Ind. Eng. Chem. Res. 1995,34, 192-201
192
PROCESS DESIGN AND CONTROL Optimal Short-Term Scheduling of Maintenance and Production for Multipurpose Plants Ilias T. Dedopoulos and Nilay Shah* Centre for Process Systems Engineering,Imperial College of Science, Technology and Medicine, London S W 7 2BY, U.K.
The increased emphasis on the production of low-volume, high-value-added products has resulted in a renewed interest in multipurpose batch plants, which tend to be the most economical facilities for the production of small amounts of many different products. In such plants, the competition between the different products for the shared production resources gives rise to complex production scheduling problems. Amongst other factors, the profitability of a plant also depends on its availability which directly depends on the maintenance work carried out. In multipurpose plants, the integrated planning of production and maintenance so that these problems are solved simultaneously should result in increased profitability compared to the case where they are solved in isolation. This may be due, for example, to the ability to exploit equiment idle time or redundancy to perform preventive maintenance while having a minimal effect on production. This paper describes a mathematical model for simultaneously scheduling production and maintenance activities within a multipurpose plant, taking account of both production resources ( e g . , equipment items) and maintenance resources (eg., crews). 1. Introduction
The contribution of maintenance to the availability of a plant, and therefore to its profitability, has always been recognized. Christer and Whitelaw (1983) estimated the annual maintenance budget for a medium sized company in the early 1980s to be over GBP 1 million, while the amount spent for maintenance purposes in the UK industry as a whole in 1991 was over GBP 14 billion (Anon., 1991). In the same reference, impressive figures on savings gained by establishing optimal maintenance policies are provided. At the level of the firm, maintenance plays a crucial role in the manufacturing process. Failure to manage maintenance can lead to losses due t o excessive downtime. The strategic importance of competent maintenance tends to increase with increasing mechanization and automation. These considerations result in the necessity of optimizing the decision-making process associated with the maintenance problem. The British Standards Institute (BS 3811:1984) defines maintenance as the combination of all technical and associated administrative actions intended to retain a n item in, or restore it to a state in which it can perform its required function. The various forms of maintenance defined by the British Standards Institute are depicted in Figure 1. In this paper, we are mainly concerned with preventive scheduled maintenance and planned corrective maintenance in multipurpose plants. Preventive scheduled maintenance is defined as the preventive maintenance carried out to a predetermined interval of time, number of operations, mileage, etc. Planned corrective maintenance is defined as the maintenance carried out
* Author to whom correspondence should be addressed. E-mail:
[email protected]. 0888-588519512634-0192$09.00/0
after a failure has occurred and intended to restore the item to a state in which it can perform its required function. Planning the execution of maintenance typically involves the solution of two different types of problems that reflect a managerial and temporal hierarchy. At the higher level, long-term maintenance planning uses information such as equipment reliability, usage, and maintenance histories and manufacturers’ recommendations to determine the approximate moments or frequencies of preventive maintenance. The solution to this problem may result in a list of equipment items that are due to be maintained over the next short-term scheduling period. Furthermore, planned corrective maintenance must be performed whenever an equipment item is known to be broken down or degraded to the extent where a maintenance intervention is reasonable. Hence, the current state of the plant may also result in the need for maintenance activities, albeit corrective ones. The short-term scheduling of maintenance, which is the problem considered in this paper, takes as input the list of necessary maintenance activities thus determined and involves the determination of the detailed timing of all the activities that must take place over a short time horizon in order that the maintenance functions be complete, taking account of the requirements placed on necessary resources (eg., taking plant equipment out of production, allocation of crews). Hence, the two problems are complementary, and both must be given due consideration when attempting to formulate an effective maintenance strategy. In multipurpose plants, the short-term scheduling of maintenance will be complicated by the need to consider production requirements as well. These plants have received much recent attention (Reklaitis, 1992; Rippin, 1993). This is due to the increased importance of the lower volume, higher value-added sector of the process
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 193 Maintenance
ki prev tive mainte ance
Scheduled maintenance
Corrective (includingemergency) maintenance
Corrective (includingemergency) maintenance
Condition based maintenance
Figure 1. Maintenance as defined by the British Standards Institute.
