Mitigating the U.S. Drug Shortages Through Pareto‐Improving Contracts

Causes of Drug Shortages

The current drug shortages primarily concern generic sterile injectable drugs, which account for 80% of the 127 shortage cases studied by the FDA in 2010 and 2011 (FDA 2011). Several characteristics of generic sterile injectable drugs make them vulnerable to shortages. First, these drugs often have low profit margins, as most of them are long off patent. Second, sterile injectable drugs require costly production lines and complex manufacturing processes that are vulnerable to quality problems violating the Current Good Manufacturing Practices (cGMPs). In the case of a cGMPs violation, depending on its severity, drug manufacturers often need to halt the production immediately, fix the problem, and wait for the FDA's inspection and approval to resume production. Some violations may even cause facility shutdowns for a substantial amount of time (FDA 2011).

The combination of low profit margins and complex production processes makes the drugs unappealing to manufacturers, leading to a high market concentration. The top three generic injectable drug manufacturers hold 71% of the entire market, and most sterile injectable drugs have one manufacturer that produces at least 90% of the drug, innovator and generic combined (FDA 2011). Further, although existing production lines can often produce multiple drugs with similar manufacturing characteristics, these sterile injectable drugs often receive low capacity allocation due to their slim margins. Manufacturers may even discontinue their production to seek higher profits from other drugs (GAO 2011).

The above characteristics of sterile injectable drugs result in industry‐wide high capacity utilization, which often implies less time for maintenance, thereby exacerbating the drug supply chain's vulnerability to quality‐related disruptions. For example, Thomas (2012) reports a case where managers order shortcuts that compromise quality and skimp on cleaning to shorten the turnaround time between batches. Such practices increase the chances of quality problems that may force a production shutdown. The high capacity utilization also makes it challenging to respond effectively to disruptions; disruptions that could otherwise be absorbed through a diversion of capacity can lead to cascading and persistent shortages (ASPE 2011, FDA 2011, GAO 2011). As a result, disruptions due to quality problems are linked to 54% of shortages (ASPE 2011), followed by the lack of capacity or other delays (21%) (Piana 2012). Furthermore, the disruption durations are highly significant and uncertain, with an average duration of over 9 months (GAO 2011).

Given that manufacturers lack incentives to maintain sufficient capacity, would they hold sufficient inventory to counter disruptions? Unfortunately, holding sterile injectable drugs is costly, as they require sterile, light‐ and temperature‐regulated storage environments. Meanwhile, failure to meet customer orders incurs very low cost: although most drug purchase contracts include failure‐to‐supply clauses (FTS) that require manufacturers to compensate buyers when failing to supply the drugs, these clauses are often very weak. For example, many contracts only compensate a buyer if the buyer can find the same drug from an alternative source, and the contracts only reimburse the price difference. However, due to the high market concentration, it is very difficult to identify an alternative source during times of shortage. As a result, drug manufacturers, on average, only compensate 10% of buyers’ losses due to FTS (ASPE 2011). Obviously, such weak FTS clauses do not help secure drug supply.

In summary, as GAO (2014) concludes, production disruptions and constrained manufacturing capacity are the key immediate causes of drug shortages, and low profit margins are the key underlying cause (cf. Figure 7 in GAO 2014). Indeed, the low prices provide “no incentive for manufacturers to produce them (the drugs) or invest in the necessary infrastructure to prevent or fix a production problem” (American Society of Clinical Oncology (ASCO) in Piana 2012). However, the weak FTS clauses, overlooked by much literature, are also detrimental, as they disincentivize drug manufacturers from maintaining sufficient inventory to satisfy customer demand.

Our Proposed Approach

Among the different perspectives that have been taken to analyze this problem, one key piece seems missing: the supply chain perspective. The drug shortage problem involves multiple supply chain parties with distinct objectives and unique relationships: while a drug manufacturer aims at maximizing its profit, the government, as the major payer (many of the drugs on shortage are covered by government programs such as Medicare), hopes to maintain its spending while reducing shortages. Healthcare providers are keen to mitigate shortages to reduce suboptimal patient outcomes, but they are also concerned about cost reduction. At the same time, the group purchasing organization (GPO), which represents its healthcare provider members and negotiates the drug purchase contract with the manufacturer, aims to improve its profit. With these different parties, the drug supply chain possesses the unique characteristics: the buyer (the provider) is not the payer (government) or the consumer (patients), the price setter (the GPO in many cases) is not the payer or the buyer or even the consumer, and both the buyer and the payer have social welfare concerns beyond cost considerations. Thus, effective shortage mitigation calls for Pareto‐improving solutions considering all the supply chain parties’ objectives and their unique characteristics.

In this study, we take the supply chain perspective and propose mitigating drug shortages through supply chain contract. As mentioned earlier, in drug purchase contracts, the contract price is often over‐emphasized, whereas the FTS clause seems nominal and is often overlooked. In this study, we propose to consider both price and FTS as key contract parameters to gain a complete understanding of the impact of the contract on shortage mitigation. Note that this impact is influenced by a price‐capacity‐inventory triangular effect. Intuitively, increasing prices can incentivize drug manufacturers to increase capacity. Given that demand is relatively insensitive to price (most of these drugs are medically necessary), the increased capacity would also reduce capacity utilization, improve quality maintenance, and potentially mitigate disruptions and, hence, shortages. In the meantime, increasing prices can also incentivize manufacturers to hold more inventory. However, capacity and inventory are strategic substitutes (cf. Angelus and Porteus 2002, Chao et al. 2009, Allon and Zeevi 2011): increasing capacity may reduce inventory because with sufficient capacity, a manufacturer can easily ramp up production if needed and does not need to keep high inventory levels. Hence, the effectiveness of the contract on shortage mitigation is not straightforward.

To evaluate the effectiveness of the Pareto‐improving contracts, we thoroughly investigate the drug supply chain by employing multiple research methodologies. First, we develop an analytical model of the drug supply chain, which captures all different parties’ objectives and decisions, and incorporates key features of the problem, such as capacity‐dependent disruption probability, uncertain recovery time, and mixed lost sales and backorders. Based on the model, we characterize the impact of the contract parameters on the objectives and explore analytical properties of the Pareto‐improving contracts. Next, by applying the widely‐used strategic decision‐making method of scenario analysis, we develop three major scenarios based on key features of the problem and realistic industry data, and test the effectiveness of our proposal on each scenario. To assure the completeness of our test, we further complement the scenario analysis with a full factorial analysis to cover a wide range of other cases. Further, we discuss our results with the FDA and ASPE staff and industry practitioners, all of whom provide insights on our proposal from various perspectives.

Our study generates several key findings. First, increasing drug prices only (referred to as IP), which has been advocated as a natural solution to the drug shortage problem in response to the current low profit margins, is unfortunately not very effective in mitigating drug shortages. Increasing drug prices must be paired with strengthened FTS clauses (referred to as IPS) to achieve consistent and significant shortage reduction. The effectiveness of this pairing is due to the triangular effect described earlier: an effective shortage mitigation requires a significant price increase to overcome the substitutional effect between capacity and inventory, but the IP approach does not render a large enough Pareto‐improving space for price increase. The IPS approach, on the other hand, adds a significant dimension to resolve this problem and is shown to be much more effective over a wide range of scenarios. For all scenarios tested, a 30% price increase under IPS leads to a minimum, average, and maximum shortage reduction of 25%, 53%, and 70%, respectively.

