Balancing learning and economies of scale for adaptive clinical trials

Operations Research for Health Care (

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Operations Research for Health Care journal homepage: www.elsevier.com/locate/orhc

Balancing learning and economies of scale for adaptive clinical trials Adam J. Fleischhacker a,∗ , Yao Zhao b a

University of Delaware, Department of Business Administration, Newark, DE 19716, United States

b

Rutgers University, The State University of New Jersey, Department of Supply Chain Management & Marketing Sciences, Newark, NJ 07102, United States

article

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Article history: Received 30 January 2013 Accepted 18 June 2013 Available online xxxx Keywords: Clinical trial supply Stochastic dynamic programming Production planning

abstract Prior to the start of an adaptive clinical trial, demand for an investigational drug can be highly uncertain. Treatment length, recommended dosages, and forecasted patient recruitment can fluctuate in response to early trial results. While initial demand forecasts can be very wrong, the factors influencing future demand can be learned during the trial. To take advantage of this learning, intra-trial production and/or packaging can be leveraged, but this is done at the expense of scale economies. In this paper, we study the balance between learning and economies of scale for adaptive clinical trials. We characterize the optimal production (or packaging) decisions and through analytical and numerical studies, we develop insights on the impact of fixed costs, learning rates in terms of forecasting future demand, inventory overage costs, and inventory underage costs on the value of having intra-trial flexibility and on decisions regarding quantity and timing of drug supply. © 2013 Elsevier Ltd. All rights reserved.

1. Introduction Before an investigational drug can obtain FDA approval, the drug must be proven both safe and effective in humans. Unfortunately, only one in five drugs that enter this human testing hurdle of the FDA approval process, also known as clinical trials, actually obtains approval [1]. This failure to obtain approval does not always mean that the drug is truly unsafe or ineffective; it may simply be that the test was not setup for success. Possibly, a different treatment lengths, dosages, or even patient populations may have led to a better trial outcome. To achieve more successes, the pharmaceutical industry is moving towards adaptive clinical trials which allow for flexibility in how a trial is conducted [2]. For example, an adaptive approach enables the review of interim trial data to modify trial parameters such as dosage, treatment duration, and study endpoints. This is in stark contrast to the traditional, more rigid approach to clinical trials where a dosage, a patient population, a length of treatment, and specific measures of success are chosen all prior to commencement of the trial. Of course, the purely rigid approach is a supply manager’s dream because demand is more certain. However, in an adaptive trial, more uncertainty about demand is introduced [3]. For example, if during an adaptive trial a biomarker is discovered that can identify patients who are more likely to respond to the investigational treatment, the study can recruit fewer patients and still preserve its statistical power [4]. Demand is effectively reduced and

∗

Corresponding author. Tel.: +1 13028316966. E-mail addresses: [email protected] (A.J. Fleischhacker), [email protected] (Y. Zhao). 2211-6923/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.orhc.2013.06.001

can be forecasted with less uncertainty. Since expected demand changes throughout an adaptive trial, the increased probability of a trial success (i.e. FDA approval) comes at the expense of the supply chain. Many of the key parameters used for forecasting required drug supply are now subject to change during the trial. While for ease of exposition, we will write from the perspective of analyzing production batching decisions, the mathematics equally applies to packaging, labeling, distribution, and other supply quantity decisions (e.g. comparator sourcing) where economies of scale may often be sacrificed to have flexibility in the clinical trial supply chain. To ensure supply for the start of a trial, clinical supply managers are forced to ‘‘forecast based on a limited number of variables before the cost of the supply chain overshadows the risk of excluded variables impacting supplies’’ [5]. In an adaptive setting, this initial forecast must include the uncertainty imposed by the adaptive approach. For example, in two types of adaptive designs, sample size re-estimation design and biomarker-adaptive design (see descriptions in [6]), a forecaster must recognize that the number of patients enrolled in the trial may increase or decrease in response to interim trial data. Once the interim trial data is analyzed, some uncertainty is removed; the trial’s target sample size will adjust accordingly. Given enough time, all of the variables impacting clinical supply requirements become less fluid and a supply manager is equipped with a significantly better forecast of demand. The need to start clinical production before demand is known leads to two possible strategies to manage the inherent forecast uncertainty. Assuming the investigational drug has sufficient shelf life, then one supply strategy is to produce sufficient material in the initial production run to accommodate any potential scenario

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A.J. Fleischhacker, Y. Zhao / Operations Research for Health Care (

for clinical trial demand. Alternatively, a supply manager can break production into two production runs. The first production ensures enough supply is available to start the trial. A second production can then ensure enough supply to end the trial and reflects quantity decisions that are made with greater forecast precision. For example, a clinical trial, with globally distributed testing centers, may start with enough supply to satisfy a launch of the trial in the United States. Once launched in the United States, international testing centers are opened and most demand for these sites will be satisfied using supply from a second production run. In an adaptive setting, supply from the second production run is committed to with greater certainty as to treatment length, dosage, and study enrollment parameters. The longer the trial continues, the longer the supply manager has to learn about the effects of trial design changes on demand and the more certain he can be about the remaining supply requirements. Unfortunately, reducing the costs of supply–demand mismatches can only be realized by increasing the number of production opportunities. The additional production runs, while costly, can be planned for using a more accurate forecast of trial demand. By incurring multiple setups, the high fixed costs of physical drug production and the associated quality control activities are amortized over less inventory and economies of scale are sacrificed. The essence of this paper is to understand this sacrifice of economies of scale and to optimally balance the benefits of learning against higher costs for physical supply. In this paper, we construct a model to plan drug supply for adaptive clinical trials. When minimizing inventory costs, the model explicitly incorporates the learning that takes place during a clinical trial. At the beginning of a clinical trial, trial parameters such as anticipated enrollment are very uncertain and thus the supply manager is making his/her best guess as to the requirements of the trial. As a trial is underway and continues, early trial results are analyzed to refine initial demand estimates. Since some trials will yield more informative early results than others, our model incorporates the rate at which demand uncertainty during the trial is reduced. In addition to modeling demand uncertainty as a function of time, the model incorporates setup costs (i.e., fixed costs of setting up production, packaging, labeling, distribution, etc.) for both pre-trial and intra-trial production. Thus, the model enables us to weigh the benefit of demand uncertainty reduction against the cost of incurring a second setup cost for production. The objective of this paper is to study the impact of fixed costs and demand learning rates (i.e. the rate of demand uncertainty reduction) on the value of producing an intra-trial batch. In addition, we look at the optimal timing of this intra-trial production with respect to both the rate of demand learning and the fixed costs associated with production. We organize the rest of the paper as follows. We review the related literature in Section 2. The model and analysis are presented in Section 3. A numerical study is presented in Section 4. Finally, we summarize the paper and discuss future research directions in Section 5. 2. Literature review In this paper, we seek to extend the existing mathematical models of the operations management literature to provide insights in the world of clinical trial supply decisions. Sophisticated models have been developed to manage inventory in other contexts and we are able to extend this work into the clinical trial space. In fact, Shah [7] advocates that many of the key challenges faced by pharmaceutical supply chains could benefit from extending the existing work of operations research academicians on capacity and inventory planning models. A good review of the existing modeling efforts specific to process industries, including clinical trial manufacturing, is found in Papageorgiou [8] and references therein.