industries, within which multipurpose plants tend to be more common. A major advantage of such plants is the relative ease of sharing the available plant resources among a number of different products produced either simultaneously or in successive time periods. However, the management of scarce resources, such as production time, processing, and storage equipment and utilities, between many competing products gives rise to difficult scheduling problems of high dimensionality. Consider the case where both maintenance and production activities must be scheduled in a plant over a relatively short time horizon of interest. A typical approach would be to fix the maintenance schedule and then to plan production around this schedule. This derives from the fact that in dedicated, continuous plants, the decisions relating to short-term maintenance scheduling are usually influenced by crew availabilities and inventories of intermediate and final product material. In multipurpose plants, however, there exists much more flexibility in the scheduling of activities to achieve certain objectives, due in particular to the multipurpose nature of the equipment and the typically batchwise mode of operation. An integrated approach to maintenance and production scheduling would be able t o exploit this flexibility by considering the interactions between the two types of activity explicitly. For example, it would be possible t o exploit the occasional processing equipment redundancy or idleness to perform opportunistic maintenance. This paper is concerned with the combined scheduling of production and maintenance in multipurpose plants. A great deal of work on the optimal scheduling of maintenance has been undertaken in the last two decades. This research has, however, in the main been confined to dedicated, continuous plants. For example, a large number of papers which address the optimal planning of maintenance within the power generating industry can be found in the literature (see, e g . , Yamayee, 19821, where typical objectives are the minimization of the total expected cost and the maximization of supply reliability. The planning horizon considered has always been relatively large (of the order of years). In contrast to this, very little research has been carried out in the area of short-term maintenance scheduling, the combined maintenance-production scheduling problem, and even less in the area of the maintenance of multipurpose batchhemicontinuous plants. In the literature, statements appear about the importance of maintenance for the profitability of such plants ( e g . ,Kniger, 1983)but little effort has been made to address this problem in detail.
In this paper, we adopt a mathematical programming approach to the solution of the combined production/ maintenance problem in multipurpose batchhemicontinuous plants. The overall objective is the ability to schedule optimally all required maintenance jobs as well as all production tasks needed to achieve the set of production goals, over a relatively short time horizon. The optimality criterion may be based on plant economics ( e g . , maximization of total net value-added) or performance ( e g . ,minization of total delay in fulfilling objectives). Constraints such as crew availability and duty periods, equipment allocation, and unavailability during maintenance, etc., are taken into account explicitly. We utilize a discrete representation of time as well as the state task network (STN) process representation of Kondili et al. (1993) that enables the consideration of process recipes of arbitrary complexity. The problem is formulated as a mixed integer linear program (MILP) and solved using a branch-and-bound method. The paper is organized as follows: the next section presents a general literature review on the topic. In section 3, we present a detailed definition of the problem and introduce the concepts utilized for its formal representation. The mathematical representation of the problem is presented in section 4,while some techniques for accelerating the solution procedure are outlined in Appendix A. An illustrative example is presented in section 5, and, finally, the major points of the paper are reiterated in a concluding section 6. 2. Review of Literature
A large amount of research work has addressed the maintenance planning problem for continuous dedicated plants, in particular for power generation plants, where objectives are usually cost- or reliability-oriented. Details on early approaches to this problem are presented by Yamayee (1982). The basic problem involves the determination of the periods of time during which the generating units are taken out of service for maintenance (this problem is often referred to as the “unit maintenance problem”) at minimum cost, while maintaining a satisfactory overall level of electricity supply. Although the two objectives of cost minimization and supply reliability maximization are identified, the problem is usually solved as a cost minimization problem with a constraint on supply reliability. Yamayee and Sidenblad (1983) address this problem using dynamic programming to minimize a combination of production and reliability cost and consider uncertainties such as forced outages and load uncertainty.
194 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995
Arueti and Okrent (1990) select the minimum total cost preventive maintenance schedule using artificial intelligence techniques. Edwin and Curtius (1990)consider simultaneous production and maintenance scheduling in power plants over a horizon of 1year divided into 52 week-long periods to minimize the expected total cost. Satoh and Nara (1991) solve a similar problem using a simulated annealing solution procedure. Yellen et al. (1992) formulate the problem in considerable detail as a mixed integer nonlinear program (MINLP). The objective is total cost minimization subject to maintenance and system constraints. The maintenance constraints include crew constraints, resource constaints, etc. System constraints include load and reliability constraints. The problem is solved by using the generalized Benders decomposition approach (Geoffrion, 1972). In this case, the decomposition is such that the master problem represents the maintenance planning problem the subproblem represents the production planning problem. All the papers above refer to the maintenance planning problem in the power generation industry. This problem differs from the scheduling problem for multipurpose plants in that power generation plants are dedicated continuous plants. They produce only one product (electricity), so that the production scheduling problem is relatively straightforward. The majority of work on maintenance as relevant t o the process industries is confined to pointing out the importance of effective management of the maintenance activities and proposing overall qualitative strategies of how a maintenance system should deal with the problem. Dumez and Frey (1987) point out the necessity of incorporating maintenance aspects in the design phase of a chemical plant and underline the importance of utilizing the knowledge of the different parties that will cope with the plant, such as the production manager, maintenance staff, etc. Kriiger (1983) states that maintenance costs are mainly personnel costs. Hence, every improvement in maintenance efficiency should lead to better utilization of the maintenance human resources. He states that 79% of the maintenance cost occurring in a large chemical company is personnel cost. He highlights the interrelation between plant availability, plant utilization, plant safety, and maintenance and recognizes the maintenance problem as an optimization problem. He also points out the impact of plant design on the maintainability of it. Since the maintenance cost in batch plants is a function of its utilization he proposes that different strategies should be implemented depending on the utilization degree of the plant. Van Rijn (1987) defines the aim of maintenance optimization as attempting to achieve a proper balance between maintenance cost and the benefit of increased process availability. He points out the necessity of solving the communication problem between maintenance and site manager and proposes a three-level hierarchical system to describe the maintenance activity, ranging from long-term goals to the management of the execution of maintenance. Kelly (1984) proposes a similar six-step procedure. Mann and Bostock (1983) use network techniques to address the short-term maintenance scheduling problem, where a number of maintenance activities must be executed over a short time horizon. The network is constructed by assigning every maintenance activity a downtime cost factor, which represents the cost which