GPO

A GPO mainly gains revenue from two sources. First, a GPO charges a contract administration fee (CAF) from manufacturers as a percentage commission (say, α percent) on sales to GPO members. The CAF is the primary source of a GPO's revenue (Burns 2002). In this study, we consider α as exogenous, since it is decided based on the sales of all drugs; the drugs in shortage, although causing serious consequences, constitute only a small portion of all drugs. Besides the CAF, a GPO also charges a membership fee to each participating healthcare provider. This fee is exogenous, flat, can be viewed as a sunk cost, and is thus not modeled in this study. It is also worth noting that a GPO does not hold any inventory. Thus, given a contract (p,s), the GPO's expected profit is

Γ O ( p , s ) = α p [ 1 − L ( p , s ) ] E ( D ) ,

(1)

Drug Manufacturer

Given a contract (p, s), the drug manufacturer makes two major decisions: the production capacity for the T periods of contract duration and the production/inventory decisions in each of the T periods. In generic drug production, manufacturers often produce multiple drugs with similar manufacturing characteristics on the same production lines and adjust capacity allocation among the drugs based on needs and contracts (Ventola 2011, GAO 2014). We assume that the capacity, once set at the beginning of the contract duration, remains unchanged over all T periods, because the capacity decision is more strategic and of longer term than inventory decisions in nature (cf. Simchi‐Levi et al. 2003, Waters 2006), and the capacity adjustment may require FDA review/approval and is hence time‐consuming (FDA, 2014, 2016b, GPO 2016).

Suppose at the beginning of the contract duration, the manufacturer has an initial capacity of

κ 0

κ 0

G ( κ − κ 0 )

κ − κ 0

Γ M ( p , s ) = max κ ≥ 0 U ( p , s , κ ) − G ( κ − κ 0 ) ,

(2)

Government

The government (via programs like Medicare) covers the reimbursement of many drugs on shortage, and bears three types of shortage‐related monetary or social welfare losses that are often much higher than the reimbursement cost. First, the reimbursement cost for alternative drugs or therapies (if any) is usually more expensive than the original drugs, incurring about $200 million extra purchase cost annually (Cherici et al. 2011a). As an example, levoleucovorin, the substitute for the oncology drug leucovorin during its shortage, is nearly 60 times more expensive (Goldberg 2009).

Second, the government must deal with social welfare losses due to suboptimal treatment outcomes. Other than some obvious cases, there has been a lack of consensus in the usage of alternative drugs and a lack of data supporting the effectiveness of alternative drugs (e.g., using doxorubicin in place of liposomal doxorubicin; see Mcbride et al. 2013). The use of substitutes also often leads to suboptimal outcomes. Metzger et al. (2012) estimate that after cyclophosphamide was substituted for mechlorethamine during its shortage, the two‐year event‐free survival for children with Hodgkin's lymphoma dropped from 88% to 75%. In a survey of oncology drug shortages by Mcbride et al. (2013), 48% of respondents reported tumor recurrences, and 40% reported reduced time to death. When there are no candidates for substitutes, drug shortages also cause delayed or canceled treatments. For example, in the aforementioned survey, 93% of institutions reported delays in chemotherapy administration or regimen changes.

Third, the shortages also incur other social welfare losses. For example, during shortages, healthcare providers may have to purchase from other sources that are generally much more expensive, and, in extreme cases, from the gray market (unofficial and unauthorized market). Cherici et al. (2011b) examine 1745 orders filled by the gray market and find an average markup of 650% and a maximum markup of 4533%. Although the government usually does not reimburse drugs purchased from the gray market (healthcare providers or patients often absorb this cost), the gray market has significantly adverse impacts, leading to significant social welfare losses.

Therefore, to fully capture government spending on drug shortages, we must consider both the monetary expenditure on reimbursement and the social welfare losses. Let r(p) represent the unit reimbursement cost for a drug with wholesale price p. For example, Medicare currently reimburses drugs at 6% above their market wholesale prices. Let

β G

β G

β G

β G ≥ r ( p )

β G

β G

β G = 58.5 × 26 % ≈ 15

By the above discussion, the total government spending under a contract (p, s) is

Γ G ( p , s ) = r ( p ) [ 1 − L ( p , s ) ] E ( D ) + β G L ( p , s ) E ( D ) ,

(3)

Healthcare Providers

Healthcare providers buy the drug at the wholesale price p and are reimbursed by Medicare at the reimbursement price r(p). Similar to the government's loss

β G

β P

β P

β P ≥ s

Γ P ( p , s ) = [ p − r ( p ) ] [ 1 − L ( p , s ) ] E ( D ) + ( β P − s ) L ( p , s ) E ( D ) ,

(4)

Sections 3.1–3.4 above have formulated the objectives of the supply chain parties involved. To evaluate these objectives, we need to further analyze the manufacturer's profit under the optimal inventory policy, U(p, s, κ), and the shortage measure, L(p, s).

Drug Manufacturer's Optimal Inventory Policy

Given a contract (p, s) and a capacity decision κ, the drug manufacturer makes inventory decisions for each of the T periods of contract duration. These decisions must consider production disruptions, which are under the influence of two types of factors: those independent of capacity utilization (e.g., a facility's specific conditions [new or old] and workforce quality assurance training) and those dependent on capacity utilization (e.g., certain aspects of maintenance and cGMPs compliance. For instance, disruptions due to managers ordering shortcuts that compromise quality and skimp on maintenance are certainly related to capacity utilization; Thomas 2012). Since our focus is supply chain analysis, we assume that the capacity‐independent factors are exogenous. In addition, we assume that the capacity‐dependent factors depend on the long‐run average capacity utilization, because a manufacturer's efforts to maintain the desired quality standard (cGMPs) are reflected in its commitment to preventive inspection and maintenance on a regular basis (FDA 2004). This requires the manufacturer to make long‐term maintenance plans based on the average capacity utilization, instead of spontaneous quality efforts based on the constantly changing capacity utilization in each period. Given a capacity κ, the long‐run average capacity utilization is

E ( D ) / ( κ T )

E ( D )

E ( D ) / ( κ T )

In practice, production disruptions lead to a mixture of backorders and lost sales. Specifically, when the production is normal (not disrupted), unmet demand is often backordered because the expected demand and supply are roughly equal and the manufacturer can mostly satisfy backorders in time. The manufacturer typically informs buyers when they will receive their backorders, and buyers typically accommodate this waiting time. Backorders usually do not cause shortages (Wellman 2001 and Tyler 2002). On the other hand, when the production is disrupted, since the disruption duration is often significant and uncertain, the manufacturer usually cannot quote a time for order delivery. In this case, buyers often resort to alternative sources, treatments, or drugs to meet their needs. Unmet demand is then lost for the manufacturer. Indeed, the survey of ASA (2012) on anesthesia drug shortages finds that 96.3% of anesthesiologists switch to alternative drugs, 50.2% change procedures, 7.0% postpone cases, and 4.1% cancel cases.