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Traditionally, most insights generated for clinical trial supply decisions are accomplished through simulation (see for example [9–11]) and by presenting analytical insights in this paper we seek to leverage the strengths of analytical models to complement this existing work. Specifically, by providing analytic results via mathematical modeling we achieve three important contributions: (1) we provide managerial insights into the tradeoffs between learning and economies of scale in the clinical trial domain, (2) we provide a starting point for simulation models as our model can generate the initial input for those models; an approach advocated by both software tools like BioClinica Optimizer and the recent literature [12], and (3) we are able to analytically characterize the optimal ordering policies in this context. Although the trade-off between economies of scale and learning in adaptive clinical trials was not studied analytically, there are noteworthy mathematical models that have been applied to other clinical trial decisions. For instance, both Colvin and Maravelias [13,14] present stochastic programming approaches to select the scheduling of investigational drugs to enter clinical trials in light of limited resources/capacity. In addition, other models exist for the capacity planning decision of how much to invest in capacity given uncertain demand (see for example [15,16]). Some other notable mathematical models that have been studied include those of You and Grossmann [17] who study the balancing of costs and responsiveness in a multi-echelon supply chain and of Kelle, Woosley, and Schneider [18] who provide a decision support methodology for optimizing inventory management within a hospital pharmacy. While portfolio selection and capacity decisions have benefited from extending the existing operations management literature to more specific process industry and clinical trial cases, to the best of our knowledge we are the first to do this for the ordering/production quantity decisions being made in an adaptive clinical trial context. We find that these decisions are similar to those faced by production planners of many short lifecycle products with uncertain demand, but the required strategies are different due to the unique aspects of adaptive trials. In one of the most cited works on production planning for short lifecycle products from the operations management literature, Fisher and Raman [19] state their key insight and the resulting strategy: The dramatic improvement in forecast accuracy after observing only 20% of initial demand suggests a strategy for reducing the cost of too much or too little inventory: commit to a modest amount of initial inventory for each product, observe initial demand, and then produce an additional amount of each product based on improved forecasts. It is worth noting that despite our rationale being similar, namely we improve forecast accuracy by providing a second opportunity for resupply, the strategy recommended in a clinical trial context is completely different. The modest initial inventory recommendation of the existing literature may no longer apply. We find that fixed and variable costs of fulfilling a supply requirement plus the rate of demand uncertainty reduction as the trial proceeds (i.e. demand learning) are key factors influencing the initial ordering decision for a specific supply requirement (e.g. API batch size for a biologic, packaging of a certain investigational dosage, initial shipment of clinical supply to a certain country depot, initial order quantity for a comparator drug etc.). And we note, the prescribed ordering decision in the clinical trial context is often vastly greater than the modest amount proposed in the operations management literature. If we abstract from adaptive clinical trials, the general idea we seek to model is that forecasts and planning decisions can be improved based on observation of a demand signal. Within this abstraction, problems similar to those faced by supply managers

A.J. Fleischhacker, Y. Zhao / Operations Research for Health Care (

for adaptive trials can be found in the operations management literature. For example, Parlar and Weng [20] research a twoperiod production decision where a second production run is possible. However, this second production run occurs after the realization of demand and thus, full learning has occurred. As an example of partial learning, Eppen and Iyer [21] study a large catalog retailer’s decisions of how much of a women’s fashion product to order and then how much to divert to outlet stores upon observation of a portion of demand. They continue a longline of studies that utilize Bayesian updating for forecast revision. A much earlier paper recognizing the importance of adaptively revising forecasts is given by Murray and Silver [22]. They employ a Bayesian methodology for updating an unknown sales probability of an item based on a known amount of potential customers. In our paper, we employ a learning curve approach to study the effect that different rates of demand learning have on the optimal initial production quantity. As detailed in the survey by Yelle [23], learning curve applications have extended far beyond the more traditional applications of modeling the decrease in per unit manufacturing costs or the increase in labor productivity due to organizational experience. To our knowledge, this paper is among the first to apply a learning curve model where forecast uncertainty is flexibly modeled as a decreasing function of the time allotted to observe demand. While this application is new,the previous studies suggest the applicability of its use. For example, [24] present an example where the coefficient of variation between forecasts and actual demand decrease as demand signals are observed, but their model of demand learning is less flexible than what is presented here. Even though our use of learning curves to model forecast uncertainty reduction is unique, the use of learning curves to understand the benefits of learning in production planning is not. Terwiesch and Bohn [25] study whether learning to increase the yield of a supply process is worth delaying time to market. In another interesting study of learning and production, Kornish and Keeney [26] study the tradeoff of learning against capacity. Their model, motivated by the annual decision of which strains of flu to vaccinate against, address ‘‘a trade-off between quantity (producing more) and quality (produce a more effective vaccine because you know more)’’. By delaying the commitment to which flu strains will be targeted by the produced vaccine, a more effective vaccine can be made, but there is less time for production. The optimal policy is discussed and its implications for choosing to commit to production or wait for more information are detailed in their study. The inclusion of a resupply opportunity for short lifecycle products is a logical extension to the classical newsvendor model [27] that has also been examined in the literature. For example, Lau and Lau [28] study a mid-season replenishment possibility (equivalent to a mid-trial resupply possibility) as do Milner and Rosenblatt [29], Fisher et al. [30], and Li, Chand, Dada, and Mehta [31]. Our paper differs from all of the extant mid-season replenishment models through the inclusion of both setup costs and demand learning in our model to make the resulting insights relevant in a clinical trial supply context. 3. The two-period model In this section, we first introduce the notation and the model in Section 3.1. We then characterize the optimal ordering policy in Section 3.2. Next we introduce the demand learning model in Section 3.3 and finally we study the impact of fixed cost in Section 3.4. 3.1. Notation and model Let the expected duration of the clinical trial (the planning horizon) be [0, T ] and assume zero lead time. We will use the following notation throughout the paper:

