Ind. Eng. Chem. Res. 2004, 43, 3799-3811
3799
Optimal Reliable Retrofit Design of Multiproduct Batch Plants Harish D. Goel,† Margot P. C. Weijnen,†,* and Johan Grievink‡ Faculty of Technology, Policy and Management and Faculty of Applied Sciences, Delft University of Technology, 2600 GA Delft, The Netherlands
The problem of the retrofit design of a multiproduct batch plant is considered from a new perspective that involves explicit consideration of inherent reliability and maintainability characteristics of existing and new equipment. Until now, in multiproduct batch plant retrofitting formulations, the production capacity has been specified by the limiting batch size and limiting cycle time. We propose a more robust retrofit solution that is obtained by defining the effective production capacity by three parameters: the limiting batch size, the limiting cycle time, and the overall plant availability. The novel simultaneous optimization framework developed in this work combines a process model and an availability model to obtain the optimal size, optimal operating mode, and optimal allocation of inherent availability for new equipment during the retrofit stage. The overall problem is formulated as a mixed integer nonlinear programming (MINLP) model, and its applicability is demonstrated through the solution of a number of examples. This framework provides the designer with the opportunity to select the initial inherent availability of new equipment during retrofitting by balancing the costs of design investments against the costs of downtime. 1. Introduction Multiproduct batch plants are designed to produce a number of related products using the same equipment in the same operationsequence. The problem of the retrofit design of a multiproduct batch plant arises, for example, when new production targets and market selling prices are specified for one or more products or when there is a need to improve the overall effectiveness of the existing plant by improving its reliability and maintainability characteristics. The retrofitting problem consists of finding those plant modifications that involve the removal of existing equipment (selling old units for salvage value) and/or the purchase of new equipment for the existing plant to maximize the net profit. Vaselenak et al.1 formulated the retrofit design of a multiproduct batch plant as a mixed integer nonlinear programming (MINLP) problem, where the new equipment is added to the existing plant and is operated either in-phase or out-of-phase with the existing units in each stage. Fletcher et al.2 extended Vaselenak et al.’s formulation by removing the restriction that any new equipment must be operated in the same manner for all products. Yoo et al.3 later generalized Fletcher et al.’s formulation by removing the difference between existing and new units and introducing the “group” concept. They defined a group as a set of units that are operated in-phase, whereas units in different groups are operated out-of-phase. Their model also allows the designer to sell old units with some salvage value. More recently, Montagna4 extended Yoo et al.’s work to include the possibility of installing storage tanks between stages. Aside from these MINLP formulations, Lee et al.5 and Lee and Lee6 presented a heuristic procedure to first determine the positions of new equipment to be added * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: ++31 15 278 8074. Fax: ++31 15 278 3422. † Faculty of Technology, Policy and Management. ‡ Faculty of Applied Sciences.
and then to solve the resulting NLP problem to obtain their optimal sizes. In all of the aforementioned problem formulations, the production capacity of the multiproduct batch plant is specified by only two parameters: the limiting batch size and the limiting cycle time. The retrofitting strategy considered in these approaches focuses on adding new equipment to either increase the limiting batch size or decrease the limiting cycle time for each product or for both. The availability characteristics of the existing plant and of the new retrofitted plant are not considered in these approaches. Because of inherent equipment failure characteristics and events such as administrative and logistical delays, human errors, etc., some unplanned shutdowns are unavoidable, which might lead to significant production losses and, accordingly, to reduced profitability. It is therefore critical to include information about existing plant availability and the possibility of improving plant availability while adding new equipment during retrofit design to obtain more robust design parameters and profitability projections. Availability, which is generally defined as the ability of an item to perform its required function at a stated instant of time or over a stated period of time, is determined by the reliability and maintainability of the item. The plant/item availability can be divided into three subtypes: operational, achievable, and inherent. Operational availability, although the most realistic of the three, is less important in design evaluations, as administrative and logistics downtime is outside the control of the designer. The achievable availability measure reflects the availability considering unplanned and planned maintenance time, and the inherent availability of a plant measures the availability to be expected when accounting for unscheduled (corrective) maintenance only. It is important to mention here that, although achievable availability is more realistic, it also introduces modeling complexity. For instance, to optimize achievable availability in the design stage, it is
10.1021/ie030656b CCC: $27.50 © 2004 American Chemical Society Published on Web 03/19/2004
3800 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 1. Illustrative example: (a) the existing plant, (b) the generalized superstructure, and (c) an (assumed) optimal solution.
