optimal reliable retrofit design of multiproduct batch plants

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 ofTechnology, 2600 GA Delft, The Netherlands

The problem of the retrofit design of a multiproduct batch plant is considered from a newperspective that involves explicit consideration of inherent reliability and maintainabilitycharacteristics of existing and new equipment. Until now, in multiproduct batch plant retrofittingformulations, the production capacity has been specified by the limiting batch size and limitingcycle time. We propose a more robust retrofit solution that is obtained by defining the effectiveproduction capacity by three parameters: the limiting batch size, the limiting cycle time, andthe overall plant availability. The novel simultaneous optimization framework developed in thiswork combines a process model and an availability model to obtain the optimal size, optimaloperating mode, and optimal allocation of inherent availability for new equipment during theretrofit 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 ofexamples. This framework provides the designer with the opportunity to select the initial inherentavailability of new equipment during retrofitting by balancing the costs of design investmentsagainst the costs of downtime.

1. Introduction

Multiproduct batch plants are designed to produce anumber of related products using the same equipmentin the same operationsequence. The problem of theretrofit design of a multiproduct batch plant arises, forexample, when new production targets and marketselling prices are specified for one or more products orwhen there is a need to improve the overall effectivenessof the existing plant by improving its reliability andmaintainability characteristics. The retrofitting problemconsists of finding those plant modifications that involvethe removal of existing equipment (selling old units forsalvage value) and/or the purchase of new equipmentfor the existing plant to maximize the net profit.

Vaselenak et al.1 formulated the retrofit design of amultiproduct batch plant as a mixed integer nonlinearprogramming (MINLP) problem, where the new equip-ment is added to the existing plant and is operatedeither in-phase or out-of-phase with the existing unitsin each stage. Fletcher et al.2 extended Vaselenak et al.’sformulation by removing the restriction that any newequipment must be operated in the same manner forall products. Yoo et al.3 later generalized Fletcher et al.’sformulation by removing the difference between existingand new units and introducing the “group” concept.They defined a group as a set of units that are operatedin-phase, whereas units in different groups are operatedout-of-phase. Their model also allows the designer tosell old units with some salvage value. More recently,Montagna4 extended Yoo et al.’s work to include thepossibility of installing storage tanks between stages.Aside from these MINLP formulations, Lee et al.5 andLee and Lee6 presented a heuristic procedure to firstdetermine the positions of new equipment to be added

and then to solve the resulting NLP problem to obtaintheir optimal sizes.

In all of the aforementioned problem formulations, theproduction capacity of the multiproduct batch plant isspecified by only two parameters: the limiting batchsize and the limiting cycle time. The retrofitting strategyconsidered in these approaches focuses on adding newequipment to either increase the limiting batch size ordecrease the limiting cycle time for each product or forboth. The availability characteristics of the existingplant and of the new retrofitted plant are not consideredin these approaches.

Because of inherent equipment failure characteristicsand events such as administrative and logistical delays,human errors, etc., some unplanned shutdowns areunavoidable, which might lead to significant productionlosses and, accordingly, to reduced profitability. It istherefore critical to include information about existingplant availability and the possibility of improving plantavailability while adding new equipment during retrofitdesign to obtain more robust design parameters andprofitability projections.

Availability, which is generally defined as the abilityof an item to perform its required function at a statedinstant of time or over a stated period of time, isdetermined by the reliability and maintainability of theitem. The plant/item availability can be divided intothree subtypes: operational, achievable, and inherent.Operational availability, although the most realistic ofthe three, is less important in design evaluations, asadministrative and logistics downtime is outside thecontrol of the designer. The achievable availabilitymeasure reflects the availability considering unplannedand planned maintenance time, and the inherent avail-ability of a plant measures the availability to beexpected when accounting for unscheduled (corrective)maintenance only. It is important to mention here that,although achievable availability is more realistic, it alsointroduces modeling complexity. For instance, to opti-mize achievable availability in the design stage, it is

* 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.

3799Ind. Eng. Chem. Res. 2004, 43, 3799-3811

10.1021/ie030656b CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 03/19/2004

essential to include detailed maintenance data reflectingthe impact of different kinds of maintenance tasks onthe total operating costs and the achievable availabilityof the system. In this work, for simplicity, only theinherent availability characteristics of the existing andnew retrofitted plant are considered in the problemformulation.

The inherent availability of a plant with given con-stant corrective maintenance time for underlying equip-ment is dictated mainly by decisions such as processsystem configuration and initial equipment reliabilitiesand can be improved by increasing the reliability of theequipment and/or adding redundancy. It must be notedhere that inherent availability is a “built-in” attributeof the equipment/system and, by definition, does notinclude the unplanned downtime caused by inherentprocess variations such as interruptions for cleaning,etc.