With the above discussion, we now describe the sequence of events. The manufacturer starts period t, 1 ≤ t ≤ T, with an initial inventory level x and a production status: normal or disrupted. If the production is normal, the manufacturer chooses a target post‐production inventory level y. The production incurs unit cost c and has a one‐period lead time that reflects the complex manufacturing process. The manufacturer then observes the demand in period t,

D t

( x − D t ) +

a + = max { a , 0 }

y − D t

( D t − x ) +

( x − D t ) +

If the manufacturer starts period t with an initial inventory x, a production status as disrupted, and the disruption has lasted i periods, then the manufacturer uses the inventory x to satisfy demand

D t

( D t − x ) +

s ( D t − x ) +

h ( x − D t ) +

λ i = P ( R ≥ i + 1 | R ≥ i )

1 − λ i

OPEN IN VIEWER

Table 1

Notation

T	Contract duration	y	Target inv. position in a period
p	Unit price	c	Unit production cost
s	Unit FTS compensation	h	Unit holding cost
α	GPO's unit commission	b	Unit backorder cost
κ 0	Original production capacity	θ (κ)	Reliability (no disruption probability) in a period
κ	Production capacity decision	R	Random disruption duration
D t	Random demand in period t	i	Number of periods the recovery has lasted
x	Initial inv. position in a period	λ i	P(R ≥ i + 1 | R ≥ i)

Under a contract (p, s) and a capacity κ, let

U t ( x )

be the manufacturer's optimal expected profit‐to‐go from the beginning of period t to the end of the planning horizon, given that the production is normal and the initial inventory is x in period t. Similarly, let

V t ( i , x )

be the manufacturer's optimal expected profit‐to‐go from the beginning of period t onward, given that the production has been disrupted for i periods, and the initial inventory is x in period t. For simplicity, we have dropped the arguments p, s, and κ in the notation of

U t ( x )

and

V t ( i , x )

. The manufacturer aims to maximize its expected profit over the contract duration. For 1 ≤ t ≤ T,

U t ( x ) = max x ≤ y ≤ x + κ ( 1 − α ) p E D t − c ( y − x ) − h E ( x − D t ) + − b E ( D t − x ) + + θ ( κ ) E U t + 1 ( y − D t ) + [ 1 − θ ( κ ) ] { − [ ( 1 − α ) p + s ] E ( D t − x ) + + E V t + 1 ( 1 , ( x − D t ) + ) } ; V t ( i , x ) = ( 1 − α ) p E min { x , D t } − s E ( D t − x ) + − h E ( x − D t ) + + λ i E V t + 1 ( i + 1 , ( x − D t ) + ) + ( 1 − λ i ) E U t + 1 ( ( x − D t ) + ) .

(5)

In the expression of

U t ( x )

, the manufacturer obtains the sales revenue less GPO's commission (1st term), and pays the production, holding, and backorder costs (2nd–4th terms). With probability θ(κ), no disruption occurs, so the production status in period t + 1 remains normal, and the manufacturer receives the expected future profit

E U t + 1 ( y − D t )

(5th term). With probability 1 − θ(κ), a disruption strikes. All unsatisfied demand, including backorders, becomes lost sales; the manufacturer returns the revenue collected from buyers, pays the FTS compensation (1st term within braces), and starts period t + 1 with an initial inventory of

( x − D t ) +

and the number of periods since disruption i = 1 (2nd term within braces).

In the expression of

V t ( i , x )

, the manufacturer uses the existing inventory x to satisfy demand (1st term) and pays the FTS compensation (2nd term) and the holding cost (3rd term). With probability

λ i

, the disruption continues in period t + 1, so the disruption duration increases to i + 1 periods and the remaining inventory is

( x − D t ) +

(4th term). With probability

1 − λ i

, the disruption ends in period t + 1, so the manufacturer starts period t + 1 with a normal production status and an initial inventory of

( x − D t ) +

(last term).

In period T + 1, the boundary condition is set to be

U T + 1 ( x ) = c x + − [ ( 1 − α ) p + s ] x −

and

V T + 1 ( i , x ) = c x

, where

x − = max { − x , 0 }

. In other words, at the end of the contract duration, if the production is normal, then the manufacturer recovers the production cost from leftover units, returns the revenue collected from all backorders, and pays the corresponding FTS compensation. If the production is disrupted in period T + 1, then the manufacturer only recovers the production cost (no backorder exists in this case). This boundary condition maintains tractability.

With this model, we now derive the structural properties of the optimal inventory decisions:

Lemma 1

Given a contract (p, s) and a production capacity κ, the maximand of

U t ( x )

in Equation 5 is concave in x and y, and both

U t ( x )

and

V t ( i , x )

are concave in x.

Given a contract (p, s) and a capacity κ, let

y t * ( x | p , s , κ )

, or simply

y t * ( x | κ )

when not confusing, denote the optimal post‐production inventory level that solves

U t ( x )

in Equation 5. Based on Lemma 1, Theorem 1 shows that the optimal inventory policy is a modified base‐stock policy.

Theorem 1

Given a contract (p, s) and a capacity κ, there exists a base‐stock level

k t

in each period t, independent of x, such that

y t * ( x | κ ) = x + κ i f x ≤ k t − κ k t i f k t − κ ≤ x ≤ k t x i f x ≥ k t .

Theorem 1 indicates that the manufacturer should not produce if the inventory level exceeds the base‐stock level

k t

and should produce up to or as close to

k t

as the capacity allows otherwise. Thus, the optimal inventory policy remains to be the standard base‐stock type, even with the additional features in the drug production.

Without loss of generality, we assume that the manufacturer starts the contract duration with zero initial inventory and normal production. Thus, the manufacturer's expected profit generated by the optimal inventory policy U(p, s, κ) in Equation 2 is simply

U 1 ( 0 )

in Equation 5.

Shortage Measure

This section derives the shortage measure L(p, s). Yurukoglu (2012) accessed the drug shortage website of the American Society of Health System Pharmacists (ASHP) and obtained the fraction of time various drugs were in shortage. However, he pointed out that the weakness of this measure is that it “does not capture the varying magnitudes of shortages.” Our study aims to capture the magnitudes. As discussed in section  , backorders that are not the result of production disruptions typically do not cause shortages, whereas lost sales during disruptions often lead to serious consequences. Thus, a critical indicator of drug shortages is the manufacturer's lost sales during production disruptions.

Given a contract (p, s), let

κ * ( p , s )

, or simply

κ *

, denote the optimal capacity solution to Equation 2. Let

J t ( x )

denote the cumulative lost sales from period t to the end of the contract duration, given that the production is normal and the initial inventory level is x in period t. Similarly, let

K t ( i , x )

denote the cumulative lost sales from period t onward, given that the production has been disrupted for i periods, and the initial inventory is x in period t. Thus, for 1 ≤ t ≤ T,

J t ( x ) = θ ( κ * ) E J t + 1 ( y t * ( x | κ * ) − D t ) + [ 1 − θ ( κ * ) ] [ E ( D t − x ) + + E K t + 1 ( 1 , ( x − D t ) + ) ] ; K t ( i , x ) = E ( D t − x ) + + λ i E K t + 1 ( i + 1 , ( x − D t ) + ) + ( 1 − λ i ) E J t + 1 ( ( x − D t ) + ) .

(6)

The boundary condition in period T + 1 is:

J T + 1 ( x ) = x −

and

K T + 1 ( i , x ) = 0

.

