Optimal Production and Maintenance Planning of

Article pubs.acs.org/IECR

Optimal Production and Maintenance Planning of Biopharmaceutical Manufacturing under Performance Decay Songsong Liu, Ahmed Yahia, and Lazaros G. Papageorgiou* Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom ABSTRACT: Considering the performance decay of the chromatography resins in the downstream purification of biopharmaceutical manufacturing, the maintenance decisions should be made at appropriate times to achieve a good performance of the whole manufacturing process. In this paper, on the basis of the literature work on production planning, a mixed integer linear programming optimization model is developed to address the production and maintenance planning of biopharmaceutical manufacturing under performance decay, in which the total operating profit is maximized. The batch production profiles are tracked by binary variables introduced in the proposed model. The proposed model is then applied to two literature-based illustrative examples. Two scenarios are investigated and compared with the optimal solutions. A discussion about the effects of the maintenance duration and cost and the resin’s lifetime is also presented.

1. INTRODUCTION Chromatography has become a very popular and common technique for purification and separation in the pharmaceutical/biopharmaceutical industry.1 Chromatography resins are typically reused multiple times, mainly due to their high cost. However, considering the limited lifetime of the resins and their performance decay,2,3 maintenance, i.e., the regeneration to restore the resins’ performances and capabilities to their initial levels, is necessary to meet predetermined quality and safety attributes of the final products. Furthermore, with the performance decay of the resins, when to perform maintenance operations is a key decision in the purification processes. An optimal decision on whether to continue using the resin with a lower yield or to regenerate the resin to its initial level with maintenance cost cannot be achieved, unless the production and maintenance planning are considered simultaneously. In the process industry, production planning and scheduling of batch processes has been extensively investigated.4−7 Medez et al.8 and Pan et al.9 provided extensive reviews in this area. A recent review by Harjunkoski et al.10 presented an overview of the existing scheduling methodologies for process industries, many of which are for batch processes. Recently, more attention has been paid to production planning and scheduling in the pharmaceutical/biopharmaceutical industry. Optimization models and techniques have been widely used in this area.11,12 Samsatli and Shah13,14 proposed a two-stage optimization procedure for biochemical processes, in which the first stage determines the processing rates, unit operation conditions, and capacities, and the second stage tackles the detailed scheduling and design adjustments. Montagna et al.15 proposed a mixed-integer nonlinear programming (MINLP) model for the design of a biotechnological multiproduct batch plant to minimize the plant capital cost. Asenjo et al.16 simultaneously optimized both the process variables and the structure of a multiproduct batch protein plant by an MINLP model. Papageorgiou et al.17 developed a mixed integer linear programming (MILP) model for the optimal strategic supply chain decision-making process for pharmaceutical industries. © 2014 American Chemical Society

The net present value (NPV) is maximized by determining the optimal product portfolio, manufacturing capacity, and production plans. Gatica et al.18 addressed the capacity planning under clinical trials uncertainty in the pharmaceutical industry and developed a multiscenario MILP model for it. Iribarren et al.19 considered the optimal synthesis problem of a plant for the production of multiple recombinant proteins expressed in several hosts. Lakhdar et al.20 formulated the medium-term planning of multiproduct biopharmaceutical manufacturing facilities as an MILP model and compared the model with the an industrial rule-based approach. Then, this work was extended to tackle the case with uncertain fermentation titers.21,22 Also, the long-term capacity planning with multiple criteria, including cost, customer service level, and utilization targets, was also addressed.23 Steansson et al.24 proposed a multiscale hierarchically structured framework for the single plant production campaign planning and scheduling for a secondary production facility with order-driven multistage, multiproduct flowshop production. Kopanos et al.25 developed an iterative two-step MIP-based solution strategy for large-scale scheduling problems in multiproduct multistage batch plants and investigated a real-world multiproduct multistage pharmaceutical batch plant. Stefansson et al.26 studied a large realworld scheduling problem from a pharmaceutical company and compared both discrete and continuous time representation formulations. Sousa et al.27 developed a decomposition-based optimization framework of the global pharmaceutical supply chain planning, including both primary and secondary manufacturing. Kabra et al.28 presented a continuous time MILP optimization model for multiperiod scheduling of biopharmaceutical plants involving multistage multiproduct Special Issue: Jaime Cerdá Festschrift Received: Revised: Accepted: Published: 17075

February 28, 2014 May 21, 2014 June 6, 2014 June 6, 2014 dx.doi.org/10.1021/ie5008807 | Ind. Eng. Chem. Res. 2014, 53, 17075−17091

Industrial & Engineering Chemistry Research

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purification processing to obtain final products. Each stage may take place on multiple suites. The existence of the storage of the intermediate products between upstream and downstream processing is considered. Also, the shelf lifetimes for both intermediate and final products are given. Between upstream and downstream processing, different throughputs are allowed. In the problem, the multiproduct production facilities operate on a campaign basis to achieve the minimum batch and cross-product contamination. Each campaign of one product includes an uninterrupted series of batches sequentially. During the changeovers between two campaigns, due to the regulation on the manufacturing specifications and product quality requirements, the changeover times for the cleaning and the new campaign setups and start-ups usually occur, which impact the efficiency and productivity of the manufacturing processes. The campaign duration include the changeover time and the time taken for the production of all batches in the campaign. In this work, we use a discrete time representation, and the whole planning horizon is divided into multiple time periods with each for one or two months long. The product demands by the end of each time period are known. It is assumed that, in each fermentation/purification suite, only one product can be produced in each time period. Thus, each campaign lasts for a number of time periods. The optimal duration of each campaign and the campaign sequences in the manufacturing facilities are to be determined in this problem. Another key issue considered in the work is the performance decay of the chromatography resin. In the downstream purification processes, usually multiple chromatography steps are required to remove the contaminants and keep a high purity level of the final products. In any chromatography purification steps in the process, a specific resin is used. With the increasing number of cycles/batches of uses, the performance of the resin becomes worse, usually in terms of the yield. In this work, to simplify the model, only one Protein A chromatography step is considered with a decaying yield of the chromatography resin. All other chromatographic or nonchromatographic steps are assumed to have no product loss. To restore the initial performance of the decayed resin, maintenance work needs to be done at an appropriate time by regenerating the column resin. The decision strategies on when to take the maintenance operations and which suite is to be maintained are to be optimized in this problem. In this optimal production and maintenance planning problem, the following are given • products; • fermentation and purification suites; • productions rates, changeover times, and production throughputs; • product demands, shelf lifetimes, and sales prices; • unit manufacturing, campaign changeover, storage, late delivery, and waste proposal costs; • minimum and maximum campaign durations; • maximum storage capacities; • duration and cost of unit maintenance operation; • decaying yield with number of batches produced; to determine (1) campaign durations and sequences, (2) number of batches produced and production amounts, (3) product sales and late deliveries, and (4) maintenance