Two approaches are predominant in accounting forthe inherent availability of a plant at the design stage:sequential and simultaneous approaches. In the moretraditional sequential approach, different reliabilityanalysis tools/methods such as reliability block dia-grams, Petri net simulations, fault tree analyses, etc.,are used to analyze the availability characteristics ofgiven design alternatives. The results of availabilityassessment provide information to improve the designby minimizing the cost of unavailability (i.e., revenueloss, maintenance costs). Although simple in application,this approach leads to expensive design iterations, andas the number of possible design alternatives increases,it becomes essentially impractical. To avoid expensivedesign iterations and to allow for the evaluation of allpossible design alternatives, a new simultaneous ap-proach has become prominent in the past decade. Thisapproach is focused on maximizing the system effective-ness measure by making reliability and maintenancedecisions simultaneously with process design/synthesisdecisions at the synthesis step.7-10

In the context of multiproduct batch plant design,Pistikopoulos et al.8 presented a simultaneous approachthat includes the reliability and maintainability aspectsof equipment while selecting optimal design parameters.They used a one-failure-mode Markovian model todescribe the inherent availability of the plant. However,as their optimization framework assumes a fixed systemstructure and initial reliability of process components,its application is limited to an advanced stage of processsynthesis when the process structure has already beenfrozen. More recently, for general process systems, Goelet al.10 proposed a new simultaneous optimization

method that provides the designer with the flexibilityto configure a process or select initial reliabilities ofequipment in a way that optimizes the inherent plantavailability together with other design parameters. Theadded flexibility in the latter work is made possiblethrough the use of a simple reliability block diagrammodel, as opposed to a Markovian model, to derive theinherent availabilities of individual units and of theoverall system.

Until now, approaches that consider reliability andmaintainability simultaneously with other design pa-rameters have been focused primarily on grassrootsdesigns and applied mainly to continuous plants. In thispaper, we extend our previous work10 to the case ofmultiproduct batch plant retrofitting and develop a newsimultaneous optimization framework that combines aprocess model and an availability model to obtainoptimal size, operating mode, and optimal allocation ofreliability for new equipment during retrofitting. Theexisting retrofitting formulation of Yoo et al.3 is ex-tended to account for production losses due to un-planned shutdown and maintenance costs and is usedas a process model in our work.

2. Illustrative Example

A small retrofit design problem of an existing two-stage multiproduct batch plant, producing products Aand B, is chosen for illustration. For a given new productdemand, the retrofit strategy of Yoo et al.3 can be usedto add new equipment to the existing plant. For a casein which only one piece of equipment can be added ateach stage of an existing plant, Figure 1a-c shows theexisting plant, the generalized superstructure (as de-scribed in Yoo et al.3), and an (assumed) optimalsolution, respectively, for this example.

Consider a case in which the reliability and maintain-ability data for existing equipment and for new equip-ment items are given. For simplicity, let us assume thatthe inherent availability (obtained from given reliabilityand maintainability data) for both existing and newequipment items is 97%. The overall plant availabilitiesof the existing and the new retrofitted plant can beestimated by constructing a reliability block diagram(RBD, as shown in Figure 2. With a simple analyticalexpression (described in eq 26 below) to estimate thesystem availability for series configurations, the plantavailabilities for both the existing and retrofitted plantsare estimated to be 94.09 and 91.26%, respectively.

It is important to explain the choice of using a seriesconfiguration to represent the reliability block diagram

Figure 1. Illustrative example: (a) the existing plant, (b) the generalized superstructure, and (c) an (assumed) optimal solution.

3800 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004

superstructure derived from the process superstructure.As pointed out by Dekker and Groenendijk,11 thereliability block diagram derived from the process flowdiagram should reflect the consequences of failures ofthe equipment. The consequences of failures are dictatedby the process interactions as defined in the processmodel. Further, the reliability block diagram should beconstructed on the basis of a specific product. In the caseof the multiproduct batch plant, different products aremanufactured using the same equipment operated inthe same number of stages. The optimal productionvalue for each product obtained with the Yoo et al.3process model is only for the system state in which allof the underlying equipment (both existing and new) isworking. Therefore, in this work, a series configurationis used to represent the reliability block diagramsuperstructure.

The consequence of assuming a series reliability blockdiagram superstructure in the present work is that aconservative set of optimal values of design parameterssuch 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 inherentavailability of a retrofitted batch plant is estimated tobe 91.26%. The 91.26% availability means here that theretrofit plant will run 91.26% of the time with “all”equipment running. In practice, the plant can run atreduced capacity (in the event of equipment failure). Forinstance, at the second stage of the retrofitted plant,we have two pieces of equipment, and in the event offailure of one piece of equipment in the second stage,depending on the type of product, the plant can still beoperational. Thus, by using a series system, we assumethat the combined production of all reduced states isnegligible. This could be true for cases in which theinherent availabilities of underlying pieces of equipmentare quite high or few of the possible reduced states areoperational. In other cases, however, the optimal solu-tion obtained with the present formulation will be onthe conservative side.

The alternative approach is to enumerate every singlepossible operational state, to assign probabilities foreach state, and then to use a Markovian model toestimate the effective production. This approach can beapplied only to a given system configuration, whereasin the present work, the system structure is to bedetermined. Further, as the number of equipment unitsincreases, the assignment of probabilities and produc-tion capacities to each state becomes a formidable task.Therefore, for those cases where it is important toconsider the production due to reduced states, a two-step approach can be undertaken. In the first step, aconservative design should be found with the presentformulation. Then, for a selected structure, a Markovianmodel can be applied to fine tune the design parameters.

It is apparent that the addition of new equipment inseries to existing equipment results in the reduction ofthe overall plant availability. The overall plant avail-ability can be improved during retrofitting by procuringmore reliable new equipment. For example, let us

consider the case in which the new equipment is alsoavailable in a different type with an inherent avail-ability of 99%. The overall plant availability of theretrofitted plant would then be 93.14%.

