Integrated Planning and Optimization of Clinical Trial Supply Chain

Article pubs.acs.org/IECR

Integrated Planning and Optimization of Clinical Trial Supply Chain System with Risk Pooling Ye Chen, Joseph F. Pekny, and Gintaras V. Reklaitis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, United States ABSTRACT: Clinical trials constitute an expensive part of the new drug development process, and thus, pharmaceutical companies are seeking ways to reduce this cost. The objective of clinical trial supply chain management is to ensure sufficient drugs for the volunteers while minimizing operational cost. Given the uncertainties in patient participation and completion of the treatment regimen, it is necessary to maintain some level of safety stock in the system, thus increasing drug inventories. However, drug packages leftover at the end of the trial must be disposed according to FDA regulations and thus constitute an important cost. In this paper, we use a simulation−optimization computational framework, which builds on our earlier work,1 to determine the appropriate level of pooled safety stock levels. The entire framework includes demand forecasting, planning and optimization, discrete event simulation, and an outer loop optimization. A case study is presented to demonstrate the utility of this proposed approach.

1. INTRODUCTION Clinical trials constitute an important and expensive part of the new drug development process, which is characterized by high failure rate, long processing time, and unpredictable enrollment of volunteers. Normally, multiple treatments (placebo, real drug, and comparator) are used in each clinical trial, and sometimes clinical trials with different testing objectives (e.g., safety, efficacy, side effects) are conducted at the same time. There is often competition for the available production and operational resources which requires resolution through effective planning and scheduling. Therefore, it is important to have a clinical trials management process which ensures minimum operation cost under constrained resources while accommodating the uncertainties inherent to the system. Given the demand uncertainties in the clinical trial, maintaining a certain level of safety stock is important to ensure there are enough inventories when a volunteer patient arrives for treatment. However, according to FDA regulations, unused materials at the end of clinical trials, called leftovers, must be disposed and cannot be recycled or reshipping to other sites, constituting a large cost for a pharmaceutical company. A key technical challenge in managing a clinical trial supply chain is how much safety stock and at what point in the supply chain should it be kept, to satisfy the stochastic demands of clinical sites and minimize total operation cost. A number of different approaches are being pursued to reduce clinical trial costs. Kimko and Duffull2 reviewed the state of the art of modern clinical trial simulation. Interactive voice response (IVR) systems have been introduced to manage the clinical trial drug delivery process,3 and the key parameters, such as replenishment trigger and supply levels, are either determined by experience or simulation techniques.4 However, these approaches do not constitute an actual optimization strategy since the simulation outcome is not used to modify the prior strategies or to make systematic improvements to the candidates in the strategy pool. Abdelkafi et al.5 are the first to use optimization techniques in the clinical trial supply chain management area. However, this contribution only focuses on © 2012 American Chemical Society

the drug supply subsystem of the clinical trial supply chain without considering the production subsystem. Clearly it is the interplay of both subsystems that determine the performance of the supply chain. The integrated clinical trial management problem is composed of the planning and scheduling of all activities, transactions, operations, and organizations during a clinical trial, beginning with active ingredient manufacturing, followed by drug product manufacturing and distribution to the clinical sites, and ending with dispensing of the drugs to patients at the clinical sites. Although this problem has not received much systematic study to date, it is related to conventional supply chain management as developed for the chemical process industries (CPI). It encompasses traditional process system engineering (PSE) issues such as the operational planning and scheduling of batch process under resource constraints and uncertainties, including the assignment of available resources, determination of the order, as well as the amount and timing of the drug products to be produced. Schmidt and Grossmann6 developed a MILP model to solve the resource-constrained scheduling problem for testing tasks in the new product development, in which uncertainties exist in the cost factors, success probabilities, and task durations. Lainez et al.7 presented an enhanced S-graph framework to handle a stochastic scheduling problem for a batch process under uncertainties. These representative references provide useful tools for planning and scheduling of batch manufacturing processes; however, most deterministic approaches deal with uncertainties in the planning models by using expected values of uncertainties to generate period plans to buffer the effects of uncertainties. The horizon of a clinical trial supply chain is only 1−2 years, at which point the trial is terminated. The horizon of Special Issue: L. T. Fan Festschrift Received: Revised: Accepted: Published: 152

March 28, 2012 July 29, 2012 July 31, 2012 July 31, 2012 dx.doi.org/10.1021/ie300823b | Ind. Eng. Chem. Res. 2013, 52, 152−165

Industrial & Engineering Chemistry Research

Article

Figure 1. Typical clinical trial supply chain.

Figure 2. Computational framework.

tion of risk pooling strategy at the supply chain tactical level, which is one point of this paper. Our work provides improved and integrated management of the entire clinical trial supply chain by using a simulation− optimization approach, which includes four modules. These consist of demand forecasting which models the stochastic patient arrival process; an integrated planning and optimization model combing a risk pooling strategy; a discrete event simulation which simulates the entire supply chain; and an outer loop search process to optimize the pooled safety stock levels. This paper extends the Sim−Opt approach1 in three elements: formulation and solution of the integrated planning model, determination of optimum safety stock levels, which balances demand satisfaction and inventory cost, including cost of leftovers, and consideration of pooling of safety stock.

traditional commercial supply chains can extend to 10 years or more, and there is no explicit point at which the supply chain is terminated and residual materials in inventory need to be discarded. Therefore, the strategy used to buffer uncertainties in commercial supply chains become ineffective as expected values cannot be effectively used as targets. Traditionally, a suitable level of safety stock is kept in a supply chain system to attain sufficiently high level of customer service level (CSL)8 under demand uncertainties. Benton9 examined the impact of uncertainty in a periodic review manufacturing environment and showed that increasing safety stock can help hedge uncertainties and provide promising service levels. Although safety stocks do increase customer service levels, this inventory does incur a holding cost. Therefore, managing inventory, especially safety stocks, is another challenging task for clinical trial supply chain management. Lagodimos and Anderson’s10 work was the first attempt to address the safety stock position problem for a two stock-point inventory system. Risk pooling strategies, under which inventory is kept at a centralized distribution center, called the pooled inventory, instead of distributing it among several retailers, constitute another approach to balancing this trade-off. Eppen11 first compared the expected holding and penalty costs in a centralized supply system with a decentralized system and demonstrated the effect of risk pooling. You and Grossmann12 also proposed a nonlinear programming model (MINLP) to simultaneously consider inventory optimization and supply chain network design under demand uncertainty. However, most work reported in the risk pooling strategy area is confined to the supply chain strategic level, and very few references discuss the implementa-

