inventory management in humanitarian operations: impact...

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Inventory Management in Humanitarian Operations:Impact of Amount, Schedule, and Uncertainty in FundingKarthik V. Natarajan, Jayashankar M. Swaminathan

To cite this article:Karthik V. Natarajan, Jayashankar M. Swaminathan (2014) Inventory Management in Humanitarian Operations: Impact ofAmount, Schedule, and Uncertainty in Funding. Manufacturing & Service Operations Management 16(4):595-603. http://dx.doi.org/10.1287/msom.2014.0497

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MANUFACTURING & SERVICEOPERATIONS MANAGEMENT

Vol. 16, No. 4, Fall 2014, pp. 595–603ISSN 1523-4614 (print) � ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.2014.0497

© 2014 INFORMS

Inventory Management in HumanitarianOperations: Impact of Amount, Schedule, and

Uncertainty in FundingKarthik V. Natarajan

Carlson School of Management, University of Minnesota, Minneapolis, Minnesota 55455, [email protected]

Jayashankar M. SwaminathanKenan-Flagler Business School, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599,

[email protected]

Funding for humanitarian operations in the global health sector is highly variable and unpredictable. We studythe problem of managing inventory in the presence of funding constraints over a finite planning period. Our

goal is to determine the optimal procurement policy given the complexities associated with funding and alsoto analyze the impact of funding amount, funding schedule, and uncertainty around the funding timing onoperations. We use a multiperiod stochastic inventory model with financial constraints and demonstrate thatdespite the funding complexities, the optimal replenishment policy is a state-independent policy that can be easilyimplemented. We also provide analytical results and several insights based on our computational study regardingthe effect of funding timing uncertainty and variability on the operating costs and fill rates. Among other results,we find that receiving funding early is beneficial in underfinanced systems while avoiding funding delays iscritical in fully financed systems. Our analysis also indicates that receiving less overall funding in a timely mannermight actually be better than delayed full funding.

Keywords : inventory management; humanitarian operations; fundingHistory : Received: June 8, 2012; accepted: May 31, 2014. Published online in Articles in Advance August 28, 2014.

1. IntroductionFinancial flows play an important role in humanitarianoperations and impact their scope, effectiveness, andefficiency. The total amount of donations received canimpact the efficacy of such operations, but the timing,predictability, and flexibility of usage around thosefunds also have a strong influence (Wakolbinger andToyasaki 2011). In a global health context, unpredictabil-ity and delays in donor funding are often cited as thereasons behind impaired supply chain managementand reduced coverage. A recent study by the BrookingsInstitution (Lane and Glassman 2008) estimates that forevery dollar received in funding, 7¢ to 28¢ is lost dueto funding delays.

Motivated by the ready-to-use therapeutic food(RUTF) supply chain in Africa, we study the problemof managing inventory in the presence of fundingconstraints over a finite horizon (Swaminathan 2010).In the problem that motivated this study, the countryoffice of UNICEF that procured RUTF was constrainedby the timing of the receipt of previously promisedfunding from donor agencies. The unpredictable natureof donor funding is typical of the funding situation inmany global health programs. Funding is received ininstallments throughout the planning period, and evenin the best of situations, there might be uncertainty

in terms of timing of receipt of those installments; inothers, both the installment amounts and timing couldbe uncertain. An important question that arises is howto effectively manage inventory, taking into accountboth the current financial position as well as the fundsthat are due to arrive in the future periods.

In this paper, we model the above situation with amultiperiod stochastic inventory model with financialconstraints. Our goal is to (i) determine the optimalordering policy given the funding complexities and(ii) characterize the impact of funding timing, fundinglevel, and funding uncertainty on the operating costs.Among other results, we show that despite the uncer-tainty in funding, the optimal replenishment policy isa state-independent modified base stock policy, whichgreatly enhances the appeal and implementation ofthe optimal policy. We also prove that uncertaintyin funding timing (in comparison to a deterministicfinancial schedule) increases operating costs and sodoes the stochastically dominated late arrival of funds.Finally, we also show that increased variability in thefunding timing (as measured by convex ordering) leadsto higher costs.

Through an extensive numerical study, we offerinsights into several issues including the impact offunding patterns, funding level (total funding received

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Natarajan and Swaminathan: Inventory Management in Humanitarian Operations596 Manufacturing & Service Operations Management 16(4), pp. 595–603, © 2014 INFORMS

as a percentage of the amount required to meet totalexpected demand) and uncertainty in funding. Ouranalysis provides the following computational insights.(1) Front-loading (where a majority of the total fundingis received in the initial periods) brings significantbenefits in underfinanced systems (< 100% fundinglevel), whereas avoiding back-loading (where a majorchunk of the total funding is received in the later peri-ods) is critical in fully financed systems (100% fundinglevel). (2) Front-loaded funding at 75% funding level isbetter than back-loaded funding at 100% funding level.Depending on the level of front- and back-loadingat 75% and 100% funding levels, the operating costsunder back-loaded funding at 100% funding level varybetween 1.5 and 5.5 times the operating costs underfront-loaded funding at 75% funding level. (3) Front-loading offers significant benefits in programs offeringproducts with high penalty costs, while the gains arerelatively modest in low-penalty environments witha low demand uncertainty. (4) Additional funding ismore valuable in operating environments with high-penalty costs, but demand uncertainty undermines thebenefit of additional funding. (5) There is a nonlinearincrease in costs with increased uncertainty in funding.Further, this effect decreases with demand uncertainty.However, the relationship between funding uncertaintyand fill rates is not monotone, as one would expect.

