Precast production scheduling with genetic algorithms - IEEE Xplore

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W.T. Chan. National University of Singapore. 10 Kent Ridge Crescent. Singapore 119260 [email protected]. Abstract: A flow shop sequencing model (FSSM) ...
Precast Production Scheduling With Genetic Algorithms W.T. Chan National University of Singapore 10 Kent Ridge Crescent Singapore 119260 [email protected]

H. Hu National University of Singapore 10 Kent Ridge Crescent Singapore 119260 [email protected]

Abstract: A flow shop sequencing model (FSSM) that incorporates actual constraints encountered in practice is proposed for the difficult case of specialized precast production scheduling. The model is solved using a Genetic Algorithm (GA). The traditional minirnim makespan and the more practical minimize. tardiness penalty objective functions are optimized separately, as well as simultaneously using a weighted approach. Experiments are conducted to investigate the effect of increasing population size. and seeding the initial population with heuristic solutions. Comparisons between the CA and classical heuristic rules show that the CA is competitive, if not better than heuristic rules in discovering a set of good solutions.

Hegazy (1999) used GAS to improve the performance of commercial off-the-shelf project management software on resource allocation and leveling tasks. Planning and scheduling are also important in precast production management and much effort has been devoted to improving the methods and techniques used in precast manufacturing. Warszawski and Ishai (1982) studied the role of the prefabrication industry in a national economy, and considered the question of determining the number, location and capacity of plants for the production of precast elements. A later paper (Warszawski 1984) discussed models to efficiently plan short (specific orders) and long (continuous demand for standard elements) production series on one or more molds. Dawood and Neale (1993) developed a computer-based capacity-planning model for precast concrete building products in order to help production managers make better planning decisions and explore options. However, none of these take into account the many constraints encountered in the industry. Most planning models use cost or makespan schedule minimization as the main objectives. However, precasters are more concemed with issues like which, when and how many elements should be produced to meet their delivery dates. Moreover, issues in production scheduling like precedence constraints between components, the organization of men and equipment have not been studied in depth. These issues are typically left to the supervisory personnel on the production floor. The result may be unnecessary idle waiting time and factory production below the optimal plant capacity. Production process scheduling models, for example, single machine sequencing, multi machine sequencing, flow shops, job shops and so on have been widely applied to manufacturing systems, computer designs, logistics and communication but are relatively unknown in the construction industry. Choo et al. (1999) pointed out that the traditional scheduling tools based on the critical path method are inadequate in expressiveness when it comes to supporting production planning and control. Production scheduling models that make use of domain knowledge in precast scheduling are not widely available in the industry. Chan and Hu (1998) investigated the appropriateness of three kinds of models, namely single machine sequencing model, parallel machine sequencing model and flow shop sequencing model to optimize the precast production process. A subsequent paper (Chan and Hu 1999) refined the ideas proposed and made a distinction between the

1. Introduction

Scheduling is known to be NP-complete and has proven to be a difficult task for human planners and schedulers particularly if optimal solutions are required. Two general classes of methods used to solve scheduling problems are based on exact mathematical programming, as in Operations Research (OR), and the use of domain knowledgel heuristic rules, as in Artificial Intelligence (AI). OR algorithms are theoretically rigorous and guarantee optimal solutions but are difficult to use in real life applications as their performance does not scale well on larger problems and it may be difficult to represent the actual situation using the mathematical formalism of the OR model. On the other hand, the use of heuristic rules to uncover good regions of the search space does not guarantee an optimal solution, although the heuristic methods are easy to apply. The Genetic Algorithm (GA) proposed by Holland (1975) is a general and robust search method that has been successhlly applied in many types of problems, including process scheduling and resource allocation (Kopfer 1995, Gen and Cheng 1996) in many disciplines. In construction management, Chan et al. (1996) proposed a GA approach for construction scheduling and resource allocation. Li and Love (1997) and Feng et al. (1 997) presented an application of GAS for time-cost trade-off optimization problems in construction project planning and controlling. Li and Love (1998) employed a GA to solve a hcility layout problem. Haidar et al. (1999) developed a decision support system for the optimization of excavating and haulage operations, and the utilization of equipment in opencast mining using a hybrid knowledge-base system and genetic algorithms.