2. PROBLEM DESCRIPTION 2.1. Production−Inventory−Distribution Clinical Trial Supply Chain. Following the work of Chen et al.,1 the clinical trial supply chain management problem can be viewed as a multiechelon production−inventory−distribution supply chain which begins with the manufacture of the active pharmaceutical ingredient (API), continues with new drug production (NDP) by adding “excipients” into active ingredient to produce bulk drug product, followed by packaging and labeling (PL) to get the final product form, which is delivered to clinical sites, using a distribution network (see Figure 1). 2.2. Activities at the Clinical Sites. When a specific clinical trial is initiated, patients are enrolled to take part at clinical sites all over the world. Unqualified patients are excluded for reasons, such as low weight, pregnancy, and so on, through a screening 153

dx.doi.org/10.1021/ie300823b | Ind. Eng. Chem. Res. 2013, 52, 152−165


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Figure 3. Computational flowchart.

Stochastic demand forecasting is an important and challenging part. Unfortunately there is not much literature available on forecasting this type of demand in detail. Some of the literature simply represents the stochastic demands by nonstationary normal distributions, and their expected values are directly used in the planning model.13 Other papers suggest forecasting the future demands based on historical data using time series techniques. However, neither of these two approaches is applicable in this study since it is difficult to get real data for a new drug that is to be clinically tested. Additionally a normal distribution is typically not adequate to represent the stochastic demand in clinical trials, which typically has been found to follow a bimodal distribution. Therefore, a simulation model which mimics the patient enrollment process was developed to determine the number of drug packages needed at the clinical sites for that trial. This model is discussed in detail in the work of Chen et al.1 With forecasted demands, available resources, and preset safety stock levels, the planning and optimization block involves formulating and solving a mixed-integer-linear-program (MILP) using the commercial software CPLEX 12.114 and the modeling software GAMS 23.215 to determine the manufacturing campaign and shipping plans. The discrete event simulation model of the entire supply chain, built using simulation software ExtendSim 7,16 captures all activities, operations, and processes involved in the clinical trial. To reduce the computation burden, the planning and simulation blocks are executed in the rolling horizon mode for each timeline, in which uncertain parameter

process. The screening process is not considered in this work: we assume that all the patients who arrive are qualified to participate in the trial. Each arrived patient is randomly assigned different clinical treatments (target drugs, placebo or comparator) with fixed treatment period: according to the clinical study design. During the treatment period, the medication kit is split into smaller dispensing packages and only one package is delivered to patients at their monthly or weekly visits. Drug inventories are kept at each clinical site to satisfy the needs of arrived patients. If there is a drug inventory shortage once a patient is enrolled or revisits to get a drug package for his/her next test period, that patient is lost to the trial. Only the patients that successfully finish the entire clinical treatment can be included in the trial data set; the others must be excluded from trial results analysis.

3. SIMULATION-BASED OPTIMIZATION APPROACH 3.1. Computational Framework. A simulation-based optimization computational framework is developed for this study to accommodate the uncertainties and reduce computational complexity of clinical trial supply chain management. The entire computational framework, shown in Figure 2, includes demand forecasting (simulation of the stochastic patients arrival process), integrated planning and optimization (solution of a MILP programming based planning and scheduling problem) with risk pooling strategy, a discrete event simulation model to simulate and evaluate the performance of the entire supply chain, and an outer loop search process to optimize the safety stock levels. 154



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Figure 4. Aggregation of unit operations in each stage.

solved with less CPU time. However, the resulting plan will in general be suboptimal. Moreover conflicts can arise among the solutions of the different submodels since each is optimized according to its own constraints. Therefore, an integrated planning model is investigated in this work, as described in the following sections. 3.2.1. Problem Statement and Assumptions. In this study the structure of the supply chain network in terms of the echelons involved is assumed to be fixed. We assume several manufacturing plants located in the US. All finished drugs (target drugs, placebo, and comparator) will be kept in the same distribution center in the US. The products have limited shelf life and must be destroyed upon expiry. The following assumptions are made to simplify the formulation of the MILP model. • Each stage, such as API, NDP, and PL, of the manufacturing process involves several unit operations or steps. However, to simplify the planning and optimization model, the sequence of operations in each stage are aggregated and treated as one task, which is conducted on an assembled production line called a process. Furthermore, the processing time and output amount are fixed and determined according to the combined operations. • The combined tasks are operated in campaign mode, where each campaign consists of multiple batches. As shown in Figure 4, unit operations of mixing, reaction 1 and reaction 2, separation, and drying are aggregated into one task conducted in the batch processing mode, where the batch processing time is represented by the cycle time of these combined unit operations. • Each campaign requires a set up time, which is determined by the process type. • The starting and ending time of each campaign should be in one time period, with no overlap allowed. • Unlimited raw materials supply is assumed and different types of intermediate materials and final products share the same inventory space with specific volume coefficients. • The unlimited intermediate storage (UIS) policy is used as the transfer policy between production stages, since the volume of drugs used in a clinical trial is small compared to commercial drug demand, so the intermediate materials could be stored and transferred between API, NDP, and PL stages without any constraints. Moreover, material transportation times between these stages are negligible.

values are obtained by sampling from appropriate distribution functions. To capture the uncertainties and effectively assess the performance of the entire supply chain, here represented by CSL, multiple optimization−simulation timelines are implemented by sampling from uncertain parameters in the Monte Carlo mode. On the basis of the estimated CSL value from the Monte Carlo simulation, an outer loop search algorithm is used to optimize the pooled safety stock levels in an iterative fashion until the optimization objective converges. The flowchart for this computational framework is shown in Figure 3. First, the stochastic demand forecasting part is executed to obtain the statistical demand values, which are used to construct the required forecast demand scenario. Some parameters are specified as inputs to the optimization− simulation process, for instance, the initial safety stock levels and preset rolling horizon mode: planning period p and execution period l. After executing the l-period plan, the system states are recorded as intermediate states and the planning− execution cycle is repeated with the planning horizon advanced l periods. This process is continued until the end of the entire horizon is reached. One such optimization−simulation run is called a timeline, and at the end of each timeline, system performance metrics, in this case the CSL, is computed and a decision is made whether CSL has converged. If not, the optimization−simulation loop is repeated for a new timeline until the CSL converges to a preset target. After the convergence of CSL, safety stock levels are updated by using a direct search algorithm based on the estimated CSL values and a new optimization−simulation inner loop is initiated. Finally, the entire process is terminated when the optimization objective is converged and the target CSL values are achieved. 3.2. Integrated Planning and Scheduling. It is important to determine the order, amount, and timing of the products to be produced since the batch facilities both at the pilot and the commercial scale are shared across different products in the pharmaceutical industry. With given available equipment, inventory capacity, and forecasted deterministic demands, the manufacturing campaign and shipping plans for the supply chain operations are determined using a deterministic MILP formulation. This integrated MILP model optimizes both the production and distribution echelons at the same time with a global objective: minimize the total operation cost. This differs from the decentralized optimization model used in our previous work.1 The decentralized model divides the entire supply chain into several submodels and each is optimized individually. Each submodel includes fewer decision variables and thus can be 155



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Figure 5. STN formulation plot for production and distribution processes.