1.1. Literature ReviewOur work is related to the following three streams ofliterature.

The Interface Between Operations and Finance. In thisstream of literature, a firm’s available capital at thestart of any given period is endogenously dependenton the revenues generated in the previous period;e.g., Archibald et al. (2002), Babich and Sobel (2004),Buzacott and Zhang (2004), Gaur and Seshadri (2005),Xu and Birge (2006), Chao et al. (2008), Hu and Sobel(2008), Caldentey and Haugh (2009), Protopappa-Siekeand Seifert (2010), and Li et al. (2013). Our work is moreclosely related to Chao et al. (2008), who also studyinventory management under financial constraints.They study the inventory replenishment problem facedby a self-financing retailer whose objective is to maxi-mize terminal wealth. For their model, they show thata capital-dependent base stock policy is optimal. Oneaspect that distinguishes our work from the existingliterature is the presence of a donor funding stream thatis exogenous to realized demand. In our work, demandfulfilled in the previous periods does not generate anyrevenue due to the nonprofit nature of the business.

Inventory Management Under Capacity Constraints.Financial constraints on procurement are somewhatsimilar to supply-capacity constraints, and a lot ofwork has been done on inventory management undercapacity constraints (see Federgruen and Zipkin 1986,

Ciarallo et al. 1994, Wang and Gerchak 1996, Aviv andFedergruen 1997, Kapuscinski and Tayur 1998). Themain difference between these models and ours is thatwhereas physical capacity constraints are rigid, financialconstraints can be made flexible. Unlike productioncapacity, unused capital does not go to waste and canbe utilized in the future periods.

Humanitarian Operations. Within humanitarian opera-tions, a majority of the papers focus on disaster relief;e.g., Beamon and Kotleba (2006) and Duran et al. (2011)focus on inventory management during emergencies.In recent years, long-term public health issues havealso received attention from the operations manage-ment community; e.g., Rashkova et al. (2011), Deo andSohoni (2014), and Taylor and Xiao (2014). Our work ismore closely related to Rashkova et al. (2011). Takinginto account the funding and procurement delays inglobal health programs, they develop a simulationmodel to predict stockouts at the country level for agiven funding schedule and inventory replenishmentpolicy. The model is then used to compare differentfunding schedules and replenishment policies. In ourwork, we identify the optimal replenishment policy forany given funding schedule, and using the optimalpolicy, we analyze how funding delays and uncertaintyin funding impact operating costs and fill rates.

The rest of this paper is organized as follows: In §2,we introduce the model and provide analytical resultsregarding the optimal policy and the impact of funding.In §3, we present computational results regarding theimpact of funding timing and funding uncertainty.In §4, we offer some managerial insights and concludethe paper.

2. ModelWe consider a finite-horizon, periodic-review inventorymodel for a single product. The planning horizon isdivided into N periods with reverse time indexing,i.e., the first period is period N , followed by N − 1,N − 2 and so on. Demands in successive periods, �t ,t =N , N − 11 0 0 0 11 are independent but not necessarilyidentically distributed with probability density functionft and cumulative distribution function Ft . Externalfunding is received in m≤N installments. We denotethe funding vector by Z = 4z11 z21 z31 0 0 0 1 zm5, where zmand z1 are the first and last installments, respectively.Therefore, the total promised funding over the horizonis TPF =

∑mj=1 zj . We do not impose any restrictions

on the installment sizes (amount received in eachinstallment), and they could be different from oneanother. The installment sizes are known beforehand,but the time of receipt of each installment could beuncertain. We refer to a scenario with uncertain fundingtiming as a stochastic funding schedule. A special caseof the stochastic funding schedule is the deterministic

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Natarajan and Swaminathan: Inventory Management in Humanitarian OperationsManufacturing & Service Operations Management 16(4), pp. 595–603, © 2014 INFORMS 597

funding schedule where both the installment amountsand timing are known beforehand. Let c be the unitpurchase cost, h denote the unit holding cost perperiod, and b be the penalty cost per unit per periodfor any unsatisfied demand. We make the followingassumptions.

1. Unmet demand is completely backlogged. Thebacklogging assumption is valid for a variety of healthcommodities, e.g., malaria bed nets and reproductivehealth supplies like contraceptives.