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• t: length of the learning period (i.e. the first period) where • • • •

• • • • • • • •

t is the time at which resupply is made available and where 0 ≤ t < T . In the case of one production batch, t = 0. D: total demand during the entire trial, i.e. period [0, T ]. D1 : demand in the 1st period, [0, t ], occurring prior to the resupply date. D2 : demand in the 2nd period, [t , T ], occurring after the resupply date. D2 |D1 =ξ : demand in the 2nd period [t , T ] given D1 = ξ where ξ is the realization. Note that the distribution of D2 |D1 =ξ varies depending on the length of the learning period t. x1 : production quantity/order for the first period and made available at time 0. x2 : production quantity/order for the second period and made available at time t. κ : setup (fixed) cost of a production run or order. c: variable/per unit manufacturing cost. πb : backorder penalty per unit short after the first period that is ultimately satisfied. πs : shortage penalty per unit of unmet demand at the end of the time horizon. r: overage penalty (destruction/recycle cost) per unit leftover at the end of the time horizon. y2 : order up to level for the second period.

To avoid trivial cases and ensure a realistic model, we make the following assumption. Assumption 1. πs > πb > c ≥ 0, πs − πb > c and demand is non-negative. This assumption is reasonable in the clinical trial practice as letting patients wait for medication is often undesirable (so πb > c > 0), and rejecting patients due to insufficient stock can be much worse (so πs > πb ). Assuming zero initial inventory, we let f1 (t ) be the optimal expected inventory cost for the two-period problem with t being the duration of the learning period and let δ(xt ) be the indicator function of xt > 0. Then, using the notation that x+ ≡ max(x, 0) the optimal cost is expressed as



f1 (t ) = min δ(x1 )κ + cx1 + πb ED1 (D1 − x1 )+





x1 ≥0

  + ED1 f2 (x1 − D1 , t ) |D1 ,

(1)

where f2 (I , t )|D1 represents the optimal expected inventory costs for the second period conditioning on D1 , given a starting inventory of I = x1 − D1 and the length of the first period, t.



f2 (I , t )|D1 =ξ = min δ(x2 )κ + cx2 − πb (−I − x2 )+ x2 ≥0

 +  + πs ED2 |D1 =ξ D2 |D1 =ξ − I − x2  +  + rED2 |D1 =ξ I + x2 − D2 |D1 =ξ .

(2)

Thus, the optimal first period cost is the sum of the first period setup costs, first period variable production costs, first period backorder penalty costs, and the expected second period costs. The second period costs consist of a setup cost for a second period production, variable production costs for the second production run, a rebate on backorder penalties charged in the first period that turn out to be supply shortages at the end of the trial (i.e. second period production does not satisfy the unmet demand of the first period), a shortage penalty for demand that is left unfilled at the end of the trial, and destruction costs for drug supply that goes unused during the trial. Note that in adding an additional replenishment option, we also must introduce an intra-period

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shortage penalty. We consider this as a backorder penalty which is typically much less severe than the shortage penalty charged at the end of the horizon. In our analysis of the scheduling of resupply, three simplifying assumptions are worthy of mention. First, our model explicitly excludes lead time considerations. Second, our model requires that the timing of the resupply date, t, be scheduled in advance of the trial’s commencement. Third, our model excludes the substantial risk of the trial being halted prior to the end of the time horizon. This risk, which we call failure risk, is due to the possibility of a trial showing that a drug is unsafe or ineffective prior to the conclusion of the trial. For certain types of clinical drug supply or other quantity decisions the zero lead time assumption may be untenable. However, for other types this assumption is valid. For example, when NeoRx Corporation outsourced clinical trial supply to International Isotopes Inc., purchase orders were only placed one week in advance and rolling forecasts were provided for 3 months of future demand [32]. These durations are much shorter than the overall duration of the trial, which may take as long as 2–3 years. Please note that these purchase orders are for the batches produced after the first batch. The lead time on the first batch may still be lengthy as manufacturing facilities are configured for initial production. Once production facilities and processes are in place, lead time on additional batches can be much shorter than that of the first batch. The timing of the intra-trial batch (i.e. resupply date) being determined prior to commencement of the actual trial is consistent with outsourcing contracts for clinical supply where the availability of manufacturing capacity is reserved in advance. It is also worth noting that Li, Chand, Dada, and Mehta [31] have found little value to dynamically determining the timing of the second production run and thus, this assumption should not lead to dramatically different results when resupply dates can be dynamically adjusted. We do not consider the effects of cancellation fees on the supply manager’s decisions, although we believe this might be an interesting area for future research. Lastly, the failure risk, although important, is excluded from our analysis and reflects the typical supply philosophy of planning for success during a clinical trial. If failure was to occur during time (0, t ), it is as if the second production option would go unutilized. By applying a discount factor to the cost of producing in the second period, this aspect of clinical trials could be captured. To keep our analysis to the balancing of economies of scale and uncertainty reduction, we propose that inclusion of failure risk in addition to uncertain demand may be an interesting area of future research. Fleischhacker and Zhao [33] demonstrate how to include failure risk for the case of deterministic demand. 3.2. Optimal ordering policy For the two-stage stochastic dynamic program developed in Section 3, we seek to characterize the optimal inventory policy. To achieve our results, some simplifying observations regarding the model developed in Section 3 will be required. We start with reframing the problem to look at order up to levels as opposed to one of choosing order quantities. For convenience, we will leverage the relationship between the second period order up to level and the second period order quantity, specifically y2 = I + x2 , and thus Eq. (2) becomes, f2 (I , t )|D1 =ξ

  = −cI + min δ(y2 − I )κ + cy2 − πb (−y2 )+ + Lξ (y2 , t ) , (3) y2 ≥I

where Lξ (y2 , t ) = πs ED2 |D

1 =ξ

D2 |D1 =ξ

+ 

.



D2 |D1 =ξ − y2

+ 

+ rED2 |D1 =ξ



y2 −

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To further analyze Eq. (3), we note that we can either produce (i.e. x2 > 0) or not produce, f2 (I , t )|D1 =ξ

   = −cI + min min κ + cy2 − πb (−y2 )+ + Lξ (y2 , t ) , y 2 >I



cI − πb (−I )+ + Lξ (I , t ) .

