across all scenariosdefined as ymaxij ¼ maxs2S ysij. If there has been a “sig-nificant” transshipment between those two clinics, i.e.,ymaxij exceeds a pre-determined threshold, we assumethose two clinics are in the same cluster. To conductcomputational experiments, we set this threshold tothe average monthly demand of the receiving clinic(i.e., j) across all scenarios, i.e.,

Ps2S psd

sj .

Note that this is not an exact clustering method(e.g., k-means). Instead, we are inferring cluster struc-tures from the results of the transshipment model.The downside to this method of developing clustersis that the cluster boundaries depend on the actualparameter values. For instance, as the cost of trans-shipment increases, the model tends to hold on tosome inventory at the district hospitals and ship themto clinics with a delay. In another extreme case, whentransshipment is very inexpensive, a clinic mayreceive a small shipment from another clinic that can-not be meaningfully assigned to the same cluster.Hence, the results of the stochastic program will notalways guarantee that we obtain mutually exclusive

Figure 8 Five Clinic Clusters in the Northern Region of Malawi [Colorfigure can be viewed at wileyonlinelibrary.com]


clusters. We emphasize, however, that we focus onidentifying mutually exclusive clusters of clinicsbased on the idea that very small or zero flowsbetween two clinics in the strategic planning modelindicate little need for short term transshipmentbetween them. So, by eliminating shipments that werelower than an empirically determined threshold, weidentified mutually exclusive clusters of clinics—seeAppendix S2 for more details. While we developed avery intuitive approach, future research would betterunderstand the power of our approach by comparingit to other optimization-based approaches or even tointegrate other holistic considerations into the pro-cess. In practice, the system designer can bring to beargood experience-based judgment.In our operational model, each clinic cluster is mod-

eled separately with a periodic review cycle in whicheach clinic’s inventory is surveyed and then a deci-sion is made as to how much to transfer to other clin-ics within the cluster. At the beginning of each period,each clinic incurs a shortage penalty for unmetdemand from the prior stage—indicated by a negativeinventory value. Next, a decision is made regardinghow much product to ship between clinics. Finally,demand arrives to each clinic within the clusteraccording to a distribution for epoch n of dn � Fnand the state is updated for the next decision epoch.The finite-horizon MDP formulation is given in Equa-tion (19) with notation in Table 3. Equation (20) limitsthe action space to allow transshipment only if inven-tory is available.

fnðNÞ ¼ PTð�NÞþ þminu2UN

�cXj2U

ðujÞþ þ Effn�1ððNÞþ þ u

� dnÞg�;

ð19Þ

where the action space is given by:

UN ¼ fu ¼ ðu1; . . .; usÞ : uj � nj andXj2U

uj ¼ 0g: ð20Þ

In the tactical model, we only focus on the clinics inone cluster, which means they are all in close proxim-ity. This makes the distance between each clinic in thecluster approximately the same, which allows us tosafely approximate cij = c, where i and j are in thesame cluster. However, an advantage of the MDPapproach is that it easily accommodates nonlinearcost functions and transport capacity limits if needed.The expected cost-to-go is based on the positive partof Ξ, because in malaria treatment, the dynamicsbehave as “lost sales,” not backorders.

4.1. Structural Properties and Insights for ClinicCluster TransshipmentIn this section, we analyze several structural proper-ties of our MDP model to gain insight into the optimaltransshipment policy for clinic clusters. We specifi-cally show that (i) the entire cluster is better off whenany clinic in the cluster increases its initial supply, (ii)balancing the inventory among clinics is optimal, and(iii) the optimal transshipment policy is of thresholdnature.Individual supply benefits the group. Theorem 1

shows that the entire cluster is always better off if anyone of its clinics receives more supplies. This theoremsupports the need for a strategic planning model thatinitially allocates ACTs to clusters effectively andequitably.

THEOREM 1. fn(Ξ) is non-increasing in ξj for all n and j.

Optimality of Inventory Balancing within a Clus-ter. In this section, we develop a model for a clusterconsisting of two clinics and show that the optimalpolicy balances the inventory between the two clinics.In section 4.2, we extend the insights regarding clusterbalancing from the analytical model to show numeri-cally that the same structure holds more generally byapplying historical data to larger clinic clusters.We begin by defining what it means for a function

to be balanced. Next, we show that the balanced prop-erty is preserved by the expectation operator inLemma 1. This lemma supports development offurther operational insights including the key result(Theorem 2 and Corollary 1) that the optimal trans-shipment policy is of threshold nature; and depend-ing on the cluster-wide inventory levels and thedisparity between the clinics the optimal action willeither (i) re-balance the inventory across the cluster sothat each clinic has the same inventory level or (ii) donothing. This result is supported by deriving ancillaryinsights that show the optimal states for a clinic clus-ter possess the property that all clinics have “roughlyequal” inventory levels (Lemma 3), and transship-ment only occurs from clinics with higher inventoryto clinics with lower inventory (Lemma 2). These last