2. All the installments are received before the endof the planning horizon. Typically, donors make acommitment based on the funding proposals, and whilethe installment sizes may vary based on the donors’budget cycles, in most cases, the committed amount isreceived in full before the end of the planning periodfor which the donation was sought.

In addition, we also assume that one dose/unit ofthe product is sufficient to meet the needs of a cus-tomer/patient. The sequence of events in any period tis as follows: (1) Funding (if any) is received at thestart of period t. (2) The procurement lead time is� ≥ 0 time periods, and procurement decisions aremade subject to the capital available on-hand and thecurrent inventory position. The procurement decisionis made before receiving any shipments. We follow thisconvention to make the description consistent withthe zero-lead time case. (3) Shipments due to arrive inperiod t are received. (4) Finally, demand is realized,and holding and backorder costs are calculated basedon the ending on-hand inventory.

Let xt denote the on-hand inventory before receivingshipments in period t, and rt be the capital available atthe start of period t, after receiving installments (if any)in period t. We denote the outstanding funding as ofperiod t (after receiving funding at the beginning of theperiod) by OFt and let OFt �OFt+1 represent the randomvariable corresponding to OFt given OFt+1. Note thatOFt+1 −OFt is the funding received at the beginningof period t. Modeling funding inflows in this fashionmakes our model general enough to capture a varietyof funding scenarios, with stochastic and deterministicfunding schedules being special cases.

The objective is to come up with an optimal order-ing policy that minimizes the total cost incurredover the planning horizon subject to the fund-ing constraints. Since we assume self-financing, theorder quantity in period t cannot exceed rt/c. LetG̃�

t 4xt1w0t 1w

1t 1w

2t 1 0 0 0 1w

�−1t 1 rt1OFt5 be the minimum

expected cost with t periods to go, given xt , rt , OFt ,and wi

t . Here, wit is the order that will be received

i periods into the future and w0t will be received in

period t. Then,

G̃�t 4xt1w

0t 1w

1t 1 0 0 0 1w

�−1t 1 rt1OFt5

= min0≤z≤

rtc

8cz+ bE�t6�t − xt −w0t 7

++hE�t

6xt +w0t − �t7

+

+EOFt−1�OFtE�t

G̃�t−14xt +w0

t − �t1w1t 1 0 0 0 1w

�−1t 1 z1 rt

− cz+ 4OFt −OFt−151OFt−1590

Because all the installments are received before theend of the horizon, OF1 = 0 always. The terminal costis G̃�

0 4 · 5= 0 ∀ 4x01w001w

101 0 0 0 1w

�−10 1 r05. In our analysis,

we find it convenient to use a modified value functionexpressed in terms of the variable Rt = rt + cIPt in placeof rt . Here, IPt = xt +

∑�−1j=0 w

jt is the inventory position

at the beginning of period t. Define

(P1) G�t 4xt1w

0t 1w

1t 10001w

�−1t 1Rt1OFt5

= min0≤z≤

rtc

8cz+bE�t6�t−w0

t −xt7++hE�t

6xt+w0t −�t7

+

+EOFt−1 �OFtE�t

G�t−14xt+w0

t −�t1w1t 10001w

�−1t 1

z1Rt−c�t+4OFt−OFt−151OFt−1590

The function G�t is jointly convex in variables xt and

wit , i = 0111 0 0 0 1 �−1 for fixed Rt and OFt (see Lemma 1

in the online appendix, available as supplementalmaterial at http://dx.doi.org/10.1287/msom.2014.0497).Using this joint convexity, it is easy to show thata 4Rt1OFt5-dependent modified base stock policy isoptimal in period t. However, we go one step fur-ther and demonstrate that the optimal replenishmentpolicy is actually simpler—the optimal policy is astate-independent modified base stock policy where theorder up-to levels depend only on t and not on Rt

or OFt .Consider problem P2, with the same cost and

demand parameters and replenishment lead time asproblem P1, but without financial constraints. LetNV �

t 4xt1w0t 1w

1t 1 0 0 0 1w

�−1t 5 be the minimum expected

cost with t periods to go for problem P2. Then,

(P2) NV �t 4xt1w

0t 1w

1t 10001w

�−1t 5

=minz≥0

8cz+bE�t6�t−xt−w0

t 7++hE�t

6xt+w0t −�t7

+

+E�tNV �

t−14xt+w0t −�t1w

1t 10001w

�−1t 1z590

Let NV �0 4 · 5= 0 ∀ 4x01w

001w

101 0 0 0 1w

�−10 5. For P2, it is

well known that there exists an optimal base stocklevel y�∗

t in each period such that if the inventoryposition in period t is below y�∗

t , it is optimal to orderup-to y�∗

t , and not order otherwise. In Theorem 1,we prove that the unconstrained base stock levelsy�∗t 1y�∗

t−11 0 0 0 1 y�∗1 , optimal for P2, are optimal for P1

with funding constraints as well.