In light of this new information gained from theseparate availability analyses of both existing and newretrofitted plants, one can observe the following: (a)Conventional retrofitting formulations (in this case Yooet al.3) obtain optimal design parameters, sizes, andoperating modes for new equipment without consideringproduction losses due to the unavailability of the exist-ing plant (5.91%) and of the new retrofitted plant(8.74%). Therefore, these production losses due tounplanned 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 werenot considered in the objective function. Hence, theopportunity to balance maintenance costs and designcosts during retrofitting is lacking in previous formula-tions.

The aforementioned shortcomings of previous formu-lations are addressed in our new retrofit problemformulation. The production losses due to unavailabilityare compensated by a more robust strategy, derived bycombining the process model with the availabilitymodel. The strategy, as illustrated in Figure 3, is tomatch the effective throughput with the projecteddemand for each product during the retrofit.

3. Modeling Framework

In this work, the effective production capacity of amultiproduct batch plant is defined by three param-eters: the limiting batch size, the limiting cycle time,and the overall plant availability in a given timehorizon. The first two parameters can be optimized inthe retrofit problem by varying the size and the operat-ing mode (in-phase or out-of-phase) of new equipment,whereas the plant availability can only be improved byselecting the appropriate plant configurations and levelsof 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 isextended here to include (a) the impact of overall systemavailability on the overall production, (b) the estimationof maintenance costs as a function of equipment avail-ability, and (c) the estimation of additional capitalinvestment needed for availability improvement of newequipment; (ii) an availability model that describes theavailability of the equipment, both existing and new,and the plant availability as a function of equipmentavailability; and (iii) an expected profit objective func-tion, which takes into account the tradeoff betweeninitial capital investment and the annual operationalcosts.

The retrofit design problem for multiproduct batchplants can be defined as follows:

Given (i) a new production target, selling price, unitcycle times, and size factors for each product; (ii) the

Figure 2. Reliability block diagram for an illustrative example.

Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3801

existing plant configuration, including the size, cost,reliability, and maintainability data for existing units;and (iii) the number, size, reliability, and maintenancecharacteristics and costs of new equipment available,determine (i) the net expected profit and the revisedplant configuration, (ii) the method of grouping parallelunits and various processing parameters for each pro-duction campaign, and (iii) the optimal inherent avail-ability for selected new equipment.

4. Problem Formulation

In this section, we describe the mathematical founda-tions and assumptions made in the development of anoptimization framework for formulating the retrofitdesign problem for multiproduct batch plants.

4.1. Process Model. The process model as describedin Yoo et al.3 is extended in this section. In particular,the model is extended here to include the impact of theoverall system availability on the overall production byconsidering HAsys as the maximum time available forproduction. Second, the capital cost estimation modelis extended to become a function of the inherent avail-ability of the equipment. Finally, the maintenance costestimation model is adapted to include the estimatedmaintenance cost as a function of the inherent reliabilityof the equipment.

The products are identified by index i, and N repre-sents the total number of products. The batch processingstages are identified by index j, and the total numberof stages in the plant is represented by the parameterM. Each stage is assumed to consist of a number ofpieces of equipment or units, and the total number ofthe existing units in a stage j is Nj

old. The total numberof new units that can be added during retrofitting instage j is Zj. The parallel units (both existing and new)

in each stage are identified by index k, and the totalnumber of existing and new units is given by Nj

total

() Njold + Zj). Index l is used to indicate the level of

inherent availability of new equipment available at theretrofit stage. The Nj

total parallel units of stage j can begrouped arbitrarily into Gj

total groups, identified byindex g.

The retrofit strategy is determined by the value ofbinary variable yijkg representing unit-to-group assign-ments, a pseudo-binary/real variable eijg indicatingwhether group g exists or not on stage j, and thevariable yjkl indicating the level of inherent availabilitychosen for new equipment at the design stage. It shouldbe noted here that the pseudo-binary/real variable eijg,as explained in Yoo et al.,3 is actually a real variablebut behaves like a binary variable when it is used inconstraints describing the condition on group existenceand 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

where tij, cij, and γj are fixed parameters and Bi is thelimiting batch size for product i. For an overlappingmode, the limiting cycle time for product i is given by

where

Figure 3. New retrofitting strategy.

Tij ) tij + cijBiγj ∀i ) 1, ..., N; j ) 1, ..., M (1)

TLi ) maxj)1,...,M(Tij

Gij) ∀i ) 1, ..., N (2)

Gij ) ∑g)1

Gjtotal

eijg ∀i ) 1, ..., N; j ) 1, ..., M (3)


The limitation on production of each product is givenby the constraint

where Qi is the upper bound on the production ofproduct i. The time available for the production of eachproduct within the given time horizon (H) is given bythe constraint

where Asys is the overall plant availability during thegiven time horizon.

Combining eqs 1-3 yields the constraint

The lower and upper bounds on the volumes of newunits are ensured by the constraint

where Vjk is the volume of the new unit k in stage j VjL

and VjU, 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

The requirement that the volume be sufficient toprocess the batch size yields the following constraint

This constraint contains the product of a real variableVjk and a binary variable yijkg, which adds difficulty tothe convergence. Introducing the continuous positivevariable Vijkg linearizes the nonlinearities of the formVjkyijkg in eq 10 by replacing it with the following set ofconstraints

Each unit k at stage j can be assigned to at most onegroup for product i

For unit k to be assigned to group g in stage j forproduct i, the unit k must be installed and the group gmust exist

Group g can exist in stage j for product i only if unitk is assigned to the group

The upper bound of the variable eijg is 1

Redundant assignment to a group with the samevalue for the objective function can be avoided byintroducing the following constraint3

The cost of a new unit k in stage j in the previousformulation is expressed by a function of the volumeVjk of the form

where K0 is the annualized fixed charge; is the annual-ized proportionality constant of a new unit in stage j;and rj is the exponential constant of a new unit in stagej, which is considered to be equal to 1 in this work.