Table 3 Clinic Transshipment Model Dynamic Program Notation

Ξ n-dimensional vector for the amount of inventory ateach clinic at the beginning of the period

ξj The j th component of Ξ, indicating how much inventory isat clinic j

UN n-dimensional integer vector space where u 2 U is defined inEquation (20), which enforces flow conservation

uj The jth component of u 2 U, which is the action describing howmuch to increase or decrease clinic j’s inventory level viatransshipment

Π n-dimensional vector of shortage penaltyc Unit cost of transshipment between clinicsdn Random variable for pharmaceutical demand in period nΦ Set of clinics in the cluster, a subset of C.


two Lemma’s also guarantee that our decision sup-port has the appealing property of being perceived asfair by implementing clinics: no clinic with less inven-tory will ship to one with higher inventory, and thegoal of the algorithm is to achieve inventory balanceamong the clinics.As a precursor to model analysis, we begin by

describing the reasonable assumption for tightly clus-tered clinics that the severity of malaria outbreak willfollow a similar pattern among the clinics of the samecluster. Mathematically, we mean that it is equallylikely to see malaria incidence of x in clinic A and y inclinic B as it is to see incidence y in clinic A and x inclinic B. We call this a symmetric demand distribution.With symmetric demand, we can prove the propertiesmentioned above. We begin with a definition of abalanced function and then proceed to show that theMDP value function is balanced, which guaranteesthe optimality of balancing inventory levels across theclinics within a given cluster.

DEFINITION 1. We call a function f : R2 ! Rbalancedif given Ξ and Ξ

0, such that n1 þ n2 ¼ n01 þ n

02, if

jn1 � n2j � jn01 � n02j then f(Ξ) ≤ f(Ξ0).

In the following lemma, we show for symmetricdemand distributions that the expected cost-to-gofunction of the MDP preserves the balanced property.We then use this lemma (with proofs for this and allresults in Online Appendix S3) to prove structuralproperties of our periodic review with transshipmentMDP and build up insights supporting the optimalityof balanced inventory levels across the cluster.

LEMMA 1. If the two clinics in a cluster have a symmetricdemand distribution and the function fn(Ξ) is balanced forall n, then gðNÞ ¼ E½fnðN � dnÞ� is also balanced.

LEMMA 2. If function fn(Ξ) is balanced for all n, theoptimal action will never transship from the clinic withlower inventory to a clinic with higher inventory.

This result, combined with the following lemma,reduces the action space significantly since the opti-mal action u� must be an element of{0, . . ., ⌊ 0.5 9 (max{ξ1, ξ2} � min{ξ1, ξ2})⌋}. u� willbe the amount of medication shipped from the clinicwith higher inventory to the one with lower inven-tory. The next lemma (proved in Online Appendix S3)demonstrates that a completely balanced inventorydistribution is the lowest cost state for a clinic cluster.Hence, each clinic cluster will desire to move towarda cluster-wide balanced inventory as long as the costof achieving the balance is not too great—which isshown by Theorem 2 and Corollary 1.

LEMMA 3. If fn(Ξ) is a balanced function for all n, thenfor any total inventory level ξ1 + ξ2, the value functionis minimized where n�1 ¼ n

�2 if ξ1 + ξ2 is even and

jn�1 � n�2j ¼ 1 if ξ1 + ξ2 is odd.

Now, we are ready for the main result, which is thatthe optimal transshipment policy follows a thresholdin which the clinics will either (i) re-balance the inven-tory across the cluster so that each clinic has the sameinventory level or (ii) do nothing. We first show thatthe MDP value function is balanced in Theorem 2.This means that the optimal solution of our MDP hasthe properties of Lemmas 2 and 3.

THEOREM 2. When the demand vector has a symmetricdistribution, the value function in (19) is balanced.