essential to include detailed maintenance data reflecting the impact of different kinds of maintenance tasks on the total operating costs and the achievable availability of the system. In this work, for simplicity, only the inherent availability characteristics of the existing and new retrofitted plant are considered in the problem formulation. The inherent availability of a plant with given constant corrective maintenance time for underlying equipment is dictated mainly by decisions such as process system configuration and initial equipment reliabilities and can be improved by increasing the reliability of the equipment and/or adding redundancy. It must be noted here that inherent availability is a “built-in” attribute of the equipment/system and, by definition, does not include the unplanned downtime caused by inherent process variations such as interruptions for cleaning, etc. Two approaches are predominant in accounting for the inherent availability of a plant at the design stage: sequential and simultaneous approaches. In the more traditional sequential approach, different reliability analysis tools/methods such as reliability block diagrams, Petri net simulations, fault tree analyses, etc., are used to analyze the availability characteristics of given design alternatives. The results of availability assessment provide information to improve the design by minimizing the cost of unavailability (i.e., revenue loss, maintenance costs). Although simple in application, this approach leads to expensive design iterations, and as the number of possible design alternatives increases, it becomes essentially impractical. To avoid expensive design iterations and to allow for the evaluation of all possible design alternatives, a new simultaneous approach has become prominent in the past decade. This approach is focused on maximizing the system effectiveness measure by making reliability and maintenance decisions simultaneously with process design/synthesis decisions at the synthesis step.7-10 In the context of multiproduct batch plant design, Pistikopoulos et al.8 presented a simultaneous approach that includes the reliability and maintainability aspects of equipment while selecting optimal design parameters. They used a one-failure-mode Markovian model to describe the inherent availability of the plant. However, as their optimization framework assumes a fixed system structure and initial reliability of process components, its application is limited to an advanced stage of process synthesis when the process structure has already been frozen. More recently, for general process systems, Goel et al.10 proposed a new simultaneous optimization
method that provides the designer with the flexibility to configure a process or select initial reliabilities of equipment in a way that optimizes the inherent plant availability together with other design parameters. The added flexibility in the latter work is made possible through the use of a simple reliability block diagram model, as opposed to a Markovian model, to derive the inherent availabilities of individual units and of the overall system. Until now, approaches that consider reliability and maintainability simultaneously with other design parameters have been focused primarily on grassroots designs and applied mainly to continuous plants. In this paper, we extend our previous work10 to the case of multiproduct batch plant retrofitting and develop a new simultaneous optimization framework that combines a process model and an availability model to obtain optimal size, operating mode, and optimal allocation of reliability for new equipment during retrofitting. The existing retrofitting formulation of Yoo et al.3 is extended to account for production losses due to unplanned shutdown and maintenance costs and is used as a process model in our work. 2. Illustrative Example A small retrofit design problem of an existing twostage multiproduct batch plant, producing products A and B, is chosen for illustration. For a given new product demand, the retrofit strategy of Yoo et al.3 can be used to add new equipment to the existing plant. For a case in which only one piece of equipment can be added at each stage of an existing plant, Figure 1a-c shows the existing plant, the generalized superstructure (as described in Yoo et al.3), and an (assumed) optimal solution, respectively, for this example. Consider a case in which the reliability and maintainability data for existing equipment and for new equipment items are given. For simplicity, let us assume that the inherent availability (obtained from given reliability and maintainability data) for both existing and new equipment items is 97%. The overall plant availabilities of the existing and the new retrofitted plant can be estimated by constructing a reliability block diagram (RBD, as shown in Figure 2. With a simple analytical expression (described in eq 26 below) to estimate the system availability for series configurations, the plant availabilities for both the existing and retrofitted plants are estimated to be 94.09 and 91.26%, respectively. It is important to explain the choice of using a series configuration to represent the reliability block diagram
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3801
Figure 2. Reliability block diagram for an illustrative example.
superstructure derived from the process superstructure. As pointed out by Dekker and Groenendijk,11 the reliability block diagram derived from the process flow diagram should reflect the consequences of failures of the equipment. The consequences of failures are dictated by the process interactions as defined in the process model. Further, the reliability block diagram should be constructed on the basis of a specific product. In the case of the multiproduct batch plant, different products are manufactured using the same equipment operated in the same number of stages. The optimal production value for each product obtained with the Yoo et al.3 process model is only for the system state in which all of the underlying equipment (both existing and new) is working. Therefore, in this work, a series configuration is used to represent the reliability block diagram superstructure. The consequence of assuming a series reliability block diagram superstructure in the present work is that a conservative set of optimal values of design parameters such as batch size, number of batches for each product, capacity of new equipment unit, etc., will be obtained. For example, in our illustrative example, the inherent availability of a retrofitted batch plant is estimated to be 91.26%. The 91.26% availability means here that the retrofit plant will run 91.26% of the time with “all” equipment running. In practice, the plant can run at reduced capacity (in the event of equipment failure). For instance, at the second stage of the retrofitted plant, we have two pieces of equipment, and in the event of failure of one piece of equipment in the second stage, depending on the type of product, the plant can still be operational. Thus, by using a series system, we assume that the combined production of all reduced states is negligible. This could be true for cases in which the inherent availabilities of underlying pieces of equipment are quite high or few of the possible reduced states are operational. In other cases, however, the optimal solution obtained with the present formulation will be on the conservative side. The alternative approach is to enumerate every single possible operational state, to assign probabilities for each state, and then to use a Markovian model to estimate the effective production. This approach can be applied only to a given system configuration, whereas in the present work, the system structure is to be determined. Further, as the number of equipment units increases, the assignment of probabilities and production capacities to each state becomes a formidable task. Therefore, for those cases where it is important to consider the production due to reduced states, a twostep approach can be undertaken. In the first step, a conservative design should be found with the present formulation. Then, for a selected structure, a Markovian model can be applied to fine tune the design parameters. It is apparent that the addition of new equipment in series to existing equipment results in the reduction of the overall plant availability. The overall plant availability can be improved during retrofitting by procuring more reliable new equipment. For example, let us
consider the case in which the new equipment is also available in a different type with an inherent availability of 99%. The overall plant availability of the retrofitted plant would then be 93.14%. In light of this new information gained from the separate availability analyses of both existing and new retrofitted plants, one can observe the following: (a) Conventional retrofitting formulations (in this case Yoo et al.3) obtain optimal design parameters, sizes, and operating modes for new equipment without considering production losses due to the unavailability of the existing plant (5.91%) and of the new retrofitted plant (8.74%). Therefore, these production losses due to unplanned downtime directly result in revenue loss. (b) Plant availability also impacts the maintenance costs, which are a significant part of the total operating costs. In previous formulations, the maintenance costs were not considered in the objective function. Hence, the opportunity to balance maintenance costs and design costs during retrofitting is lacking in previous formulations. The aforementioned shortcomings of previous formulations are addressed in our new retrofit problem formulation. The production losses due to unavailability are compensated by a more robust strategy, derived by combining the process model with the availability model. The strategy, as illustrated in Figure 3, is to match the effective throughput with the projected demand for each product during the retrofit. 3. Modeling Framework In this work, the effective production capacity of a multiproduct batch plant is defined by three parameters: the limiting batch size, the limiting cycle time, and the overall plant availability in a given time horizon. The first two parameters can be optimized in the retrofit problem by varying the size and the operating mode (in-phase or out-of-phase) of new equipment, whereas the plant availability can only be improved by selecting the appropriate plant configurations and levels of initial reliability of new equipment. The key elements of our approach are as follows: (i) a process model as described in Yoo et al.3, which is extended here to include (a) the impact of overall system availability on the overall production, (b) the estimation of maintenance costs as a function of equipment availability, and (c) the estimation of additional capital investment needed for availability improvement of new equipment; (ii) an availability model that describes the availability of the equipment, both existing and new, and the plant availability as a function of equipment availability; and (iii) an expected profit objective function, which takes into account the tradeoff between initial capital investment and the annual operational costs. The retrofit design problem for multiproduct batch plants can be defined as follows: Given (i) a new production target, selling price, unit cycle times, and size factors for each product; (ii) the
3802 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 3. New retrofitting strategy.