production. Liu et al.29 optimized the chromatography column sizing design in the antibody purification processes using MINLP optimization by minimizing the total cost of goods per gram. Later, the chromatography sequencing decisions were incorporated in the proposed mixed integer linear fractional programming (MILFP) model.30 Siganporia et al.31 developed a discrete-time MILP model for capacity planning of multiple biopharmaceutical products with either batch or perfusion bioprocesses across multiple facilities. A rolling time horizon methodology was implemented in conjunction with the MILP model to overcome computational expense. Meanwhile, lots of literature work considered preventive maintenance together with production planning and scheduling.32−40 Also, maintenance planning under performance decay has received lots of attention. Georgiadis and Papageorgiou41 proposed an MINLP model for the optimization problem of cleaning and energy management in a complex heat exchanger network under fouling. Then, the heat integration and fouling issues were incorporated into the production scheduling optimization problem.42 Alle et al.43 addressed the cyclic scheduling of cleaning and production operations in multiproduct multistage plants with performance decay and developed an MINLP model to optimize the production and cleaning schedules. The same authors44 proposed an MINLP model and an MILP model for the economic lot scheduling problem under performance decay. Then, the above work was extended by considering the exponential performance decay.45 In the work of Schulz et al.,46 the optimal scheduling of cracking furnace shutdowns was integrated to short-term production planning by a discrete time multiperiod MINLP model. Jain and Grossmann47 developed a mixed-integer nonlinear programming (MINLP) model for the cyclic scheduling of continuous parallel-process units regarding the exponential decay in performance. Heluane et al.48 optimized the operation of continuous evaporation units presenting a decrease in efficiency with time by an MINLP model. Heluane et al.49 addressed the scheduling of production and cleaning operations in a sugar plant with performance decay. Later, this work was extended by considering simultaneous redesign and scheduling of multiple effect evaporation networks.50 To the best of our knowledge, no literature work has considered the production planning of biopharmaceutical manufacturing under performance decay. In this work, we address the production and maintenance planning of biopharmaceutical manufacturing, where the yield of resins in downstream purification decreases with the number of batches produced. A mathematical programming model is aimed to be developed to tackle this problem. The remaining of this paper is structured as follows: in Section 2, the problem description is presented; the proposed production and maintenance planning model is presented in Section 3; Section 4 describes two illustrative examples; their computational results are discussed in Section 5; concluding remarks are given in Section 6.

2. PROBLEM STATEMENT In this work, we consider the production and maintenance planning problem in the biopharmaceutical primary manufacturing. The biopharmaceutical manufacturing processes considered involve two stages: upstream fermentation and downstream purification. During the manufacturing, all products undergo upstream fermentation processing first with intermediate products produced and then downstream 17076

dx.doi.org/10.1021/ie5008807 | Ind. Eng. Chem. Res. 2014, 53, 17075−17091


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Figure 1. Black-box model representation.

schedules of the chromatography column, so as to maximize the total operating profit.

FBjpt = Wjpt + FR jp·(FTjpt − βp ·Wjpt ), ∀ j , p , t

where βp denotes the duration of the first batch of a campaign plus any start-up time required for setup/cleaning; FTjpt is the corresponding production time; FRjp is the continuous downstream production rates; Wjpt indicates the changeover time is considered for the start of a new downstream campaign. Binary variable Yipt is introduced to indicate whether product p is manufactured in suite i during time period t, while Ujpt is introduced to indicate whether product p is manufactured in suite j during time period t. If product p is not produced in the previous time period t − 1, which means the campaign starts at time period t, Zipt or Xjpt will only be equal to 1.

3. MATHEMATICAL FORMULATION In this section, we propose a discrete-time MILP model of the production and maintenance planning of biopharmaceuticals, on the basis of the literature work on medium-term production planning model by Lakhdar et al.,20 which is described in the Appendix. In the proposed model, one main assumption is that a continuous rate of production was introduced to represent the batch manufacturing of biopharmaceuticals. Here, each batch is treated as a “black box”, whose detail level is required by the considered problem’s medium-term planning horizon. At the start of each production campaign of a product, there should be changeover times for the setup, cleaning, sterilization, etc. The total campaign duration, T, with n batches can be expressed by the following equation: T = α + BT ·(n − 1)

Zipt ≥ Yipt − Yip , t − 1 , ∀ i , p , t

(4)

Xjpt ≥ Ujpt − Ujp , t − 1 , ∀ j , p , t

(5)

If crude product p is not kept in storage before downstream production, the changeover time should be taken into account. If any crude product p is held in storage prior to downstream production, then the inclusion of a changeover time βp to the downstream production time will not happen. Thus, Wjpt = 1 only if both Zipt and Xjpt are equal to 1.

(1)

where α is the changeover time, including duration of the first batch of a campaign plus any start-up time required for setup/ cleaning; BT is the effective batch time, repeated within the campaign. Also, the feature of intermediate storage between the upstream fermentation and downstream purification is incorporated in the black box representation, as shown in Figure 1. Due to the decaying chromatography resin yield, the product mass in each batch can be lower than that in a full batch, which means a batch produced when there are no decays in the chromatography resin yield, i.e., yield = 100%. In this work, the production amount is expressed by the number of full batch equivalent. The production amount of one batch is ε ≤ 1 when the chromatography resin yield is ε·100%. 3.1. Batch Number Constraints. At the upstream stage, the number of batches produced, CBipt, equals continuous production rates for crude/upstream, CRip, multiplied by its respective production time, CTipt. If product p is produced at given suite i during time period t, changeover time, αp, denoting the duration of the first batch of a campaign plus any start-up time, will only be deducted from the production duration, if binary variable Zipt is equal to 1; i.e., it is the start of a new upstream campaign. As αp includes the duration of the first batch of a campaign, the production should be plus one after αp is removed from the production duration. CBipt = Zipt + CR ip·(CTipt − αp·Zipt ), ∀ i , p , t

(3)

Wjpt ≥

∑j Zipt card(j)

+ Xjpt − 1, ∀ j , p , t

(6)

In each fermentation/purification suite and in each time period, at most, one product can be produced, in order for the production constraints to capture the required campaign changeover considerations.