would incur should the production unit shut down. It is proposed that activities with similar shut down factors should be performed concurrently. Ulosoy et al. (1992)address the problem of scheduling a number of planned maintenance tasks on a daily basis. Due to limited resources, the problem of ranking the maintenance tasks occurs. This problem is solved by assigning a priority function t o each task, based on the maintenance frequency, the maintenance criticality rating, and the maintenance delay. Tan et al. (1992) developed a method to propose longterm preventive maintenance strategies which make the optimal use of process down time to repair and replace aged components (“opportunistic maintenance”). The employment of neural networks is proposed, in order to calculate the minimum cost for opportunistic maintenance policies. The data for training the neural network are provided by Monte Carlo simulation. The aim of grouping similar maintenance activities so that they are executed more or less contemporaneously is termed “maintenance activity combination”. This problem has been studied, e g . , by Mann and Bostock (1983) and Dekker et al. (1992). The major advantage of this approach is better utilization of the available human resources which itself results in reduction of the maintenance cost. In general, the work described in the literature takes little account of the trade-offs between maintenance and production goals, and few procedures for the production of short-term maintenance schedules are described. In multipurpose plants, the production scheduling problem is itself of a very complicated nature. Therefore, any attempt to address the maintenance and production scheduling problems concurrently will lead to very complicated models. This work addresses the combined maintenanceproduction scheduling problem in multipurpose batch/ semicontinuous plants. The overall objective is the ability to schedule optimally all maintenance jobs which have to be carried out within a certain time horizon and all production tasks which are needed to achieve the set of production goals. The optimality criterion is some measure of profitability ( e g . , added-value, minimum delay, etc.),and a mathematical programming approach is utilized.
3. Description of the Problem For the formal description of this problem we introduce the terms crew, crew set, and maintenance task defined as follows. Crew. A crew is a set of persons who are qualified to perform a job. The persons forming the crew need not have the same qualifications. A crew incurs a fxed cost (call-out charge) whenever it performs a maintenance task and a variable hourly cost depending on the nature of the maintenance task that has to be performed and the qualification of the crew to do that particular job. Since the crew consists of people there is a maximum period of time the crew can work without interruption (the “duty period”). This period is followed by a compulsory rest phase, during which the crew cannot be employed to perform any job. Crew Set. As the name suggests a crew set is an assembly of crews. It is necessary to introduce this concept in order to describe the fact that there is, in general, more than one way to perform the same maintenance task. Each of these different ways may be characterized by a different crew set, cost, and
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 19s duration. This also allows a flexible approach to the employment of maintenance personnel by simply assuming that a crew set comprises people with the necessary tasks to perform the task. Maintenance Task. A maintenance task is fully described by the list of units on which the task has to be done, its timing, and the alternative human resources that are needed t o accomplish the task. Associated with every maintenance task is an earliest possible start denoting when it is allowed to start being performed and a latest possible finishing time denoting the time at which the maintenance task must be finished. Since the same maintenance task can be performed by different crew sets the actual value of the duration may depend on which crew set is performing the maintenance task. The allocation of a crew set to a maintenance task enforces the allocation of all the crews that belong to this crew set to the same maintenance task. Furthermore, it may be desirable that a maintenance task precedes another or that it has to be performed concurrently with another maintenance task. A maintenance task may be either preventive, in that it is being performed in order t o ensure a future level of equipment reliability, or corrective, in that it must be undertaken to restore an item of equipment to normal working order. The production scheduling part of the problem involves the determination of the allocation of multipurpose equipment to production tasks over time, the detailed task timings and batch-sizes, and the utilization of other resources such as storage (dedicated or flexible) and utilities. The problem to be solved may now be described by the following: Given 1. product recipes, in terms of state task networks (STNs); 2. equipment details (functionalities, capacities, etc.); 3. a list of equipment in different degrees of degradation that should be repaired; 4.production targets; 5. a set of preventive maintenance or inspection tasks that have to be carried out on certain items of equipment and their characteristics (e.g., duration, maintenance window, precedence relations, etc.); 6. information on crew availabilities and costs; 7. a horizon of interest. Determine 1.the allocation of equipment to production tasks over the horizon; 2. the timing of the maintenance tasks; 3. the flow of material through the plant, so as to maximize the value-added over the time horizon. The value-added objective function includes cost terms such as call-out charges, maintenance costs, and lost production, as well as revenue terms such as contributions to stock or customer deliveries. The callout charge concept is explained below. A call-out charge is incurred each time a maintenance crew must be called to the plant to perform one or more maintenance tasks. The inclusion of this charge in the objective function will avoid the generation of schedules where the maintenance tasks are spaced out over the horizon such that many call-outs will occur, all other things being equal. This may of course reflect the actual callout cost of an external crew or simply a measure of inconvenience for an internal crew.