existing plant configuration, including the size, cost, reliability, and maintainability data for existing units; and (iii) the number, size, reliability, and maintenance characteristics and costs of new equipment available, determine (i) the net expected profit and the revised plant configuration, (ii) the method of grouping parallel units and various processing parameters for each production campaign, and (iii) the optimal inherent availability for selected new equipment. 4. Problem Formulation In this section, we describe the mathematical foundations and assumptions made in the development of an optimization framework for formulating the retrofit design problem for multiproduct batch plants. 4.1. Process Model. The process model as described in Yoo et al.3 is extended in this section. In particular, the model is extended here to include the impact of the overall system availability on the overall production by considering HAsys as the maximum time available for production. Second, the capital cost estimation model is extended to become a function of the inherent availability of the equipment. Finally, the maintenance cost estimation model is adapted to include the estimated maintenance cost as a function of the inherent reliability of the equipment. The products are identified by index i, and N represents the total number of products. The batch processing stages are identified by index j, and the total number of stages in the plant is represented by the parameter M. Each stage is assumed to consist of a number of pieces of equipment or units, and the total number of the existing units in a stage j is Nold j . The total number of new units that can be added during retrofitting in stage j is Zj. The parallel units (both existing and new)
in each stage are identified by index k, and the total number of existing and new units is given by Ntotal j () Nold + Zj). Index l is used to indicate the level of j inherent availability of new equipment available at the parallel units of stage j can be retrofit stage. The Ntotal j groups, identified by grouped arbitrarily into Gtotal j index g. The retrofit strategy is determined by the value of binary variable yijkg representing unit-to-group assignments, a pseudo-binary/real variable eijg indicating whether group g exists or not on stage j, and the variable yjkl indicating the level of inherent availability chosen for new equipment at the design stage. It should be noted here that the pseudo-binary/real variable eijg, as explained in Yoo et al.,3 is actually a real variable but behaves like a binary variable when it is used in constraints describing the condition on group existence and defining an upper bound on its value. For product i in a unit of stage j, the unit cycle time, Tij, is conventionally expressed as
∀i ) 1, ..., N; j ) 1, ..., M (1)
Tij ) tij + cijBγi j
where tij, cij, and γj are fixed parameters and Bi is the limiting batch size for product i. For an overlapping mode, the limiting cycle time for product i is given by
TLi ) max
j)1,...,M
( ) Tij Gij
∀i ) 1, ..., N
(2)
where Gtotal j
Gij )
∑ eijg
g)1
∀i ) 1, ..., N; j ) 1, ..., M
(3)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3803
The limitation on production of each product is given by the constraint
∀i ) 1, ..., N
Each unit k at stage j can be assigned to at most one group for product i
(4)
Gjtotal
where Qi is the upper bound on the production of product i. The time available for the production of each product within the given time horizon (H) is given by the constraint
g)1
niBi e Qi
N
niTLi e HAsys ∑ i)1
(5)
∑ yijkg e 1 ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal (13) j
For unit k to be assigned to group g in stage j for product i, the unit k must be installed and the group g must exist
yijkg e yjk ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal ; j
where Asys is the overall plant availability during the given time horizon. Combining eqs 1-3 yields the constraint
tij + cijBγi j TLi
yijkg e eijg
Gtotal j
g
∑ eijg g)1
total ∀j ) 1, ..., M, k ) Nold (7) j + 1, ..., Nj
VLj
where Vjk is the volume of the new unit k in stage j and VU j , respectively, are the lower and upper limits on the volume for the chosen new unit. To ensure the distinct assignment of new units, the following constraints are included
yjk g yj,k+1 ∀j ) 1, ..., M; k )
Nold j
+ 1, ...,
Ntotal j
+ 1, ...,
Ntotal j
- 1 (8)
Vjk g Vj,k+1 Nold j
- 1 (9)
The requirement that the volume be sufficient to process the batch size yields the following constraint Nold j
Ntotal j
- eijg) +
∑ Vjkyijkg g SijBi k)1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal (10) j This constraint contains the product of a real variable Vjk and a binary variable yijkg, which adds difficulty to the convergence. Introducing the continuous positive variable Vijkg linearizes the nonlinearities of the form Vjkyijkg in eq 10 by replacing it with the following set of constraints Nold j
(
∑ Vjk +
Ntotal j
ZjVU j )(1
Group g can exist in stage j for product i only if unit k is assigned to the group
eijg e
- eijg) +
k)1
∑ Vijkg g SijBi
k)1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal (11) j Vijkg e VU j yijkg, Vijkg e Vjk ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal ; j (12) g ) 1, ..., Gtotal j
∑ yijkg
k)1
∀i ) 1, ..., N, j ) 1, ..., M, g ) 1, ..., Gtotal (16) j The upper bound of the variable eijg is 1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal j (17)
eijg e 1
Redundant assignment to a group with the same value for the objective function can be avoided by introducing the following constraint3 Ntotal j
∀j ) 1, ..., M; k )
∑ Vjk + k)1
(15) g ) 1, ..., Gtotal j
Njtotal
VLj yjk g Vjk g VU j yjk
(
∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal ; j
∀i ) 1, ..., N; j ) 1, ..., M (6)
The lower and upper bounds on the volumes of new units are ensured by the constraint
ZjVU j )(1
(14) g ) 1, ..., Gtotal j