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specialized and the comprehensive precast production. It also proposed a model based on flow shop sequencing for the specialized situation. Minimization of makespan length and total earliness / tardiness penalty incurred were considered separately in the evaluation of schedule quality and only simple production constraints were included. Two of the major shortcomings which prevented the model fiom being readily used in actual production scheduling were the inability to express precedence constraints and the pursuit of a single objective where multi-objective solutions were required. This paper describes hrther development of the model to address these shortcomings.

(usually heating) or a natural process. (7) Demolding: stripping the side frame and taking out the components. (8) Finishing, patching and repairing of components. ( 1 0) Placing the completed components in the stockyard to achieve the delivery strength. (11) Transporting the components to the construction site.

2.2 Two Methods of Crew Organization Besides production line organization, the other dimension is that of production crew organization. According to Warszawski (1990), there are two altematives - under the first altemative, the same crew performs all operations outlined above. After casting and completing the processing of one component, the crew moves to the next mold and starts to work on the second component, repeating the same 2. Modelling Precast Production Scheduling sequence of actions. Under the second altemative, the total 2.1 Precast Production process is broken into several activities performed by Two basic types of component production systems may be different crews with specialized tools and work methods. employed in a prefabrication plant, namely the stationary Although the first alternative (which may be called the system and the traveling system (Warszawski 1990). With comprehensive method) is easy to employ, it is usually less the stationary system, all basic production activities are efficient than the other in terms of labor, tools and performed in one place and horizontal or battery molds are workspace utilization. It also creates problems of usually used. In the traveling system shown in Fig. 1, molds coordinating access to common resources by the different are moved on a rolling line or a conveyor fiom one work teams. For example, under a casting cycle of I day, all workstation to another, with different activities being teams will want to use crane for demolding hardened performed at each workstation. Almost always, a movable elements in the moming when they start their shifts. The comold line is connected to a curing chamber where the ordination of concrete supply may pose another problem, concrete mix undergoes an accelerated but controlled curing again for the same reason, when all crews progress at process. approximately the same rate. The comprehensive organization has been found to be more suitable for the stationary mold system. The second method of production organization (which may be called the specialized method) requires division of the total work into several tasks such as demolding, mold preparation, casting and so on, with each task being performed by a different crew. This method is very efficient in terms of crew and equipment usage since there is little . idle waiting time when all the crews work at a balanced rate. , I , The system is particularly efficient with a movable mold system, that is, molds moving between workstations with tasks performed at each one of them, eliminating the need for crews to walk between and set up at different stations. Although this method is well suited to a movable production line, it can also be employed in a production line using static M: mould; WS: workstation; molds where crews move fiom one mold to another. This EM: empty mould; CO: completed component. system will work smoothly if all components are identical so Fig. 1 Production system: travelling molds that a balanced rhythm of production can be maintained. However, when heterogeneous elements are involved with Typically, the production process in a precast factory different production times, there is a loss of efficiency due consists of the following sequence: (1) Mixing of concrete to the difficulty of balancing the work at all the workstations (including aggregates and cement storage and handling, and (crews). Some components have to wait for the next operation and some workstations (crews) have to be waiting moving of concrete &om mixing center to molds). (2) Setting of molds: cleaning and oiling of mold surface, and idly for the arrival of new components. fastening of side fkames. (3) Placing of reinforcements, Under the specialized method of organization, scheduling fixtures, electrical conduits, and inserts etc. to be contained is simple when identical components are involved; however, in the components. (4) Casting: pouring, compacting and it is rather difficult to determine an efficient schedule if the leveling of concrete. ( 5 ) Curing: through an artificial components are heterogenous. Unfortunately, this is the