An Optimal Threshold Policy for Inventory Bal-ancing. The lowest cost state for the clinic cluster is abalanced inventory level. However, to achieve abalanced state in each epoch requires paying a trans-shipment cost, so it may not be optimal to re-balancethe cluster in every epoch. This section provides thekey insight that the clinics should follow a thresholdpolicy that re-balances inventory when the differencebetween inventory levels is above a certain thresh-old, but will not re-balance if both clinics have eithertoo little inventory or a surplus of inventory. Figure 9provides a typical example of the optimal transship-ment areas. In Area 1 there is not enough inventorywithin the cluster (shortages being likely at bothclinics) and in Area 3 there is sufficient inventory inthe cluster (shortages being unlikely at either clinic);hence no transshipment occurs. In Area 5, Clinic 1has surplus inventory while Clinic 2 does not haveenough, with the reverse occurring in Area 4, and sore-balancing occurs in both Area 4 and Area 5. Thestructure demonstrated in Figure 9 is guaranteed bythe following Corollary, which follows directly fromTheorem 2.

COROLLARY 1. Under a non-decreasing shipping cost,the optimal policy is of threshold nature with stage-dependent thresholds. Depending on the shipping cost,the optimal action will perform the minimal amount oftransshipment necessary to balance the inventory (in thesense of Definition 1) or do nothing.

As an example of Corollary 1, consider the casewhere there is a fixed cost per shipment. The optimalpolicy balances the inventories between the twoclinics in the following way: u = 0 ifc [ Effn�1ððNÞþ � dnÞg; otherwise u is the optimalaction that brings the inventory levels of the clinics tobn1 þ n22 c and n1 þ n2 � b

n1 þ n22 c. Any analytical proof

regarding the structure of an optimal policy for


clusters consisting of more than two clinics can becomplex.

4.2. Illustrative Example of the Optimal Area-Based Transshipment PolicyIn this section, a numerical example is used to gaininsight into state-specific optimal actions. For the pur-pose of exposition, we begin with an example of acluster consisting of two clinics. We also scale theunits of demand and supply to obtain the followingrestricted state space:

N ¼nðn1; n2Þ 2 R2 : �5� ni � 9; 8i

2 f1; 2g and n1 þ n2 � 9o:

We solve the two-dimensional MDP under threedifferent parameter settings where the ratio of short-age penalty to unit transportation cost was either: (1)low, (2) moderate, or (3) high and solve it for sixstages (one stage for each month of the malaria sea-son). Figure 9 illustrates the optimal actions at stage 5(i.e., n � 1) for each state for cases (1), (2), and (3).The optimal actions in Figure 9 are identified by fiveareas, 1 through 5, described in more detail in Table 4.Figure 9 shows that the ratio of shortage penalty to

transportation cost (p/c) plays an important role. Asthis ratio increases, there is more transshipmentbetween clinics; transshipment Areas 4 and 5 becomeslarger while no action Areas 2 and 3 shrink. As the

p/c ratio decreases we observe less transshipment,which has the opposite effect on the areas. Thisbehavior demonstrates the importance of low costand accessible shipping options for short distancetransport as this leads to more effective transshipmentpolicies.In section 4.1, we found the structure of an optimal

transshipment policy with two-clinic clusters andsymmetric demand. While the setup considered wasreasonable both in the demand assumption (asargued previously) and size (a number of clustersfrom the strategic model contained only two clinics),we can further extend the analytical results to clusterscontaining three clinics through numerical analysis.The insights are summarized below:

INSIGHT 1. Corollary 1 extends to clusters of size greaterthan two. When the demands of all the clinics in a clusterare symmetrically distributed, the optimal action is tobalance the inventory between the clinics in the cluster.

INSIGHT 2. As the ratio of the shortage penalty to thetransshipment cost increases, it is optimal to ship moreunits between the clinics; increasing the effectiveness oftransshipment in preventing ACT shortage.

Similar to the previous example, we scale the unitsof demand and supply to obtain the followingrestricted state space:

N ¼nðn1; n2; n3Þ 2 R3 : �5� ni � 9; 8i 2 f1; 2; 3g and

n1 þ n2 � 9o:

The system state has three dimensions, so we onlyillustrate the optimal actions for three inventory levelsat Clinic 3: 0, 2, and 4. For ease of comparison, wechose a moderate ratio of shortage penalty to trans-portation cost, i.e., p/c = 10. The optimal actions in