In this work, to estimate the extra investment neededto improve the availability of new equipment, we needto extend the conventional cost model to make it afunction of the inherent availability of equipment. Thereare two alternatives for extending the existing costmodels that would describe the cost-reliability functionof equipment in an objective function. The alternativesare (i) using exponentially increasing closed-form func-tions to relate the cost and the reliability/availabilityof the equipment10 or (ii) directly using the discrete setof cost and reliability data of an equipment unit in thedesign problem.12

niBi e Qi ∀i ) 1, ..., N (4)

∑i)1

N

niTLi e HAsys (5)

tij + cijBiγj

TLi

g ∑g)1

Gjtotal

eijg ∀i ) 1, ..., N; j ) 1, ..., M (6)

VjLyjk g Vjk g Vj

Uyjk

∀j ) 1, ..., M, k ) Njold + 1, ..., Nj

total (7)

yjk g yj,k+1

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total - 1 (8)

Vjk g Vj,k+1

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total - 1 (9)

(∑k)1

Njold

Vjk + ZjVjU)(1 - eijg) + ∑

k)1

Njtotal

Vjkyijkg g SijBi

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal (10)

(∑k)1

Njold


k)1

Njtotal

Vijkg g SijBi

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal (11)

Vijkg e VjUyijkg, Vijkg e Vjk

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (12)

∑g)1

Gjtotal

yijkg e 1

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal (13)

yijkg e yjk

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (14)

yijkg e eijg

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (15)

eijg e ∑k)1

Njtotal

yijkg

∀i ) 1, ..., N, j ) 1, ..., M, g ) 1, ..., Gjtotal (16)

eijg e 1 ∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal

(17)

∑k)1

Njtotal

2Nj

total-kyijkg g ∑k)1

Njtotal

2Nj

total-kyijk,g+1

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal - 1

(18)

f(Vjk) ) K0 + Kj1 Vjk

rj (19)


In principle, either of the above two alternatives canbe used in the problem formulation. Actually, the closed-form exponentially increasing function for reliability andcost is derived from the discrete sets and is mainly usedin problem formulation to minimize the total numberof binary variables in the final problem. In industry,discrete sets of reliability-cost data can be eithergathered in-house from purchase and maintenancedepartments or requested externally from equipmentsuppliers. Once the discrete sets are available forequipment units, the continuous function can be esti-mated by plotting the data on a reliability-cost dia-gram, as shown in Figure 4.

It is important to note here that the choice ofdescribing the relation between cost and reliability bya continuous function or by discrete sets has a signifi-cant impact on the complexity and computationalburden of the resulting problem. For example, describ-ing the relation as a continuous exponential functionintroduces nonlinearity into the objective function,whereas using discrete sets increases the total numberof binary variables in the problem. In this work, discretesets of reliability-cost data are used directly to avoidnonlinearity in the final problem. The new extended costestimation model can be given as

where Kjl2 is the annualized fixed charge associated

with the selection of alternative l for the new unit instage j. For example, for a piece of equipment used inFigure 4, the value of Kjl

2 can be estimated (consideringtype A as base case) as 0, 1000, and 3000, respectively,for types A, B, and C.

The maintenance costs constitute a significant portionof the total operating costs. In the previous formulation,these costs are not included in the objective function.The maintenance costs consist of both corrective andpreventive maintenance costs. Because only inherentavailability is considered in this work, preventivemaintenance costs are not included in the objectivefunction. The corrective maintenance cost is dictated bythe inherent reliability of each unit j and can beestimated as a function of Ajk

13

where Cjc and ∆j

c are the cost and duration of correctivemaintenance for unit k in stage j.

4.2. Availability Model. In this section, we developan availability model to estimate the total plant avail-ability (Asys) and equipment availability (Ajk), which isused in the development of eqs 5 and 21 in the processmodel described earlier.

The inherent availability (Ajk) of new unit k in stagej is given by

where the parameter Ah jl describes the inherent avail-ability of alternative l available for the new unit in stagej. The following constraint ensures that only one alter-native is selected if new unit k is selected in stage jduring the retrofitting process

The inherent availability for existing units is givenby

where Ajkold is the parameter describing the inherent

availability of existing units.The parameters Ah jl and Ajk

old can be estimated fromhistoric reliability and maintainability data. For in-stance, for a given constant failure rate λjk

old and repairrate µjk

old for existing unit k in stage j, Ajkold can be

estimated from

The total plant availability is estimated from theinherent availabilities of units by using a reliabilityblock diagram (RBD). The generic reliability blockdiagram superstructure is illustrated in Figure 5. Thetotal plant availability can be expressed as

where variable A′jk in eqs 26 and 27 is a dummyvariable, which is described by the relation explainedin eq 27. Constraint 27 ensures that only the availabili-ties of equipment selected at each iteration are consid-ered in the estimation of the overall system availability(Asys) and sets the availability of nonexisting equipmentto unity. Constraint 27 contains the product of a realvariable, Ajk, and a binary variable, yjk, which addsdifficulty to the convergence. Introducing a continuouspositive variable, A′′jk, linearizes the nonlinearities ofthe form Ajkyjk in eq 27 by replacing it by the followingset of constraints