Figure 6. Campaign formulations in in the integrated model.

• The starting point for the shelf life clock is at the completion time of the packing and labeling stage. • To reduce the shipping cost, all the materials shipped from site f to site f ′ by the same shipping method are combined into one shipment. The major differences of this planning model from traditional supply chain models are as follows:

The uniform discrete time representation is used in this model. As shown in Figure 6, the whole plant is operated in campaign mode. For each production campaign, several batches of task i are produced within each campaign, and the batch processing time of task i (or the processing time of aggregated unit operations) on production line j is a fixed parameter. A setup time is required to start a campaign, which is only determined by the equipment type of a certain site. For a distribution campaign, only one batch is included without setup time and its batch size is variable. The batch processing time is determined by the distance and shipping method and is fixed as a parameter related to the shipping type. Overall, the objective of this planning and optimization model is to minimize the total operation cost while satisfying demand requirements under certain resource constraints. The following data is assumed to be known in advance: • The structure of supply chain network; • A fixed time horizon H; • A set of tasks i; • A set of potential production lines or equipment j which could conduct task i; • Task recipes (mass balance coefficients, setup time, processing time, input and output material state); • Upper bounds of production capacity of equipment or production line j; • A set of clinical sites, in which products are provided to patients, and their forecasted demands scenarios Dsf t; • Shelf life sls for material s; • Inventory capacity of the different sites; • Safety stock level of material s at site f; • Cost parameters such as production, holding, shipping, and penalty costs for unsatisfied safety stocks and unsatisfied demands. The goal is to determine:

• Demands are not repeated over many consecutive periods, and there is no pattern for demands. • Product shelf life is tracked and considered in the planning model, and expired products must be disposed. • Leftovers are be treated as wastage since materials in one clinical site cannot be reshipped to another site or reused at the end of clinical trials according to FDA regulations. 3.2.2. Integrated MILP Planning Model. An integrated MILP planning model is formulated using the state−task network (STN) representation.17 As shown in Figure 5, the production and distribution processes are each aggregated and treated as a combined task i, which is conducted on equipment or production line j. The difference between the production and distribution tasks are that production tasks only occur at one site, and their input and output materials are different; while the distribution task relates to two sites (start place and destination), and its input and output materials are of the same type. Thus, variable Pijft, is used to represent the production amount of task i performed in production line j in site f during time period t; while the corresponding decision variable related to the distribution process is Psjf f′te, which represents the amount of state s shipped by method j from site f to site f ′ during time period t which will expire at e. One more index e is added to the distribution variable since an expiration date must be considered after the packaging and labeling stage and before the shipping operations. 156



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• Whether there is a production or distribution campaign started in time period t; • The start time and duration of each campaign; • The total number of batches manufactured in each production campaign, for the predetermined batch size; • Production amount and shipping amount of each campaign type during time period t. The detailed MILP formulation is outlined in the following section: the terms can be found in the Nomenclature section. A. Objective. Minimize the total operation cost, including the setup cost, processing cost, inventory cost, wastage cost, penalty cost for unsatisfied safety stock levels, and penalty cost for missed demand at the clinical sites. min TC =

Ssft − Ssft − 1 = R sft −

i ∈ Is j ∈ Ji

∀ s ∈ SRM, f ∈ FP, t > 1

(2). Material Balance for Intermediate States in the Production Sites. The inventory of material s at the end of time period t is equal to its inventory at the end of time period t − 1 (its initial inventory when t = 1), plus the produced amount and minus the amount consumed to produce other material. Also, the amount of material s consumed by tasks should be less than the previous inventory of this state, since we assume the consuming tasks start at the beginning of time period t and there are should be enough inventory to enable this consuming task. ⎧ S − SINI = ∑ ∑ α P ̅ ijft sf sij ⎪ sft i ∈ Is̅ j ∈ Ji ⎪ ⎪ − ∑ ∑ αsijPijft , ⎪ i ∈ Is j ∈ Ji ⎪ ⎨ ∀ s ∈ SIS, f ∈ FP, t = 1 ⎪ ⎪ ⎪ ∑ ∑ αsijPijft ≤ SINIsf , ⎪ i ∈ Is j ∈ Ji ⎪ ∀ s ∈ SIS, f ∈ FP, t = 1 ⎩

∑ ∑ ∑ ∑ ∑ CsrawαsijPijft s ∈ SRM i ∈ Is

+

∑∑∑ i

+

j

∑ Pijft )

f

t

f

f′

f

t

C jffdp′(

f′

∑ ∑ ∑ PEsjff ′te) s

t

⎧S − S ̅ Pijft − ∑ ∑ αsijPijft , sft − 1 = ∑ ∑ αsij ⎪ sft i ∈ I j ∈ J i ∈ Is j ∈ Ji s i ⎪ ⎪ ⎪ ∀ s ∈ SIS, f ∈ FP, t > 1 ⎨ ⎪ ∑ ∑ αsijPijft ≤ Ssft− 1, ⎪ i ∈ Is j ∈ Ji ⎪ ⎪ ∀ s ∈ SIS, f ∈ FP, t > 1 ⎩

e

f

t

∑ ∑ Csfw(∑ Esft + SsfH) f

t

d ∑ ∑ ∑ [Cpenalty (Dsft − SDsft ) s

+

Cijfp (

∑ ∑ Csfh (∑ Ssft ) s

+

∑ xijft ) t

∑ ∑∑ s

+

t

∑ ∑ ∑ C jffds ′(∑ XDjff ′ t ) j ∈ Jdc

+

Cijfs (

∑∑∑

j ∈ Jdc

+

f

f

i ∈ Ip j ∈ Jp

+

j

f

t

ss Cpenalty (SSf

(3). Material Balance at the Packaging Stage (Also the Main Distribution Center). Shelf life consideration starts after this stage, and the risk pooling strategy is applied here if this main distribution center has downstream clinical sites. So the distribution center and its downstream clinical sites are combined as one node, and the demands from its downstream clinical sites should be satisfied at this distribution center. • If e = t + SLs, only the newly produced material at time period t will expire at time e = t + SLs. Consequently the inventory of material s in site f which will expire at e = t + SLs at time period t is equal to the sum of new production amount in time period t of all tasks i which produce material s.