∑ Yipt ≤ 1,

∀ i, t (7)

p

∑ Ujpt ≤ 1, p

∀ j, t (8)

3.2. Maintenance and Batch Tracking Constraints. After each maintenance operation, a resin restores its performance to the initial level. To model the maintenance operations, including the regeneration of the aged resins, we introduce a binary variable Mjt to indicate whether the maintenance is performed in suite j during time period t. In order to track the production profile of each batch produced, we label each produced batch by the order in the batch production sequence since the last maintenance. Thus, the first batch, after each maintenance operation, is labeled as 1, and the second one is labeled as 2 and so on. Here, a binary variable, Ljnt, is used to express whether a batch labeled as n (index n is used to indicate that a batch is the nth batch since the last maintenance) is produced in suite j in time period t.

(2)

Similarly, at the downstream stage, if product p is produced at suite j during time period t, the number of batches produced, FBjpt, can be denoted as below: 17077



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NMj , t − 1 ≤ n − 1 + NM jmax ·(1 + Mjt − Ljnt ), ∀ j , n , t

The value of n starts from 1 but cannot exceed the maximum number of batches allowed between two maintenance operations, NMmax j , when the chromatography resin reaches the minimum yield authorized. Table 1 shows an example of

(14)

If the batch labeled n is produced in time period t (Ljnt = 1), then there are at least n batches produced since the last maintenance by the end of that time period.

Table 1. Example of the Values of Variables Mjt and Ljnt

NMjt ≥ n − NM jmax ·(1 − Ljnt ), ∀ j , n , t

time period variable

t1

t2

t3

Mj1,t Lj1,1,t Lj1,2,t Lj1,3,t Lj1,4,t Lj1,5,t Lj1,6,t

1 1 1 0 0 0 0

0 0 0 1 1 1 0

1 1 1 1 1 0 0

As the batches are labeled consecutively in purification suite j, in each time period, the value of variable Ljnt should be 1 or 0 in consecutive order of batch label n. If there is maintenance in time period t, then the values Ljnt should start with a number of consecutive ones. Otherwise, the values Ljnt should start with a number of consecutive zeros, followed by a number of consecutive ones. As shown in Table 1, in time period t1, Lj1,n,t1 = 1 for n = 1 and 2 and Lj1,n,t1 = 0 for n = 3, 4, 5, and 6, while in time period t2, when n = 1 and 2, Lj1,n,t1 = 1 and Lj1,n,t1 = 1 for n = 3, 4, and 5, before it becomes 0 for n = 6. For a feasible and correct label, in one suite and one time period, the values of Ljnt change at most twice between 0 and 1 with the values of n. We let Qjnt = |Ljnt − Lj,n−1,t|, to track the change of the values between Ljnt and Lj,n−1,t. The values of Qjnt should be 1 for at most two ns in each time period, e.g., time periods t1 and t2 in Table 2. It is required that the batches should be labeled in the

the values of Mjt and Ljnt in a production and maintenance schedule. In purification suite j1, the maintenance is done in time period t1, i.e., Mj1,t1 = 1. Considering the case that two batches are produced in t1 and three batches are produced in t2 after the maintenance, we let the two batches produced in time period t1 labeled as 1 and 2, i.e., Lj1,1,t1 = Lj1,2,t1 = 1, while the batches produced in t2 are labeled as 3, 4, and 5, i.e., Lj1,3,t2 = Lj1,4,t2 = Lj1,5,t2 = 1. In the time period t3, another maintenance operation is taken (Mj1,t3 = 1), and the four batches produced in that time period are labeled from 1 to 4; then, we have Lj1,1,t3 = Lj1,2,t3 = Lj1,3,t3 = Lj1,4,t3 = 1. Note that the label is not product specific. The labels used here only present how many batches have been produced since the last maintenance, and these batches could be for the same product or for different products. In this case, in the example shown in Table 1, suite j1 could produce one product in time period t1 and another product in time period t2. Here, an integer variable, NMjt, is introduced to express the number of batches produced since the last maintenance by the end of time period t. The value of NMjt is equal to the total number of batches labeled in time period t, plus the number of batches produced by the end of the previous time period, NMj,t−1, if there is no maintenance in time period t. NMjt = NMj , t − 1·(1 − Mjt ) +

∑ Ljnt ,

Table 2. Feasible Example of the Values of Variables Mjt, Ljnt, and Qjnt time period t1

n n n n n n

NMMjt ≤ NMj , t − 1 , ∀ j , t

(11)

NMMjt ≥ NMj , t − 1 −

NM jmax ·(1

− Mjt ), ∀ j , t

∑ Ljnt , n

Qjn,t1

Mj,t2

= = = = = =

1 2 3 4 5 6

Ljn,t2

Qjn,t2

0 0 0 1 1 0

0 0 0 1 0 1

0 1 1 1 0 0 0

1 0 0 1 0 0

Table 3. Infeasible Example of the Values of Variables Mjt, Ljnt, and Qjnt time period t1

(12)

t2 variable

Thus, eq 9 is equivalent to the following linear constraint: NMjt = NMj , t − 1 − NMMjt +

Ljn,t1

order of the production sequence. However, in Table 3, the batches produced in the same time period are not consecutively labeled, which is not allowed. For example, in time period t1, the two batches produced are labeled as 1 and 3, and the first batch produced in time period t2 is labeled as 2. In this case,

The above constraint is nonlinear, which needs to be linearized. Here, we introduced an auxiliary variable NMMjt ≡ NMj , t − 1·Mjt and the following constraints: (10)

Mj,t1 1

(9)

NMMjt ≤ NM jmax ·Mjt , ∀ j , t

t2 variable

no. of batches

∀ j, t

n

(15)

no. of batches

∀ j, t

Mj,t1

Ljn,t1

Qjn,t1

1 0 1 0 0 0

1 1 1 1 0 0

1

(13)

n n n n n n

If a maintenance operation does not take place in suite j at the beginning of time period t (Mjt = 0) and the batch labeled n is produced in that time period (Ljnt = 1), then there are at most n − 1 batches produced since the last maintenance by the end of the previous time period t − 1. 17078

= = = = = =

1 2 3 4 5 6

Mj,t2

Ljn,t2

Qjn,t2

0 1 0 1 1 0

0 1 1 1 0 1

0



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amount, FPtot jt , is defined as the summation of the production amount, FPjpt, of all products.

the values of Qjnt are 1 for four ns in both two time periods in Table 3 (n = 1, 2, 3, and 4 in time period t1 and n = 2, 3, 4, and 6 in time period t2). To prevent the occurrence of the infeasible case in Table 3, we need the following constraint to force that the summation of Qjnt should be no greater than 2 for each suite and time period:

∑ Q jnt ≤ 2,

FPjttot =

∀ j, t (22)

p

Also, the production amount of each product cannot exceed its number of batches produced in each time period.