(4)

Observation 1. If y2 > I, then the optimal order-up-to level for the second period y∗2 ≥ 0. Proof. If I ≥ 0, y∗2 ≥ 0 by definition. If I < 0, consider a y2 ≤ 0. By Assumption 1 and Eq. (3),

κ + cy2 − πb (−y2 )+ + Lξ (y2 , t )   = κ + cy2 − πb (−y2 ) + πs ED2 |D1 =ξ D2 |D1 =ξ − y2   = κ + cy2 + ED2 |D1 =ξ πs D2 |D1 =ξ + (πs − πb )(−y2 ). By Assumption 1, the cost function is decreasing in y2 for y2 ≤ 0. Thus y∗2 ≥ 0.  By Observation 1, Eq. (4) can be reduced to,







f2 (I , t )|D1 =ξ = −cI + min min κ + cy2 + Lξ (y2 , t ) , y 2 >I



cI − πb (−I )+ + Lξ (I , t ) .

(5)

The following observation shows that the second period cost function is not convex and thus we cannot directly apply the classical result of (s, S ) policy (e.g., see [34, Section 9.5]) to this problem. Observation 2. The second period cost function, cI − πb (−I )+ + Lξ (I , t ), is not convex in I, but it is unimodular in I and approaches infinity as I → ±∞. Proof. First, we note that −πb (−I )+ is concave in I. For I < 0, it follows by Assumption 1 that the second period cost function, cI − πb (−I )+ + Lξ (I , t ), reduces to cI − πb (−I ) + πs ED2 |D

1 =ξ

[D2 |D1 =ξ − I ]

= πs ED2 |D1 =ξ [D2 |D1 =ξ ] − (πs − πb − c )I , which is linear (thus convex) in I. For I ≥ 0, the second period cost function reduces to cI + Lξ (I , t ), which is also convex in I because Lξ (I , t ) is convex. However, the left derivative of the cost function at I = 0 equals −(πs − πb − c ) < 0 (by Assumption 1), which is greater than the right derivative of the cost function at I = 0, −(πs − c ). Thus, the cost function is not convex in I for I ∈ (−∞, ∞). To show the cost function is unimodular, we note that it is convex and decreasing in I for I ∈ (−∞, 0]. Because it is also convex in I for I ∈ [0, ∞), it must be unimodular. Finally, as I → −∞, the slope of the cost function is −(πs − πb − c ); as I → ∞, the slope approaches c + r. The proof is now completed.  Now we are ready to identify the optimal ordering policy for the second period given D1 = ξ . Let S2 (ξ ) be the smallest global minimizer of cy + Lξ (y, t ), and s2 (ξ ) be the largest I (but smaller than S2 (ξ )) such that cI − πb (−I )+ +  Lξ (I, t ) = κ +

c ≥ 0, where cS2 (ξ ) + Lξ (S2 (ξ ), t ). Indeed, S2 (ξ ) = ΦD−21|D =ξ ππs − s +c 1 ΦD2 |D1 =ξ (·) is the probability density function of D2 |D1 =ξ . s2 (ξ ) must exist by the unimodularity and asymptotic properties shown in Observation 2.

Theorem 1. The optimal ordering policy for the second period is a (s, S ) type of policy depending on D1 = ξ , where s = s2 (ξ ) and S = S2 (ξ ). In other words, if the beginning inventory position I < s2 (ξ ), we order up to S2 (ξ ); otherwise, we do not order.

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Proof. The proof follows directly from the definition of s2 (ξ ), S2 (ξ ) and Observation 2.  Note that s2 (ξ ) and S2 (ξ ) are dependent on D1 = ξ but independent of second period starting inventory I. By Theorem 1, we can write f2 (I , t )|D1 =ξ as follows, f2 (I , t )|D1 =ξ = −cI +



κ + cS2 (ξ ) + Lξ (S2 (ξ ), t ), I ≤ s2 (ξ ) (6) cI − πb (−I )+ + Lξ (I , t ), I > s2 (ξ ).

We now show f2 (I , t )|D1 =ξ is κ -convex for any ξ . By [34, Section 9.5], we have the following definition. Definition 1. We call a function f (x) κ -convex if for any x, and non-negative u and v, f (x) satisfies f (x) + v

f (x) − f (x − u) u

≤ f (x + v) + κ.

Lemma 1. f2 (I , t )|D1 =ξ is κ -convex in I for any ξ . Proof. For simplicity, we drop ξ from our notation. We also define f˜ (I , t ) = f2 (I , t ) + cI. Clearly if f˜ (I , t ) is κ -convex, f2 (I , t ) is also κ -convex. If s2 ≥ 0, then f˜2 (I , t ) =



κ + cS2 + L(S2 , t ), cI + L(I , t ),

I ≤ s2 I > s2 .

By [34, Section 9.5], f˜2 (I , t ) and thus f2 (I , t ) is κ -convex. For s2 < 0, we first consider any I ≤ s2 . In this case, an order is placed regardless of the value of u. By the definition of the ˜ ˜ (s2 , S2 ) policy, we must have f˜2 (I , t ) + v f2 (I ,t )−uf2 (I −u,t ) = f˜2 (I , t ) = f˜2 (S2 , t ) + κ ≤ f˜2 (x, t ) + κ for all x. Next we consider I ≥ S2 . The κ -convexity inequality must hold because f˜2 (I , t ) is convex and increasing. Finally, we consider s2 < I ≤ S2 . By Eq. (6), f˜2 (I , t ) = cI − πb (−I )+ + Lξ (I , t ) which is the second period cost function studied in Observation 2. By Observation 2, f˜2 (I , t ) is unimodular in I for I > s2 and decreasing in I for I < 0. Together with the definition of S2 and the fact that S2 ≥ 0, S2 is also the smallest global minimizer of f˜2 (I , t ). Thus f˜2 (I , t ) is decreasing in I for f˜ (I ,t )−f˜2 (I −u,t ) s2 < I ≤ S2 , and therefore, f˜2 (I , t ) + v 2 ≤ f˜2 (I , t ) ≤ u f˜2 (S2 ) + κ ≤ f˜2 (I + v) + κ . The proof is now completed. 