∑2 k)1
Njtotal-k
Ntotal j
yijkg g
∑2 k)1
Njtotal-k
yijk,g+1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal -1 j (18) The cost of a new unit k in stage j in the previous formulation is expressed by a function of the volume Vjk of the form
f(Vjk) ) K0 + K1j Vrjkj
(19)
where K0 is the annualized fixed charge; is the annualized proportionality constant of a new unit in stage j; and rj is the exponential constant of a new unit in stage j, which is considered to be equal to 1 in this work. In this work, to estimate the extra investment needed to improve the availability of new equipment, we need to extend the conventional cost model to make it a function of the inherent availability of equipment. There are two alternatives for extending the existing cost models that would describe the cost-reliability function of equipment in an objective function. The alternatives are (i) using exponentially increasing closed-form functions to relate the cost and the reliability/availability of the equipment10 or (ii) directly using the discrete set of cost and reliability data of an equipment unit in the design problem.12
3804 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
4.2. Availability Model. In this section, we develop an availability model to estimate the total plant availability (Asys) and equipment availability (Ajk), which is used in the development of eqs 5 and 21 in the process model described earlier. The inherent availability (Ajk) of new unit k in stage j is given by P
Ajk )
A h jlyjkl ∑ l)1 total ∀j ) 1, ..., M; k ) Nold (22) j + 1, ..., Nj
Figure 4. Plot of inherent availability versus initial investment costs.
In principle, either of the above two alternatives can be used in the problem formulation. Actually, the closedform exponentially increasing function for reliability and cost is derived from the discrete sets and is mainly used in problem formulation to minimize the total number of binary variables in the final problem. In industry, discrete sets of reliability-cost data can be either gathered in-house from purchase and maintenance departments or requested externally from equipment suppliers. Once the discrete sets are available for equipment units, the continuous function can be estimated by plotting the data on a reliability-cost diagram, as shown in Figure 4. It is important to note here that the choice of describing the relation between cost and reliability by a continuous function or by discrete sets has a significant impact on the complexity and computational burden of the resulting problem. For example, describing the relation as a continuous exponential function introduces nonlinearity into the objective function, whereas using discrete sets increases the total number of binary variables in the problem. In this work, discrete sets of reliability-cost data are used directly to avoid nonlinearity in the final problem. The new extended cost estimation model can be given as
f1(Vjk,Ajk) ) K0 + K1j Vrjkj + K2jlyjkl
(20)
where K2jl is the annualized fixed charge associated with the selection of alternative l for the new unit in stage j. For example, for a piece of equipment used in Figure 4, the value of K2jl can be estimated (considering type A as base case) as 0, 1000, and 3000, respectively, for types A, B, and C. The maintenance costs constitute a significant portion of the total operating costs. In the previous formulation, these costs are not included in the objective function. The maintenance costs consist of both corrective and preventive maintenance costs. Because only inherent availability is considered in this work, preventive maintenance costs are not included in the objective function. The corrective maintenance cost is dictated by the inherent reliability of each unit j and can be estimated as a function of Ajk13
f2(Ajk) ) Ccj H(1 - Ajk)/∆cj
(21)
where Ccj and ∆cj are the cost and duration of corrective maintenance for unit k in stage j.
where the parameter A h jl describes the inherent availability of alternative l available for the new unit in stage j. The following constraint ensures that only one alternative is selected if new unit k is selected in stage j during the retrofitting process P
yjkl ) yjk ∑ l)1
total ∀j ) 1, ..., M; k ) Nold j + 1, ..., Nj
(23)
The inherent availability for existing units is given by
Ajk ) yjkAold jk
∀j ) 1, ..., M; k ) 1, ..., Nold (24) j
where Aold jk is the parameter describing the inherent availability of existing units. The parameters A h jl and Aold jk can be estimated from historic reliability and maintainability data. For instance, for a given constant failure rate λold jk and repair old rate µold jk for existing unit k in stage j, Ajk can be estimated from
Aold jk )
µold jk µold jk
+
λold jk
∀j ) 1, ..., M; k ) 1, ..., Nold (25) j
The total plant availability is estimated from the inherent availabilities of units by using a reliability block diagram (RBD). The generic reliability block diagram superstructure is illustrated in Figure 5. The total plant availability can be expressed as j M Ntotal
Asys )
∏ ∏ A′jk j)1 k)1
(26)
A′jk ) Ajkyjk + (1 - yjk) (27) ∀j ) 1, ..., M; k ) 1, ..., Ntotal j where variable A′jk in eqs 26 and 27 is a dummy variable, which is described by the relation explained in eq 27. Constraint 27 ensures that only the availabilities of equipment selected at each iteration are considered in the estimation of the overall system availability (Asys) and sets the availability of nonexisting equipment to unity. Constraint 27 contains the product of a real variable, Ajk, and a binary variable, yjk, which adds difficulty to the convergence. Introducing a continuous positive variable, A′′jk, linearizes the nonlinearities of the form Ajkyjk in eq 27 by replacing it by the following set of constraints
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3805
Figure 5. Reliability block diagram superstructure.