Again assuming that the production is normal and the initial inventory is zero at the beginning of the contract duration, we define the shortage measure as the cumulative lost sales ratio over the contract duration:

L ( p , s ) = J 1 ( 0 ) / E ( D )

, where

D = ∑ t = 1 T D t

.

Pareto‐Improving Contracts

In sections 3 and 4, we have formulated the objectives of supply chain parties and derived the shortage measure, all as functions of the contract parameters (p, s). In this section, we explore properties of the Pareto‐improving contracts. Let

( p 0 , s 0 )

denote the current contract, then

Γ O ( p 0 , s 0 )

,

Γ M ( p 0 , s 0 )

,

Γ G ( p 0 , s 0 )

, and

Γ P ( p 0 , s 0 )

represent the disagreement points for GPO, drug manufacturer, government, and providers, respectively, and

L ( p 0 , s 0 )

is the current shortage measure. As the successful implementation of a proposal requires the support of all supply chain parties, we focus on the Pareto‐improving contracts that improve/maintain the objectives of all parties from their respective disagreement points and mitigate the shortages.

Definition 1

The set of Pareto‐improving contracts is

Ψ = { ( p , s ) : Γ O ( p , s ) ≥ Γ O ( p 0 , s 0 ) , Γ M ( p , s ) ≥ Γ M ( p 0 , s 0 ) , Γ G ( p , s ) ≤ Γ G ( p 0 , s 0 ) , Γ P ( p , s ) ≤ Γ P ( p 0 , s 0 ) , L ( p , s ) ≤ L ( p 0 , s 0 ) }

.

To study the Pareto‐improving contracts, we first derive properties of the supply chain measures. Since Medicare currently reimburses drugs at 6% above their market wholesale prices, in this analysis, we consider a reimbursement price r(p) such that r(p + δ) − r(p) ≥ δ, ∀δ > 0. This condition means that when the wholesale price p increases, the reimbursement (retail) price increases by a larger amount, which encompasses Medicare's current policy as a special case. In addition, for generality, in Appendix S3, we also consider a fixed‐price policy (i.e., r(p) = a for some a) that was used by Medicare before 2005.

Next, we characterize the impacts of the contract parameters on supply chain measures

Proposition 1

Γ O ( p , s )

increases in p and s.

Γ M ( p , s )

increases in p and decreases in s.

Γ G ( p , s )

decreases in s but is non‐monotone in p.

Γ P ( p , s )

and L(p, s) decrease in p and s.

y t * ( x | κ )

increases in p and s for any given capacity κ.

With these properties, we are ready to explore the Pareto‐improving contracts. For a given p, let

s ι ( p )

be such that

Γ ι ( p , s ι ( p ) ) = Γ ι ( p 0 , s 0 )

, where the subscript ι = O, P, M, or G, represents GPO, providers, manufacturer, and government, respectively. Similarly, let

s L ( p )

be such that

L ( p , s L ( p ) ) = L ( p 0 , s 0 )

. The

s ι ( p )

's have the following properties:

Lemma 2

s O ( p )

,

s P ( p )

and

s L ( p )

decrease in p;

s M ( p )

increases in p;

s G ( p )

is generally non‐monotonic in p.

Since Proposition 1 has shown that for any given p, strengthening s reduces the manufacturer's profit but improves all other measures (note that decreasing

Γ G ( p , s )

,

Γ P ( p , s )

, and L(p, s) means improvement), the Pareto‐improving s value must be above

s O ( p )

,

s P ( p )

,

s L ( p )

,

s G ( p )

and below

s M ( p )

. In addition, since the price p and FTS compensation s should only be adjusted within reasonable ranges, we allow upper bounds for them, denoted by

p ¯

and

s ¯

. When no upper bounds are necessary, we simply let

p ¯

or

s ¯ = ∞

. Further, we say an interval [a, b] is empty if a > b. Thus,

Theorem 2

The set of Pareto‐improving contracts is: 

Ψ = { ( p , s ) : p ∈ [ p 0 , p ¯ ] , s ∈ [ max { s 0 , s G ( p ) } , min { s ¯ , s M ( p ) } ] }

.

Theorem 2 indicates that among the lower bounds

s O ( p )

,

s P ( p )

,

s L ( p )

, and

s G ( p )

,

s G ( p )

is the tightest, so other bounds become redundant. Theorem 2 provides an expression of the Pareto‐improving contracts. Unfortunately, further properties of the contracts, especially how the bounding curves cross (never cross, cross once or multiple times), are analytically intractable. The difficulty lies in the fact that both of the lower‐ and upper‐bound curves for s involve the shortage measure L(p, s), which has no analytical expression and has to be evaluated numerically by Equation 6. In view of the analytical difficulty, we conduct an extensive numerical study in section 6 to explore further properties. We find that the bounding curves cross only once on

[ p 0 , p ¯ ]

in most scenarios, and the Pareto‐improving contracts exist in all scenarios. Appendix S9 contains more details.

The Pareto‐improving contracts serve as a feasible region for contract selection and provide supply chain parties with options and common ground for determining a contract under a specific situation. Notably, for generic drugs, GPOs typically have high bargaining power: the top five GPOs in the U.S. command 85–90% of the market (FDA 2011, ASPE 2011). Thus, we next specify the Pareto‐improving contract when the GPO is the contract designer, that is, the contract that maximizes the GPO's profit.

Theorem 3

Let

p ^ = max { p : max { s 0 , s G ( p ) } ≤ min { { s ¯ , s M ( p ) } , 0 ≤ p ≤ p ¯ } }

. Then the optimal contract for the GPO is

( p ^ , min { s ¯ , s M ( p ^ ) } )

.

In Theorem 3,

p ^

is the largest value on the range

[ 0 , p ¯ ]

where the lower bound curve (the government's iso‐spending curve) is below the upper bound curve (the manufacturer's iso‐profit curve). Thus,

p ^

is the largest feasible price for Pareto improvement. By Proposition 1, the GPO's profit is maximized at the largest feasible p and s, that is,

( p ^ , min { s ¯ , s M ( p ^ ) } )

.

So far, we have derived some properties of the Pareto‐improving contracts. Based on these properties, we next conduct an extensive scenario analysis with realistic industry data to evaluate the effectiveness of our proposal and its impact on the different supply chain parties.

Scenario Analysis

In section 6.1, we construct three important scenarios that correspond to a major portion of the drugs on shortage and then in section 6.2, we transform the scenarios to quantitative decision scenarios. We further discuss our data source and parameter setting in section 6.3 and evaluate our proposal under each scenario in section 6.4. Finally, for greater generality, we complement the three major scenarios by a full factorial analysis of a wide collection of scenarios in section 6.5.

Scenario Construction

We generally follow the procedure suggested by Schoemaker (1995) for the scenario analysis. Based on the scenario‐axes technique (Schwarz 1991, van't Klooster and van Asselt 2006), we aim to identify the driving forces that have the highest impact on the shortages and exhibit considerable variation across the drugs on shortage (as discussed in section 2, variation plays a key role in determining the effectiveness of a proposal). Note that from 2010 to 2014, the therapeutic classes that experienced the most serious shortages included anesthetic, anti‐infective, cardiovascular, nutritive, oncology, and endocrine (ASPE 2011, FDA 2011, GAO 2011, 2014). As discussed in section 1, the root driving forces that differentiate these drugs from other drugs are their low profit margins and special manufacturing characteristics. Since the profit margins are already captured in the contract parameters, manufacturing characteristics seem to be the proper criterion for scenario construction. With respect to this criterion, the drugs in the therapeutic classes with critical shortages naturally fall into three scenarios.