Theorem 2. The optimal ordering policy for the first period is a (s, S ) type of policy. Proof. By Lemma 1, f2 (I , t )|D1 =ξ is κ -convex. By Lemma 9.5.1 of [34], ED1 [f2 (x1 − D1 , t )|D1 ] is also κ -convex, and so is cx1 +

    πb ED1 (D1 − x1 )+ + ED1 f2 (x1 − D1 , t ) |D1 . By Theorem 9.5.2 of [34], the proof is completed.



Let (s1 , S1 ) be the optimal (s, S ) policy for the first period. Since there is no inventory prior to the first production run, as long as s1 > 0 then one would produce up to S1 in the first period. Otherwise, no production will be available at t = 0. 3.3. Demand learning model The optimal first and second period order policies are established in Section 3.2 without making assumptions regarding the relationship between first and second period demand. However, as previously motivated, we are very much interested in this relationship and require a model for the amount of demand uncertainty that can be reduced prior to the second production run. For our analysis, we start with an initial belief of total demand during

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the planning horizon, D, that we assume to be normally distributed with mean µ and standard deviation σ . To connect demand in the first period to demand in the second period, we will also want to assume dependent demand in periods 1-2, D1 and D2 , such that D = D1 + D2 . Following Fisher and Raman [19] and Fisher et al. [30], we assume (D1 , D2 ) follows a bivariate normal distribution so that our belief in total demand is consistent with how we split the total demand between the period before and the period after resupply. To study the demand uncertainty that remains as the trial progresses, we extend the model of Fisher and Raman to include a correlation coefficient ρ(t ) that is a function of time and where ρ(t ) depends on the amount of learning that can take place by time t. In Section 4, we borrow methodology from the learning curve literature to model this function, ρ(t ), and use it to study various rates of demand learning. To accommodate these assumptions regarding demand, it follows that the marginal distribution D1 is also normal with mean µ1 and standard deviation σ1 . For t ∈ [0, T ], we define the first period demand with respect to our beliefs regarding total demand: E (D1 ) = µ1 = α(t )µ,

σ 2 (D1 ) = σ12 = β(t )σ 2 ,

where α(t ) and β(t ) are fractions increasing from 0 to 1 as t increases from 0 to T . For example, one could choose α(t ) = t /T and β(t ) = t /T so that both expected demand and demand uncertainty in the first period are proportional to the percentage of the planning horizon that is defined as the first period. Similarly, the marginal distribution of D2 is normal with E (D2 ) = µ2 = (1 − α(t ))µ,

σ (D2 ) = σ2 = −ρ(t )σ1 +



ρ 2 (t )σ12 + σ 2 − σ12 .

Conditioning on D1 = ξ , D2 follows a normal distribution with the following parameters (Fisher and Raman [19]):

µ2 (ξ ) = µ2 + ρ(t )σ2

ξ − µ1 , σ1

 σ2 (ξ ) = σ2 1 − ρ 2 (t ).

(7)

To highlight the effect of correlation, one notes that for a given t, as ρ increases, σ2 decreases and thus σ2 (ξ ) decreases. We model demand learning through correlation between first and second period demand. Because of this correlation, supply requirements for second period doses/packages is better predicted after seeing early trial results. Without learning, the uncertainty in second period demand is represented by variance in second period demand σ22 . From squaring the standard deviation in Eq. (7), we see that only a fraction of this variance remains after observing D1 = ξ : Fraction of Unexplained Variance=

σ22 (ξ ) σ22

= 1 − ρ 2 (t ).

Consistent with this mathematical interpretation, we will model demand learning as a reduction in unexplained variance. Intuitively, the fraction of unexplained variance in second period demand should be close to one when t is early in the time horizon and closer to zero when t is near the end of the horizon. To study different rates of learning, we will assume that the amount of learning, more specifically the percentage reduction in uncertainty surrounding second period demand, 1 −ρ(t )2 , follows a power law form. This is very similar to the notion of a learning curve model introduced by Wright [35]. Instead of modeling reduction in the required time to produce airplane parts as Wright did, we use a power law form to model a level of uncertainty in second period demand as proportional to the percentage of the trial that  T −is yet to t t γ 2 and be completed (i.e. T − ). As such, we have 1 − ρ ( t ) = T T thus

 ρ(t ) =

 1−

T −t T

γ (8)

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A.J. Fleischhacker, Y. Zhao / Operations Research for Health Care (

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Table 1 Baseline parameter values. Parameter

Description

Baseline value

t T

γ

Length of first (learning) period Length of planning horizon Rate of demand learning

5 10 0

D(0, t )

First period demand N (µ, σ )

N (1000 ∗

D( t , T )

Second period demand N (µ, σ )

κ πb πs

 5 , 3002 ∗ 10 )  10−5 2 N (1000 ∗ 10 , 300 ∗ (1 −

Setup cost of a production run Backorder penalty End of horizon shortage penalty Variable production cost Destruction cost

0 20 50 2 1

c r

5 10

5 10

))

where γ ≥ 0 is the shape parameter of the learning curve. When γ = 1 the amount of learning is linear in time, when γ < 1, learning is slow and the uncertainty parameter, 1 − ρ(t )2 , will be a concave function of time. At γ = 0, there is no learning and demand is independent across periods. Lastly, when γ > 1, learning occurs more rapidly and 1 − ρ(t )2 is a convex function of time. 3.4. Impact of the fixed cost The effects of learning and a reduction in demand uncertainty can only be taken advantage of if production of an intra-trial batch (or planning for an intra-trial order) is both planned and economically viable. Intuitively, as the fixed cost of production (ordering) in the second period, κ , increases, we might expect that our ability to take advantage of demand learning is diminished. This is indeed the case for the two-period model. In fact, the following proposition confirms this intuition and shows how the two-period model reduces to a one period newsvendor model [27] for high fixed costs associated with a second production opportunity. Proposition 1. S1 approaches to the newsvendor quantity as κ →

∞. Proof. For simplicity, we drop the dependence on D1 for (s2 , S2 ). Suppose that we produce in the first period, the total cost function can be expressed as follows,

 κ + min cx1 + πb ED1 [(D1 − x1 )+ ] x 1 >0

+ ED1 [−c (x1 − D1 ) + κ + cS2 + LD1 (S2 , t )|D1 ≥ x1 − s2 ]  + ED1 [−πb (D1 − x1 )+ + LD1 (x1 − D1 , t )|D1 < x1 − s2 ] . As κ → ∞, s2 → −∞ (by Observation 2) for each realization of D1 . Thus, the cost function tends to