∀ j, t (16)

n

∑ FPjpt ,

FPjpt ≤ FBjpt , ∀ j , t

The following linear formulation is used to formulate Qjnt as the absolute value function:

(23)

3.4. Timing Constraints. In the following constraints, minimum and maximum production times are enforced if a product is produced on a suite in either the upstream (eq 24) or downstream stage (eq 25).

Q jnt ≥ Ljnt − Lj , n − 1, t , ∀ j , n , t

(17)

Q jnt ≥ Lj , n − 1, t − Ljnt , ∀ j , n , t

(18)

CT pmin·Yipt ≤ CTipt ≤ CT pmax·Yipt , ∀ i , p , t

For any purification suite and any given time period, the total batches of all products produced should be the summation of all batches labeled.

(24)

FT pmin·Ujpt ≤ FTjpt ≤ FT pmax·Ujpt , ∀ j , p , t

(25)

∑ Ljnt = ∑ FBjpt , n

The total production time on each upstream/downstream tot suite in each time period, CTtot it /FTjt , is equal to the summation of each product’s production time, CTipt/FTjpt.

∀ j, t (19)

p

3.3. Production Constraints. In the upstream processing, the production amount, CPipt, is calculated in terms of the number of batches produced; thus, we have the following equation: CPipt = CBipt , ∀ i , p , t

CTittot = FT jttot =

Ljn,t1

1 n=1 n=2 n=3 n=4 n=5 total

100 100 95 90 85

Mj,t2 Ljn,t2

1 1 0.95 0 0 2.95

0 0 0 1 1 2

production

∑ ydjn·Ljnt , n

(29)

3.5. Storage Constraints. The inventory amount of crude product, p, by the end of the time period t, CIpt, equals the inventory in time period t − 1, plus the production amount during time period t, CPipt, minus the amount processed at downstream stage, FBjpt/λp, and the wasted amount, CWpt, when the product’s shelf life is expired. Here, λp is the production correspondence factor for crude to final production of product p.

0 0 0 0.90 0.85 1.75

calculation of the total production amount. As the yield decays, in time period t1, 3 batches are produced with production amounts of 1, 1, and 0.95, respectively, and the total production amount is 2.95. In time period t2, two batches are produced, and their production amounts are 0.9 and 0.85. Thus, the production amount in time period t2 is 1.75. Thus, we calculate the total production amount in each suite and in each time period, FPtot jt , as the total product of the yield and the binary variable: FPjttot =

(28)

FT jttot ≤ H − γj·Mjt , ∀ j , t

0 1 1 1 0 0 3

(27)

If the maintenance operation occurs in time period t in a purification suite, i.e., Mjt = 1, the total available production time should be the available production time, H, minus the duration for maintenance, γj, which is the upper bound of the total production time in purification suites, FTtot jt .

t2 production

∀ j, t

CTittot ≤ H , ∀ i , t

time period

Mj,t1

∑ FTjpt ,

The total production time in upstream suites, CTtot it , in each time period should be limited by the given available production time, H.

Table 4. Example of the Calculation of Production Amounts

ydjn, %

(26)

p

In the downstream processing, as mentioned earlier, the production amount of a batch is expressed as the faction of a full batch and is equal to the chromatography resin’s yield for the batch. Thus, the total production amount of all products should be equal to the summation of the corresponding yield of each batch produced. Here, the chromatography resin’s yield of the batch labeled as n is given. Table 4 gives an example of the

no. of batches

∀ i, t

p

(20)

t1

∑ CTipt ,

CIpt = CIp , t − 1 +

∑ CPipt − i

1 ·∑ FBjpt − CWpt , ∀ p , t λp j (30)

Similarly, the inventory amount of final product, p, by the end of the time period t, FIpt, equals the inventory in time period t − 1, plus the production amount during time period t, FPjpt, minus the sales, Spt, and the wasted amount, FWpt. FIpt = FIp , t − 1 +

∀ j, t

∑ FPjpt − Spt − FWpt , j

(21)

∀ p, t (31)

The inventory amount at both upstream and downstream stages, CIpt and FIpt, should not exceed their maximum available inventory capacities, Cp and Fp, respectively.

where ydjn is the yield of resin in suite j of the nth batch produced since the last maintenance and the total production 17079



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CIpt ≤ Cp , ∀ p , t

(32)

FIpt ≤ Fp , ∀ p , t

(33)

Both crude and final products are constrained by limited product shelf life (ζp for crude product p and σp for final product p). The crude product stored cannot be processed downstream, and the final product cannot be sold, after its shelf life expires.

Figure 2. Multisuite biopharmaceutical facility of Example 1.

t + ζp

1 ·∑ ∑ FBjpθ , ∀ p , t λp j θ = t + 1

CIpt ≤

Table 5. Product Demand in Number of Batches of Example 1

(34)

t1

t + σp

∑

FIpt ≤

Spθ , ∀ p , t

P1 P2 P3

(35)

θ=t+1

3.6. Late Delivery Constraints. Late deliveries/backlogs occur when a given batch of product is late to meet a product demand. The backlog amount of product p by the end of time period t, Δpt, is equal to its backlog amount in time period t − 1, plus its demand in time period t, Dpt, minus its sales in time period t, Spt. Δpt = Δp , t − 1 + Dpt − Spt , ∀ p , t

(36)

∑ ∑ υp·Spt − ∑ ∑ ∑ ηp·CPipt p

−

p

p

−

t

∑ ∑ ∑ ψp·Zipt

t

i

p

t

∑ ∑ ∑ ψp·Xjpt − ∑ ∑ ρp ·CIpt − ∑ ∑ ωp·FIpt j

−

i

∑ ∑ ∑ ηp·FBjpt j

−

t

p

t

p

t

p

t

∑ ∑ δp·Δpt − ∑ ∑ τp·(CWpt + FWpt ) − p

t

p

t

∑ ∑ φj ·Mjt j

t

t3

t4

t5

6

t6 6

6 8

8

production, respectively. Table 8 presents the price and cost data. The production factor, λp, is 1 for each product. The larger Example 2 considers three fermentation suites (I1−I3), two purification suites (J1−J2), and four products (Figure 3). The upstream fermentation suites are product specific: suites I1 can process both products P1 and P2, while products P3 and P4 can be processed in suites I2 and I3. The production factor, λp, is 1 for P1 and P2 and 0.5 for P3 and P4. Also, a larger 1.5-year planning horizon is investigated, which is equivalent to nine two-month long time periods. More details of the data are shown in Tables 9−12. It is assumed that each batch involves five cycles in the chromatography purification. On the basis of the experimental work of Jiang et al.,2 we approximate that the yield of a Protein A resin (named resin A) decays with the increasing number of batches produced, as shown in Figure 4. The yield decreases to 65% for the 11th batch after maintenance, which is the minimum yield authorized through a chromatography column. Then, a maintenance operation must be done. The unit maintenance cost is set to be 15 relative monetary units (rmu), and the maintenance duration is ignored here due to its short time period. The optimization models are implemented in GAMS51 using a CPLEX MILP solver on a 64-bit Windows 7-based machine with a 3.20 GHz six-core Intel Xeon processor W3670 and 12.0 GB RAM. The optimality tolerance is 0%, and the CPU limit is set to 3600 s.