Theorem 1. Let y�∗t 1y�∗

t−11 0 0 0 1 y�∗1 be the optimal base

stock levels corresponding to problem P2. Let 4xt1w0t 1w

1t 1

0 0 0 1w�−1t 1Rt1OFt5 be the system state in problem P1 at the

beginning of period t. Then, the optimal ordering policy forproblem P1 has the following simple structure:

Order up-to Rt/c if Rt/c ≤ y�∗t 1

Order upto y�∗t if Rt/c > y�∗

t 1 IPt <y�∗t 1

Do not order if IPt ≥ y�∗t 0

(1)

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Theorem 1 raises an important question: Why arethe base-stock levels independent of OFt and Rt? Recallthat OFt keeps track of the outstanding amount, anddetermines the level of uncertainty in future funding.However, because of backlogging, funding uncertaintydoes not impact the total demand met between t andthe end of the horizon. The total demand satisfieddepends only on the sum: c × (inventory position) +

capital available on-hand + outstanding funding. Sincean increase in OFt implies a corresponding decreasein on-hand funding, the sum is independent of OFt .Hence, the outstanding funding and future fundinguncertainty impact only the actual costs incurred butnot the incremental difference in costs obtained bychanging the order quantity.

A related question is as follows: Why is the base-stock level independent of Rt (= rt + cIPt), which actslike a capacity constraint? Under capacity constraints, itis well known that the (capacity-dependent) order up-tolevels are higher than the corresponding unconstrainedbase stock levels (see, e.g., Federgruen and Zipkin1986). Although both funding and capacity constraintsplace an upper bound on the order quantity, there is afundamental difference between the two. While unusedcapacity goes to waste, unused capital can be used inthe later periods, i.e., it acts like transferable capacity.Intuitively, this is the reason why the unconstrainedbase stock levels are optimal even under fundingconstraints.

2.1. Impact of Funding TimingConsider funding scenarios 1 and 2 such that

OF 2t � 4OF 2

t+1 = i5≥st OF 1t � 4OF 1

t+1 = i5

∀ t ∈ 82131 0 0 0 1N − 19 and (2)

OF nt � 4OF n

t+1 = i′5≥st OF nt � 4OF n

t+1 = i5

∀ t = 21 0 0 0 1N − 11n ∈ 811291 i′ > i1 (3)

where ≥st means first-order stochastic dominance (seeShaked and Shanthikumar 2007). Condition (2) impliesthat the outstanding funding at the beginning of anyperiod t is (stochastically) larger under funding sce-nario 2. Condition (3) says that, under both fundingscenarios, the outstanding funding at the beginning oft stochastically increases in the outstanding funding atthe beginning of t + 1. We denote the value functionsunder scenarios 1 and 2 (satisfying conditions (2) and(3) stated above) by G�11

t and G�12t , respectively. In the

following theorem, we demonstrate that (stochastically)early funding leads to lower costs due to increasedprocurement flexibility.

Theorem 2. If conditions (2) and (3) hold, then G�12t 4xt1

w0t 1 0 0 0 1w

�−1t 1Rt1 j5 ≥ G�11

t 4xt1w0t 1 0 0 0 1w

�−1t 1Rt1 j5 for

every j ∈ 601TPF 7.

2.2. Impact of Variability in Funding TimingNext, we focus on the variance aspect of the uncertaintyin funding. For fixed total funding, we consider twofunding scenarios such that

4OFt −OF 2t−15 �OFt ≥cvx 4OFt −OF 1

t−15 �OFt

∀ t = 3141 0 0 0 1N and (4)

4OFt −OF nt−15 �OFt is SSCV in OFt for n= 1120 (5)

Condition (4) states that for fixed OFt , the fundingreceived in period t−1 (or equivalently the outstandingfunding as of t− 1) is more variable under fundingscenario 2 than under scenario 1. The convex orderingimplies that the mean funding received (or equivalentlythe mean outstanding funding) remains the same acrossthe two scenarios. Condition (5) states that the fundingreceived in period t − 1 is strong stochastically concave(SSCV) in OFt under both funding scenarios. Giventhat the expected funding received remains the same,the following theorem shows that a higher variabilityaround the funding received in any given period drivesup the operating costs.

Theorem 3. Let conditions (3), (4), and (5) hold. Then,G�12

t 4xt1w0t 1 0 0 0 1w

�−1t 1Rt1 j5 ≥ G�11

t 4xt1w0t 1 0 0 0 1w

�−1t 1

Rt1 j5 for every j ∈ 601TPF 7.

3. Computational StudyFor our computational study, we first consider deter-ministic funding schedules and analyze how fundingpatterns impact operating costs. Recall that under deter-ministic funding, both the installment amounts and thetiming are fixed. Subsequently, we consider stochasticfunding schedules to understand how uncertainty infunding timing affects system performance. Throughoutthe numerical study, we relate our results to commonoperating environments seen in humanitarian programs.For simplicity, we assume that the replenishment leadtime is zero throughout the numerical study.