The low‐adjustability scenario (L‐scenario). This scenario typically fits sterile drugs that require highly specialized or even dedicated facilities, including, for example, some anti‐infective and oncology drugs. The production of these drugs must take place in “clean room” environments with well‐defined manufacturing processes and controls. The production often requires a complex, specialized process of aseptic processing and terminal sterilization. Some drugs requires additional lyophilization (Woodcock and Wosinska 2013). Certain drugs, such as penicillin (anti‐infective), are “highly sensitizing and can trigger serious allergic reactions at very low levels and as a result, may be limited to specific manufacturing lines” (GAO 2014). Examples of other drugs that require highly specialized lines include cephalosporins (antibiotics) and some cytotoxics (oncology) (Woodcock and Wosinska 2013). Thus, for the drugs in the L‐scenario, changing capacity usually incurs a high cost (e.g., involving complex reconfiguration of lines, deep cleaning, and substantial effort to assure the cGMPs). However, once the capacity is increased to a sufficient level, many of the currently observed disruptions, such as contaminations from microorganisms and visible particulate matter, can be significantly mitigated. Thus, the cost and the gain from capacity adjustment often have a positive relationship: the higher the cost, the less likely the drug manufacturer has already adjusted the capacity to a desirable level to mitigate disruptions, and the more likely a significant reliability improvement will occur once capacity adjustment is stimulated by a proposal. The low‐adjustability scenario then corresponds to high cost and high gain.

The medium‐adjustability scenario (M‐scenario). This scenario applies to sterile drugs that require moderately specialized facilities, including, for example, some anesthetic, cardiovascular, nutritive, and endocrine metabolic drugs. These drugs still require the complex aseptic processing and sterilization, but they share the production lines with other drugs with similar manufacturing characteristics. The production of multiple drugs on the same lines (via different batches) is common in practice (Ventola 2011). In an example from GAO (2014), 30–50 different drugs are manufactured on a given line. Changing the capacity of these drugs requires the reallocation of the line capacity, certain cleaning, and reconfiguration. However, the cost and gain from changing the capacity are often lower than the highly specialized line case in the L‐scenario.

The high‐adjustability scenario (H‐scenario). This scenario mostly includes orally‐administered drugs, which require much less specialized facilities and less complex processing (from 2011 to 2013, orally‐administered drugs represented 27% of the shortages, GAO 2014). These drugs often share production lines with other drugs, and the capacity adjustment is less costly (although still costly compared with the drugs without serious shortage issues). However, because of the low capacity adjustment cost, the capacity may have already been at a reasonable level, meaning a relatively low gain in reliability from the capacity adjustment.

The above three scenarios correspond to three distinct types of manufacturing characteristics, which, together with the contract parameters, capture the key driving forces of the problem. Next, we translate these scenarios to quantitative decision scenarios for evaluating our proposal.

Evolving Toward Decision Scenarios

In line with Schoemaker (1995), after constructing narrative scenarios, we need to translate them to decision scenarios to quantitatively test our strategies and gain insights. We do this by mapping the three scenarios to different sets of parameter values in the previously developed supply chain model. As discussed, the three scenarios mainly differ in the cost and gain (reliability improvement) from capacity adjustment, for which we provide further details next.

Capacity Adjustment Cost

As discussed in section 3.2, given a contract, the manufacturer adjusts its capacity from

κ 0

to κ for the contract duration with a cost of

G ( κ − κ 0 )

. Prior literature often models the capacity adjustment cost by

G ( κ − κ 0 ) = − g 1 ( κ − κ 0 ) − + g 2 ( κ − κ 0 ) +

, where

g 1

(

g 2

) is the return (cost) from using one less (more) unit of capacity (cf. Angelus and Porteus 2002 and Allon and Zeevi 2011). We adopt this structure, as it provides a tractable and practical way to model the cost. Further, since manufacturers often produce multiple drugs on the same production line, increasing (decreasing) κ by a unit means decreasing (increasing) a unit capacity for another drug on the same line. Thus, we consider

G ( κ − κ 0 ) = g ( κ − κ 0 )

, that is,

g 1 = g 2 = g

, where g represents the opportunity loss (gain) from reducing (increasing) a unit capacity for another drug. This consideration is also innocuous in our scenario analysis, because in all scenarios examined, our proposal always incentivizes the manufacturer to increase its capacity, so

g 1

does not take effect (more details in Appendix S8). When

G ( κ − κ 0 ) = g ( κ − κ 0 )

, in Equation 2,

κ 0

is additively separable from κ. The optimal capacity decision is then independent of the original capacity

κ 0

, allowing us to normalize

κ 0

to 100, the same as the average monthly demand (see “Demand Parameters” in section 6.3). Such a value reflects the current industry situation that without disruptions, demand is roughly met (no excess capacity). Data on the unit capacity adjustment cost, g, are scarce. From industry practice, we make a reasonable estimate that g = 3000 (L‐scenario), 2000 (M‐scenario), and 1000 (H‐scenario). Appendix S8 provides details of this estimation.

Capacity Adjustment Gain: Estimation of Reliability Function

The second critical difference among the scenarios is the impact of capacity adjustment on disruption mitigation. As discussed, capturing the exact relationship between the capacity and disruption probability is very difficult. Fortunately, as in all scenario analyses where approximation is often used for future projection, we have developed the following approximation to estimate the reliability function (definition in section 4.1) based on a simple model and the data from industry reports and previous research.

In each period, the manufacturer's production can be either normal or disrupted. Following a common assumption in literature (cf. Chapter 9 in Ross 2014), we assume that the normal production duration (time until disruption, or up time) follows an exponential distribution with mean γ(κ) under a given capacity κ. By the memoryless property of exponential distribution, the reliability (the probability that no disruption occurs in a period) is then the probability that the system up‐time lasts longer than one period, that is,

θ ( κ ) = e − 1 / γ ( κ )

, which is the same in every period. Recall from section 4.1 that the reliability consists of a capacity‐independent part and a capacity‐dependent part that is a function of the long‐run average capacity utilization

E ( D ) / ( κ T )

, or

E ( D t ) / κ

for i.i.d. demands in this analysis (see “Demand Parameters” in section 6.3 for details). Thus, we use a simple model:

γ ( κ ) = γ 1 + γ 2 [ 1 − E ( D t ) / κ ] , γ 1 , γ 2 ≥ 0

, where

γ 1

represents the capacity‐independent part, and

γ 2 [ 1 − E ( D t ) / κ ]

represents the capacity‐dependent part with

1 − E ( D t ) / κ

capturing the unutilized portion of time that can be used for maintenance. Clearly, as κ → ∞,

γ ( κ ) → γ 1 + γ 2

, which is the upper bound of the mean up‐time. The existence of the upper bound means that the manufacturer cannot arbitrarily extend the mean up‐time by increasing capacity. Thus,

γ 2

captures the maximum amount of reliability improvement by capacity adjustment.