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prevailing situation in the industry where it is very common to produce many different types of elements to satisfir several contract orders at the same time. We focus on this latter problem and propose a scheduling model and the corresponding algorithms for this difficult but very practical situation encountered in the industry. 2.3 A Flow Shop Sequencing Model for Specialized Precast Production Based on our investigation and observation of precast plants, precast production under the specialized method possesses many of the characteristics of the traditional flowshop sequencing problem (FSSP). The FSSP is generally described as follows: There are m machines and n jobs; each job consists of m operations, and each operation requires a different machine. The n jobs have to be processed in the same sequence on all the machines. The objective is to find the sequence of jobs minimizing the maximum flow time, which is technically called the makespan of the production schedule. However, Dudek et al. (1992) pointed out there was little descriptions on the applications of algorithms minimizing schedule makespan for the static FSSP in the literature. They pointed out that the modeling assumptions made in the standard flow shop sequencing model are overly restrictive and idealistic; practical constraints are not considered in the model and the criterion of makespan minimization is not the primary consideration in industry where due dates are more important. Lack of interest fiom industry because of these shortcomings may have caused researchers to direct their effort elsewhere. Chan and Hu (1999) developed a modified Flow Shop Sequencing Model (FSSM) which attempted to address some of these issues. The FSSM can be summarized as follows: There are n jobs (precast components) to be machines processed in the same task sequence by m (4) (workstations or crews, that is, mold setting, reinforcement setting, casting, curing, demolding and finishindrepairing). A distinction is made among normal working time, offnormal working time and overtime. All operations can be pre-empted with the exception of casting and curing. Every job is processed on one machine at a time, and every machine processes one job at a time except for curing, which is a parallel process. Rescheduling is possible after the curing operation if it is advantageous to do so. The objective is to find a schedule (Le the job processing sequence) that gives the minimum makespan, meets delivery dates or satisfies other quantifiable criteria Precedence constraints between jobs have recently been added to the FSSM; these are explaind in more detail in Section 3.2. The processing time of job i on machine k is given by irk (i=l, ...n; k=l, ...m). Let CO,, k) denote the completion time of job j , on machine k, fi,, j2, ...j,) a job permutation, and let To and TN denote the daily normal-working time and offnormal working time respectively. T e 2 4 - T D . Let O T represent overtime. The completion time of pre-emptible operations and non-pre-emptible operations can be computed as Equation (1) and Equation ( 2 ) respectively.

CQ,,k)=

i'

if t I 2 4 . d + TD

t+TN

where t = max/CQ

ift>24-d+TD

,-,,k), CQ

I

.k - 1)) + t,,

;

d = int(U24).

i = 2,3,-,,n; k = 2;..,m.

1'

rft124 d+T, or rft>24d+T, and OT > 0 If i > 24 d + T, andOT =0

C68,k)=

24(d+ I)+tJ,k

where t = max{C'lr_,,k),C6,,k - l))+iJ,k, d=mt(t/24) 1=2,3;..,n,k = 2 , ..,m

(2)

And the makespan is calculated as: makespan = C,, = C ( j n , m ) (3 1 Let d, denote the due date, and c, the completion time for job j,. Associated with each job is a unit earliness penalty a,>o and a unit tardiness penalty fl > 0. Tardiness TI is defined as T, = max{O, c, - d,) while earliness E, is defined as E, = max{O, d,-c,). A job i can be given a time window a, within which no penalties are incurred if its completion time is within the interval (d,-a,, d,+aJ Let I7be the set of all possible job sequences and o be an arbitrary sequence. Assuming that the penalty functions are linear, a generic expression for the Earliness / Tardiness (E/T) penalty can be formulated as Equation (4).