Once a particular crew has been called out, its members can only work for a maximum fEed amount of time (egg.,a shift) before retiring for a compulsory rest period. During this period, no maintenance activity can take place unless a replacement crew is called out. Of course, once this period is over, the same crew may be called out again. The proposed model takes into account the key complexities associated with the definition of a maintenance scheduling problem, while retaining all the generality of the original short-term scheduling fonnulation of Kondili et al. (1993). The model is described in the next section. 4. Mathematical Model 4.1. Notation. We shall use the following notation for the problem data:
Cm: set of maintenance tasks which have to be performed concurrently with m
tit: cost of calling out crew g at time t CZ,: cost of performing maintenance task m by crew g at time t d:: duration of maintenance task m when performed by crew g dk: duration of maintenance task m when performed by crew set k E Umk d?: minimum duration of maintenance task m H. the planning horizon M: set of maintenance tasks which have to be performed within the planning horizon Mg:set of maintenance tasks that crew g can perform Mj: set of maintenance tasks concerning unit j Pm:set of maintenance tasks which can only start being performed if m is finished Pi: set of maintenance tasks which can only start being perfrmed if m has started tEs: earliest start time of maintenance task m tiE: latest finishing time of maintenance task m U m k : set k of crew sets which can perform maintenance task m rj: set of conditions that u n i t j can be in 0;: maximum number of successive time periods during which crew g may perform maintenance tasks 0;: number of rest periods that must elapse between successive call-outs of the same crew Furthermore, the following indices are introduced:
g: standard subscript for maintenance crews j: standard subscript for processing units k: standard subscript for crew sets m : standard subscript for maintenance tasks t: standard subscript for time The formulation is based on a discrete time representation. The horizon of interest is divided into a number of time periods of equal duration. The length of these intervals is determined by the greatest common divisor of the duration of all maintenance and processing tasks. Thus, the number of time periods employed is bounded by what is sufficient for the accurate description of the problem. The key problem variables are described in the next subsection. 4.2. Key Problem Variables. For the description of the problem, the definition of the following decision variables is necessary:
196 Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995
Mmgt: binary variable indicating whether or not maintenance task m starts being performed by crew g at the beginning of time period t Ygt: binary variable indicating whether or not a crew of type g is called out a t time t Wqt: binary variable indicating whether or not unit j is allocated to task i a t the beginning of time period t Umkt: binary variable indicating whether or not maintenance task m starts being performed by crew set k at time t Zmt: binary variable indicating whether or not maintenance task m has started being performed between tandH+ 1 Wjft: binary variable indicating whether or not unit J in condition c at time t The key constraints are described in the next subsection. 4.3. Key Constraints. The following mathematical programming formulation describes the problem. 4.3.1. Objective Function. The objective is to maximize value-added, which may be defined as the difference between the value of the inventories at time H 1and the value of the inventories at the beginning of the planning horizon, minus the storage cost, minus the value of any material received from external sources, plus the value of any deliveries, minus the cost of the utilities consumed, minus the maintenance cost. In general, the total maintenance cost will comprise three terms: (i) a cost relating to how often, and when, each crew is called out (if necessary); (ii) a cost relating to when each item of equipment is maintained; and (iii) a cost relating to the crew that performs each maintenance task. These three factors are taken into account by the objective function form described below:
+
max value added due to production
-
The value-added due to production term includes all the revenues and expenditures other than those associated with maintenance. The first summation involves the costs of the maintenance tasks themselves and the second the call-out costs. 4.3.2. Maintenance Constraints. Several constraints must be formulated to model maintenance scheduling in the generality described above. First, we must take into account the fact that some of the maintenance tasks will have t o be performed concurrently and some will have to precede others. The former situation can be expressed as follows:
Zmrt= Zmt
kf t , m' E C,