3.1. Deterministic Funding Schedules

3.1.1. Experimental Setup. We consider planninghorizon, N , of different lengths, N = 2, 4, 6, 12, and 24.Purchase cost c was normalized to 1. For each N , wevaried the following parameters.

Holding Cost: We chose four values for holding cost—h= 0001, 0.05, 0.1, and 0.25.

Penalty Cost: For each h, the penalty cost b was variedso that the critical ratio (CR), 4b− c5/4b+h5, took onvalues 0.2, 0.4, 0.6, 0.8, 0.9, and 0.95, respectively.

Demand: We consider uniform and truncated normaldemand. To test the impact of demand variability, weconsidered U ∼ 67011307 and U ∼ 62511757 for uniformdemand. For normal demand, the mean was fixed at100 units and we used coefficient of variation values of

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0.1 and 0.25. The normal distribution was truncated atthree standard deviations. Thus, for each N , we have4 ∗ 6 ∗ 4 = 96 problem instances.

Funding Patterns: For each combination of N1h1 b anddemand, we consider five funding patterns and fourfunding levels. The funding patterns are extremelyfront-loaded (EFL) funding, moderately front-loaded(MFL) funding, evenly spread (ES) funding, moderatelyback-loaded funding (MBL) funding, and extremelyback-loaded (EBL) funding. The holding cost andfunding patterns were chosen to be consistent withan earlier study that analyzes the RUTF supply chainin the Horn of Africa (Swaminathan et al. 2009). Forbackorder costs, because of lack of precise estimates,we carry out a sensitivity analysis over a wide rangeof critical ratios. For all the funding vectors, the totalfunding received is equal to N ∗ funding level ∗ meandemand. In our experiments, we consider 25%, 50%,75%, and 100% funding levels and we label them asseverely underfinanced, moderately underfinanced,mildly underfinanced, and fully financed systems,respectively. For example, a severely underfinancedsystem receives N ∗ 0025 ∗ mean demand over theplanning period. We use N = 4, U ∼ 67011307 demandand 100% funding to explain the difference betweenthe funding patterns.

EFL: The entire funding (= N ∗ mean demand) isreceived upfront; i.e., the funding vector is40101014005. Recall that we count time in thereverse order.

MFL: In the first N/2 periods, the installment sizeis 105 ∗ mean demand followed by 005 ∗ meandemand in the last N/2 periods. For the spe-cific case considered, the funding vector is450150115011505.

ES: Every installment is equal to mean demand;i.e., the funding vector is 41001100110011005.

MBL: In the first N/2 periods, the installment sizeis 005 ∗ mean demand followed by 105 ∗ meandemand in the last N/2 periods. The fundingvector would be 415011501501505.

EBL: The entire funding is back-loaded to the lastperiod; i.e., the funding vector is 44001010105.

Note that as we move from EBL funding to EFLfunding, additional funds are received in the initialperiods. Compared to ES funding, front-loading can beviewed as a funding advance and back-loading canbe considered a funding delay. Using ES funding asa benchmark, we investigate how back-loading andfront-loading the funding impacts operating costs. Wecompute the relative percentage cost difference for aparticular funding pattern, say EBL funding, as follows:100 ∗ 4costEBL − costES5/costES.

Table 1 Average Percentage Cost Difference for Different FundingPatterns Relative to ES Funding

EBL MBL MFL EFL

948.56 248010 −29076 −3105

3.1.2. Impact of Funding Pattern. From Table 1, wesee that operating costs increase almost exponentiallywith funding delays 4back-loading5. However, contraryto common perception, ES funding is not the optimalfunding pattern since, under ES funding, there is littleflexibility to deal with large demand surges upfront.Having flexibility upfront is valuable, and we seethat even moderate front-loading results in savingsof 29.8%. However, pushing additional funds to theinitial periods yields little to no return and, even incase of EFL funding, the savings is only 31.5%.

Next, we analyze if the insights obtained from Table 1change with the operating environment. We focus ontwo parameters, critical ratio and demand uncertainty.We visualize different operating environments as beingelements of the 2 × 2 matrix in Table 2, correspondingto high and low levels of demand uncertainty andcritical ratio, respectively. Some illustrative examples ofproducts that fit into each category are provided in thetable (see Swaminathan et al. 2009; USAID 2009, 2010).For illustration, we label the demand distributions withrange 67011307 and 62511757 as low variability andhigh variability demand, respectively. We pick CR= 002and CR= 0095 to represent a low and high critical ratio,respectively.