− SINVsft)]

The first term represents the raw material cost. The second term represents the fixed setup cost associated with starting a campaign for task i on production line j in site f, and the third term is the variable cost associate with production amount in that campaign. The fourth and fifth terms represent the fixed and variable shipping costs. The sixth term represents the holding cost, and sixth term is associated with the wastage cost for expired products and leftovers at the end of the entire horizon. The eighth and ninth term represent the penalty cost for missed demand and unsatisfied safety stock requirements. Several constraints, shown below, are included in this MILP planning model to describe the entire supply chain, including the production and distribution part. B. Constraints. a. Material Balance. (1). Material Balance for Raw Material. Inventory of raw material s at the end of each time period t is equal to its inventory at the end of time period t − 1 (its initial inventory when t = 1), plus the purchased amount of material s at time period t and minus the amount of material s consumed in conducting task i in this time period. Ssft − SINIsf = R sft −

∑ ∑ αsijPijft ,

Ssfte =

∑ ∑ αsij̅ Pijft ,

∀ s ∈ SFP, f ∈ FP, t

i ∈ Is̅ j ∈ Ji

• If e − SLs < t < e, no newly produced material satisfies the expiry condition but consumption of this material does occur. So the inventory of material s at the end of time period t which will expire at e equals to the inventory of material expiring at e at the end of time period t − 1, minus the amount consumed in conducting task i which uses material s as its input material, and minus the demand satisfied. Also, the total consumed amount and satisfied demand should be less than the available inventory at the end of previous time period t − 1.

∑ ∑ αsijPijft , i ∈ Is j ∈ Ji

∀ s ∈ SRM, f ∈ FP, t = 1 157



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⎧ S − SINI = − ∑ ∑ ∑ P sfe ijff ′ te − SDsfte , ⎪ sfte f i I j J ′ ∈ ∈ s i ⎪ ⎪ ⎪ f ∈ FP, e − SLs < t < e , t = 1 ⎨ ⎪ ∑ ∑ ∑ Pijff ′te + SDsfte ≤ SINIsfte , ⎪ f ′ i ∈ Is j ∈ Ji ⎪ ⎪ ∀ f ∈ FP, e − SLs < t < e , t = 1 ⎩

t + SLs

Ssft ≥ STsf ,

0≤

∑ υsSsft ≤ Cf ,

∀ t < H, f

s

d. Campaign Output Constraints. (1). Production Part. Total produced amount of the campaign for task i on production j in site f, Pijft, should be less than or equal to its batch size Bif t multiplied by the number of batches in that campaign Nijf t. The constraint of Pijf t ≥ Nijf t is added which forces Nijft to zero when Nijf t is zero; otherwise, Nijf t will not be zero since it is not included in the objective function.

∀ s ∈ SFP, f ∈ FP, t < e − SLs

Nijft ≤ Pijft ≤ Bijf Nijft , Xijft < Nijft ,

∀ i ∈ Ip , j ∈ Ji , f ∈ FP, t

∀ i ∈ Ip , j ∈ Ji , f ∈ FP, t

(2). Distribution Part. Total shipped amount from site f to f ′ at time period t should be less than its upper bound. Also, only the products within their shelf life could be shipped to their downstream sites.

⎧ S − SINI = ∑ ∑ ∑ P sfe ijf ′ fte − SDsfte , ⎪ sfte f ′ i ∈ Is j ∈ Ji ⎪ ⎪ ∀ s ∈ SFP, f ∈ FCS, e − SL < t < e , t = 1 s ⎨ ⎪ ⎪ SDsfte ≤ SINIsfe , ⎪ ∀ s ∈ SFP, f ∈ FCS, e − SL < t < e , t = 1 ⎩ s

t + SLs

XDjff ′ t ≤

∑ ∑ ∑ Pijff ′te ≤ PCUjff ′tXDjff ′ t , s

e = t + 1 i ∈ Is

∀ s ∈ SFP, j ∈ Jdc , f ′ ∈ FDC, t

e. Batch Processing Mode Constraints. (1). Production Line Capacity. Each production line has its own operational capacity, so the total processing time for the campaigns conducted on this production line should satisfy its operation capacity Tift in each time period t. The processing time of each campaign includes its setup time xijf tTsjf if there is a campaign started in this time period and its processing time for batch processing.

⎧S − S sf , t − 1, e = ∑ ∑ ∑ Pijf ′ fte − SDsfte , ⎪ sfte f ′ i ∈ Is j ∈ Ji ⎪ ⎪ ∀ s ∈ SFP, f ∈ FCS, e − SL < t < e , t > 1 s ⎨ ⎪ ⎪ SDsfte ≤ Ssf , t − 1, e , ⎪ ∀ s ∈ SFP, f ∈ FCS, e − SL < t < e , t > 1 ⎩ s

∑ (xijftT jfs

(5). Satisfied Demand Constraints. Total satisfied demand of material s in site f in time period t should be less than total forecasted demands Dsf t. Here Dsf t is the aggregated demand at the distribution center from the demands of its downstream clinical sites.