999888777666555444333222111000

-1 -1 -1-2 -2 -2-3 -3 -3-4 -4 -4-5 -5 -5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Clinic 1Cl

inic

2Clinic 1

Clin

ic 2

Area 1No Ac�on

Area 2No Ac�on

Area 3No Ac�on

Area 42 1

Area 51 2

Clinic 1

Clin

ic 2

Case (2): 50 Case (3): = 100Case (1): /c = 10

Area 42 1

Area 42 1

Area 51 2

Area 51 2

Area 1No Ac�on

Area 1No Ac�on

Area 2No Ac�on Area 2

No Ac�on

Area 3No Ac�on

Area 3No Ac�on

/c = /c

Figure 9 Optimal Actions in Period 5 for Three Parameter Settings

Table 4 Five Areas in the Two-Dimensional Illustrative Example

Area Description Actions

1 Clinics 1 and 2 face shortage No action is possible2 Clinics have low inventories No action is recommended3 Both clinics have high surplus No action is required4 Clinic 1 is facing shortage

while Clinic 2 has surplusClinic 2 transships to clinic 1

5 Clinic 2 is facing shortagewhile Clinic 1 has surplus

Clinic 1 transships to Clinic 1


Figure 10 are identified by five areas, detailed inTable 5. Figure 10, shows that the optimal policy bal-ances the inventory between the three clinics whendemand is symmetric. As the inventory level of clinic3 increases, that clinic ships more pharmaceuticalunits to Clinics 1 and 2 if needed.

4.3. Clinic Clustering vs. Fully IntegratedOptimizationIn section 3.6, we introduced a method for integratingthe strategic and tactical levels of supply allocationthrough the decomposition of the strategic model intoclinic clusters based on the structure of the solution.While this approach has significant computationaladvantages (the tactical MDP is intractable at the levelof the full-scale problem), a question remains regard-ing loss of optimality stemming from the cluster-based decomposition. In this section, we address thisissue by studying four of the larger groups of clinicsfor which the strategic model recommended clusterdecomposition. Through numerical analysis, we com-pare for each group (i) the fully integrated solution,(ii) the cluster decomposition solution suggested insection 3.6, (iii) and the solution of the baseline (i.e.,naive) model from section 3.1.For the fully integrated solutions, we solve a MDP

that performs both the initial allocation of inventory(stage 1) and the transshipment between any pair of

clinics within the group of clinics studied (stage 2).This is done in reverse, by solving the MDP for allpossible initial inventory allocations, and then select-ing the optimal initial inventory to minimize the totalcost using an exhaustive search.For the cluster decomposition solution, we first run

the optimization that solves the strategic planningproblem to identify the optimal clusters within thelarger group and initial allocation of inventory withineach cluster (as described in section 3.6). Next, wesolve an MDP for each cluster separately, only allow-ing transshipment between clinics within the samecluster. This approach bridges the strategic and thetactical/operational models.For the baseline model, we simply solve the base-

line optimization and allocate inventory accordingly.In solving the MDP, we use the actual demand pat-terns (scaled down for tractability) at each clinic toincorporate (i) non-stationary demand by month dur-ing the six-month malaria season, and (ii) demandvariability by year by including very high, high, med-ium, low, and very low years.To select the four larger groups of clinics to study,

we first identified groups of four and five clinics that(i) were all in close proximity to one another and (ii)were split into two clusters based on the global strate-gic solution (containing all 290 clinics). We cappedthe group size at five clinics because analyzing anylarger group of clinics causes the fully integratedsolution to be intractable due to the curse of dimen-sionality. Furthermore, including larger groups usu-ally entails groups of clinics with significant distancebetween clusters, in which case the optimal solutionwould almost never utilize shipping routes not avail-able in our cluster topology—as seen in the strategicsolution. Hence, this analysis should be sufficient tocapture the key comparison of full integration vs. thedecomposition approach. Table 6 shows distances inkilometers between clinics in each of the four mastergroups.

999888777666555444333222111000

-1 -1 -1-2 -2 -2-3 -3 -3-4 -4 -4-5 -5 -5

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Clin

ic 2

Clin

ic 2

Area 1No Ac�on

Area 2No Ac�on

Area 31,2 3

2 1,3

Area 4

Area 5

1 2,3

Clinic 1Clinic 3 Inventory Level = 0


Area 3No Ac�on


Clin

ic 2

13

2 3

2 1,3

2 1

12

Area 5

1 2,3

Area 31,2 3

Area 42 1

Area 5

1 2

Area 1No Ac�on

Area 13 1,2

Area 2No Ac�on

Area 2

Area 4

3 1

32

Figure 10 Optimal Actions in Period 5 for a Cluster Consisting of Three Clinics

Table 5 Five Areas in the Three-Dimensional Illustrative Example

Area Description Actions

1 Clinics 1 & 2 face shortage If 3 has surplus, transships to1 and 2, no action otherwise

2 Clinics 1 & 2 havelow inventories

If 3 has surplus, transships to1 or 2, no action otherwise

3 Clinics 1 & 2 havehigh surplus

1 & 2 transship to 3 if needed

4 Clinic 2 has surplus Clinic 2 transships to Clinics1 and/or 3 if needed

5 Clinic 1 has surplus Clinic 1 transships to Clinics1 and/or 3 if needed


The clusters derived from the strategic solutions foreach group were as follows. Group 1: Cluster1 = Clinic 274 and 285; Cluster 2 = Clinic 139, 142,and 143. Group 2: Cluster 1 = Clinic 249 and 252; Clus-ter 2 = Clinic 1, 2, and 263. Group 3: Cluster 1 = Clinic12 and 13, Cluster 2 = Clinic 256 and 265. Group 4:Cluster 1 = Clinic 3 and 4; Cluster 2 = Clinic 225 and252.We now present the optimality gap of both the clus-