3.7. Objective Function. The objective is to maximize operating profit, which is total sales revenue (υp), minus the operating cost, including batch manufacturing cost (ηp), changeover cost (ψp), storage cost (ρp and ωp), late delivery penalty cost (δp), waste disposal cost (τp), and maintenance cost (φj): PR =

t2

(37)

5. RESULTS AND DISCUSSION 5.1. Optimal Results. The summary of the model statistics of both examples is shown in Table 13. The model size of Example 2 is about double that of Example 1, while the computational time of Example 2 is 1 order of magnitude larger than that of Example 1. The optimal sales revenue of Example 1 is 675 rmu, and the optimal sales revenue of Example 2 is 1054.25 rmu. Figure 5 shows the breakdown of the optimal sales revenue. In both examples, the largest cost proportions come from the manufacturing cost, maintenance cost, and late delivery cost, while storage cost, changeover cost, and waste disposal cost are relatively smaller. Especially, there is no waste disposal cost for Example 1. The optimal schedule of Example 1 is show in Figure 6, and the resins’ yield profiles are given in Figure 7. In Figure 6 and the following figures for production and maintenance schedules in this paper, M indicates that maintenance is done in the

Overall, the above proposed MILP model for biopharmaceutical production and maintenance planning includes eqs 2−8 and 10−36 as constraints and eq 37 as objective function.

4. TWO ILLUSTRATIVE EXAMPLES In this section, we apply the developed model to two illustrative examples, extended from the literature examples,20 both based on industrial information from real case studies in a biopharmaceutical company. Example 1 involves a multiproduct facility with two fermentation suites (I1−I2) and two purification suites (J1−J2), producing three different products (P1−P3). As shown in Figure 2, each product can be produced in all suites. A 1-year planning horizon is considered, which is divided into six time periods with each for two months long. The demands in number of batches are given in Table 5. The production rates, changeover times, product lifetimes, storage capacities, minimum campaign lengths, and storage costs are shown in Tables 6 and 7 for upstream and downstream 17080



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Table 6. Upstream Production Data of Example 1 production rate (batch/day)

changeover time (day)

product lifetime (time period)

storage capacity (batch/time period)

minimum campaign length (day)

storage cost (rmu/ day)

0.05 0.045 0.08

30 32 22.5

1 1 1

10 10 10

20 21 12.5

5 5 5

P1 P2 P3

Table 7. Downstream Production Data of Example 1 production rate (batch/day)






0.1 0.1 0.1

40 42 34.5

3 3 3

40 40 40

10 10 10

1 1 1

P1 P2 P3

purification suite J1, while fermentation suite I2 and purification suite J2 involve the production of all three products. The maintenance operations take place in time periods t1 and t4 for purification suite J1. By the end of time period t3, after producing six batches, the resin’s yield is down to 90% before the maintenance operations in time period t4. The maintenance of purification suite J2 occurs four times, in time periods t1, t2, t3, and t5, respectively. Although, by the end of time period t1, the yield is still 100% after producing three batches, it is still necessary to perform maintenance operation in time period t2 due to the forthcoming high production. Then, the yield becomes 95% after producing five batches in time period t2. Then, after the maintenance in time period t3, the regenerated resin is used for five batches in two time periods. When the yield reaches 95% at the beginning of time period t5, another maintenance work is implemented after five batches of production. A total of six maintenance tasks is implemented in both of the two suites. In suite J1, a total of 14 batches are produced, while suite J2 produces more, 21 batches. The production, sales, inventory, and late deliveries in the optimal solution of Example 1 are shown in Figure 8. The storage of intermediate products of P1 and P2 are always 0, while the storage of P3’s intermediate product exists in both time periods t1 and t4. The storage of product P1 occurs in time periods t3 and t5, both less than two batches. As to product P2, its storage reaches four batches by the end of time period t2. The storage of product P3 in time periods t3 and t5 are no more than thee batches. Because of the resin’s performance decay, by time period t2, the production of product P3 is only 7.95 batches, which is less than its demand of 8 batches and leads to 0.05 batches of late deliveries until time period t3. Similarly, by time period t5, the backlog of product P3 is 0.15 batches. Also, the demand of product P1 cannot be met by time periods t4 and t6. Overall, the late deliveries in the optimal solution are very small. Figure 9 presents the optimal production and maintenance schedules of Example 2. As to Example 2, the fermentation suites can only produce specific products, as given in Figure 3. Because of a smaller number of purification suites, each one is

Table 8. Price and Cost Data of Example 1

P1 P2 P3

sales price (rmu/ batch)

manufacturing cost (rmu/ batch)

waste disposal cost (rmu/ batch)

lateness penalty (rmu/ batch)

changeover cost (rmu/ day)

20 20 20

2 2 2

5 5 5

20 20 20

1 1 1

Figure 3. Multisuite biopharmaceutical facility of Example 2.

Table 9. Product Demand in Number of Batches of Example 2 t1 P1 P2 P3 P4

t2

t3

t4

t5

6

t6

t7

t8

4

4

4 4

10 6

t9

10

8

indicating suite and time period; for fermentation suites, the numbers indicate the number of batches produced and production time in days (number in brackets); for purification suites, the numbers indicate the production amount, the number of batches produced, and production time in days (number in brackets). As shown in Figures 6 and 7, products P1 and P3 share fermentation suite I1, and products P1 and P2 share Table 10. Upstream Production Data of Example 2

P1 P2 P3 P4

production rate (batch/day)






0.05 0.045 0.08 0.08

30 32 22.5 22.5

1 1 1 1

10 10 10 10

20 21 12.5 12.5

5 5 5 5

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Table 11. Downstream Production Data of Example 2 production rate (batch/day)