+ Tijfp Nijft ) ≤ Tjft ,

∀ j ∈ Jp , f ∈ FP, t

i ∈ Ij

(2). Batch Number Constraint. The number of batches in each campaign should satisfy a specified upper bound, shown as follows. Xijft < Nijft ≤ NBmax Xijft ,

t + SLs

∀ s ∈ SFP, f ∈ F , t , e > SL(s)

∀ i ∈ Ip , f ∈ FP

Overall, this integrated MILP planning and scheduling model is formulated for the entire clinical trial supply chain by making some assumptions which help to simplify the model, and allow it to be solved in reasonable time. The scale of this model is related to the number of material s, number of task i, number of production line j, number of site t, and number of time period t. The production campaigns and distribution plans obtained by solving the MILP serve as the driver for the simulation model. 3.3. Discrete Event Simulation. A simulation model of the entire supply chain was constructed using the discrete event simulation software ExtendSim. The details of this model are discussed in Chen et al.1 Monte Carlo simulation is implemented

e=t+1

b. Shelf Life and Expired Material Calculation. Inventory of material s at site f, whose expiration date is t should be counted as expired material at the end of time period t. Esft = Ssftt ,

∀ s ∈ SFP, f ∈ FCS, t ≤ H

(3). Storage Capacity Constraint. Total amount of material s stored in location f at any time period t should not exceed the maximum storage capacity Cf.

(4). Material Balance at the Intermediate Distribution Sites. The risk pooling strategy is also applied here by combining this intermediate distribution center and its downstream clinical sites together and maintaining the safety stock at this node. The material balance relationship is similar to the material balance in the main distribution center by considering production, consumption, and satisfied demands.

SDsfte ≤ Dsft ,

∀ s ∈ SFP, f , t ≤ H

(2). Safety Stock Constraint. Safety stock constraints are imposed at the main distribution center and intermediate distribution centers when the risk pooling strategy is applied.

• If t < e − SLs, no such material exists since the maximum shelf life of material s is SLs.

∑

Ssfte ,

e=t+1

⎧S − S sft − 1, e = − ∑ ∑ ∑ Pijff ′ te − SDsfte , ⎪ sfte f ′ i ∈ Is j ∈ Ji ⎪ ⎪ s SFP, f FP, e − SLs < t < e , t > 1 ∀ ∈ ∈ ⎪ ⎨ ⎪ ∑ ∑ ∑ Pijff ′te + SDsfte ≤ Ssf ,t − 1,e , ⎪ f ′ i ∈ Is j ∈ Ji ⎪ ⎪ ∀ f ∈ FP, e − SLs < t < e , t > 1 ⎩

Ssfte = 0,

∑

Ssft =

∀ s ∈ SFP, f , t ≤ H

c. Inventory Calculation. (1). Inventory Balance. Inventory of material s in site f at the end of time period t is the sum of all the inventories of material s in site f which is within its expiration date. 158



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Figure 7. Risk pooling strategy analysis.

the normal progression of the simulation, we will thus find that the safety stocks kept at the distribution center will never be used since all the distribution operations are determined in the planning model to satisfy the forecasted deterministic demands. To resolve this conflict, a pull supply component must be added into the simulation model between the distribution center and its downstream clinical sites, which will serve to replenish clinical site inventory from the distribution center to buffer against the demand uncertainty. To accomplish this, the (Q, r) policy was inserted into the simulation model at the clinical sites. Initially, Q and r parameters are set according to the model proposed in Brandimarte and Zotteri,18 shown in eq 2, where E (demand) is the expected value of demand and STD (demand) is the estimate of the standard deviation of the demand. Given these initial value, a set of initial simulations are conducted to ensure that the selected r value is large enough to cover the demands over the shipment lead time but not too large, because otherwise, the shipment order will be initialized too frequently and the shipping cost will be large. The Q value must be large enough to ship the inventories at the intermediate distribution center to the final clinical sites in time. On the basis of our case studies, eq 2 provides good estimates of these two values although for some cases small adjustments (1% to the estimated Q and r values) were beneficial.

by sampling from the uncertain parameters. In this simulation model, the final tracked simulation result is customer service level (CSL), as defined in eq 1. CSL =

∑f ∑s ∑t satisfied_demands(s , f , t ) total_demand

(1)

3.4. Implementation of Risk Pooling Strategy. As mentioned earlier, centralized safety stocks, a form of risk pooling strategy, could reduce the total inventory of the entire supply chain and thus the leftovers for a clinical trial supply chain. Considering the high cost of clinical trials, especially the leftover cost, this strategy is potentially useful providing it does not significantly reduce the customer service level. First of all, drugs shipped to one clinical site cannot be reshipped to another clinical site. Consequently any oversupply including safety stock at the clinical site would be treated as wastage. Second, leftovers at the end of the clinical trial life must be treated as wastage and disposed, so reducing total safety stocks by using a risk pooling strategy could help reduce the cost of conducting the trial. 3.4.1. Risk Pooling Strategy on the Supply Chain Tactical Level. In the literature, there is not much discussion of how to implement risk pooling on the supply chain tactical level, that is, to address questions such as how much safety stocks should be kept in the distribution center and when and how much safety stock should be allocated to the final retailer to buffer the demand uncertainty. The key challenge in applying a risk pooling strategy within our simulation−optimization framework is how to solve the conflict between “pull” and “push” supply systems, as shown in Figure 7. Under the risk pooling strategy, the distribution center and its downstream clinical sites are treated as one node in the optimization model, where the safety stocks and aggregated external demands are treated as the parameters of this node. Thus, the production plans and distribution plans will be determined for this aggregated node, without reflecting the required internal transfers between distribution center and clinical sites. Under this framework, the production campaign and distribution plans generated by the optimization part, consisting of items such as the shipping time and shipping amount from the production site to the distribution center, serve to drive the simulation. However, in the simulation model, both the distribution center and clinical sites function as independent nodes of the supply chain, and the entire simulation model functions as a push system with the plans as its input or “orders”. Extra safety stocks are only kept in the distribution center as a buffer under the risk pooling strategy, while external stochastic demands are realized at the final clinical sites. Consequently, there is a gap between the inventory buffer and demand uncertainty, which makes the buffering function ineffective. In

⎧ ⎪Q = 2.33 × STD(demand) ⎨ ⎪ ⎩ r = Q × E(demand)

(2)

In this way, the conflict between planning model and simulation model is resolved and a risk pooling strategy can be successfully implemented in the simulation−optimization computation framework. 3.4.2. Safety Stock Optimization in the Outer Loop. Although safety stock is useful to reduce the stock out risk and increase customer service level, it introduces extra holding cost into the supply chain system and thus the safety stock levels must be carefully optimized. This can readily be achieved within the simulation−optimization framework through an outer loop optimization using a suitable direct search algorithm. Given the monotonically increasing relationship between customer service level and safety stock level,13 a direct search algorithm can be effectively used to adjust the safety stock levels, to reduce the deviation between output variables and target performances. Zapata et al.19 observed that although this approach to the adjustment of safety stocks does not consider the constraints related with the embedded planning and optimization model, the monotonic nature of CSL guarantees convergence to a global optimal solution if the estimator of customer service level from 159