Effect of the Operating Environment on the RelativeCosts of Different Funding Patterns. In Table 3, we pickN = 24 and normal demand to present the insights,but the results hold for other N and uniform demandas well. From the table, we see that the benefits offront-loading increase with the critical ratio, and thiseffect is amplified under demand uncertainty. Underback-loading, the results are different. We see thatcritical ratio amplifies the negative impact of back-loading, whereas demand uncertainty has a mitigatingeffect. Overall, Table 3 illustrates that avoiding back-loading is paramount to performance improvementin all operating environments. On the other hand,front-loading offers significant gains for products with

Table 2 Matrix of Different Operating Environments

Critical ratio

L H

Demand uncertaintyL Essential medicines,

e.g., aspirin tabletsDepo-Provera

contraceptive injectionsH Disposable gloves,

cotton wool packsReady-to-use

therapeutic foods

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Table 3 Effect of Demand Variability and Critical Ratio on thePercentage Cost Difference (Relative to ES Funding) forDifferent Funding Patterns at 100% Funding Level, N = 24

EBL MBL MFL EFL

CR = 002 N ∼ 67011307 11239010 312082 −12069 −12084N ∼ 62511757 999043 240082 −24075 −25084

CR = 006 N ∼ 67011307 21222064 561051 −23039 −23067N ∼ 62511757 11566074 378013 −39090 −41061

CR = 0095 N ∼ 67011307 61325019 11598094 −68061 −69044N ∼ 62511757 21936011 709073 −76094 −80033

high critical ratios but the gains are relatively modestat low critical ratios, especially in environments withlow demand uncertainty. Hence, the decision to investin front-loading initiatives needs to be reconciled withthe nature of the product being offered by the program.

3.1.3. Impact of Additional Funding. In Figure 1,we use the cost incurred under no funding constraints(NFC) as the benchmark. Note from Figure 1 that thebenefits of additional funding remain fairly constantat all funding levels under evenly spread and back-loaded funding. However, the benefits of additionalfunding reduce significantly with the funding levelfor front-loaded funding, and the rate of reductionincreases with the degree of front-loading.

Figure 1 also demonstrates that the losses fromback-loading (gap between the lines corresponding toMBL/EBL and ES funding) are monotone increasing inthe funding level, whereas the benefits of front-loadingfollow a U-shaped pattern, with maximum benefitsseen either in moderately or mildly underfinanced

Figure 1 (Color online) Average Percentage Cost Difference forDifferent Funding Patterns at Different Funding LevelsRelative to NFC

EBL

MBL

ES

MFL

EFL

EBL

MBL

ES

MFL

EFL

N = 4

N = 24

25 50 75 100

25 50 75 100

Funding level (%)

Funding level (%)

1,400

1,200

1,000

800

600

400

200

0

9,0008,0007,0006,0005,0004,0003,0002,0001,000

0

(a)

(b)

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Table 4 Effect of the Operating Environment on the Incremental Benefitsof Additional Funding (Between 75% and 100% Funding Level,Relative to NFC) for Different Funding Patterns, N = 24

EBL MBL ES MFL EFL

CR = 002 N ∼ 67011307 6045 272039 350007 141008 85018N ∼ 62511757 5083 262054 316072 138039 83055

CR = 0095 N ∼ 67011307 488028 41984015 61307062 21774044 11825021N ∼ 62511757 440032 41680053 51597083 21656082 11737029

systems. The intuition is that in severely underfinancedsystems, the amount pushed to the initial periods isrelatively less, and in fully financed systems thereis sufficient cash already available for the additionalfunding to make a large impact.

Table 4 provides insights regarding how the bene-fits of additional funding change with respect to theoperating environment. From Table 4, we see that thebenefit of additional funding increases with the criticalratio but demand uncertainty undermines the benefits.Furthermore, the table also reveals that for productswith a low critical ratio, additional funding has lowimpact when funding is extremely back-loaded. In thiscase, it might be more beneficial for an organization tofirst work on reducing funding delays before trying tosecure more funding.

3.1.4. Funding Level vs. Funding Pattern. We referback to Figure 1 to understand the relative importanceof funding level vis-à-vis funding pattern. From thefigure, we see that at low funding levels (25% and50% funding levels), the funding pattern is almostinconsequential—except for extreme back-loading, allother funding patterns at 100% funding level performbetter than extreme front-loading at 25% and 50%funding levels. For a mildly underfinanced system,the results are drastically different. From Figure 1,we see that back-loaded funding at 100% fundinglevel performs significantly worse compared to front-loaded funding at 75% funding level. However, ESfunding at 100% funding level outperforms even EFLfunding at 75% funding level. This demonstrates that,at reasonably high funding levels, funding pattern iscritical and an increase in overall funding should notbe traded for a delay in funding.

3.2. Stochastic Funding Schedules

3.2.1. Experimental Setup. Except for funding pat-terns, the experimental setup remains unchanged fromdeterministic funding. As we mentioned in §2, fundingis received in m≤N installments. For a fixed N , weconsider several values of m for the stochastic fundingcase, details of which are given in Table 5.