0.1 0.1 0.1 0.1

40 42 44.5 44.5

3 3 3 3

40 40 40 40

10 10 10 10

1 1 1 1

P1 P2 P3 P4

look at the number of batches produced between two maintenance periods, we can see that there are at most 10 batches produced, so the minimum yield is 70% (Figure 10). Comparing the number of batches produced in two purification suites, 31 batches and 26 batches are produced in suites J1 and J2, respectively. The optimal production, sales, storage, and late deliveries of Example 2 are presented in Figure 11. Among all intermediate products, only those of products P1 and P2 have storage, which can be up to 3 batches. As to the final products, product P1’s storage occurs from time period t6 to t8, while P3 is stored in time periods t3, t4, t7, and t8. Especially in t8, product P3’s storage is 6.5 batches. Product P4 is stored in the first three time periods, while the storage of product P2 is zero in all time periods. The production of product P1 by time period t4 is only 4.95 batches, while the corresponding demand is 6 batches, so there are 1.05 batches delivered late. While product P2’s demand is four batches at time period t7 and there are two separate deliveries in both time periods t7 and t8, product P3’s late delivery is charged in time periods t5 and t6, in which 4.3 and 0.45 batches are late, respectively. 5.2. Scenario Analysis. Here, we consider two more scenarios and compare them with the optimal solutions. In Scenario A, maintenance operation decisions are optimized by solving the reduced optimization model with fixed optimal production schedules without performance decay. In Scenario B, chromatography resin is only allowed to be maintained whenever its yield decays at the minimum authorized level, which is 65% in the examples. 5.2.1. Scenario A. If we solve the two examples, ignoring the performance decay of chromatography resin (Figure 4) and the maintenance time and cost, i.e., ydjn = 1 and γj = φj = 0, the obtained optimal profits of both examples are 490 and 539 for Examples 1 and 2, respectively. It should be noted that the optimal profit of Example 2 is the same as that of the literature model.20 However, for Example 1, the optimal profit is slightly higher than that reported in the literature,20 487, because a 5%

Table 12. Price and cost data of Example 2

P1 P2 P3 P4

sales price (rmu/ batch)

manufacturing cost (rmu/ batch)

waste disposal cost (rmu/ batch)

lateness penalty (rmu/ batch)

changeover cost (rmu/ day)

25 20 17 17

5 2 1 1

5 5 5 5

20 20 20 20

1 1 1 1

Figure 4. Decaying yield profile of resin A with the number of batches produced, ydjn.

Table 13. Model Statistics of Examples 1 and 2 no. of equations no. of continuous variables no. of discrete variables optimal profit (rmu) CPU time (s)

example 1

example 2

1219 291 550 375.6 98

2296 642 1060 334.4 814

used to produce all products. The maintenance takes place in time periods t1, t2, t4, t5, and t8 in purification suite J1 and in time periods t1, t2, t3, t4, t6, and t7 in purification suite J2. Thus, there are 11 maintenance tasks in total. With a detailed

Figure 5. Optimal breakdown of the sales revenue of both Examples 1 and 2. 17082



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Figure 6. Optimal production and maintenance schedule of Example 1.

without decays produces one batch of crude product P1 fewer than the optimal schedule (Figure 6) in the upstream fermentation in Example 1. Similarly, in Example 2, the optimal schedule (Figure 9) produces four more batches of crude product P3 in time period t7 than the case without considering decays. Now, we consider the decay of the chromatography resin (Figure 4) and solve the reduced proposed production and maintenance planning model with fixed production schedules given in Figures 12 and 13 to optimize the maintenance operation schedule. On the basis of the fixed production schedules, seven maintenance operations are decided by the optimization model in time periods t1, t3, and t5 in purification suite J1 and in time periods t1, t2, t3, and t6 in purification suite J2, respectively, for Example 1. Meanwhile, for Example 2, there are maintenance operations in purification suite J1 in time periods t1, t2, t4, t5, t7, and t9 and in purification suite J2 in time periods t1, t2, t3, t4, t6, and t7. There are a total of 12 maintenance operations.

Figure 7. Yield profiles with the number of batches produced in the purification suites.

optimality tolerance was used in the literature, while in this work the global optimal solution is found with 0% optimality tolerance. The optimal schedules without decays are shown in Figures 12 and 13. It should be noted that the optimal schedule

Figure 8. Optimal solution of Example 1: (a) production; (b) sales; (c) storage; (d) late deliveries. 17083



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Figure 9. Optimal production and maintenance schedule of Example 2.

produced, which lead the chromatography resin yield to be a minimum authorized level, before the next maintenance. For the two examples investigated in this work, 11 batches need to be produced to achieve the minimum 65% yield. To guarantee it, the following constraint is introduced: Ljn , t − 1 ≥ Mjt , ∀ j , n = NM jmax , t > 1

(38)

By solving the proposed model for the two illustrative examples, the optimal production and maintenance schedules are presented in Figures 14 and 15. In Example 1, only two maintenance operations take place in each purification suite, in time periods t1 and t5. Thus, there are a total of four maintenance operations. In Example 2, there are three maintenance operations in each purification suite: purification suite J1 takes maintenance in time periods t1, t4, and t7, while purification suite J2 takes maintenance in time periods t1, t6, and t9.

Figure 10. Yield profiles with the number of batches produced in the purification suites.

5.2.2. Scenario B. In the Scenario B, we force that, after each maintenance, a maximum number of batches, NMmax j , must be

Figure 11. Optimal solution of Example 2: (a) production; (b) sales; (c) storage; (d) late deliveries. 17084



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Figure 12. Optimal production schedule without decay of Example 1.

Figure 13. Optimal production schedule without decay of Example 2.

Figure 14. Optimal production and maintenance schedule with minimum yield of Example 1.

5.2.3. Comparison. In this section, we compare the profits under Scenarios A and B with the optimal one. As shown in Table 14, the optimal profits of Examples 1 and 2 are 4.3% and 8.5% higher than those of Scenario A, respectively. Moreover, in Scenario B, the profit of Example 1 is 33.8% lower and that of Example 2 is 69.2% lower than the optimal ones. The profits of Scenario A are smaller than the optimal solution, due to the increased late delivery cost and maintenance cost. In Example 1, the difference between the two schedules mainly comes from the maintenance cost. For the optimal schedule, there are six maintenance tasks costing 90 rmu, while in Scenario A, the cost of for seven maintenance tasks is 105 rmu. In Example 2, there is the higher cost of maintenance and late deliveries in Scenario A. The 12 maintenance operations in Scenario A generate a cost of 180

rmu, while the optimal schedule has 11 maintenance tasks and a 165 rmu corresponding cost. Also, the optimal schedule of Scenario A has a sales amount of product P3 by time period t9, 0.6 batches shorter than the demand, which leads to a 12 rmu late delivery cost. In Scenario B, the profits for both examples are significantly reduced. Although Scenario B has fewer maintenance operations, leading to less maintenance costs for both examples, there are much higher late delivery costs. In Example A, the late delivery cost in Scenario B is 148 rmu, compared to 33 rmu in the optimal solution, while in Example 2, the late delivery cost in Scenario B is 438 rmu, much higher than that in the optimal solution, 204 rmu. Also, the sales revenues in Scenario B are also much less than the optimal solutions. The reason for that is the reduced production amount from each batch in Scenario B, 17085