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Figure 8. Network of clinical trial supply chain case study.

the simulation model is unbiased. Therefore, a direct search algorithm is capable of optimizing the safety stocks using the simple gradient logic that if CSL improves in a particular direction, then one can continue to seek improvement by moving in that direction. The optimization problem solved in the outer loop can thus be mathematically described by eq 3. The main objective is to minimize the sum of the deviations from the target customer service levels. ⎧ min J(SS) = ∑ |ΔCSL(SSi)| ⎪ i ⎨ ⎪ Tar ⎩CSL(SSi ) = ΔCSL(SSi ) = CSLi

Table 1. Information of Clinical Trial Study duration treatments treatment period clinical sites patient dropout rate

4.1.2. Process Information for the Planning and Optimization. Process information related to the production and distribution parts used in the integrated MILP based planning model is shown in Tables 2 and 3, which includes campaign setup

(3)

Here CSL(SSi) is the estimated customer service level of product i from the current simulation with the current level of SSi safety stock in the system, and CSLTar is the target customer i service level. ΔCSL(SSi) is the deviation of current customer service level from the target customer service level, which is used to update the current safety stock level setting, as shown in eq 4. SSin + 1

=

SSin

+

αinΔCSL(SSi )n

Table 2. Campaign Related Information of Each Stage

(4)

Where αin =

24 month placebo:high dosage drug:low dosage drug:comparator = 1:2:2:2 6 weeks country A (36%), country B (24%), country C (21%), country D (19%) 45%

SSin − SSin − 1 J(SS)n + J(SS)n − 1

The entire search process is driven by the sum of the deviations of target customer service level until the final customer service levels and safety stock levels converge. These converged safety stock levels are the final amount of safety stocks that should be kept in a supply chain system. It should be noted that for given supply chain capacities and demand uncertainties the search may converge but fail to meet the target CSL levels. The CSL levels at convergence will be the highest that the supply chain can achieve under the given system conditions.

setup time

batch processing time

fixed cost

unit

month

month

$/camp

API node NDP node PL node normal dist USDC− EUDC express dist USDC− EUDC normal dist DC−CS express dist DC−CS

0.1 0.1 0.1 0

0.4 0.5 0.3 1

0

variable cost

maximum batches

5000 2500 1250 50

$/unit batch 90 60 3 9

4 4 4 1

0.5

200

15

1

0

0.2

5

1

1

0

0.1

20

4

1

Table 3. Mass Balance Relationship of Each Stage

4. CASE STUDY To demonstrate the utility of the proposed simulation optimization approach, we employ an industrial motivated case problem which has been previously reported.1 We also use this problem to compare supply chain performance with and without risk pooling. 4.1. Case Problem. 4.1.1. Clinical Trial Design. The clinical trial supply chain network and clinical trial parameters from Chen et al.1 are employed in this case study and are shown in Figure 8 and Table 1, respectively.

batch size

input material

API node NDP node

25 kg 25 kg

25 kg 25 kg

PL node

50

70000 drugs

distribution (normal) distribution (express) 160

output material 25 kg 150000 drugs 1500 packages

input coefficient

output coefficient

1 1

1 6000

1400

30

1

1

1

1



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time, batch processing time, fixed cost, variable cost of production, mass balance relationship, and so on. For the distribution part, the shipping time and cost from the US distribution center to EU distribution center are higher than they are from distribution center to the corresponding clinical sites. Drug shelf life used in the following case studies is 8 months, and the total enrollment period is 24 months. 4.2. Results. 4.2.1. Performance of Planning and Optimization Model. Parameters used in the rolling horizon strategy are a 4 month planning period, where only the plans of the first month are executed. The final performance metrics such as amount of drug leftovers and CSL are recorded at the end of the 24 month horizon. A comparison of the performance of the integrated MILP planning model when executed in a single step for entire 24 month horizon and when executed using the 4 month rolling horizon mode are shown in Table 4. As expected,

Figure 9. Combined demand scenarios used in the planning model.

Figures 10 and 11 show the progression of the outer loop optimization process for pooled safety stock levels at the US distribution center and EU distribution center. The objective values also are shown on the corresponding figures. We note that after around 13 iterations, the optimization objective starts to converge, and the search process is terminated after 18 iterations. The final safety stocks are the levels that should be maintained in this case. Table 5 records the statistical results of demand satisfaction, which includes both the missed demand and satisfied demand for each type of drug in every country. Table 6 contains the corresponding customer service level based in Table 5. It is evident that all the final estimated CSLs are close to 90%, the target CSL we set for this case study. With the closed loop computational framework, which includes demand forecasting, planning and optimization, simulation, and outer loop optimization, we could find reasonable safety stock values for the preset target CSL. Table 7 shows the final total leftover amount (including expired material and oversupplied drugs within their shelf life) and the amount of expired drugs. From this table, we can observe that the expired drugs only constitute a small part of total leftovers, and most leftovers are the due to oversupply. 4.2.3. Comparison of Risk Pooling Case with Nonrisk Pooling Case. In this section, case studies with risk pooling strategy and without risk pooling strategy are compared to demonstrate the utility of the risk pooling strategy. In the risk pooling case study, safety stocks are only kept in the US distribution center and EU distribution center and optimized in the outer loop; while in the nonrisk pooling case study, safety stocks are kept in the final four countries. The search results show that the target 90% CSL cannot be satisfied at all these four countries at the same time by increasing the safety stock. This occurs because of production resource constraints. Specifically, the bottleneck exists in the production of drug 1 and drug 2, which require the material from the API process as their input material. To compare with risk pooling strategy with the same production resources, the target CSL is relaxed from 90% to 85%. The results of the outer loop optimization on safety stocks in four countries are shown in Figure 12. We find that after 15 iterations, safety stock levels do converge. Table 8 shows the comparison of leftovers and safety stocks obtained in the outer loop optimization for the case studies with risk pooling strategy and without risk pooling strategy. For ease of comparison, the safety stocks kept in the supply chain system and leftovers at the end of the clinical trial are combined together

Table 4. Performance of MILP Planning Model total variables integer variables iteration number relative gap execution time

whole horizon

rolling horizon (4:1)