horizon. That means that variable Zmt must take the 1 - d:'". Expressed value 1 at least at time H mathematically:
+
'm,t,-d"
=
Qm
It may be the case that certain maintenance jobs are not vital, but should be performed if possible. In this case, it is possible to associate a penalty l l m with a job m and to penalize its absence from the final schedule - Z m , t L E - din~the ) objective functhrough terms llm(l tion. In the case of corrective maintenance, we use the concept of the state of a unit (see Pantelides (1993) and Crooks (1992))to model its condition at any time. This allows us to generalize corrective maintenance to any action that changes the condition of the unit. We can therefore consider equipment at various stages of repair, rather than simply operating or not. In any particular condition, there may be a restriction on the production tasks that can be performed or on the batchsize or throughput. The condition of a unit j at time t depends on its condition at time t - 1 or on the task that just has finished being performed on unit j or/and on the task which starts being performed a t time t on unit j . This can be expressed mathematically as follows: c' ieI,c,c
Qj, c
E
rj,t = 1,H + 1 (6)
Additional initial and final conditions may be specified, leading to the equations
Wjco0= 1,Wjct= 0
Q c f co and
wjcp+l= 1
For a comprehensive discussion of the concept above see Crooks (1992) or Pantelides (1993). Furthermore, we must express the fact that if a set of crews is allocated to a maintenance task, then all the crews that belong to this set are allocated to this maintenance task as well. Expressed mathematically:
Furthermore, we must assure that every maintenance task will be performed only once within the time horizon. This leads to the following expression:
(2)
The latter t-dk,
if m can only start if m' has finished, or if m must wait until m' starts being performed.
We must assure that every planned preventive maintenance activity will be performed within the planning
Note that Zm,tEs - 1is defined to be 0, and Zma+l = 1. Assuming that every crew can perform only one maintenance task at a time, the following constraint must be satisfied: k-1
Q m,k
E Km,g E
Umk,t
E
[tEs
- tkE - d P 1 (9)
The fact that a crew can be called out only once within
Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 197 Productl1 Product12
Packing13
/
Packing14
-3
Productl3 Product14
Product21 Product22 Blending2 Product23 Product24
\
Product31 Product32 Product33 Packing34
ProduclM
Figure 2. State task network for example problem.
its working cycle can be expressed as follows:
This fact may be expressed mathematically as follows
ei+e;-i
e=o A maintenance task m can only be performed if the required crew g has been called out, and if it can be assured that the maintenance task can be finished before crew g enters the resting phase. This implies that a maintenance task of duration longer than the typical work period should be divided into a number of shorter ones with appropriate precedence relations. In order t o ensure that the equipment is not used for production in between maintenance activities, unit condition constraints of type (6)may be utilized to insist that the unit is in a nonfunctional condition over such intervals. The first condition leads to following expression:
(14)
0;
hiwt5 CY,,,-, e=o
Vm
E Mg,g,t
(lla)
Since crew g may be able to perform more than one task within its assigned working interval e,: this expression can be aggregated t o
The second condition leads to following expression:
ei+ei-i y@+ ksEUmk
um&,t+e ,g=&dm+l
where tl and t 2 are the starting and finishing periods of the interval of unavailability. 4.3.3. Equipment Allocation Constraints. At any given time t, an equipment item cannot perform more than one task. Assuming that the mode of operation is nonpreemptive the same item cannot perform any othertask until the current one is finished. This is valid for both production and maintenance tasks. This leads t o the following mathematical expression (see Shah et al. (1993)):
vgg,m E Mg,t
(12) Finally, it must be taken into account that a crew g may be unavailable for a certain time period (Le., weekends).
It should be noted here that these constraints provide the interface between the production scheduling problem and the maintenance problem. The production scheduling problem is described as presented by Kondili et al. (1993)and includes capacity, material balance, and utility constraints. The interested reader is directed to the original reference. The above mathematical model is implemented within a general multipurpose plant optimization system based on an engineering-oriented input language and results display (Papageorgiou et al., 1992). A number of efficiency-enhancing features are added to the model; these are summarized in Appendix A. 5. An Illustrative Example The following is a case study based on a representative process that illustrates the integration of both corrective and preventive maintenance with production scheduling.