Next, we estimate the distribution of the disruption duration (down time) based on the data in GAO (2011). The data well fit an exponential distribution with a mean of 9 months (see more details in Appendix S8). Thus, given that the production is disrupted in a period, the production recovers in the next period with probability

1 − e − 1 / 9 ≈ 0.11

, which means that

λ i

, defined in section 4.1, is about 1−0.11 = 0.89 for all i.

Now we are ready to obtain some estimates of the parameters

γ 1

and

γ 2

in the reliability function. Since both of the up and down times follow exponential distributions, the two production status, normal and disrupted, constitute two states of a Markov chain. The long run fraction of time that the system is disrupted can then be calculated as

( 1 − e − 1 / γ ( κ ) ) / ( 1 − e − 1 / γ ( κ ) + 0.11 )

. According to the drug shortage data at ASHP (measured by Yurukoglu 2012), the average fraction of time that the listed drugs are in shortage is 0.1713. This statistic indicates that

( 1 − e − 1 / γ ( κ ) ) / ( 1 − e − 1 / γ ( κ ) + 0.11 ) = 0.1713

, which implies γ(κ) = 43.477. This estimate is derived from the data from 2005 to 2012, which reflects the current situation where manufacturers’ capacities are barely sufficient to meet demand, that is,

κ = κ 0 ≈ E ( D t )

. Hence,

γ ( κ 0 ) = γ 1 + γ 2 [ 1 − E ( D t ) / κ 0 ] = γ 1 = 43.477

, which provides an estimate for

γ 1

. The reliability function is then

θ ( κ ) = e − 1 / γ ( κ ) = e − 1 43.477 + γ 2 ( 1 − 100 / κ ) .

Further estimation of

γ 2

requires data on various levels of capacity and the corresponding reliability, which are scarce, confidential, and markedly different for different manufacturers. Fortunately, the three scenarios constructed correspond to various levels of

γ 2

and cover a reasonable range of the parameter value. In this analysis, we choose to let

γ 2 = 200 , 500 , 800

correspond to the H‐, M‐, and L‐scenarios where the gain by capacity increase is low, moderate, and high, respectively. Appendix S8 illustrates θ(κ) for the three levels of

γ 2

.

Data Source and Parameter Setting

Data used in the analysis come from the following three sources:

SEER 18 and SEER‐Medicare Link, two premium research databases on cancer treatments maintained by National Cancer Institute. SEER 18 provides data on cancer patients’ demographic characteristics and drugs used in treatments, from which we estimate the demand parameters. SEER‐Medicare Link provides data related to cancer drug reimbursement, from which we estimate the original and substitute drug prices and substitution rate.

Data derived from white papers, industrial reports, academic research, etc., as detailed later.

In addition, for the parameters that are unavailable from databases and reports, unattainable from industry (non‐existent or confidential, e.g., capacity and its impact on disruptions), or different from drug to drug (e.g., production cost), for generality, we test multiple levels of these parameters in a wide collection of scenarios in our scenario and full factorial analysis.

We next provide details on our parameter setting.

Contract Parameters

We set the contract duration to be 2 years or T = 24 months, a typical contract length in practice (cf. the discussion on page 382 of Navarro 2009). For the current contract (as a benchmark for the proposed contracts), we normalize the price to

p 0 = 100

, and assume the FTS compensation

s 0 = 1.1

based on the following estimation: many contracts in current practice do not require a manufacturer to compensate all lost sales but only the price difference for the units filled by alternative sources. On average, alternative sources satisfy about 10% of a manufacturer's lost sales during shortages (ASPE 2011), and the average price difference is about 11% of the contract price (Cherici et al. 2011a). Thus, the manufacturer on average compensates

s 0 = 10 % × 11 % × 100 = 1.1

for each unit of lost sales. This weak compensation reflects the current situation.

Demand Parameters

Since the generic drugs under investigation typically have been on the market for a long time with relatively steady demand distributions, we assume that the demands in each period,

D t

's, are i.i.d.. In this analysis, we estimate the demand distribution from the data on three typical drugs in critical shortage: fluorouracil, cytarabine, and bleomycin, which treat Stage III colon cancer, acute myeloid leukemia, and Hodgkin's lymphoma, respectively. We fit the demand distributions of the three drugs and find that the data are well fit by normal distributions (the Anderson‐Darling normality tests fail to reject the null hypothesis of normality). Without loss of generality, we normalize the mean demand to 100 and assume the standard deviation is 10. The 10% standard deviation to mean ratio (coefficient of variation) is estimated from the three drugs, whose coefficients of variation are 7%, 11% and 11%, respectively, with an average of about 10%.

Cost Parameters

We assume that the unit monthly holding cost is h = 3, and the unit monthly backorder cost is b = 4. The 3% monthly (or 36% annually) holding cost factor is consistent with the high cost of storing sterile injectable drugs, which requires sterile, light‐ and temperature‐regulated environments. Based on our interactions with the industry, we learn that the unit production cost often varies from 20% to 60% of the contract price for different generic drugs on shortage. In our scenario analysis, we first set the unit production cost to the average value, c = 40. We then broaden our coverage by testing three levels, c = 20, 40, 60, for drugs with low, medium, or high production cost in the factorial analysis. In fact, our analysis shows that the effectiveness of the proposed contracts does not vary significantly with the production cost.

GPO Commission

The GPO charges a percentage commission, α, to the manufacturer for sales to the GPO's members. The “safe harbor” provisions of the 1987 amendment nominally limit α to 3%. Since GPOs have high bargaining power, we assume α = 3% in this analysis.

Proposal Evaluation under the Three Major Scenarios

We are now ready to evaluate the proposed solution. In this evaluation, we discuss the results under the M‐scenario in detail since it corresponds to a large class of drugs. We also briefly discuss the corresponding results under the L‐ and H‐scenarios, since the results in all scenarios are similar qualitatively but differ somewhat in quantity, as reported later.

We first demonstrate the set of Pareto‐improving contracts. In Figure 1, the shaded region shows the Pareto‐improving contracts under the M‐scenario. The horizontal axis represents the contract price as multiples of the current price

p 0 = 100

(e.g., a value of 1.2 on the axis means a price increase to p = 120). The vertical axis represents the FTS compensation as multiples of

p 0

. By Theorem 2, the upper curve of the region is the manufacturer's iso‐profit curve

s M ( p )

, which is the highest FTS compensation the manufacturer is willing to pay to ensure the same profit as in the current contract

( p 0 , s 0 )

. The lower curve is the government's iso‐spending curve

s G ( p )

, representing the lowest FTS compensation to maintain government spending as in the current contract. The curves

s O ( p )

,

s P ( p )

and

s L ( p )

do not appear on the figure, since increasing drug prices and strengthening FTS always benefit the GPO, healthcare providers and shortage mitigation (Proposition 1). These features (shapes) of the Pareto‐improving region remain the same under the L‐ and H‐scenarios.

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Figure 1

Pareto‐Improving Contracts

Figure 1 shows that in some Pareto‐improving contracts, s may be 50 times the original drug price, which appears difficult to implement. Similarly, since government spending is sensitive to price increase, doubling the price may also cause concerns. For demonstration, we restrict the price increase to 30% of the original price (i.e.,

p ¯ = 1.3 p 0

) and examine the effectiveness of the Pareto‐improving contracts with this moderate price increase. In Figure 1, the 30% price increase corresponds to the vertical line at p = 1.3, and the corresponding Pareto‐improving contracts are then the shaded area to the left of this line, referred to as the “Pareto triangle.” Figure 2 provides details of the Pareto triangle, with data for the figure presented in Appendix S10.