  κ + min cx1 + ED1 [LD1 (x1 − D1 , t )] , x 1 >0

where ED1 [LD1 (x1 − D1 , t )] = ED1 [ED2 [πs (D1 + D2 − x1 )+ + r (x1 − D1 − D2 )+ |D1 ]] = ED1 +D2 [πs (D1 + D2 − x1 )+ + r (x1 − D1 − D2 )+ ]. The last equality comes from the definition of conditional expectation. Note that ED1 +D2 [πs (D1 + D2 − x1 )+ + r (x1 − D1 − D2 )+ ] represents the cost function of the newsvendor model without the second period, the proof is now completed.  In general, as the fixed cost κ increases, S1 will be more likely used to cover both D1 and D2 , and thus S1 typically increases. 4. Numerical analysis The objective of this section is to quantify the effects of setup costs, learning rates, and inventory penalty costs on the value of the second production/ordering/packaging/etc. option, the optimal timing of learning (i.e. first period length), and the optimal first

Fig. 1. Expected costs when κ = $0.

batch size. The penalty costs include the overage penalty cost associated with too much inventory, r + c, and the underage penalty costs, πb − c and πs − c, of not being able to satisfy demand during the first and second periods, respectively. For clinical trials, stockouts are particularly costly if they delay the trial and both πb and πs would represent significant penalty costs. As a starting point for the analysis, a baseline problem is created using values that a clinical supply manager might face. We assume that mean demand over the time horizon is 1000 units and the standard deviation of demand is 300 units. We assume that demand for the time horizon (i.e. anticipated duration of trial) is normally distributed and since this distribution is divisible, we can mathematically divide demand between two periods. Thus, the fraction, Tt , represents the percentage of total demand expected to occur in the first period. The parameters of our baseline model, in the absence of demand learning (i.e. γ = 0 in Eq. (8)), are shown in Table 1. In Fig. 1, we compare the expected costs of our baseline model with the expected costs of a newsvendor model (i.e. a single production run or order at t = 0 with no resupply available). For our baseline model, we have arbitrarily scheduled an additional replenishment option mid-way through the planning horizon. As can be seen from the graph, the additional replenishment opportunity leads to a greater than 10% reduction in costs. It is also interesting to note that the expected costs of the baseline model are less sensitive than the newsvendor model to first period order quantity. At this point, one might conclude that a mid-trial replenishment can reduce the expected costs and lead to a decision that is less sensitive to model parameters. However, the comparison we made ignores several key components that constitute the uniqueness of clinical trial supply decisions. First, setup costs are rarely zero whether they represent the production of clinical trial drug supply, the paperwork/time costs associated with sending resupply through customs, or any of the other aforementioned quantity

A.J. Fleischhacker, Y. Zhao / Operations Research for Health Care (

Fig. 2. Expected costs when κ = $1600.

decisions faced by clinical supply managers. And second, our comparison fails to account for demand learning; as the trial progresses the supply manager is able to reduce uncertainty surrounding demand and the scheduling of resupply, represented by the variable t, is not restricted to be only halfway through the clinical trial. For expositional purposes, we first introduce setup costs into our analysis without considering demand learning. From the proof of Proposition 1, we know that as setup costs increase the twoperiod model reduces to the newsvendor model. We can see this effect by comparing Fig. 2 which includes a setup cost (κ = $1600) to Fig. 1 which assumes there are zero fixed costs when producing. With setup costs introduced, we can see that the value of an additional replenishment mid-way through the planning horizon yields minimal savings of 2% of the newsvendor solution’s inventory costs. To possibly counteract this minimization of value offered by a resupply opportunity, the economic effects of scheduling drug resupply at different points of the clinical trial are examined. To analyze this, we graph the expected costs as a function of the first period length (t) as seen in Fig. 3. For every choice of first period length, t, the optimal first period order quantity has been numerically determined. We see the optimal scheduling of resupply is close to the entire planning horizon (t ≈ 8.5) and not the arbitrarily chosen mid-horizon production (t = 5). As a result of our modeling choices, we note that when t = T , the achieved solution outperforms the newsvendor solution. In this case, the value of the additional replenishment at t = T enables one to essentially exchange end of horizon shortage penalty costs with backorder penalty costs in cases of high demand (note: all considered scenarios lead to a first period production where x1 > 0). In practice, many realistic scenarios exist where this advantage of setting t = T disappears and we comment on how these scenarios are reflected by the model. First, as setup costs increase beyond the κ = $1600 depicted in Fig. 3, the exchange of shortage costs for backorder costs will no longer be economically justified. The expected costs when t = T will be identical to that of the newsvendor solution. Additionally, backorder and shortage costs may be essentially equal in many realistic cases. In practice, clinicians may not enroll patients in a trial with unreliable supply and thus, the advantage offered by setting t = T again disappears as the costs of backorders and shortage costs are essentially indistinguishable. While setup costs reduce some of the benefit of the intra-trial replenishment option, demand learning creates greater incentives to plan an additional replenishment and counter-balance the costs imposed by an additional setup. Intuitively, faster rates of demand learning (higher γ in Eq. (8)) encourage earlier scheduling of the potential second production run. To see this in our example, we now analyze the baseline model with setup costs for various rates of demand learning. We pick various values of our learning

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Fig. 3. Expected costs versus timing of second replenishment with setup costs.

Fig. 4. Expected costs versus timing of second replenishment with setup costs and demand learning.

parameter, γ , to represent different rates of uncertainty reduction that may be experienced by clinical supply managers facing a quantity decision. The expected costs of our baseline model with setup costs (κ = $1600) for various learning rates are in Fig. 4. We see from this graph that faster learning leads to both an earlier scheduling for the second production run and larger cost reductions versus the single newsvendor production. More importantly, the cost benefit of learning has effectively nullified the substantial setup cost. We see that when the rate of learning is simply linear in time (γ = 1), we can achieve savings of greater than 10% over the single production solution. This linear learning rate would yield a correlation coefficient of 0.82 at the optimal duration of the first period (t ≈ 6.75) which is consistent with the correlations used in [30]. While the scheduling decision is important, it is not made in isolation. The optimal supply strategy will simultaneously consider the timing of resupply and the sizing of the first batch. In Fig. 5, we analyze the interplay of production scheduling and optimal initial order quantity for various setup costs and linear learning (γ = 1). In the absence of setup costs (i.e. κ = 0) and when replenishment is planned after observing a small fraction (roughly 10%) of demand, we observe that the optimal initial order quantity is less than half of what would be ordered without the opportunity of resupply (i.e. the newsvendor batch size). However, with even modest setup costs of $100, the initial order quantity is much larger and closer to the newsvendor quantity. This is a key observation that the suggestion of a ‘‘modest amount of initial inventory’’ (Fisher and Raman [19]) is less appropriate when setup costs are factored into the decision making. The benefits of ample inventory, including avoiding additional setup costs and first period backorder costs, outweigh the benefits of uncertainty reduction afforded by a second production.