198 Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 Multipurpose Intermediate Storage
0 0
Blender
~
Storage
r
7
' i
1-
U -
Product Storage
U
1
Packline4
I
[PacklineBI Dedicated Intermediate Storage
Figure 3. Material flow through example plant. Table 1. Production Rsquirements for Illustrative Examde
a
product product-1-la product-1-2 product-1-3 product-1-4
minimum demand (te) 40.0 200.0 100.0 500.0
maximum demand (te) 60.0 250.0 120.0 550.0
product-2-1 product-2-2 product-2-3 product-2-4
40.0 120.0 50.0 500.0
60.0 150.0 60.0 600.0
product-3-1 product-3-2 product-3-3 product-3-4
70.0 270.0 300.0 500.0
90.0 300.0 350.0 600.0
Table 2. Details of Maintenance Jobs for Illustrative Examde earliest latest crew set hourly job start end alternatives duration cost MBlender2" 4 pm day 1 12 8 125 193 4 180 2,3 4 180 MPacklinel 4 pm day 1
-
192 13 2,3
8 4 4
125 180 180
MPackline2 4 p m d a y 1
-
12 1,3 2,3
8 4 4
125 180 180
12 1,3 2,3
8 4 4
125 180 180
MPacklinel 4 p m d a y 2
Product-x_y denotes base powder 2 , pack size y.
We consider a lubricant blending plant that is able to produce a number of different products according to the following simple recipe: Recipe 1. Blend 3 different grades of lubricant. 2. Pack the grades in 4 different pack sizes. The plant is able to produce different products according to the recipes described by the STN depicted in Figure 2. The material flow through the plant is depicted in Figure 3. Equipment 1. Two Blenders, of capacity 70 te. 2. Three dedicated tanks of capacity 50 te, each suitable for storing a different grade. 3. Four multipurpose storage tanks of capacity 50 te, suitable for all grades. 4. Five packaging lines with rates between 10 and 20 t e h . 5. Three warehouses for the storage of the final products. Complications 1. Blender 2 is in a degraded state where only 60% of its capacity can be utilized. Corrective maintenance may be used to improve this to 100% if necessary. 2. Each packaging line must be supervised by two operators; there are eight operators available for two 8-h shifts a day for lines 1-4 and two operators for 24 h a day for line 5. 3. When switching between some pack sizes, the packaging lines must be retooled; this takes 8 h and is
a
-
This is the optional corrective maintenance task for blender
2.
Table 3. Crew Details for Illustrative Example crew
max duty period
min rest period
call out cost
availability
1 2 3
12 12 12
8 8 8
200 150 400
all shifts all shifts all shiffs
performed by a single tool fitter, available for two shifts a day. The tool fitter can only perform one changeover a t a time. The Production Planning Part The plant must be scheduled over a 5-day week and must fulfill the production requirements specified in Table 1 at the end of this period. All the demands are assumed to have a value of 10 unitslte, and changeovers to cost 10 unitsh. The Maintenance Planning Part The processing unitspacking lines 1 , 2 ,and 4 require preventive maintenance during the time horizon, and an option of corrective maintenance for blender 2 also exists. The details of the maintenance jobs are given in Table 2. The details of the available crews are in Table 3. Note that, in this case, the hourly cost depends on the crew set used. Each crew set comprises two crews. Crews 1and 2 are assumed to be in-house crews, while crew 3 represents specialized contact labor. Hence, the sets that include crew 3 are able to perform the maintenance jobs more quickly, albeit at a higher cost. Both in-house and external crews incur call-out charges, but those for the external crew are considerably
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 199
Production & Maintenance Schedule
Unit
Tasks I ..BUND1 2 ..BLEND2
rms(a)
3 ..BLEND3 4
TANKS@) TANKS@)
.. BUNDI-DBG
5
.. BCXhDZ-DEG
6
.. BLWM-DEG
I
.. PACK11
.. PACK12 .. PACK13 10 .. PACK14 I I .. PACKZI 12 .. PACK22 13 .. PACK23 14 .. PACK24 15 .. PACK31
8
9
TANKS(d)
BLENDER I BLENDER2
16.. PACK32
PACKLINE1
.. PACK33 .. PACK34 19 .. CHANGE-1-2
17
18
PACKLINE2
20
21
PACKLINE3
.. CHANGE-2-1 .. CHANGE-3-2
22 .. CHANGE-2-3 23
PACKLINE4
.. 1S"
24
.. STBLENM
25
.. snLEND3
PACKLINES 0
10
20
30
40
M
60
70
EO
90
1W
I10
I 20
Time -->
Figure 4. Gantt chart for example problem. Table 4. Problem and Solution Statistics for Illustrative Example no. of integer variables 1979 total no. of variables 3924 no. of constraints 5070 no. of non-zero elements 24 358 total no. of search LP calls total CPU requirement (SUN SPARC1041,s) optimal value of the objective function fully relaxed objective value
200 340 28 940 30 200