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Figure 2

Pareto Triangle Curves [Color figure can be viewed at wileyonlinelibrary.com]

Specifically, Figure 2a shows the Pareto triangle. As predicted in Lemma 2, the manufacturer's iso‐profit curve

s M ( p )

increases in p, and the government's iso‐spending curve

s G ( p )

is non‐monotonic in p. Notably, while the iso‐profit curve also corresponds to our proposal of IPS (increasing p and s together) with the manufacturer's profit staying the same as in the current contract, the iso‐spending curve here coincides with the solution of IP (increasing p only), because the change in s is very small on this curve. Thus, examining the two curves also serves the purpose of comparing IPS and IP. Figures 2b–d illustrate how the optimal capacity decision

κ *

, the base‐stock level in the first period

k 1

, and the shortage measure L change as p increases. In Figure 2b, both IPS and IP can incentivize the manufacturer to increase capacity. However, IPS is much more effective since it leads to continuous capacity increase. Figure 2c shows that under IPS, the manufacturer slightly reduces stock level initially but continuously stocks more after the price is above a certain level. Under IP, however, the manufacturer stocks the same or even less when the price increases. This decreasing inventory is due to the price‐capacity‐inventory triangular effect discussed earlier: a price increase would increase capacity and stock, but capacity and stock are substitutes. Clearly, IPS incentivizes the manufacturer to increase capacity and stock more significantly, hence overcoming the triangular effect. Consequently, as evidenced in Figure 2d, IPS significantly mitigates the shortages.

The Pareto triangle contains all implementable Pareto‐improving contracts. To select a specific contract for demonstration, we again focus on the GPO‐designed contract. As the GPO's profit

Γ O ( p , s )

increases in p and s (Proposition 1), the GPO's optimal contract corresponds to the highest point of the Pareto triangle in Figure 2a, that is,

( p , s ) = ( 1.3 p 0 , s M ( 1.3 p 0 ) )

(Theorem 3). In addition, since the drug shortage measure L(p, s) decreases in p and s (Proposition 1), the GPO‐designed contract also maximizes shortage reduction, that is, shortage mitigation is well aligned with GPO's incentives. Thus, hereafter we use the GPO‐designed contract to evaluate the shortage mitigation by our proposal and the impacts of model parameters.

Note that similar results and patterns hold under the L‐ and H‐scenarios. We thus omit the details therein and summarize the shortage mitigation under the three scenarios. We find that our proposal of IPS can reduce the drug shortages by 60%, 58%, and 52% in the L‐, M‐, and H‐scenarios, respectively. Thus, the proposal is very effective and robust. Interestingly, the shortage reduction in the low‐adjustability scenario is even more significant than the other two scenarios. This is due to the complementary effect between the cost and gain from capacity adjustment, which will be detailed in section 6.5. This result is of considerable practical importance because the drugs with low‐adjustable capacities often present the most severe shortages.

Full Factorial Analysis for Other Scenarios

The above scenario analysis tests our proposal under three major scenarios, which are derived from industry practice and capture a wide variety of drugs on shortage. However, under rarer situations, some drugs may have high cost and low gain, or low cost and high gain from capacity adjustment, which are not captured in the three scenarios. Thus, for greater generality, we complement the scenario analysis with a full factorial analysis, where we permutate parameter values to test a wider collection of scenarios. Specifically, we test three parameters for multiple levels: g = 1000, 2000, 3000,

γ 2 = 200 , 500 , 800

, and c = 20, 40, 60, totaling 27 combinations. These parameter values are the same as in the scenario analysis, but here we consider their full factorial combinations. For each combination, we vary the contract price p from

p 0 = 100 to 250

with various steps as needed (32 values total) and compute the corresponding curves

s ι ( p ) , ι = O , P , M , G , L

. Thus, in total, we compute 27 × 32 × 5 = 4320 contracts. In computing the contracts, we apply the analytical properties derived as a guideline. For example, we apply the base‐stock policy in Theorem 1 to evaluate the optimal inventory decisions. We apply Proposition 1 and use a bisection search to find the values of

s ι ( p ) , ι = O , P , M , G , L

.

We examine the shortage reductions in all of the 27 scenarios. We find that across all scenarios, the 30% price increase leads to a minimum, average, and maximum shortage reduction of 25%, 53%, and 70%. Clearly, the worst (best) shortage reduction 25% (70%) appears when the capacity adjustment cost is very high (low) but reliability improvement is very low (high). Appendix S10 presents details of the shortage mitigation results.

In addition, to understand what parameters have significant impacts on the shortage mitigation, we further conduct an ANOVA of the shortage mitigation results. The ANOVA identifies three terms with significant influence: g,

γ 2

, and

g γ 2

interaction with p‐values of  < 0.0001,  < 0.0001, and 0.005, respectively. The production cost c, though very different for different drugs, does not significantly influence shortage mitigation. Figure 3 plots the interaction effect

g γ 2

. Consistent with our intuition, the shortage reduction increases when the capacity adjustment cost g decreases and the reliability improvement parameter

γ 2

increases. Somewhat non‐intuitively, when g is larger, as

γ 2

increases, the shortage reduction increases faster, which means that g and

γ 2

have some complementary effects on shortage mitigation. The explanation is that when capacity adjustment incurs low cost, the manufacturer may have already adjusted capacity desirably. Thus, incentivizing the manufacturer to further adjust capacity may not yield significant benefit. However, when capacity adjustment is expensive, the manufacturer is less willing to adjust the current capacity to a desirable level. In this case, a high

γ 2

can overcome the status quo, incentivize the manufacturer to move, and thus achieve effective shortage mitigation. This complementary effect also explains why the shortage mitigation in the L‐scenario (60%) is even better than those in the M‐ and H‐scenarios (58% and 52%), where the

g γ 2

complementary effect dominates the single effect of the cost g. The practical meaning is that the drugs that experience the most severe shortages may benefit most from our proposal.

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Figure 3

g γ 2

Interaction Effect [Color figure can be viewed at wileyonlinelibrary.com]

Finally, as a demonstrative example, we apply our solution to a specific oncology drug, leucovorin, whose shortages have been highly influential. We collect the demand, price and reimbursement data on this drug, fine‐tune the model parameters, and evaluate our solution. As a result, our solution recommends increasing the price by 30%, from 89.4 cents to $1.16 for a 50‐mg vial, and increasing the FTS penalty from about 1 cent to $4.04 (roughly 3.5 times the drug price). These changes will reduce the shortages by 60.3%. In comparison, only increasing the price to $1.16 without increasing the FTS penalty can reduce the shortages by 12.8%. This example again shows the effectiveness of our solution. Appendix S10 provides more details of this example.

Government Spending and Policy Implication

Although not directly involved in the negotiation of drug purchase contracts, the government, as a payer, may have significant impacts on the contracts through policies (e.g., reimbursement policy). In this section, we derive additional insights on the impact of our proposal on government spending.