66076 1584 47242 0.90% 33.434 s

24170 392 160 0.65% 2.328 s

under the rolling horizon mode, the planning model could be solved much faster and with fewer decision variables compared to one-pass solution of the full 24 month model. 4.2.2. Results of Case Study with Risk Pooling Strategy. When applying the risk pooling strategy on the proposed clinical trial supply chain network, country A and country B are combined with the US distribution center as one node, and country C and country D are combined with the EU distribution center as one node in the optimization model. Safety stocks are kept in these two combined nodes to serve as the buffers for the stochastic demands from its downstream clinics. Their levels are optimized via the outer loop direct search. The variation of demand (Cv) used in this case study is 0.8. To reduce the final leftovers and wastages, only the lowest forecasted enrollment rate demand scenario is used in the planning model, which is discussed in the work of Chen et al.1 In this case, the demands of country A and B are combined together to be satisfied from the US distribution center, and likewise the demands of country C and D are combined to be satisfied from the EU distribution center. Combined demands of placebo and drug 1 in US distribution center and EU distribution center are shown in Figure 9. Monte Carlo simulation is applied by sampling from the distributions of the uncertain parameters to compile the final statistical simulation results. The replications continue until the relative error of CSL is less than or equal than 2%. At the end of each set of Monte Carlo simulations, the estimated expected value of CSL for each drug is obtained and used to initiate the next outer loop optimization step. The target CSL set for this case study is 90%. The initial safety stock of each drug type kept in both US and EU distribution center is 10 packages. The outer loop optimization stops when conditions 5 are satisfied. |J(SS)n − J(SS)n − 1| ≤ 0.03 |ssns − ssns − 1| ≤ 5,

s = 1...8

(5) 161



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Figure 10. Optimization of pooled safety stock in US distribution center.

Figure 11. Optimization of pooled safety stock in EU distribution center.

Table 5. Demand Satisfaction (Drug Packages) country placebo low dosage high dosage comparator

missed satisfied missed satisfied missed satisfied missed satisfied

Table 7. Leftovers and Expired Drug Packages

A

B

C

D

drug type

total leftovers

expired drug package

190 3569 781 5770 754 5948 476 6900

117 2683 539 4314 508 4336 303 5207

167 2119 449 3731 476 3678 420 3844

153 1938 413 3493 360 3507 440 3320

US placebo US drug 1 type US drug 2 type US comparator EU placebo EU drug 1 type EU drug 2 type EU comparator

1840 1062 1036 1078 3068 1592 1901 2987

46 5 4 11 375 11 6 22

Table 6. Customer Service Level country

A

B

C

D

placebo dose 1 dose 2 comparator

94.9% 88.1% 88.7% 93.5%

95.8% 88.9% 89.5% 94.5%

92.7% 89.3% 88.5% 90.2%

92.7% 89.4% 90.7% 88.3%

converged amount of safety stock setting. Compared to the risk pooling case, shown in Table 6, lower CSL is achieved by these converged safety stock settings, which are even higher than the safety stock amount in risk pooling case. From these two comparison tables, we can conclude that the level of total safety stocks could be reduced by applying risk pooling strategy, resulting in fewer leftovers at the end of clinical trials.

5. CONCLUSION A simulation-based optimization computational framework for the determination of safety stock levels via an outer optimization loop has been presented. Case studies discussed in this paper

based on the same drug type. Even with lower CSL requirement, the nonrisk pooling case requires more safety stocks to be kept in the supply chain, and thus more leftovers remain at the end of clinical trials. Table 9 presents the CSL achieved with the 162



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Figure 12. Outer loop optimization process in different countries.

stocks due to production constraints. This reaffirms that production resources must be carefully managed to achieve the highest CSL. The simulation based framework presented in this work provides a very flexible tool for balancing financial factors against customer service levels. Moreover, the Monte Carlo treatment of the various uncertain parameters offers a convenient mechanism for dealing with a comparatively large number and variety of these uncertain parameters. The direct treatment of the supply chain problem with a large number of uncertain parameters via the typical scenario based MILP model generally leads to very large dimensionality MILPs which can be challenging to solve. By the same token, the requirements of Monte Carlo sampling of the uncertain system parameters which is sufficient to meet statistical tests can also result in large computation times. The crossover point at which stochastic MILP becomes preferable to the Sim−Opt approach is difficult to assess a priori. One straightforward approach which can be used to reduce the Sim− Opt computational burden is to execute the different timelines in parallel on multiple processors. Because of minimal communication overheads, the clock time saved will be proportional to the number of processors used. The use of the proposed framework does allow some other beneficial studies, such as exploration of the specific relationship between the effect of risk pooling strategy and shipping lead time. Simulations with different shipping lead time and the risk pooling effect could be readily conducted, which will be helpful in making decisions on whether it is necessary to conduct risk pooling and where the pooled safety stocks should be kept in the supply chain system. Moreover, the proposed integrated framework also

Table 8. Comparison of Leftovers and Safety Stocks with risk pooling

nonrisk pooling

drug type

leftovers

safety stock

leftovers

safety stock

US placebo US drug 1 type US drug 2 type US comparator EU placebo EU drug 1 type EU drug 2 type EU comparator

1839.7 1062.2 1036.1 1078.4 3068.2 1592.35 1900.9 2986.9

0 83 60 10 384 913 934 323

2518.7 4004.7 3653.6 3927.1 3253.8 3829.9 3297.9 3720.4

281 1856 1977 556 253 2039 2275 553

Table 9. CSL of Nonrisk Pooling Case with Converged Safety Stocks country

A

B

C

D

placebo dose 1 dose 2 comparator

88.7% 84.1% 84.4% 86.8%

87.1% 84.2% 85.6% 85.6%

87.1% 84.7% 87.7% 87.3%

87.1% 84.8% 88.2% 85.8%

demonstrated the utility of this simulation−optimization computation framework to handle the features of the clinical trial supply chain management problem. By applying the risk pooling strategy at the supply chain tactical level, the total safety stocks kept in the system to hedge the demand uncertainties and their holding cost could be reduced, compared to the nonrisk pooling strategy. Moreover, the case study demonstrates that arbitrary CSL levels may not be achievable even with high safety 163



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provides considerable flexibility in addressing interesting design problems, such as the selection of clinical sites based on historical performance in patient enrollment and trial completion as well as the development of incentives to improve patient persistence to the end of the trial.