A′jk ) A′′jk + (1 - yjk)
∀j ) 1, ..., M; k ) 1, ..., Ntotal j (28)
Ajk - max{A h jl}(1 - yjk) e A′′jk e Ajk ∀j ) 1, ..., M; k ) 1, ..., Ntotal (29) j
Equations 24-29) constitute the availability model. 4.3. Objective Function. The objective function of the problem, which is to be maximized, is the expected annualized net profit. Expected net profit is defined here as the net income minus the annualized investment and operating costs. The objective function can be represented as M Njold
N
∑ i)1
piniBi +
M
-
∑ ∑ Rjk(1 - yjk) j)1 k)1
Ntotal j
∑ ∑ j)1
(K0j yjk + K1j Vjk) -
k)Nold j +1 M
P
Ntotal j
∑ ∑ ∑ j)1 l)1
K2jlyjkl
k)Nold j +1
M
-
stage 1
stage 2
Ntotal j
∑ ∑ Ccj H(1 - Ajk)/∆cj j)1 k)1
A B
4.0 5.0
6.0 3.0
A B
2.0 1.5 1 4000 0.97 2 0 4000 30 560 32.54 10 200 0, 1000, 2100 0.97, 0.98, 0.99
1.0 2.25 1 3000 0.97 2 0 3000 30 560 32.54 10 250 0, 1000, 2100 0.97, 0.98, 0.99
Sij
l
max
parameter tij
h jl}yjk, A′′jk e max{A l
Table 1. Input Data for Example 1
(30)
The first term of the objective function is the revenue from product sales. The second term corresponds to the income from disposed batch units, and the third and fourth terms correspond to investment costs and the cost of increasing the inherent availability of new batch units, respectively. The last term corresponds to the accumulated corrective maintenance costs. The problem described by eqs 1-30 corresponds to an MINLP problem and can be solved by the outer approximation (OA) algorithm of Duran and Grossmann.14 The MINLP problem described above contains several nonconvex terms in the constraints and in the objective function. The exponential transformation of nonconvex terms, as described in Vaselenak et al.,1 is used to remove nonconvexities. The resulting set of transformed equations is given in Appendix A. 5. Examples Three examples are presented to demonstrate that the new retrofit strategy provides greater flexibility and more robust solutions as compared to the conventional formulations. The first two examples are taken from previous works, Vaselenak et al.1 and Yoo et al.3, respectively. For comparison with the previously published results, these two examples are solved first for the case in which maintenance costs are considered in the objective function and second for the case in which
Nold j Vold j Aold jk Zj VLj VU j K0 K1j ∆cj Ccj K2jl A h jl pi A B Qi A B
1.0 2.0 1 200 000 1 000 000
they are included in the objective function. The third example is added to demonstrate the sensitivity of the results to the new cost parameters (K2jl and Ccj ) introduced in the formulation presented in this work. The examples are solved using the DICOPT2+ solver in the GAMS environment on an AMD athlon processor. 5.1. Example 1. An existing multiproduct batch plant consisting of two stages is considered to produce products A and B. The process data for this example are taken from Vaselenak et al.1 and are given in Table 1. Table 1 also includes three potential alternatives for new equipment units with different inherent availabilities and capital costs that are considered available at the retrofitting stage. The relationship between the inherent availability and the costs reflects the commonly used exponential relationship between reliability and capital cost. The example is solved for two different cases: a formulation in which the maintenance cost model is excluded (case 1) and a formulation in which it is included (case 2). The optimal structures and groupings for products A and B for the new retrofitted plant obtained for the two cases with the present formulation are similar to those obtained by Yoo et al.3 Table 2 shows the results obtained using the model by Yoo et al.3 and the results obtained with the model developed in this work. It is interesting to note the following from Table 2: (1) The proposed formulation results (in both cases) in a lower net profit than reported by Yoo et al.3 This can be explained by the fact that Yoo et al.3 did not account for the revenue loss caused by unplanned downtime and the maintenance costs (in case 2).
3806 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 2. Results for Example 1 this work Yoo et TLi Bi ni niBi new units
al.3
case 1
case 2
product A
product B
product A
product B
product A
product B
6 2679 448 1 200 000
3 905 1104 1 000 000
6 2779 431 1 200 000
3 1039 961 1 000 000
6 2755 435 1 200 000
3 1007 992 1 000 000
Vjk stage 1 stage 2 overall availability maintenance costs ($) profit ($)
1358
Ajk
Vjk
Ajk
Vjk
Ajk
-
1559
0.97
1511
0.99
0.913
0.931 9,300
3,125,236
3,118,698
3,118,698
Table 3. Input Data for Example 2 parameter
stage 1
stage 1
tij A B
1.0 2.0
1.0 1.0
A B
Zj VLj
4.0 1.0 2 2000, 3000 0.96, 0.96 3 1000
2.0 2.0 3 1000, 2000, 3000 0.96, 0.96, 0.96 2 1000
VU j Rold jk K0 K1j ∆cj Ccj
3000 24 000, 32 000 10 000 10 10 100
3000 16 000, 24 000, 32 000 10 000 10 10 120
K2jl A h jl pi
0, 300, 800 0.96, 0.98, 099
0, 300, 800 0.96, 0.98, 0.99
Sij Nold j Vold j Aold jk
A B
Figure 6. Optimal structure for example 2 obtained with the formulation of Yoo et al.3
0.15 0.10
Qi A B
2 000 000 4 000 000
(2) In case 2, the extra capacity needed because of the unavailability of the existing and retrofitted plant is compensated partly by an increase in the volume of new equipment and partly by the selection of new equipment with a better inherent availability (option 3). The choice between increasing the volume versus increasing the inherent availability of new equipment is dictated by the marginal costs for capacity (K1j ) and inherent availability (K2jl) and the maintenance data (Ccj and ∆cj ). Thus, it is important to note that the optimal solution is sensitive to the values chosen for K1j , K2jl, Ccj , and ∆cj . (3) Further it is important to compare the case 1 and case 2 results. The equipment chosen in case 2 has a better inherent availability (option 3) and reduced size is chosen as compared to that chosen in case 1. This could be explained by the fact that maintenance costs are a function of inherent availability only and, therefore, to reduce maintenance costs, more reliable equipment is chosen as an optimal solution with a corresponding capacity compensation.