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Figure 15. Optimal production and maintenance schedule with minimum yield of Example 2.

the optimal production and maintenance schedule is shown in Figure 16. Using the original maintenance cost, there are 11 maintenance tasks (Figure 9), while with the doubled unit maintenance cost, there are only 9 maintenance tasks remaining in Figure 16. The production schedules in fermentation suites are exactly the same. The difference in the production schedules in the downstream purification suites occurs after time period t4, when the two purification suites switch their schedules. Because of the different maintenance strategies taken, although the number of batches produced is the same, the production amounts are different. In both time periods t2 and t4, suite J2 is not maintained, and its production is 3.85 batches, less than what is shown in Figure 11. In time period t5, instead of the maintenance in suite J1, J2 is maintained. Thus, the total production in time period t5 is 3.75 batches, less than 3.85 batches in the optimal schedule with a smaller unit maintenance cost. Therefore, we can see that, with the increased unit maintenance cost, the number of maintenance operations could become fewer and lead to a different production and maintenance schedule. Next, we investigate the effects of the maintenance time by considering a scenario with nonzero maintenance duration, i.e., we let γj = 1 for Example 2, whose optimal production and maintenance schedule is shown in Figure 17. Still, we have the same production schedules in fermentation suites as in Figure

Table 14. Comparison of Profits of the Optimal Solution, Scenarios A and B example 1 profit optimal scenario A scenario B

375.6 359.4 248.8

example 2

difference, %

profit

difference, %

4.3 33.8

334.4 306.0 103.0

8.5 69.2

as the chromatography resins decay. The strategy in Scenario B causes a significant loss of final products in each batch, especially after many batches are produced, when the yield becomes very low. Thus, the above comparisons show that the production and maintenance planning should be considered together with the performance decay of chromatography resin, which is an important issue in the biopharmaceutical industry and should not be neglected. Also, the timing of when to perform maintenance operations is a key decision and should be determined in a systemic method. The proposed model can solve these issues efficiently and successfully. 5.3. Effects of Maintenance Cost and Duration. In this section, first, the effect of the maintenance cost is examined through the study of Example 2. Here, we consider a scenario with the maintenance cost doubled, i.e., we let φj = 30. Then,

Figure 16. Optimal production and maintenance schedule with the double maintenance cost of Example 2. 17086



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Figure 17. Optimal production and maintenance schedule with 1 day maintenance duration of Example 2.

Figure 18. Decaying yield profile of resin B with the number of batches produced, ydjn.

9. In the purification suites, the total number of maintenance operations is the same, but the last two operations are different in terms of both time period and suite. The downstream production schedules are different from time periods t6 to t9. In the original optimal schedule, the purification suite J2 in time period t6 involves maintenance operation, while its total production duration is 60 days, the upper limit, so when the maintenance duration increases from 0 to 1, the schedule in time period t6 becomes infeasible; thus, all plans in the following week should be rescheduled. The production amounts of products P2, P3, and P4 are unchanged, while only product P1 has less production, 7.5 batches, decreased from 7.7 batches. The above study shows that the maintenance duration also has an impact on the optimal production and maintenance schedule. 5.4. Effects of Resin Performance. In this section, we investigate the impact of the resin performance on the production and maintenance schedules. On the basis of the literature work,2 we consider an alternative Protein A resin, resin B, with a different decreasing yield, as shown in Figure 18. For this resin, the resin has a longer lifetime, whose yield is kept at 100% for the first 8 batches and decreases to 65% at the 15th batch produced. By solving the proposed model with the resin B’s performance, we achieve the optimal profits for both examples (Table 15). Due to the better performance of resin B, lower late delivery costs and maintenance costs are generated, which leads to higher profits than the case using resin A.

Table 15. Model Statistics of Examples 1 and 2 with Resin B no. of equations no. of continuous variables no. of discrete variables optimal profit (rmu) CPU time (s)

example 1

example 2

1411 291 646 424.1 136

2584 642 1204 428.0 748

Also, the optimal production and maintenance schedules of the two examples are obtained, shown in Figures 19 and 20. Comparing them to the solutions using resin A, in the upstream fermentation suites for Example 1, production of crude product A is one batch less using resin B than resin A. In Example 2, the two resins generate the same upstream schedules. In the downstream, it is obvious that the resin B is less frequently maintained, due to its longer lifetime. There are four maintenance operations for Example 1 and six for Example 2, much fewer than the required 6 and 11 maintenance operations using resin A, respectively. Because of the maintenance operations’ difference when using two resins, the resulting yields of the resins in each time period affect the production schedules. For Example 1, when using resin B, the downstream purification schedule becomes different from that of resin A from time period t3 to t6. Although the products produced in each time period from t3 to t6 switch between the two purification suites, their production amounts are different, when using the two resins, due to the resin’s performance affected by the maintenance schedules. Similarly, in Example 2, the 17087



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Figure 19. Optimal production and maintenance schedule of Example 1 with resin B.

Figure 20. Optimal production and maintenance schedule of Example 2 with resin B.

downstream schedules become different after the first time periods between the two resins used. Thus, we can conclude that, if other parameters remain unchanged, a longer lifetime resin generates a higher profit and needs fewer maintenance operations, which affect the production strategies accordingly.

chromatography step. This work can be extended by considering the performance decay of all resins involved in the downstream process. Another future direction of this work is to address the uncertainties. In reality, some parameters used in the model are uncertain, e.g., demand, resin’s yield, changeover time, and shelf life. A stochastic optimization model could be developed to tackle these issues.