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Table 3 Table of savings for various setup costs and learning rates.

κ – 100 200 400 800 1,600 3,200 6,400 12,800

Learning rate (γ ) 0

0.5

1

2

4

16.5% 14.7% 13.1% 10.9% 7.8% 4.6% 1.9% 0.4% 0.0%

28.8% 26.5% 24.4% 20.9% 15.8% 9.8% 4.5% 1.3% 0.2%

31.7% 29.1% 26.9% 23.1% 17.4% 10.7% 4.9% 1.4% 0.2%

34.8% 32.1% 29.7% 25.5% 19.1% 11.6% 5.2% 1.5% 0.2%

38.2% 35.3% 32.7% 28.0% 20.8% 12.4% 5.5% 1.6% 0.2%

Table 4 Expected savings over a newsvendor solution for increasing underage costs. Fig. 5. Optimal first batch size versus timing of second replenishment with setup costs and demand learning. Table 2 Probability of Mid-Trial replenishment for various setup costs and learning rates.

κ – 100 200 400 800 1,600 3,200 6,400 12,800

Learning rate (γ ) 0

0.5

1

2

4

66.7% 50.2% 41.1% 30.5% 19.8% 10.4% 4.8% 1.6% 0.1%

66.3% 59.4% 52.1% 43.3% 31.9% 18.7% 9.2% 3.4% 0.7%

70.6% 64.5% 57.5% 48.8% 36.8% 21.4% 10.2% 3.6% 0.7%

74.1% 69.1% 63.0% 54.4% 40.0% 24.4% 11.0% 3.8% 0.8%

77.4% 73.6% 68.1% 60.6% 47.5% 27.2% 11.9% 4.0% 0.8%

In the absence of setup costs and in the presence of demand learning, a second production run is a likely event. For our baseline model with linear learning and zero setup costs, the second production run is optimally scheduled at around t ≈ 5.5 and it is expected that 71% of the time the production run will be utilized. The other 29%, demand is so low in the first 55% of the planning horizon that a second production is not needed. Even though overage risk is present, this risk is offset by having enough inventory to avoid intra-period backorder costs. As soon as we introduce setup costs, we also introduce a notion of economies of scale in production. A manager’s expectation of producing more than once reflects his willingness to sacrifice scale economies to achieve savings. The tradeoff between sacrificing scale economies to better match supply and demand is summarized in Table 2. We see from this table that setup costs significantly decrease the probability of a second production. For example, in the case of linear learning (γ = 1), the introduction of setup costs of $1600 reduces the likelihood of producing a second time from 71% to 21%. Further increases in setup costs drastically reduce the likelihood of producing a second time. From a planning perspective, scheduling resupply in the presence of high setup costs is really an emergency supply option for cases of unforeseen high demand. Even though the likelihood of leveraging a drug resupply option can be small, the value of this option remains significant when trial requirements are more predictable once partial trial results are observed. This can be seen in Table 3 which shows the expected savings over the newsvendor model when optimally scheduling potential replenishment. From this table, we can see that with linear learning and setup costs of $1600, a 10.7% reduction in costs can be expected by just having a resupply option available even though this resupply option will only be exercised about 21% of the time (from Table 2). Another consideration in this mid-trial replenishment environment is how the penalty costs (i.e. inventory overage, inventory

πb /πs

% Savings

20/50 40/100 80/200 160/400 320/800

10.7% 13.5% 15.8% 17.7% 19.35%

Table 5 Expected savings over newsvendor solution varying destruction and backorder costs. r

1 2 4 8 16 32 64 128

% Savings (πb /πs =) 10/100

40/100

80/100

17% 20% 26% 33% 42% 51% 59% 65%

14% 17% 22% 30% 38% 45% 51% 54%

12% 15% 17% 22% 27% 36% 40% 45%

underage, and intra-period backorder penalties) impact our decisions of replenishment timing and initial order quantities. And even more importantly, how do changes in these parameters affect the magnitude of savings over a simpler newsvendor solution? The clinical trial supply environment is driven by a fear of delaying a trial due to insufficient supply and intuitively, one would think increasing underage penalties (πs or πb ) would lead to greater expected savings of a second production. In studying this numerically, we surprisingly find the advantage of having an intra-trial replenishment option is not dramatically improved by dramatically increased underage penalties. For example, our numerical study has found that doubling the two underage penalties of our baseline model with setup costs (κ = $1600, πb = 40, πs = 100) only increases expected savings over the newsvendor solution an additional 2.8% from 10.7% to 13.5%. Further increases to these underage penalties, as shown in Table 4, yield similarly modest results with the reason being that avoiding these underage penalties is relatively inexpensive; overage costs are only $3 which is small in comparison to the end of horizon shortage penalty of $50 of our baseline model. Basically, it is cheap to hedge against having too little inventory by simply producing more. In studying the effect of changes to the overage penalty, we find that increasing the overage penalty (via increases in destruction costs, r) leads to greater jumps in savings magnitude than increasing underage costs. For example, if we look at the case (Table 5) where πb = 40, πs = 100, r = $64 and κ = $1600, one can see that savings of greater than 50% over the newsvendor solution can be achieved. Table 5 also shows the interplay between varying backorder costs along with overage costs. While increasing backorder costs diminishes the value of intra-season replenishment, it does not erase the savings completely. Large initial orders and

A.J. Fleischhacker, Y. Zhao / Operations Research for Health Care (

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Table 6 Effect of parameter increases on performance and decision variables. Parameter ↑

Relative tonewsvendor solution % Savings

Learning (γ ) Setup costs (κ ) Backorder penalty (πb ) Shortage penalty (πs ) Destruction costs (r)

↑ ↓ ↓ ↑ ↑

f1 (0)−f1 (topt ) f1 (0)