5.2. Example 2. This example is taken from Yoo et al.,3 and it illustrates the disposal of an existing unit. The input data for this example are listed in Table 3. Figures 6 and 7 show the optimal plant structure and groupings for both products A and B. As shown in Figure 7 (for case 1), the revised plant obtained by the proposed formulation disposes of the existing equipment units of volume 3000 L in stage 1 and of volumes 1000 and 2000 L in stage 2, while adding a new unit of volume 1327 L with an inherent availability of 98% in stage 1. Other optimal design parameters such as batch size, number of batches, limiting cycle time, etc., for each product are summarized in Table 4. The extra capacity needed because of the unavailability characteristics of the existing and retrofitted plant in example 2 is compensated partly by increasing the volume of new equipment and partly by increasing the total plant availability. In Table 4, the new equipment chosen in case 2 has a better inherent availability (option 3) and reduced size as compared to that chosen in case 1. 5.3. Example 3. In the previous two examples, it is shown that adding maintenance costs in the objective function mainly influences the inherent availability and capacity of the new equipment. This example is intended
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3807 Table 4. Results for Example 2 this work Yoo et TLi Bi ni niBi new units
al.3
case 1
case 2
product A
product B
product A
product B
product A
product B
1 1000 2000 2 000 000
1 1000 4000 4 000 000
1 831 2404 2 000 000
1 1326 3014 4 000 000
1 827 2417 2 000 000
1 1308 3056 4 000 000
Vjk stage 1 stage 2 sold units stage 1 stage 2 overall availability maintenance costs ($) profit ($)
Ajk -
1358 2000 1000, 3000 752,000
Vjk
Ajk
1327
0.98
3000 1000, 2000 0.903 748,430
Vjk 1308
Ajk 0.99
3000 1000, 2000 0.912 5,318 742,512
Figure 8. Optimal structure for example 3, scenario 1.
Figure 7. Optimal structure for example 2 obtained with the proposed formulation (case 1).
to show the sensitivity of the optimal solution to the new cost parameters K2jl and Ccj . The example8 considers the retrofitting of an existing plant that produces two products to be processed in three stages. The input data for this example are given in Table 5. The example is solved first for the values of K2jl and c Cj given in the Table 5, and then for two different scenarios. In the first scenario, K2jl remains the same, but the value of Ccj is increased by 50%. Conversely, in the second scenario, Ccj remains the same, but the value of K2jl is increased by 50%. Figures 8 and 9 show the revised plants obtained for the different cases. As shown in Figure 8 (for the nominal and scenario 1 cases), the existing equipment units of volume 2000 L in stage 1 and 2500 L stage 2 are discarded, and new units of volumes 2500 and 1875 L are added in stages 1 and 2, respectively, in the new
Figure 9. Optimal structure for example 3, scenario 2.
retrofitted configuration. For scenario 2, the optimal structure is shown in Figure 9, where the existing equipment units of volume 2000 L in each stage 1 and stage 2 are discarded, and new units of volumes 2500
3808 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 5. Input Data for Example 3 parameter
stage 1
stage 2
stage 3
tij A B
8 16
20 4
8 4
A B
2 4 2 2500, 2000 0.96, 0.96 1 500
3 6 2 2000, 2500 0.96, 0.96 1 500
4 3 1 2500 0.96 2 500
VU j Rold jk K0 K1j ∆cj Ccj
2500 75 000, 85 000
2500 80 000, 95 000
2500 90 000
35 000 30 15 100
35 000 35 10 100
40 000 40 15 150
K2jl A h jl pi
0, 3000, 6000 0.96, 0.98, 0.99
0, 3000, 6000 0.96, 0.98, 0.99
0, 3000, 6000 0.96, 0.98, 0.99
Sij Nold j Vold j Aold jk Zj VLj
A B
5.5 7.0
Qi A B
250 000 250 000
Table 6. Results Obtained in This Work for Example 3 nominal TLi Bi ni niBi new units stage 1 stage 2 stage 3 sold units stage 1 stage 2 stage 3 overall availability maintenance costs ($) profit ($)
scenario 1
scenario 2
product A
product B
product A
product B
product A
product B
10 625 229 194 700
8 635 364 240 700
10 625 229 194 700
8 635 364 240 700
10 625 224 192 600
8 635 356 238 700
Vjk 2500 1875
Ajk 0.99 0.99
Vjk 2500 1875
Ajk 0.99 0.99
Vjk 2500 1875
Ajk 0.98 0.98
2000 2500
2000 2500
2000 2000
0.867 7,400 2,705,810
and 1875 L are added in stages 1 and 2, respectively, in the new retrofitted configuration. It is interesting to note here that, in all three cases, the same numbers of groups are obtained for products A and B and two new equipment units of similar capacities are added. The results for the nominal case and for the two scenarios are summarized in Table 6. It is interesting to note the sensitivity of the optimal results to the values of K2jl and Ccj . The points of deviations are different profit projections in each case, and in the case of scenario 2, different existing pieces of equipment are disposed. The difference in profitability between the nominal case and scenario 1 can be explained by the increment in the maintenance cost, and similarly, the selection of the lower inherent availability in case 2 is dictated by the increased incremental cost, K2jl. From Table 6, it can be observed that the optimal design parameters are sensitive to K2jl in this particular example, and therefore, uncertainty in these data should be minimized by requesting cost and reliability data for different equipment types from suppliers.