higher. The optimization procedure will select the most economic alternative, simultaneously taking account of the production aspects. The versatility of the crew concept is exploited to present the maintenance alternatives to the optimizer. The Solution The problem was solved through the gBSS package, interfaced with the CPLEX optimization package from CPLEX Optimization Inc. The objective function was the maximization of overall value-added, defined in this case the value of all demands fulfilled, less the cost of changeovers and maintenance. The problem and solution statistics are outlined in Table 4. It is interesting to note that a solution with an objective function within about 4% of that of the fully relaxed problem is obtained after only 200 branch-andbound nodes. The delivery of the products is detailed in Table 5. The optimal schedule is depicted in Figure 4. This Gantt chart depicts both production and maintenance tasks. For production tasks, the number above the horizontal bar denoting unit occupation is the task number (cross-referenced in the Key to Tasks section) and the number below the batch size. Maintenance jobs
Table 5. Product Delivery Profile for Illustrative Example demand demand product met (te) product met (te) product-1-1 60.0 product-1-3 100.0 product-1-4 550.0 product-1-2 250.0 product-2-1 product-2-2
60.0 150.0
product-2-3 product-2-4
60.0 600.0
product-3-1 product-3-2
90.0 280.0
product-3-3 product-3-4
350.0 600.0
are indicated by hatched boxes. The assignment of the crews is depicted in Figure 5. The results exhibit some interesting features. The production schedule is arranged to avoid pack size transitions that lead to changeovers. Corrective maintenance is undertaken on the blender, blender 2, to allow it to operate with a higher capacity. This is undertaken using in-house crews. Packaging lines 1 and 2 are maintained sequentially, where the external crew is called in to assist the in-house crew with the second line. This enables the completion of the maintenance of line 2 within the duty period of the in-house crew 2, avoiding an additional call-out cost and, to a certain extent, consolidating the maintenance activities. Finally, line 4 is maintained by the in-house crews. 6. Conclusions
A mixed integer linear programming formulation is presented which addresses the combined production and maintenance scheduling problem for multipurpose planb. While the production scheduling problem is based on the formulation Kondili et al. (19931, a new formulation is presented that accommodates the characteristics of
200 Ind. Eng. Chem. Res., Vol. 34, No. 1, 1995 Maintenance Plan
line 2
H
Crew 3
w
Blender 2
Crew 2
Crew 1
w
w
line 2
0
line 4
H
H
Blender 2
40
20
line 4
H
60
80
100
120
Time (hrs)
Figure 5. Crew assignment for example problem.
the maintenance scheduling problem, at a very low level of detail, resulting in hour-by-hour integrated schedules of production and maintenance. The formulation takes into account the features that arise from the fact that maintenance is performed by people that may be both internal or external to the production site. Such features are, for example, callout charges, limited duty periods, and the need for rest, different qualifications for different maintenance tasks, etc. It is recognized that these features and the interaction with production planing strongly influence the profibability of a plant, resulting in a need for simultaneous optimization. The formulation results in large scale mixed integer linear programming models that can be solved using standard MILP packages, as well as tailored branch and bound techniques. In addition to the set of constraints which are needed to accurately model the problem, additional redundant constraints are added in order to make the solution more efficient. An example was used to illustrate the ability of the model to assess and select from different maintenance options as well as to integrate maintenance and production scheduling.
Appendix A: Techniques for Improving Solution Efficiency From the mathematical formulation described in section 4 of the paper, it is clear that the inclusion of maintenance aspects in the scheduling problem increases its size significantly. This appendix describes measures taken t o reduce the size of the problem and to accelerate the solution algorithm by adding additional redundant constraints and fixing variables during the branch-and-bound process. A.1. Adding Constraints. The following additional, redundant constraints can be employed in order t o help the algorithm to find the optimal solution, by reducing the region of integer infeasibility and by expressing the constraints of the model in a different way. (1)Relation between the 2 and U variables 'mt
=c
t umk,O ES
e=tm
V m, t
E
-
[t"," tkEl (A.l)
(2) Relation between the M and the U variables k :gC u.k
(A.2) (3) A crew can work on m at maximum once in the planning horizon LE
tm
ES
t=tm
(4) A job has to be performed using exactly one crew set t?-&
(5) Since variable Zmt maintains the value 1 after having obtained it, the following relation holds:
Whenever a crew is called out, it should start performing a maintenance task immediately.
Constraint A.6 has the desirable property of breaking degeneracy and forces the results to be more acceptable (eg.,it avoids the situation where a crew has been called out but does nothing for the first few periods). A.2. Fixing Variables. Finally, efficiency may be increased by intervening in the branch-and-bound procedure and fixing variables to values implied t o the values of other variables fixed during the branch-andbound procedure. This also reduces the dimensionality of the problem.