For stochastic funding, we assume 100% fundinglevel. To capture the uncertainty in the funding timing,we vary the number of installments. Depending onm, the amount received in each installment varies

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Table 5 Number of InstallmentsConsidered for Each N

N m

2 1, 24 1, 2, 3, 46 1, 2, 3, 4, 612 1, 2, 3, 4, 6, 1224 1, 2, 3, 4, 6, 12, 24

(=N/m∗ mean demand in each period). We assumethat the number of installments received in a periodis uniformly distributed between 0 and the numberof outstanding installments. To understand what hap-pens when we increase the number of installments,consider the example of N = 4, m= 1 and U ∼ 6701 1307demand. When m= 1, the four extreme funding sce-narios, namely 44001010105, 40140010105, 40101400105and 40101014005, are equally likely. When we increasem to 2, the probability of extreme funding scenarioslike 44001010105 reduce from 1/4 to 1/16 while theprobability of a more evenly spread funding vector like4200101200105 increases to 1/8. Hence, the probabilityof the funding being more evenly spread out increaseswith the number of installments, thereby reducing thevolatility in funding received until any given period.

3.2.2. Impact of Funding Uncertainty. From Fig-ure 2, we see that reducing the funding volatility, andmaking the funding more smooth and evenly spreadout, lowers operating costs. However, such benefitsof reducing funding uncertainty show diminishingrates of return, i.e., as m increases, the marginal valueof receiving the funding in an additional installmentdecreases.

To understand why reducing funding uncertaintyleads to lower costs, consider the case where m= 1 andN = 6. The single installment could be received in anyof the six periods with equal probability. Of course,there is nothing better than receiving the installmentin the first period (probability 1/6), but we also need

Figure 2 (Color online) Average Percentage Cost Difference Due toFunding Timing Uncertainty Relative to a Funding Schedulewith m = N

N = 2

N = 4

N = 6

N = 12

N = 24

1 2 3 4 5 126 24

100

90

80

70

60

50

40

30

20

10

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Number of installments (m)

Table 6 Effect of Demand Variability and Critical Ratio on theAverage Relative Percentage Cost Difference Due toFunding Uncertainty (Relative to a Funding Schedulewith m = N), N = 24

m = 1 m = 2m = 3m = 4m = 6m = 12m = 24

CR = 002 N ∼ 67011307 78082 33050 19027 14091 8086 2069 0N ∼ 62511757 76054 32020 18033 14012 8025 2038 0

CR = 0095 N ∼ 67011307 106045 45029 26020 20027 11094 3052 0N ∼ 62511757 102078 43029 24080 19009 11005 3009 0

to take into account the other possibilities includingan extreme back-loading with probability 1/6. Con-sidered together, receiving funding in one installmentis no longer ideal—in fact, it is the worst. In general,when the number of installments decreases, it increasesthe possibility of both front- and back-loading. How-ever, the nonlinear increase in costs as we move fromfront-loaded to back-loaded funding implies that inexpectation, the operating costs increase.

From Table 6, we see that the benefit of reducingfunding uncertainty increases with the critical ratiobut demand uncertainty undermines the benefits. Theundermining effect is more pronounced at highercritical ratios. Table 6 further illustrates that to achievecomparable performance (in terms of relative costs),humanitarian organizations offering products with highpenalty costs need to reduce funding uncertainty evenfurther when compared to low critical ratio products.This highlights the greater need to work with donors,financial institutions, and other stakeholders to reducefunding uncertainty in high-penalty environments likeRUTF programs.

3.2.3. Funding Level vs. Funding Uncertainty. Inthis section, we aim to address the following question:Which of the two has a greater impact on systemperformance—funding level or funding uncertainty?Comparing rows 1 and 2 in Table 7 with Table 8,we see that at low (25% and 50%) funding levels,the funding pattern is inconsequential. Receiving lessoverall funding severely hurts performance, makingeven the most uncertain funding (m = 1) at 100%funding level attractive in comparison in almost allcases (an exception being EFL funding at 50% fundinglevel).

For 75% funding level, the results are very different(compare row 3 in Table 7 with Table 8). While the

Table 7 Average Relative Percentage Cost Difference4100 ∗ 4Cost−CostNFC5/CostNFC) for Different DeterministicFunding Patterns at Different Funding Levels

Funding level (%) EBL MBL ES MFL EFL

25 81357025 61894014 61377074 51861040 4183708450 81208085 51282063 41250012 31220079 2120007375 81060046 31671019 21125099 923060 596072100 71917089 21086040 213073 26065 16080

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Table 8 Average Relative Percentage Cost Difference4100 ∗ 4Cost−CostNFC5/CostNFC5 Due to FundingTiming Uncertainty

m = 1 m = 2 m = 3 m = 4 m = 6 m = 12 m = 24

2,800.83 11968097 11710029 11630068 11518065 11407039 11363

back-loaded vectors at 75% funding level performsignificantly worse than even the most uncertain fund-ing at 100% funding level, front-loaded funding at75% funding level outperforms uncertain funding at100% funding level, even when the uncertainty isconsiderably reduced (m= 24). This demonstrates thatat relatively high funding levels, the choice betweendeterministic funding and an even larger overall butuncertain funding is not straightforward.