0.867 11,100 2,702,110
0.850 8,400 2,667,663
Table 7. Computational Performance number total CPU of binary number of number of time example formulation variables variables constraints (s) 1 2 3
Yoo et al.3 this work Yoo et al.3 this work this work
60 80 130 160 110
151 188 295 356 257
280 317 592 653 451
1.2 2.0 7.9 14.2 11.6
The computational statistics are summarized in Table 7. The number of binary variables in the third column also includes the binary variables needed to represent piecewise linearization of the negative exponential term in the objective function. It should be noted that the computational burden of the proposed formulation is of the same order of magnitude as the computational burden of Yoo et al.’s formulation. In Table 7, the computational details of only one of the cases are reported, as there is a very slight difference in computational burden for different cases for the same example.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3809
6. Conclusions
) total number of parallel units in stage j Ntotal j
A new optimal retrofit method is presented for multiproduct batch plant design. The key improvements of this method over previous methods are as follows: (i) This approach considers the reliability and maintainability of existing and new equipment units and uses this information to quantify the costs of unavailability (revenue loss due to production loss and maintenance costs due to increased unplanned shutdowns). (ii) It performs a tradeoff between the cost of unavailability and the extra capital investment needed to increase the size and/or inherent availability of new equipment while maximizing the overall expected net profit, thus providing a more robust retrofit solution. As compared to previous formulations, the proposed new method requires additional data to be specified, 2 c c such as the values of A h jl, Aold jk , Kjl, Cj , and ∆j . The authors found that these data can be easily obtained from internal sources such as the company log book, purchase department, maintenance department, etc., or they can be requested from external sources such as vendors. In cases in which the data are not readily available, the cost of obtaining these data can be included in the objective function. The effectiveness of the proposed method is demonstrated by means of three examples. These examples clearly demonstrate that the proposed method provides greater flexibility to the designer to obtain a more robust and reliable retrofit strategy with only a moderate increase in computational time.
VU j ) maximum volume of new units in stage j
Acknowledgment The authors dedicate this paper to Art Westerberg to whom they are indebted for his visionary perspective, enthusiastic involvement, and invariably challenging comments on a wide range of engineering research projects at TU Delft since 1997. Moreover, they express their deep appreciation for the stimulating mix of wit, insight, and creativity that Art always brought to Delft. Nomenclature Indexes i ) products j ) stages k ) units g ) groups l ) design alternatives for inherent availability improvement Parameters N ) number of products manufactured M ) number of batch processing stages in the process P ) number of available design alternatives for inherent availability improvement Gtotal ) total number of groups in stage j j H ) operating time period Sij ) size factor of product i in stage j Tij ) operation time for product i in stage j cij ) parameter in the expression for Tij γj ) parameter in the expression for Tij tij ) processing time of product i in stage j Zj ) maximum number of new units that can be added to stage j Nold ) number of existing units in stage j j
VLj ) minimum volume of new units in stage j Qi ) upper bound on the production of product i pi ) expected net profit per unit of product i A h jl ) inherent availability of design alternative l for a unit in stage j Aold jk ) inherent availability of existing units Rold jk ) annualized capital cost returned when the existing unit k in stage j is sold rj ) exponential constant for a new unit in stage j K0 ) annualized fixed charge for a new unit in stage j K1j ) annualized proportionality constant for a new unit in stage j K2jl ) annualized fixed charge associated with the selection of alternative l for a new unit in stage j Ccj ) cost of corrective maintenance for a unit in stage j ∆cj ) duration of corrective maintenance for a unit in stage j λj ) constant failure rate of unit k in stage j µjk ) constant repair rate of unit k in stage j Variables Continuous Variables ni ) number of batches of product i Bi ) batch size of product i TLi ) limiting cycle time of product i Vjk ) volume of new unit k in stage j Vijkg ) volume of unit k in stage j for the use of product i in group g Ajk ) inherent availability of unit k in stage j A′jk ) dummy variable used in eq 26 A′′jk ) dummy variable used in eq 28 Asys ) total plant availability eijg ) indication of whether group g exists on stage j Binary Variables yjk ) binary variable for unit k in stage j yjkl ) binary variable for the selection of availability improvement alternative l of unit k in stage j yijkg ) binary variable for the inclusion of unit k in stage j for the use of product i in group g