■

Parameters

Tp = length of one planning time period Rsft = external purchase of raw material state s in location f during time period t SINIsf SINIsfe = initial inventory of state s (which will expire at e) in location f NBmax = maximum number of production batches of task i in one campaign performed in equipment j in location f TPijf = fixed batch processing time of production task i performed on equipment j in location f Bijf = batch size of production task i on equipment j in site f during time period t (irrelevant to time period t) PUCiff ′t = upper bound of shipping capacity for method j from f to f ′ in time period t Tsjf = setup time for a campaign started on equipment j in site f Tjf t = total available processing time of equipment j in site f during time period t SLs = shelf-life time of state s (for final products) Cf = total storage room in site f STsf = safety stock of state s (for final products) in site f

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

NOMENCLATURE

Indices

i = tasks j = production line (equipment) or shipping method: express or normal speed s = states (raw material, intermediate material and final product) f, f ′ = different sites in supply chain t = planning periods e = expiration time period

Cost Coefficients

Craw s = cost of raw material per amount Csijf = setup cost for a campaign CPijf = unit production cost for a campaign Cds sif f′ = start cost for a shipment of method j from f to f ′ Cdp jf f′ = unit cost for a shipment of method j from f to f ′ Chsf = holding cost Cwsf = waste cost Cdpenalty = penalty cost for unsatisfied demand Csspenalty = penalty cost for unsatisfied safety stock requirement

Sets

Is = set of tasks producing state s Is = set of tasks consuming state s Ij = set of tasks could be conducted on equipment j (production line) If = set of tasks could be conducted in site f Jf = set of equipment that can be implemented in site f Ip = production tasks Jp = set of production lines Jdc = set of distribution methods (express or normal) FP = set of production sites (including the packaging and labeling sites) FDC = set of intermediate distribution sites SRM = set of raw material states SIS = set of intermediate states

Greek Symbols

■

αsij = the mass fraction coefficients of state s for task i consumed in equipment j α̅sij = the mass fraction coefficients of state s for task i produced by equipment j υs = specific volume of state s

REFERENCES

(1) Chen, Y.; Mockus, L.; Orcun, S.; Reklaitis, G. V. SimulationOptimization Approach to Clinical Trial Supply Chain Management with Demand Scenario Forecast. Comput. Chem. Eng. 2012, 40, 82−96. (2) Kimko, H. C.; Duffull, S. B. Simulation for Designing Clinical Trials; Marcel Dekker: New York, 2003. (3) McEntegart, D.; Barry, O. G. The Impact on Supply Logistics of Different Randomization and Medication Management Strategies Using Interactive Voice Response (IVR) Systems. Pharma. Eng. 2005, No. September/October, 36−44. (4) Peterson, M.; Byrom, B.; Dowlman, N.; McEntegart, D. Optimizing clinical trial supply requirements: simulation of computercontrolled supply chain management. Clin. Trials 2004, 1 (4), 399−412. (5) Abdelkafi, C.; Beck, B.; David, B.; Druck, C.; Horoho, M. Balancing Risk and Costs to Optimize the Clinical Supply ChainA Step Beyond Simulation. J. Pharma. Innov. 2009, 4 (3), 96−106. (6) Schmidt, C. W.; Grossmann, I. E. Optimization Models for the Scheduling of Testing Tasks in New Product Development. Ind. Eng. Chem. Res. 1996, 35 (10), 3498−3510. (7) Lainez., J. M.; Hegyhati, M.; Friedler, F.; Puigjaner, L. Using Sgraph to address uncertainty in batch plants. Clean Technol. Environ. Pol. 2010, 2 (12), 105−115. (8) Stevenson, W. J. Production/operations Management; Irwin/ McGraw-Hill: Boston, 1999. (9) Benton, W. C. Safety Stock and Service Levels in Periodic Review Inventory Systems. J. Oper. Res. Soc. 1991, 42 (12), 1087−1095. (10) Lagodimos, A. G.; Anderson, E. J. Optimal positioning of safety stocks in MRP. Int. J. Prod. Res. 1993, 31 (8), 1797.

Binary Variables

Xijft = 1 if task i starts a production campaign on equipment j in site f during time period t, 0 otherwise XDijf ′t = 1 if there is a shipment from site f to f ′ using shipping method j, 0 otherwise Integer Variables

Nijf t = number of batches of production task i on equipment j in site f during time period t Continuous Variables

Xijft = amount of production task i produced on equipment j in site f during time period t (for production part) PEsif f′te = amount of state s by method j is shipped from site f to site f ′ during time period t which will expire at e (for distribution part) Esft = amount of expired product s in site f at the end of period t Ssft = inventory amount of state s in place f at the end of period t Ssfte = inventory amount of state s which will expire at e in place f at the end of period t (begin from the storage room in packaging and labeling site) (s ∈ Sfp) Dsf t = external demand of state s in site f during time period t SDsfte = demand satisfied of state s in site f during time period t using product expired at time 164



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(11) Eppen, G. D. Effects of Centralization on Expected Costs in a Multi-Location Newsboy Problem. Manage. Sci. 1979, 25 (5), 498−501. (12) You, F.; Grossmann, I. E. Mixed-Integer Nonlinear Programming Models and Algorithms for Large-Scale Supply Chain Design with Stochastic Inventory Management. Ind. Eng. Chem. Res. 2008, 47 (20), 7802−7817. (13) Jung, J. Y.; Blau, G.; Pekny, J. F.; Reklaitis, G. V.; Eversdyk, D. A simulation based optimization approach to supply chain management under demand uncertainty. Comput. Chem. Eng. 2004, 28 (10), 2087− 2106. (14) IBM ILOG CPLEX 12.1 User’s Manual; IBM Co.: Armonk, NY; available from www.ibm.com; 2009. (15) GAMS 23.1 Users Guide; GAMS Development Corporation: Washington, DC; available from www.gams.com; 2008. (16) ExtendSim 7 User Guide; Imagine That Inc.: San Jose, CA.; available from www.extendsim.com; 2007. (17) Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A general algorithm for short-term scheduling of batch operationsI. MILP formulation. Comput. Chem. Eng. 1993, 17 (2), 211−227. (18) Brandimarte, P.; Zotteri, G. Introduction to Distribution Logistics; John Wiley & Sons: New Jersey, 2007. (19) Zapata, J. C.; Pekny, J.; Reklaitis, G. V. Planning Production and Inventories in the Extended Enterprise; Springer: US, 2011; pp 593−627.

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Integrated Planning and Optimization of Clinical Trial Supply Chain

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