1st batch size (%)



x1 xnewsv



↓ ↑ ↓ ↑↓ ↑↓

an intra-season replenishment ensure sufficient time to receive demand information and order resupply while still avoiding backorder costs. While not shown in Table 5, for the majority of overage costs explored, optimal batch sizes and optimal timing values fall within small ranges. Optimal batch sizes fall between 71% and 79% of their respective newsvendor optimal order sizes and optimal timing for resupply falls between 62%- and 68% of the planning horizon. While increases to both overage and underage penalties will always increase the expected savings over a newsvendor solution, the magnitude of percentage savings is parameter dependent. Percentage savings increases (per dollar of increased penalty cost) are fastest when overage and underage costs are highly unbalanced and increases are made to the lower of the two costs. Conversely, when increases are made to the higher of the two costs, only marginal benefits will be realized. In a pharmaceutical setting where underage costs far exceed the overage costs, the observation on the effects of increasing overage penalties suggests that increases in variable production costs greatly increase the attractiveness of planning for intra-trial replenishment. In summary, our numerical study pursued an understanding of the value of an intra-trial replenishment over the newsvendor solution. The value that is created depends on two key decisions in planning for this intra-trial option. First, how is the first batch size decision affected by the presence of a resupply option and second, when should the potential resupply be planned for. These two decisions, and the potential to achieve meaningful savings are affected by a multitude of parameters and for simplicity, we now summarize our findings in Table 6. 5. Conclusions The inclusion of setup costs and learning curves in our study leads us to extend the existing body of literature dealing with an additional replenishment options for short lifestyle products to the decision making required in adaptive clinical trials. In addition to characterizing the optimal inventory policies to be used in this setting, we discover meaningful insights that differ from those in the existing operations research literature. Most notably, due to high setup costs in this setting, the initial order quantity remains large as compared to the newsvendor batch size. In essence, the large inventory potentially avoids one incurring setup costs a second time and also, avoiding costly intra-trial shortages/delays. Through our model, we discover how the selection of learning period length, or equivalently the scheduling of resupply, is driven by both changes in setup costs and learning. We observe that increasing setup costs initially increases the optimal learning period length and then decreases it as setup costs exceed a certain threshold. At lower levels of setup costs, when these costs increase, it is advantageous to have a longer learning period to permit greater observation of demand and a more certain second period forecast. Eventually, further increases in setup costs decrease the learning period as to avoid intra-trial shortages in the rare cases of extremely high demand where incurring an expensive second setup becomes economically necessary. Our study is the first to look at the tradeoff between economies of scale offered by

1st batch size (x1 )

1st Period length (t )

↓ ↑ ↓ ↑ ↓

↓ ↑↓ ↓ ↑ ↓

infrequent/large batching decisions and benefiting from demand learning through more frequent production/packaging batches. We have found that the ideal conditions for consideration of an additional production run are when setup costs are low, learning is fast, and both overage and underage penalties are significant. In certain examples, we find savings to exceed 50% and our model can be used by clinical supply managers for insight into assessing both the economic benefit and the timing offered by intra-trial resupply opportunities. Acknowledgement The second author was supported in part by grant no. 0747779 from the National Science Foundation. References [1] J.A. DiMasi, The price of innovation: new estimates of drug development costs, Journal of Health Economics 22 (2003) 469–479. [2] L.J. Lesko, Paving the critical path: how can clinical pharmacology help achieve the vision? Clinical Pharmacology & Therapeutics 81 (2007) 170–177. [3] W. He, O.M. Kuznetsova, M. Harmer, C. Leahy, K. Anderson, N. Dossin, L. Li, J. Bolognese, Y. Tymofyeyev, J. Schindler, Practical considerations and strategies for executing adaptive clinical trials, Drug Information Journal 46 (2012) 160–174. [4] W. Jiang, B. Freidlin, R. Simon, Biomarker-adaptive threshold design: a procedure for evaluating treatment with possible biomarker-defined subset effect, Journal of the National Cancer Institute 99 (2007) 1036–1043. [5] C. Hall, Clinical supplies on the front lines: adapting to trial demand, in: Clinical Supply Management for Pharmaceuticals Conference, Philadelphia, PA, 2008. [6] S.-C. Chow, M. Chang, et al., Adaptive design methods in clinical trials—a review, Orphanet Journal of Rare Diseases 3 (2008). [7] N. Shah, Pharmaceutical supply chains: key issues and strategies for optimisation, Computers and Chemical Engineering 28 (2004) 929–941. [8] L.G. Papageorgiou, Supply chain optimisation for the process industries: advances and opportunities, Computers & Chemical Engineering 33 (2009) 1931–1938. [9] M. Peterson, B. Byrom, N. Dowlman, D. McEntegart, Optimizing clinical trial supply requirements: simulation of computer-controlled supply chain management, Clinical Trials 1 (2004) 399–412. [10] C. Abdelkafi, B. Beck, B. David, C. Druck, M. Horoho, Balancing risk and costs to optimize the clinical supply chain—a step beyond simulation, Journal of Pharmaceutical Innovation 4 (2009) 96–106. [11] G. Nicholls, N. Patel, B. Byrom, Simulation: a critical tool in adaptive, Applied Clinical Trials (2009). [12] Y. Chen, L. Mockus, S. Orcun, G. Reklaitis, Simulation-optimization approach to clinical trial supply chain management with demand scenario forecast, Computers & Chemical Engineering 40 (2012) 82–96. [13] M. Colvin, C. Maravelias, A stochastic programming approach for clinical trial planning in new drug development, Computers & Chemical Engineering 32 (2008) 2626–2642. [14] M. Colvin, C. Maravelias, Modeling methods and a branch and cut algorithm for pharmaceutical clinical trial planning using stochastic programming, European Journal of Operational Research 203 (2010) 205–215. [15] G. Gatica, L. Papageorgiou, N. Shah, Capacity planning under uncertainty for the pharmaceutical industry, Chemical Engineering Research and Design 81 (2003) 665–678. [16] E. George, N. Titchener-Hooker, S. Farid, A multi-criteria decision-making framework for the selection of strategies for acquiring biopharmaceutical manufacturing capacity, Computers & Chemical Engineering 31 (2007) 889–901. [17] F. You, I. Grossmann, Balancing responsiveness and economics in process supply chain design with multi-echelon stochastic inventory, AIChE Journal 57 (2010) 178–192. [18] P. Kelle, J. Woosley, H. Schneider, Pharmaceutical supply chain specifics and inventory solutions for a hospital case, Operations Research for Health Care 1 (2012) 54–63.

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