Appendix A: Convexification of the MINLP To ensure that the global optimum of the MINLP problem described by eqs 1-27 is obtained, the following exponential transformations are introduced (Vaselenak et al.1)
x1i ) ln ni x2i ) ln Bi x3i ) ln TLi i ) 1, ..., N
(A1)
3810 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Using these transformations, the following formulation is obtained
Ntotal j
eijg e
M Nold j
N
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal (A15) j
pi exp(x1i + x2i) - ∑ ∑ Rold ∑ jk (1 - yjk) i)1 j)1 k)1
min -
∑ ∑ j)1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal j (A16)
eijg e 1
Ntotal j
M
+
∑ yijkg
k)1
(K0j yjk + K1j Vjk) +
Njtotal
k)Nold j +1 M
Njtotal
P
∑ ∑ ∑ j)1 l)1
K2jlyjkl
∑ k)1
Njtotal total-k
2Nj
j M Ntotal
Ccj H(1 - Ajk)/∆cj ∑ ∑ j)1 k)1
(A2)
yijk,g+1
A h jlyjkl ∑ l)1 total ∀j ) 1, ..., M; k ) Nold (A18) j + 1, ..., Nj
∀i ) 1, ..., N
(A3)
P
yjkl ) yjk ∑ l)1
Gtotal j
tij exp(-x3i) + cij exp(γjx2i - x3i) e
total-k j
P
Ajk )
subject to the following constraints
x1i + x2i e ln Qi
2N ∑ k)1
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal -1 j (A17)
k)Nold j +1
+
yijkg g
∑ yijg
total ∀j ) 1, ..., M; k ) Nold j + 1, ..., Nj
(A19)
g)1
∀i ) 1, ..., N; j ) 1, ..., M (A4) N
exp(x1i + x3i) e HAsys ∑ i)1 exp(x2i) eBi Nold j
(
(A5)
M Ntotal j
Asys )
∀i ) 1, ..., N
(A6)
∏ ∏ A′jk j)1 k)1
(A21)
A′jk ) Ajkyjk + (1 - yjk)
Ntotal j
∑ Vjk + ZjVUj )(1 - eijg) + k)1 ∑ Vijkg g SijBi k)1
(A22) ∀j ) 1, ..., M; k ) 1, ..., Ntotal j
∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal (A7) j Vijkg e VU j yijkg, Vijkg eVjk ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal ; j g ) 1, ...,
Gtotal j
total ∀j ) 1, ..., M; k ) Nold (A9) j + 1, ..., Nj
A′′jk e max{A h jl}yjk, Ajk - max{A h jl}(1 - yjk) e A′′jk e Ajk l
l
j ) 1, ..., M; k ) 1, ..., Ntotal (A24) j In addition, the following constraints are added by Montagna4 to reduce the computational burden Gtotal Ntotal j j
yjk g yj,k+1 total ∀j ) 1, ..., M; k ) Nold - 1 (A10) j + 1, ..., Nj
∑ ∑ yijkg ) g)1 ∑ k)1 ∑ yi'jkg
∀i, i′ ) 1, ..., N, i * i′; j ) 1, ..., M (A25)
j N Gtotal
yjk e
Gtotal j
∑ yijkg e 1 g)1
Gtotal Ntotal j j
g)1 k)1
Vjk g Vj,k+1 total ∀j ) 1, ..., M; k ) Nold - 1 (A11) j + 1, ..., Nj
∀j ) 1, ..., M; k ) 1, ..., Ntotal j (A23)
A′jk ) A′′jk + (1 - yjk)
(A8)
VLj yjk g Vjk g VU j yjk
∑ ∑ yijkg i)1 g)1
∀j ) 1, ..., M; k ) 1, ..., Ntotal j (A26)
(A12) ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal j
Gtotal j
∑ yijkg e yjk
g)1
yijkg e yjk ∀i ) 1, ..., N; j ) 1, ...,
∀j ) 1, ..., M; k ) 1, ..., Nold (A20) j
Ajk ) yjkAold jk
M; k ) 1, ..., Ntotal ; j total (A13) g ) 1, ..., Gj
yijkg e eijg ∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal ; j (A14) g ) 1, ..., Gtotal j
∀ i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Ntotal (A27) j yij,g+1 e yijg ∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gtotal (A28) j Ntotal Gtotal j j
∑ ∑ yijkg g 1
k)1 g)1
∀i ) 1, ..., N; j ) 1, ..., M (A29)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3811
Literature Cited (1) Vaselenak, J. A.; Grossmann, I. E.; Westerberg, A. W. Optimal retrofit design of multiproduct batch plants. Ind. Eng. Chem. Res. 1987, 26, 718-726. (2) Fletcher, R.; Hall, J. A. J.; Johns, W. R. Flexible retrofit design of multiproduct batch plants. Comput. Chem. Eng. 1991, 15, 843-852. (3) Yoo D. J.; Lee H.; Ryu J.; Lee I. Generalized retrofit design of multiproduct batch plants. Comput. Chem. Eng. 1999, 23, 263695. (4) Montagna, J. M. The optimal retrofit of multiproduct batch plants. Comput. Chem. Eng. 2003, 27, 1277-1290. (5) Lee, H.; Lee, I.; Yang, D. R.; Chang, K. S. Optimal synthesis for the retrofitting of multiproduct batch plants. Ind. Eng. Chem. Res. 1993, 32, 1087-1092. (6) Lee, H.; Lee, I. A synthesis of multiproduct batch plants considering both in-phase and out-of-phase modes. Comput. Chem. Eng. 1996, 20, S195-S200. (7) Grievink, J.; Smit, K.; Dekker: R.; Van Rijn, C. F. H. Managing reliability and maintenance in the process industry. In Proceedings of the Conference on the Foundation of Computer Aided Operations, FOCAPO; CACHE/CAST Division of AIChE: New York, 1993; pp 133-157. (8) Pistikopoulos, E. N.; Thomaidis, T. V.; Melin, A.; Ierapetritou, M. G. Flexibility, reliability and maintenance considerations
in batch plant design under uncertainty. Comput. Chem. Eng. 1996, 20, S1209-S1214. (9) Vassiliadis, C. G.; Pistikopoulos, E. N. Maintenance scheduling and process optimization under uncertainty. Comput. Chem. Eng. 2001, 25, 217-236. (10) Goel, H. D.; Grievink, J.; Herder, P. M.; Weijnen M. P. C. Integrating Reliability Optimization into Chemical Process Synthesis. Reliab. Eng. Sys. Saf. 2002, 78, 247-258. (11) Dekker: R.; Groenendijk, W. Availability assessment methods and their application in practice. Microelectron. Reliab. 1995, 1257-1274. (12) Majety, S. R. V.; Dawande, M.; Rajgopal, J. Optimal reliability allocation with discrete cost-reliability data for components. Oper. Res. 1999, 47, 899-906. (13) Dedopoulos, I. T.; Shah, N. Long-term maintenance policy optimization in multipurpose process plants. Chem. Eng. Res. Des. 1995, 74, 307-320. (14) Duran, M. A.; Grossmann, I. E. An outer approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 1986, 36, 307-339.
Received for review August 11, 2003 Revised manuscript received December 12, 2003 Accepted December 17, 2003 IE030656B