We focus on the Pareto‐improving contracts that minimize government spending under Medicare's current reimbursement policy, r(p) = 1.06p, and under the M‐scenario for illustration (other scenarios exhibit similar patterns). Since government spending decreases in s (Proposition 1), minimizing the spending entails selecting contracts on the upper curve of the Pareto‐improving region,

s M ( p )

. Thus, the optimal price that minimizes the spending is

p * = arg min p Γ G ( p , s M ( p ) )

from Equation 3. As the government's shortage loss

β G

varies from drug to drug, we test multiple levels of

β G

and report the corresponding

p *

in Table 2. Note that in this table,

p *

increases in

β G

because the shortage measure

L ( p , s M ( p ) )

decreases in p (Proposition 1) and thus

Γ G ( p , s M ( p ) )

is a submodular function of

β G

and p.

OPEN IN VIEWER

Table 2

Price Minimizing Government Spending

β G	≤ 1.94 p 0	1.95– 7.09 p 0	7.10– 8.78 p 0	8.79– 14.51 p 0	≥ 14.52 p 0
p *	100	102	116	124	130

Notes. The current price

p 0

is normalized to 100. Since we only test discrete levels of price p ∈ [100, 130] with an increment of 2,

p *

here only takes multiples of 2 within [100, 130].

p *

may go higher if we allow larger price increase.

In addition, to demonstrate the robustness of the IPS solution, we further examine the price range that reduces government spending from the current contract for different values of

β G

. We find that the price range is [100, 100] for

β G ≤ 1.94 p 0

, meaning that if the shortage loss is small, then no price beyond 100 can reduce the spending. However, this range expands quickly as

β G

increases, and reaches [100,130] when

β G ≥ 6.11 p 0

(more details in Appendix S11). In other words, at

β G = 6.11 p 0

, although the optimal price that minimizes government spending is

p * = 102

(Table 2), any price on the entire tested range

[ p 0 , 1.3 p 0 ]

reduces government spending compared with the current contract. Thus, when

β G

is sufficiently large (as in many practical cases), raising the drug price according to IPS by any percentage below 30% would not increase government spending.

Ending Remarks

Drug shortages have posed significant public health threats in recent years. Addressing the problem remains a top priority for the FDA, and a focus of attention of government agencies such as the ASPE and the Government Accountability Office (GAO) and organizations such as the ASCO. As one of the very few academic studies examining the drug shortage problem, our research takes a supply chain perspective, a piece that is currently missing in the literature, and proposes to mitigate shortages through supply chain contracts. By modeling the unique characteristics of the drug supply chain, we investigate the Pareto‐improving contracts that mitigate drug shortages, improve the manufacturer's profit, cut government spending and providers’ cost, and ensure the GPO's profit. We explore analytical properties of these contracts and show their effectiveness in shortage mitigation through scenario analysis based on realistic industry data. Our results suggest an important implication: increasing price only (IP), implied by many (e.g., ASCO in Piana 2012) as a natural cure for the drug shortage problem, is not very effective in shortage mitigation. Rather, increased price must be paired with strengthened FTS compensation to achieve very effective shortage reduction. This is because IP does not provide sufficient Pareto‐improving space to counter the substitutional effect between capacity and inventory when price increases. Our analysis provides the range of effective price and FTS values under a wide collection of scenarios and the corresponding optimal production/inventory policies for drug manufacturers.

Throughout our research, we have actively pursued interactions with government agencies and industry practitioners. We made a presentation to key staff of the FDA and ASPE in the U.S. Department of Health and Human Services, who later presented our proposal in the national ASHP/Pew drug shortage stakeholders’ meeting and helped us to connect with industry practitioners (GPO) to discuss the potential of implementation. We also made connections with manufacturers interested in our proposal through industry forums. These interactions help us not only to improve our model and analysis but also to obtain additional insights.

As advocated in the FDA's Strategic Plan for Preventing and Mitigating Drug Shortages (FDA 2013), one important aspect of preventing and mitigating drug shortages is to be able to reward quality/reliability. In the pharmaceutical industry, while differentiating product quality is often difficult since all marketed drugs are supposed to have met the same quality standards, rewarding reliability of drug supply, “an important, but largely unrecognized aspect of quality” (Woodcock and Wosinska 2013), is much easier to implement. It could be argued that shortcuts may be taken to meet a deadline that may hurt quality, as in the aforementioned case discussed in Thomas (2012). Anand et al. (2012) provide logic that quality requires constant attention; and other pressures may distract from this. To the extent though that manufacturers understand that attention to compliance with quality will result in better reliability, our proposal of IPS provides the right incentives. It does so by granting a higher reward (i.e., lower FTS penalty) to manufacturers who are able to provide a more reliable drug supply (or fewer manufacturing disruptions). Our quantitative analysis also confirms the effectiveness of this approach.

While the government is an integral part of the drug shortage problem, drug purchase contracts are between the manufacturer and the GPO, which are naturally profit‐aiming entities. The FDA always encourages achieving desirable results through financial incentives, which is also the focus of this study. While increasing prices to secure supply may be common in many industries, it seems to be an “intriguing” concept for the pharmaceutical industry (particularly for buyers represented by GPOs) because: (1) the leaders of the industry, brand drug manufacturers, almost never run into shortages due to high profitability; (2) generic drugs may have insufficient supply due to low profitability, but the prices of these drugs are largely influenced by GPOs, which are often concerned about price increases since their function is to pool demand for lower prices. Indeed, while price increase is considered an immediate cost to buyers, more reliable supply is a long‐term benefit. Our proposal calls for buyers’ willingness to pay a higher price to hedge against future uncertainty. While this is a difficult task for GPOs which, in the current system, are largely evaluated by prices but not explicitly penalized by supply shortage, it is critical that GPOs take the responsibility to convince buyers about the tradeoff between the short‐term cost and long‐term benefit in order to implement our proposal to address the shortage problem. We believe our quantitative analysis and the potential benefit demonstrated in the analysis help with this task and contribute to the advancement of drug purchase contracting.

While we believe our study captures the major tradeoffs in the drug shortage problem, we have made some simplifying assumptions as laid out in the study and used mean values and industry norms to estimate a few parameters. With the FDA striving to improve data sharing across the drug supply chain, an updated evaluation with future data would be helpful to refine the effect of the proposal. It is also worth noting that our proposal is most useful for drugs prone to shortages and those whose shortages cause significant losses. Our analysis does not intend to cover all drugs. For example, most brand drugs and orally‐administered drugs (whose shortages are uncommon due to high profit margins or less complex manufacturing processes) and drugs that are not medically necessary or have inexpensive substitutes are not the target of our proposal. Finally, although we focus on the specific context of drug shortages (see the discussion of context‐specific industry studies in Joglekar et al. 2016), our analysis provides a possible framework for industries where product shortages adversely impact social welfare. These industries are very different from those where the main theme is revenue management rather than shortage mitigation.

Footnotes

Acknowledgments

We are grateful to Aaron Yao for providing support and access to medical and reimbursement data. We express our gratitude to the staff of FDA and ASPE for their valuable comments and suggestions. We sincerely thank the department editor, Nitin Joglekar, the senior editor, and the two anonymous referees who helped improve this study immensely. The first author gratefully acknowledges support from the Doug and Maria DeVos Faculty Summer Research Award at Purdue University.

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