minf(cr)=z(a,(E,-~,YJ(E,- ~ , ) + P , ( T- ~ , w ( T - a , ) ) ,=I

VEn

(4)

where U(x-c)is the unit step function defined as U ( x - a , )=

0 rfxa,

The E/T penalties are reduced to a tardiness penalty if only tardiness is considered. 2.4 Heuristic Rules and Genetic Algorithms for Flow Shop Sequencing Problems The flow shop sequencing problem has proven to be NPcomplete in general and most researchers have focused on approximate algorithms employing heuristics to provide good and quick solutions. Gen and Cheng (1 997) described some of the well-known heuristics in their survey which included Parmer's, Gupta's, CDS, RA as well as the NEH heuristics. However, these heuristics are developed for minimizing the schedule makespan and few algorithms have been published for optimizing the earliness / tardiness (E/T) criterion. In industry, the commonly used heuristic for the tardiness problem is the earliest due date (EDD) rule which sequences the jobs in ascending order of their due dates. Genetic algorithm have been successfully applied to solve the FSSP. Reeves (1995) proposed one such implementation and concluded that simulated annealing algorithms and GAS produce comparable results for most sizes and types of problem, but GAS will perform relatively better for large problems and will reach a near-optimal solution more quickly. Chen et al. (1995) generated a GA based heuristic for the FSSP with makespan as the criterion. Murata et al. (1996) applied a multi-objective GA to the FSSP.

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Input

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FSSM Model

processing order of jobs in a schedule. The final gene value represents the weight of the makespan criterion that provides one way to set the weights in the weighted twoobjective function. We will compare the algorithm performance with different weight settings in the following section. In practical precast scheduling, it is sometimes required to assign precedence constraints among the jobs in the production sequence. This can be easily done in the random key representation. If element i must be prefabricated before elementj, we check and enforce the requirement that generi] genefi] . Crossover and mutation: The traditional crossover can be used on the random key representation. However, after the crossover and mutation we ne& to check (and adjusted, if necessary) to guarantee that all precedence constraints have been satisfied. Decoding: decoding the random key representation to build a schedule is simple. The first n gene values are used as priorities (usually interpreted in ascending order) and used to sort the activities so as to get a legal schedule. The @+I)-th gene value is equal to the weight of the makespan criterion whilst the weight of the E/" criterion is set equal to (I-weight of makespan). For example, if we have a chromosome of I I jobs ( n = I f ) that has the following gene values: [0.11 0.03 0.70 0.66 0.45 0.38 0.97 0.72 0.22 0.84 0.59 0.251, when sorted, the first I I values become: 0.03- 0.11+ 0.22- 0.38- 0.45- 0.59- 0.66- 0.700.72- 0.84- 0.97. Comparing the new list with the initial permutation, we get the order in which to consider the jobs when building the schedule: 2-+ 1+ 9- 6-+ 5- 11- 4-, 3- 8- 10- 7. The weight of the makespan criterion is 0.25whilst that of the E/T criterion is I. 0 - 0.25 = 0.75. The schedule can then be evaluated to determine its fitness. Obiectivdevaluation functions: Makespan is calculated using Equations (l), (2) and (3), whilst the penalty for earliness / tardiness is calculated using Equation (4). For the makespan-tardiness optimization, Murata et al. (19%) suggested a weighted sum approach to combine the two objective functions into a scalar fitness solution as follows: f(x) = -WMakXpn.Makespan (x)

heuristics Decoded solutions

- WTardrnrss .Tardiness (x)

Gene Dosition

1

2

...

...

1

...

... n n+l

Fig. 3 Chromosome representation where n is the number of jobs; altogether, there will be @ + I ) genes. The former n gene values determine the

(5)

where WMaksw, is a random real number over the closed interval [0,11, WTadiness =I- WmhsPn. However, the makespan and the tardiness penalty may not be in same magnitude in some problems. As a result, it is possible that the contribution of the small evaluation value will be dominated by that fiom the large one. We propose two linear normalization equations to remedy this: ifm,