Figure 4. Plot of inherent availability versus initial investmentcosts.

f1(Vjk,Ajk) ) K0 + Kj1 Vjk

rj + Kjl2yjkl (20)

f2(Ajk) ) CjcH(1 - Ajk)/∆j

c (21)

Ajk ) ∑l)1

P

Ah jlyjkl

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total (22)

∑l)1

P

yjkl ) yjk ∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total

(23)

Ajk ) yjkAjkold ∀j ) 1, ..., M; k ) 1, ..., Nj

old (24)

Ajkold )

µjkold

µjkold + λjk

old∀j ) 1, ..., M; k ) 1, ..., Nj

old (25)

Asys ) ∏j)1

M

∏k)1

Njtotal

A′jk (26)

A′jk ) Ajkyjk + (1 - yjk)

∀j ) 1, ..., M; k ) 1, ..., Njtotal (27)


Equations 24-29) constitute the availability model.4.3. Objective Function. The objective function of

the problem, which is to be maximized, is the expectedannualized net profit. Expected net profit is defined hereas the net income minus the annualized investment andoperating costs. The objective function can be repre-sented as

The first term of the objective function is the revenuefrom product sales. The second term corresponds to theincome from disposed batch units, and the third andfourth terms correspond to investment costs and the costof increasing the inherent availability of new batchunits, respectively. The last term corresponds to theaccumulated corrective maintenance costs.

The problem described by eqs 1-30 corresponds toan MINLP problem and can be solved by the outerapproximation (OA) algorithm of Duran and Gross-mann.14 The MINLP problem described above containsseveral nonconvex terms in the constraints and in theobjective function. The exponential transformation ofnonconvex terms, as described in Vaselenak et al.,1 isused to remove nonconvexities. The resulting set oftransformed equations is given in Appendix A.

5. Examples

Three examples are presented to demonstrate that thenew retrofit strategy provides greater flexibility andmore robust solutions as compared to the conventionalformulations. The first two examples are taken fromprevious works, Vaselenak et al.1 and Yoo et al.3,respectively. For comparison with the previously pub-lished results, these two examples are solved first forthe case in which maintenance costs are considered inthe objective function and second for the case in which

they are included in the objective function. The thirdexample is added to demonstrate the sensitivity of theresults to the new cost parameters (Kjl

2 and Cjc) intro-

duced in the formulation presented in this work. Theexamples are solved using the DICOPT2+ solver in theGAMS environment on an AMD athlon processor.

5.1. Example 1. An existing multiproduct batch plantconsisting of two stages is considered to produce prod-ucts A and B. The process data for this example aretaken from Vaselenak et al.1 and are given in Table 1.Table 1 also includes three potential alternatives fornew equipment units with different inherent availabili-ties and capital costs that are considered available atthe retrofitting stage. The relationship between theinherent availability and the costs reflects the commonlyused exponential relationship between reliability andcapital cost.

The example is solved for two different cases: aformulation in which the maintenance cost model isexcluded (case 1) and a formulation in which it isincluded (case 2). The optimal structures and groupingsfor products A and B for the new retrofitted plantobtained for the two cases with the present formulationare similar to those obtained by Yoo et al.3 Table 2shows the results obtained using the model by Yoo etal.3 and the results obtained with the model developedin this work. It is interesting to note the following fromTable 2:

(1) The proposed formulation results (in both cases)in a lower net profit than reported by Yoo et al.3 Thiscan be explained by the fact that Yoo et al.3 did notaccount for the revenue loss caused by unplanneddowntime and the maintenance costs (in case 2).

Figure 5. Reliability block diagram superstructure.

A′jk ) A′′jk + (1 - yjk) ∀j ) 1, ..., M; k ) 1, ..., Njtotal

(28)

A′′jk e maxl

{Ah jl}yjk,

Ajk - maxl

{Ah jl}(1 - yjk) e A′′jk e Ajk

∀j ) 1, ..., M; k ) 1, ..., Njtotal (29)

max ∑i)1

N

piniBi + ∑j)1

M

∑k)1

Njold

Rjk(1 - yjk)

-∑j)1

M

∑k)Nj

old+1

Njtotal

(Kj0yjk + Kj

1Vjk) -

∑j)1

M

∑l)1

P

∑k)Nj

old+1

Njtotal

Kjl2yjkl

-∑j)1

M

∑k)1

Njtotal

CjcH(1 - Ajk)/∆j

c (30)

Table 1. Input Data for Example 1

parameter stage 1 stage 2

tijA 4.0 6.0B 5.0 3.0

SijA 2.0 1.0B 1.5 2.25

Njold 1 1

Vjold 4000 3000

Ajkold 0.97 0.97

Zj 2 2Vj

L 0 0

VjU 4000 3000

K0 30 560 30 560Kj

1 32.54 32.54

∆jc 10 10

Cjc 200 250

Kjl2 0, 1000, 2100 0, 1000, 2100

Ah jl 0.97, 0.98, 0.99 0.97, 0.98, 0.99pi

A 1.0B 2.0

QiA 1 200 000B 1 000 000


(2) In case 2, the extra capacity needed because of theunavailability of the existing and retrofitted plant iscompensated partly by an increase in the volume of newequipment and partly by the selection of new equipmentwith a better inherent availability (option 3). The choicebetween increasing the volume versus increasing theinherent availability of new equipment is dictated bythe marginal costs for capacity (Kj

1) and inherent avail-ability (Kjl

2) and the maintenance data (Cjc and ∆j

c).Thus, it is important to note that the optimal solutionis sensitive to the values chosen for Kj

1, Kjl2, Cj

c, and ∆jc.