ter-based decomposition strategy and the baselinemodeling approach compared to the fully integratedoptimal solution. We analyze different starting levelsof total inventory for the entire group. We start at thehighest levels that can potentially satisfy the fulldemand in most scenarios and decrease the total ini-tial inventory in the cluster by two until reaching avery low level of 10. This allows us to study the solu-tion at different levels of inventory relative to demandto capture the impact of inventory scarcity (or lackthereof) on the optimal solution as well. Since each ofthe clusters was normalized to have similar averagedemand (although different dispersion of demandacross clinics in the cluster), we use the same rangefor all clusters. Table 7 presents the results for the fourgroups for very low initial inventory levels (10) up tohigh initial inventory levels (32). Beyond this size ofinitial inventory the fully integrated model for fiveclinic groups (group 1 and 2) became intractable.There are several key observations from the table.First is that the percent optimality gap for the pro-posed decomposition heuristic is very small for both 4and 5 clinic groups, typically between 0%–1%, with

the average gap being 0.5% and the maximum onlyreaching 3.2%. Second, the gap for the decompositionheuristic remains stable as the initial inventoryincreases, whereas the gap for the baseline modelgrows monotonically at an increasing rate. When theinitial inventory is low, all models can use nearly allof the initially allocated demand. The decompositionheuristic continues to track the fully integrated modelclosely because the clinic clusters created by thedecomposition heuristic have the property that cross-cluster shipping is generally undesirable so the fullyintegrated model rarely uses these shipping lanes.Hence, the fully integrated model behaves like theclustered model in most cases, which leads to thesmall optimality gap.

4.4. Computational Bounds for the FullyIntegrated ModelIn section 4.3, we show that the cluster-based modelbehaves like the fully integrated model for a limitedset of clinics. However, the fully integrated model canbecome intractable (due to the increase in the state-space) as the number of clinics increases. In this sec-tion we calculate lower bounds on the fully integratedmodel for larger problem instances to describe theperformance of the cluster-based model for a largerset of clinics.Karmarkar (1987) develops a Lagrangian relaxation

approach (by relaxing inventory balance constraintsand adding them with a penalty to the objective func-tion) to calculate a lower bound for multi-location,multi-period inventory problems. This work also

Table 6 Intraclinic Distances for Four Larger Groups of Clinics that can be Decomposed into Clinic Clusters

Group 1 Group 2 Group 3 Group 4

139 142 143 274 285 1 2 249 252 263 12 13 256 265 3 4 225 232139 – 27 13 12 13 1 – 5 15 19 16 12 – 8 30 32 3 – 5 15 12142 27 – 21 15 20 2 5 – 16 20 18 13 8 – 27 31 4 5 – 16 14143 13 21 – 11 19 249 15 16 – 4 6 256 30 27 – 4 225 15 16 – 6274 12 15 11 – 9 252 19 20 4 – 6 265 32 31 4 – 232 12 14 6 –285 13 20 19 9 – 263 16 18 6 6 –

Note: Data in matrix form with clinic number as the row and column headers.

Table 7 Optimality Gap (%) for Decomposition Heuristic and Baseline Optimization vs. the Fully Integrated Optimization for Clinic Groups 1–4

Initial inventory in group

10 12 14 16 18 20 22 24 26 28 30 32

Group 1 Decomp 0.0 0.0 0.0 0.0 0.0 0.1 0.4 1.1 1.1 0.8 0.8 1.6Baseline 3.5 4.8 7.4 10.6 14.4 19.0 26.3 37.7 52.1 70.6 95.1 126.9





proposes a method for calculating upper bounds bydecomposing the problem by location—similar to ourcluster-based approach. In this vein, we develop aneasily implementable lower bound for our problem.Instead of dualizing the inventory balance constraints,we adapt a version of the two-stage stochastic pro-gram in which all the demand is realized in twostages. We consider a set of 14 clinics—groups 1,2,and 3 as described in Table 6 and compare the lowerbound (based on this stochastic program adaptation)and the upper bound (results of

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