Ind. Eng. Chem. Res., Vol. 34,No. 1, 1995 201 Fixing a MmBt variable to the value 1 leads to the following relations:
Geofion, A. M. Generalized Benders Decomposition. JOTA 1972,
10,237-260. Kelly, A. Maintenance Planning and Control; Butterworth: London, 1984. Kondili, E.; Pantelides, C. C.; Sargent, R. W. H. A General Algorithm for Short-Term Scheduling of Batch Operations. Part I - Mathematical Formulation. Comput. Chem. Eng. 1993,17, 211-227. Kriiger, H.-G. Kostensenkung durch Planmlssige Instandhaltung unter Beriicksichtigung der Erforderlichen Verfiigbarkeit (Cost Reduction by Planned Maintenance, Paying Attention to Required Plant Availability). Chem. Zng. Techn. 1983,55, 625-
629. Mann, L., Jr.; Bostock, H. H. Short-range Maintenance Planning/ Scheduling Using Network Analysis. Hydrocarbon Process.
1983,62,97-101.
The following relations also hold: [xUm*kt*
=
Fixing a Ygt variable'to the value 1 leads to the following relations:
v e E [t + 1- t
Y~ = 1 -yg0 = o
+ e; + e; - 13
Furthermore, crew unavailability implies
Mwt = 0
V m, t
E
[tl- d r
+ 1 ...t21,
g the unavailable crew
Ygt= 0
If t E [tl - d,"'"
+ 1 ..t21, g the unavailable crew
Literature Cited Arueti, S.; Okrent, D. A Knowledge-Based Prototype for Optimization of Preventive Maintenance Scheduling. Rel. Eng. System Safety 1990,30,93-114. Christer, A. H.; Whitelaw, J. A n Operational Approach to Breakdown Maintenance: Problem Recognition. J . ORs 1983,34,
1041-1052. Crooks, C. A. Synthesis of Operating Procedures for Chemical Plants. PhD Thesis, University of London, 1992. Dekker, R.; Smit, A.; Losekoot, J. Combining Maintenance Activities in an Operational Planning Phase: a Set Partitioning Approach. ZMA J . Math. Appl. Business Znd. 1992,3, 315331. Dumez, F.; Frey, W. Instandshaltungsaspekte bei der Anlagenplanung (Maintenance Aspects of Plant Design). Chem. Zng. Techn. 1987,59,789-792. Edwin, K. W.; Curtius, F. New Maintenance-scheduling Method with Production Cost Minimization via Integer Linear Programming. Elect. Power Energy Syst. 1990,12,165-170.
Pantelides, C. C. Unified Frameworks for Optimal Process Planning and Scheduling. Proc. Znt. Conf. Found. Comput.-Aided Proc. Ops., 2nd 1993. Papageorgiou, L. G.; Shah, N.; Pantelides, C. C. A Software System for Optimal Planning and Scheduling of General Batch Operations. Proc. Adv. Progress Control, 3rd 1992,161-170. Reklaitis, G. V. Overview on Scheduling and Planning of Process Operations. Presented at the NATO Advanced Study Institute on Batch Processing Systems Engineering, Antalya, Turkey,
1992. Rippin, D. W. T. Batch Process Systems Engineering: a Retrospective and Prospective Review. Comput. Chem. Eng. 1993,
S17,S1-S13. Satoh, T.; Nara, K. Maintenance Scheduling by Using Simulated Annealing Method. ZEEE Trans. Power Syst. 1991, 6,476-
483. Shah, N.; Pantelides, C. C.; Sargent, R. W. H. A General Algorithm for Short-term Scheduling of Batch Operations-11. Computational Issues. Comput. Chem. Eng. 1993,176,229-244. Softening the Cost of Maintenance. Process Erg. 1991,Oct, 35. Tan, J. S.; Kramer, M. A. A Reliability Approach to Safety Monitoring and Maintenance Advising. Presented at AIChE Annual Meeting, Miami Beach, FL, 1992. Ulosoy, G.; Ilhan, 0.;Soydan, N. An Approach for Assigning Task Priorities in Maintenance Scheduling. Internal Report, Bogazici University, Istanbul, Turkey, 1992. Van Rijn, F. H. A Systems Engineering Approach to Reliability, Availability and Maintenance. Proc. Znt. Conf.Found. Cornput.Aided Process Ops., 1st 1987. Yamayee, Z. Maintenance Scheduling: Description, Literature Survey, and Interface with Overall Operations Scheduling. ZEEE Trans. Power App. Syst. 1982,PAS-101,2770-2779. Yamayee, Z.; Sidenblad, K. A Computationally Efficient Optimal Maintenance Scheduling Method. ZEEE Trans. Power App. Syst. 1983,PAS-102,330-338. Yellen, J.;Al-Khamis, T. M.; Vemuri, S.; Lemonidis, L. A Decomposition Approach to Unit Maintenance Scheduling. ZEEE Trans. Power Sys. 1992,7,726-733. Received for review April 20, 1994 Revised manuscript received August 23, 1994 Accepted August 31, 1994*
IE9402590
* Abstract published in Advance A C S Abstracts, November 15, 1994.