(3) Further it is important to compare the case 1 andcase 2 results. The equipment chosen in case 2 has abetter inherent availability (option 3) and reduced sizeis chosen as compared to that chosen in case 1. Thiscould be explained by the fact that maintenance costsare a function of inherent availability only and, there-fore, to reduce maintenance costs, more reliable equip-ment is chosen as an optimal solution with a corre-sponding capacity compensation.

5.2. Example 2. This example is taken from Yoo etal.,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 andgroupings for both products A and B. As shown inFigure 7 (for case 1), the revised plant obtained by theproposed formulation disposes of the existing equipmentunits of volume 3000 L in stage 1 and of volumes 1000and 2000 L in stage 2, while adding a new unit ofvolume 1327 L with an inherent availability of 98% instage 1. Other optimal design parameters such as batchsize, number of batches, limiting cycle time, etc., for eachproduct are summarized in Table 4. The extra capacityneeded because of the unavailability characteristics ofthe existing and retrofitted plant in example 2 iscompensated partly by increasing the volume of newequipment and partly by increasing the total plantavailability. In Table 4, the new equipment chosen incase 2 has a better inherent availability (option 3) andreduced size as compared to that chosen in case 1.

5.3. Example 3. In the previous two examples, it isshown that adding maintenance costs in the objectivefunction mainly influences the inherent availability andcapacity of the new equipment. This example is intended

Table 2. Results for Example 1

this work

Yoo et al.3 case 1 case 2

product A product B product A product B product A product B

TLi 6 3 6 3 6 3Bi 2679 905 2779 1039 2755 1007ni 448 1104 431 961 435 992niBi 1 200 000 1 000 000 1 200 000 1 000 000 1 200 000 1 000 000new units

Vjk Ajk Vjk Ajk Vjk Ajk

stage 1 1358 - 1559 0.97 1511 0.99stage 2 -overallavailability

0.913 0.931

maintenancecosts ($)

9,300

profit ($) 3,125,236 3,118,698 3,118,698


parameter stage 1 stage 1

tijA 1.0 1.0B 2.0 1.0

SijA 4.0 2.0B 1.0 2.0

Njold 2 3

Vjold 2000, 3000 1000, 2000, 3000

Ajkold 0.96, 0.96 0.96, 0.96, 0.96

Zj 3 2Vj

L 1000 1000

VjU 3000 3000

Rjkold 24 000, 32 000 16 000, 24 000, 32 000

K0 10 000 10 000Kj

1 10 10

∆jc 10 10

Cjc 100 120

Kjl2 0, 300, 800 0, 300, 800

Ah jl 0.96, 0.98, 099 0.96, 0.98, 0.99pi

A 0.15B 0.10

QiA 2 000 000B 4 000 000

Figure 6. Optimal structure for example 2 obtained with theformulation of Yoo et al.3


to show the sensitivity of the optimal solution to thenew cost parameters Kjl

2 and Cjc. The example8 consid-

ers the retrofitting of an existing plant that producestwo products to be processed in three stages. The inputdata for this example are given in Table 5.

The example is solved first for the values of Kjl2 and

Cjc given in the Table 5, and then for two different

scenarios. In the first scenario, Kjl2 remains the same,

but the value of Cjc is increased by 50%. Conversely, in

the second scenario, Cjc remains the same, but the

value of Kjl2 is increased by 50%.

Figures 8 and 9 show the revised plants obtained forthe different cases. As shown in Figure 8 (for thenominal and scenario 1 cases), the existing equipmentunits of volume 2000 L in stage 1 and 2500 L stage 2are discarded, and new units of volumes 2500 and 1875L are added in stages 1 and 2, respectively, in the new

retrofitted configuration. For scenario 2, the optimalstructure is shown in Figure 9, where the existingequipment units of volume 2000 L in each stage 1 andstage 2 are discarded, and new units of volumes 2500

Table 4. Results for Example 2

this work

Yoo et al.3 case 1 case 2


TLi 1 1 1 1 1 1Bi 1000 1000 831 1326 827 1308ni 2000 4000 2404 3014 2417 3056niBi 2 000 000 4 000 000 2 000 000 4 000 000 2 000 000 4 000 000new units

Vjk Ajk Vjk Ajk Vjk Ajk

stage 1 1358 - 1327 0.98 1308 0.99stage 2 -sold unitsstage 1 2000 3000 3000stage 2 1000, 3000 1000, 2000 1000, 2000overall availability 0.903 0.912maintenance costs ($) 5,318profit ($) 752,000 748,430 742,512

Figure 7. Optimal structure for example 2 obtained with theproposed formulation (case 1).

Figure 8. Optimal structure for example 3, scenario 1.

Figure 9. Optimal structure for example 3, scenario 2.


and 1875 L are added in stages 1 and 2, respectively,in the new retrofitted configuration. It is interesting tonote here that, in all three cases, the same numbers ofgroups are obtained for products A and B and two newequipment units of similar capacities are added.