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6. CONCLUDING REMARKS In this work, we have developed an MILP optimization model to determine the optimal production and maintenance schedule of a biopharmaceutical manufacturing process, considering the performance decay of the chromatography resin, on the basis of a literature optimization model20 for the production planning. In the developed model, the production batches produced after each maintenance operation are tracked by introduced binary variables. By maximizing the total operating profit, the optimal production schedules and maintenance operations are determined. Then, two illustrative examples have been investigated to show the applicability of the proposed model. In the Results and Discussion, the optimal schedules, sales, storage, and late deliveries are analyzed. The importance of this work is shown by comparing two scenarios. Also, the effects of the maintenance duration, maintenance cost, and resin’s lifetime on the optimal solutions are discussed. From the above studies, it is shown that the maintenance issue is important in biopharmaceutical manufacturing, and the maintenance duration/cost and resin’s lifetime affect the optimal production schedules. In this work, we consider a black-box presentation of the downstream purification processes, and the performance decay is assumed to only happen to one resin in the Protein A

APPENDIX The literature MILP model20 for production planning of biopharmaceutical manufacturing is presented here. A.1. Production Constraints

CPipt = Zipt + CR ip· (CTipt − αp· Zipt ), ∀ i , p , t

(A.1)

FPjpt = Wjpt + FR jp· (FTjpt − βp · Wjpt ), ∀ j , p , t

(A.2)

Zipt ≥ Yipt − Yip , t − 1, ∀ i , p , t

(A.3)

Xjpt ≥ Ujpt − Ujp , t − 1, ∀ j , p , t

(A.4)

Wjpt ≥

∑i Zipt card(i)

∑ Yipt ≤ 1,

+ Xjpt − 1, ∀ j , p , t

∀ i, t (A.6)

p

∑ Ujpt ≤ 1, p

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(A.5)

∀ j, t (A.7)



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A.2. Timing Constraints

CT pmin· Yipt ≤ CTipt ≤ CT pmax· Yipt , ∀ i , p , t

(A.8)

FT pmin· Ujpt ≤ FTjpt ≤ FT pmax· Ujpt , ∀ j , p , t

(A.9)

CTittot =

∑ CTipt ,

■

i = fermentation suite j = purification suite n = number of batches p = product t, θ = time periods

(A.10)

∑ FTjpt ,

∀ j, t (A.11)

p

CTittot ≤ H , ∀ i , t

(A.12)

FT jttot

(A.13)

≤ H , ∀ j, t

Parameters

Cp = storage capacity of crude product p, batches CRip = production rate of crude product p in suite i, batches per unit time CTmin = minimum production time for crude product p at p time period t CTmax = maximum production time for crude product p p Dpt = demand of final product p at time period t Fp = storage capacity of final product p, batches FRjp = production rate of final product p in suite j, batches per unit time FTmin p = minimum production time for final product p at time period t FTmax = maximum production time for final product p p H = available production time over time period t NMmax = maximum number of batches of final products j produced between two maintenance operations in suite j ydjn = yield of chromatography resin for the batch labeled n maintenance operation in suite j αp = changeover time for production of first batch of crude product p βp = changeover time for production of first batch of final product p γj = duration of a maintenance operation in suite j δp = unit cost charged as penalty for each late batch of final product p ζp = shelf lifetime of crude product p, in number of time periods ηp = unit cost for each batch produced of product p λp = production correspondence factor for crude to final production of product p ρp = unit cost for each stored batch of crude product p σp = shelf lifetime of final product p, in number of time periods τp = unit cost of disposing a batch of product waste p υp = unit sales price for each batch of final product p φj = unit cost for a maintenance operation in suite j ψp = unit cost for each new campaign of product p ωp = unit cost for each stored batch of final product p

A.3. Storage Constraints

1 · ∑ FPjpt − CWpt , ∀ p , t λp j

∑ CPipt −

CIpt = CIp , t − 1 +

i

(A.14)

FIpt = FIp , t − 1 +

∑ FPjpt − Spt − FWpt ,

∀ p, t (A.15)

j

CIpt ≤ Cp , ∀ p , t

(A.16)

FIpt ≤ Fp , ∀ p , t

(A.17)

t + ζp

1 CIpt ≤ · ∑ ∑ FPjpθ , ∀ p , t λp j θ = t + 1

(A.18)

t + σp

∑

FIpt ≤

Spθ , ∀ p , t

(A.19)

θ=t+1

A.4. Late Deliveries Constraints

Δpt = Δp , t − 1 + Dpt − Spt , ∀ p , t

(A.20)

A.5. Objective Function

PR =

∑ ∑ υp·Spt − ∑ ∑ ∑ ηp ·CPipt p

−

t

Xjpt −

p

t

t

i

p

t

j

p

t

∑ ∑ ρp ·CIpt − ∑ ∑ ωp·FIpt − ∑ ∑ δp·Δpt p

■

p

∑ ∑ ∑ ηp ·FPjpt − ∑ ∑ ∑ ψp·Zipt − ∑ ∑ ∑ ψp· j

−

i

t

p

t

p

t

∑ ∑ τp·(CWpt + FWpt) p

t

NOMENCLATURE

Indices

∀ i, t

p

FT jttot =

acknowledged. Financial support from the consortium of industrial and governmental users is also acknowledged.

(A.21)

Continuous Variables

AUTHOR INFORMATION

CIpt = amount of crude product p stored over time period t CPipt = amount of crude product produced in suite i over time period t CTipt = production time for product p on suite i over time period t CTtot it = total production time for suite i over time period t CWpt = amount of crude product p wasted over time period t FIpt = amount of crude product p stored over time period t FPjpt = amount of final product p produced in suite j over time period t FPtot jt = total amount of final products produced in suite j over time period t

Corresponding Author

*Tel.: +44-20-7679-2563. Fax: +44-20-7383-2348. E-mail: l. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding from the UK Engineering and Physical Sciences Research Council (EPSRC) for the EPSRC Centre for Innovative Manufacturing in Emergent Macromolecular Therapies hosted by University College London is gratefully 17089



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FTjpt = production time for product p on suite j at time period t FTtot jt = total production time for suite j over time period t FWpt = amount of final product p wasted over time period t K = large number PR = profit, the optimization objective Spt = amount of product p sold over time period t Δpt = amount of product p that is late over time period t Binary Variables

Ljnt = 1 if a batch labeled n is produced in suite j over time period t; otherwise 0 Mjt = 1 if suite j is maintained over time period t; otherwise 0 Qjnt = 0 if two consecutive batches labeled n and n − 1 in suite j are both produced over time period t; otherwise 1; ≡ | Ljnt − Lj,n−1,t| Ujpt = 1 if final product p is produced in suite j over time period t; 0 otherwise Wjpt = 1 if one or more campaigns are starting in suites i and j over time period t; 0 otherwise Xipt = 1 if a new campaign of final product p is started in suite i over time period t; 0 otherwise Yipt = 1 if crude product p is produced in suite i over time period t; 0 otherwise Zipt = 1 if a new campaign of crude product p is started in ite i over time period t; 0 otherwise Integer Variables

CBipt = number of batches of final product p produced in suite i over time period t FBjpt = number of batches of final product p produced in suite j over time period t NMjt = number of batches of final product p produced in suite i since last maintenance by time period t NMMjt = auxiliary variable, ≡ NMj,t−1·Mjt

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REFERENCES

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Optimal Production and Maintenance Planning of

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