3.3. Fill RatesOperational efficiency is paramount to improving aideffectiveness, but meeting demand in a timely fashionis also an important consideration in humanitariansettings. We define fill rate as the percentage of demandthat is met on time without being backlogged. FromTable 9, we see that under deterministic funding,front-loading holds the key to increasing fill rates inunderfinanced systems while avoiding back-loadingis important in fully funded systems. This result isconsistent with our earlier discussion with respect tooperating costs. Interestingly, under stochastic funding,we see from Table 10 that the relationship betweenfunding uncertainty and fill rate is not monotone asone would expect. The probability of both front- andback-loading increase with funding uncertainty and theresults demonstrate that, in expectation, the positiveimpact of front-loading could dominate the negativeeffects of back-loading when it comes to fill rates.However, fill rates only tell one part of the story—what fraction of the demand is met on-time withoutbacklogging. For backlogged demand, an analysis ofour data shows that the average waiting time actually

Table 9 Average Fill Rates for Different Funding Patterns at SeveralFunding Levels, N = 24

EBL MBL ES MFL EFL

Funding level 0025 0000 0057 1018 1087 25013005 0000 1018 2087 8059 500090075 0000 1089 8077 59096 75009100 2076 10044 76006 96091 97089

Table 10 Average Fill Rates for Different Number of Installments forN = 24

m = 1 m = 2 m = 3 m = 4 m = 6 m = 12 m = 24

50.06 51014 51024 52043 51069 53072 53070

decreases with the number of installments and so doesthe cumulative waiting time for the population.

4. Conclusions and Managerial InsightsIncorporating funding flows into operational decisionsis necessary for making optimal and operationallyfeasible decisions. In this paper, we study the problemof managing inventory of a health commodity subjectto variable funding constraints. Our work brings outseveral important insights that would be valuable tohumanitarian supply chain managers. To begin with,we find that preventing funding delays should be thetop-most priority for humanitarian organizations. Ourstudy suggests that front-loading initiatives need to bereconciled with the system funding level. Moderatefront-loading is beneficial at all funding levels, butextreme front-loading brings little to no additionalbenefits in a fully financed system. Moreover, for bothmoderate and extreme front-loading, the benefits offront-loading are nonmonotone in the funding level.Our study also demonstrates that the benefits of front-loading are significantly lower for products with a lowcritical ratio, especially in low-demand uncertaintyenvironments. Managers need to exercise caution anduse careful judgment when deciding the level of front-loading in such situations.

Oftentimes, humanitarian organizations make anall-out effort to raise as much funding as possible tosupport the various programs, but our analysis showsthat such an approach is not the most effective one.Our results indicate that even if the funding level islower, performance may be better if the funding isreceived earlier or in a steady fashion.

Finally, managers also need to pay close attentionto their operating environment when taking steps toimprove the funding situation. One such aspect isthe volatility of the underlying demand. Our analy-sis shows that demand uncertainty undermines thebenefits of additional funding. Moreover, whereas themagnitude of the benefits of front-loading increasewith demand volatility, the opposite is true regard-ing the savings resulting from reducing the fundinguncertainty. Front-loading initiatives are likely to yieldsignificant benefits in highly unpredictable environ-ments like RUTF programs, while reducing the fundinguncertainty can be expected to result in substantialsavings in case of health commodities like reproductivesupplies, which have a relatively more stable demandpattern.

Although we have several interesting computationalresults in this paper, some of those are worthy of deeperanalytical analysis. In particular, our results show thatthe benefits of front-loading follow a U-shaped patternwith respect to the funding level, with maximumbenefits seen in moderately or mildly underfinanced

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systems. A potential avenue for future work would beto analytically characterize the threshold funding levelat which the benefits of front-loading start to decline.Such a characterization could be useful in identifyingthe tipping point at which humanitarian supply chainmanagers can expect to see a “diminishing rate ofreturn” on initiatives to secure additional funding fromdonors. Furthermore, our results demonstrate that thetrade-off between early funding and additional fundingis not straightforward at relatively high funding levels.It would be potentially useful to develop an analyticalmodel to more completely characterize the benefits ofearly vis-à-vis additional funding. Finally, since demanduncertainty undermines the benefits of reducing thefunding uncertainty, it could be potentially valuableto model demand and funding uncertainty in greaterdetail and explore the right combination of efforts todirect towards reducing demand uncertainty vis-à-visreducing funding uncertainty.

Supplemental MaterialSupplemental material to this paper is available at http://dx.doi.org/10.1287/msom.2014.0497.

AcknowledgmentsThe authors are grateful to the editor-in-chief, the associateeditor, and the three anonymous referees for their construc-tive feedback, which has helped in improving the contentsand presentation of this paper significantly. Furthermore,the authors thank the seminar participants at Drexel Univer-sity, University of Notre Dame, Indian School of Business,University of Minnesota, Georgia Institute of Technology,University of California, Berkeley, Utah Winter OperationsConference 2012, 2012 MSOM Conference, 2012 AnnualPOMS Conference, and 2012 and 2013 INFORMS AnnualMeetings for valuable feedback.

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