The results for the nominal case and for the twoscenarios are summarized in Table 6. It is interestingto note the sensitivity of the optimal results to thevalues of Kjl

2 and Cjc. The points of deviations are

different profit projections in each case, and in the caseof scenario 2, different existing pieces of equipment aredisposed. The difference in profitability between thenominal case and scenario 1 can be explained by theincrement in the maintenance cost, and similarly, theselection of the lower inherent availability in case 2 isdictated by the increased incremental cost, Kjl

2. FromTable 6, it can be observed that the optimal designparameters are sensitive to Kjl

2 in this particular ex-ample, and therefore, uncertainty in these data shouldbe minimized by requesting cost and reliability data fordifferent equipment types from suppliers.

The computational statistics are summarized in Table7. The number of binary variables in the third columnalso includes the binary variables needed to representpiecewise linearization of the negative exponential termin the objective function. It should be noted that thecomputational burden of the proposed formulation is ofthe same order of magnitude as the computationalburden of Yoo et al.’s formulation. In Table 7, thecomputational details of only one of the cases arereported, as there is a very slight difference in compu-tational burden for different cases for the same example.


parameter stage 1 stage 2 stage 3

tijA 8 20 8B 16 4 4

SijA 2 3 4B 4 6 3

Njold 2 2 1

Vjold 2500, 2000 2000, 2500 2500

Ajkold 0.96, 0.96 0.96, 0.96 0.96

Zj 1 1 2Vj

L 500 500 500

VjU 2500 2500 2500

Rjkold 75 000, 85 000 80 000, 95 000 90 000

K0 35 000 35 000 40 000Kj

1 30 35 40

∆jc 15 10 15

Cjc 100 100 150

Kjl2 0, 3000, 6000 0, 3000, 6000 0, 3000, 6000

Ah jl 0.96, 0.98, 0.99 0.96, 0.98, 0.99 0.96, 0.98, 0.99pi

A 5.5B 7.0

QiA 250 000B 250 000

Table 6. Results Obtained in This Work for Example 3

nominal scenario 1 scenario 2


TLi 10 8 10 8 10 8Bi 625 635 625 635 625 635ni 229 364 229 364 224 356niBi 194 700 240 700 194 700 240 700 192 600 238 700new units

Vjk Ajk Vjk Ajk Vjk Ajkstage 1 2500 0.99 2500 0.99 2500 0.98stage 2 1875 0.99 1875 0.99 1875 0.98stage 3sold unitsstage 1 2000 2000 2000stage 2 2500 2500 2000stage 3overall availability 0.867 0.867 0.850maintenance costs ($) 7,400 11,100 8,400profit ($) 2,705,810 2,702,110 2,667,663

Table 7. Computational Performance

example formulation

numberof binaryvariables

totalnumber ofvariables

number ofconstraints

CPUtime(s)

1 Yoo et al.3 60 151 280 1.2this work 80 188 317 2.0

2 Yoo et al.3 130 295 592 7.9this work 160 356 653 14.2

3 this work 110 257 451 11.6


6. Conclusions

A new optimal retrofit method is presented for multi-product batch plant design. The key improvements ofthis method over previous methods are as follows:

(i) This approach considers the reliability and main-tainability of existing and new equipment units anduses this information to quantify the costs of unavail-ability (revenue loss due to production loss and main-tenance costs due to increased unplanned shutdowns).

(ii) It performs a tradeoff between the cost of unavail-ability and the extra capital investment needed toincrease the size and/or inherent availability of newequipment while maximizing the overall expected netprofit, thus providing a more robust retrofit solution.

As compared to previous formulations, the proposednew method requires additional data to be specified,such as the values of Ah jl, Ajk

old, Kjl2, Cj

c, and ∆jc. The

authors found that these data can be easily obtainedfrom internal sources such as the company log book,purchase department, maintenance department, etc., orthey can be requested from external sources such asvendors. In cases in which the data are not readilyavailable, the cost of obtaining these data can beincluded in the objective function.

The effectiveness of the proposed method is demon-strated by means of three examples. These examplesclearly demonstrate that the proposed method providesgreater flexibility to the designer to obtain a more robustand reliable retrofit strategy with only a moderateincrease in computational time.

Acknowledgment

The authors dedicate this paper to Art Westerbergto whom they are indebted for his visionary perspective,enthusiastic involvement, and invariably challengingcomments on a wide range of engineering researchprojects at TU Delft since 1997. Moreover, they expresstheir deep appreciation for the stimulating mix of wit,insight, and creativity that Art always brought to Delft.

Nomenclature

Indexes

i ) productsj ) stagesk ) unitsg ) groupsl ) design alternatives for inherent availability improve-

ment

Parameters

N ) number of products manufacturedM ) number of batch processing stages in the processP ) number of available design alternatives for inherent

availability improvementGj

total ) total number of groups in stage jH ) operating time periodSij ) size factor of product i in stage jTij ) operation time for product i in stage jcij ) parameter in the expression for Tij

γj ) parameter in the expression for Tij

tij ) processing time of product i in stage jZj ) maximum number of new units that can be added to

stage jNj

old ) number of existing units in stage j

Njtotal ) total number of parallel units in stage j

VjU ) maximum volume of new units in stage j

VjL ) minimum volume of new units in stage j

Qi ) upper bound on the production of product ipi ) expected net profit per unit of product iAh jl ) inherent availability of design alternative l for a unit

in stage j

Ajkold ) inherent availability of existing units

Rjkold ) annualized capital cost returned when the existingunit k in stage j is sold

rj ) exponential constant for a new unit in stage jK0 ) annualized fixed charge for a new unit in stage j

Kj1 ) annualized proportionality constant for a new unitin stage j

Kjl2 ) annualized fixed charge associated with the selec-tion of alternative l for a new unit

in stage j

Cjc ) cost of corrective maintenance for a unit in stage j

∆jc ) duration of corrective maintenance for a unit instage j

λj ) constant failure rate of unit k in stage jµjk ) constant repair rate of unit k in stage j

Variables

Continuous Variables

ni ) number of batches of product iBi ) batch size of product iTLi ) limiting cycle time of product iVjk ) volume of new unit k in stage jVijkg ) volume of unit k in stage j for the use of product i

in group gAjk ) inherent availability of unit k in stage j

A′jk ) dummy variable used in eq 26

A′′jk ) dummy variable used in eq 28Asys ) total plant availabilityeijg ) indication of whether group g exists on stage j

Binary Variables

yjk ) binary variable for unit k in stage jyjkl ) binary variable for the selection of availability

improvement alternative l of unit k in stage jyijkg ) 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 MINLPproblem described by eqs 1-27 is obtained, the followingexponential transformations are introduced (Vaselenaket al.1)

x1i ) ln ni

x2i ) ln Bi

x3i ) ln TLi

(A1)

i ) 1, ..., N


Using these transformations, the following formulationis obtained

subject to the following constraints

In addition, the following constraints are added byMontagna4 to reduce the computational burden

min -∑i)1

N

pi exp(x1i + x2i) - ∑j)1

M

∑k)1

Njold

Rjkold(1 - yjk)

+ ∑j)1

M

∑k)Nj

old+1

Njtotal

(Kj0yjk + Kj

1Vjk) +

∑j)1

M

∑l)1

P

∑k)Nj

old+1

Njtotal

Kjl2yjkl

+ ∑j)1

M

∑k)1

Njtotal

CjcH(1 - Ajk)/∆j

c (A2)

x1i + x2i e ln Qi ∀i ) 1, ..., N (A3)

tij exp(-x3i) + cij exp(γjx2i - x3i) e ∑g)1

Gjtotal

yijg

∀i ) 1, ..., N; j ) 1, ..., M (A4)

∑i)1

N

exp(x1i + x3i) e HAsys (A5)

exp(x2i) eBi ∀i ) 1, ..., N (A6)

(∑k)1

Njold


k)1

Njtotal

Vijkg g SijBi

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal (A7)

Vijkg e VjUyijkg, Vijkg eVjk

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (A8)

VjLyjk g Vjk g Vj

Uyjk

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total (A9)

yjk g yj,k+1

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total - 1 (A10)

Vjk g Vj,k+1

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total - 1 (A11)

∑g)1

Gjtotal

yijkg e 1

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal (A12)

yijkg e yjk

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (A13)

yijkg e eijg

∀i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal;

g ) 1, ..., Gjtotal (A14)

eijg e ∑k)1

Njtotal

yijkg

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal (A15)

eijg e 1 ∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal

(A16)

∑k)1

Njtotal

2Njtotal-kyijkg g ∑

k)1

Njtotal

2Njtotal-kyijk,g+1

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal - 1

(A17)

Ajk ) ∑l)1

P

Ah jlyjkl

∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total (A18)

∑l)1

P

yjkl ) yjk ∀j ) 1, ..., M; k ) Njold + 1, ..., Nj

total

(A19)

Ajk ) yjkAjkold ∀j ) 1, ..., M; k ) 1, ..., Nj

old (A20)

Asys ) ∏j)1

M

∏k)1

Njtotal

A′jk (A21)

A′jk ) Ajkyjk + (1 - yjk)

∀j ) 1, ..., M; k ) 1, ..., Njtotal (A22)

A′jk ) A′′jk + (1 - yjk) ∀j ) 1, ..., M; k ) 1, ..., Njtotal

(A23)

A′′jk e maxl

{Ah jl}yjk, Ajk - maxl

{Ah jl}(1 - yjk) e A′′jk e Ajk

j ) 1, ..., M; k ) 1, ..., Njtotal (A24)

∑g)1

Gjtotal

∑k)1

Njtotal

yijkg ) ∑g)1

Gjtotal

∑k)1

Njtotal

yi'jkg

∀i, i′ ) 1, ..., N, i * i′; j ) 1, ..., M (A25)

yjk e ∑i)1

N

∑g)1

Gjtotal

yijkg ∀j ) 1, ..., M; k ) 1, ..., Njtotal

(A26)

∑g)1

Gjtotal

yijkg e yjk

∀ i ) 1, ..., N; j ) 1, ..., M; k ) 1, ..., Njtotal (A27)

yij,g+1 e yijg

∀i ) 1, ..., N; j ) 1, ..., M; g ) 1, ..., Gjtotal (A28)

∑k)1

Njtotal

∑g)1

Gjtotal

yijkg g 1 ∀i ) 1, ..., N; j ) 1, ..., M (A29)


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Received for review August 11, 2003Revised manuscript received December 12, 2003

Accepted December 17, 2003

IE030656B


optimal reliable retrofit design of multiproduct batch plants

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