Jobshop scheduling in a shipyard

Transcription

1 Jobshop scheduling in a shipyard Thomas Stidsen 1, Lars V. Kragelund and Oana Mateescu Abstract. Jobshop scheduling is considered a standard problem to solve by means of Genetic Algorithms (GA) and a number of articles have been written about this subject [1]. In spite of this, only few applications of GA have been published [], [] on real jobshop scheduling problems. Even though some applications may not be published for business related reasons, more practical applications seem necessary, in order to convince engineering and planning people of the use of GA for jobshop scheduling. A shipyard is a large and complex organization, including many employees with many specialized functions, many different workshops differentiated over a large area and all dependent on each other in complex ways. Because ships are usually built in small series, the organization needs to be flexible in order to adapt to new needs and for the planning people the demands are huge. If it is possible to establish efficient tools for jobshop scheduling in such an organization, for at least parts of the planning, it will be of significant practical value. Problem definition The scheduling task, which is to be performed, is the scheduling plan for a workshop situated at the shipyard. The setup is explained in the following subsections..1 The workshop The actual workshop produces large parts of the ship. These parts are called blocks and they usually weigh more than 0 tons. The blocks are all built of steel plates, which are welded together Introduction Initial stations Injection gate Handwelder stations Shipbuilder stations Robot station This article presents a pilot project (based on a cooperation between the companies Mærsk Data (MD), Odense Steel Shipyard (OSS) and the authors) in the use of a GA for jobshop scheduling for one large workshop situated at the largest shipyard in Denmark, OSS. The shipyard mainly produces supertankers and containerships and is currently producing a series of the largest containerships ever built. Nowadays ships are not produced from scratch in a dry dock. Instead, a lot of different workshops/assemblylines are used to construct large parts of the ship, which are then transported to the dry dock and mounted. In this way the production is highly parallel, thus significantly reducing the time needed to produce a ship. To avoid waiting for different parts of the ship at the dry dock and to ensure an efficient use of the resources, good scheduling is paramount. There are different levels of scheduling, where the top level deals with the problem of dividing the ship into large parts. Each of these parts are to be produced at different workshops and then are to be delivered at the dry dock at a given time. A lower level scheduling is then performed at each workshop in order to ensure efficient use of the resources and at the same time to fulfill the requirements of the macro scheduling plan. 1 Department of Computer Science, University of Aarhus, stidsen@daimi.aau.dk, WWW: Department of Computer Science, University of Aarhus, larsvk@daimi.aau.dk, WWW: Department of Computer Science, University of Aarhus, savanna@daimi.aau.dk, WWW: Figure 1. The B4-workshop at Odense Steel Shipyard. The workshop consists of one large assembly line divided into 1 different parts, called stations, which is where the actual production is performed (figure 1). At the first three stations the large steel plates are placed on the assembly line. These three stations are not included in the scheduling plan. At stations 4, 5, and 8 the different parts of the blocks are mounted and point-welded by workers called shipbuilders. Station number acts as a gate where new blocks can be injected into the line; no work is performed at this station. Station number is a robot station where welding robots perform the welding. At the three remaining stations, i.e., 11 and 1, welding is done by handwelders. This is necessary, since not all welding can be performed by robots and because the welding performed by the robots need to be checked and occasionally repaired by human welders. Each block to be produced is rolled through the workshop, thereby getting assembled. It is not possible to roll a block from one station into an occupied station. To ensure optimal use of the production facility, it is obviously necessary to limit the number of empty stations. To grasp these two requirements in the scheduling planning, it is assumed that all stations roll at the same time. This is obviously very critical for the scheduling, since a roll can only happen when all the stations have finished their jobs. Each roll takes a certain amount of time, where work cannot be performed on the blocks. Since the blocks seldom occupy the entire space of each station, blocks are placed together at each station, forming a so called package. In this c 1 Stidsen, Kragelund and Mateescu ECAI. 1th European Conference on Artificial Intelligence Edited by W. Wahlster Published in 1 by John Wiley & Sons, Ltd.

2 way the number of rolls needed to be performed is reduced when producing a larger number of blocks. A package may hold up to blocks, if this is geometrical possible, meaning that they can be placed in some way inside the area of the stations. These geometrical restrictions only apply at the two dimensional horizontal level. It is important to note that the term package is only used in the planning to describe a group of blocks, which are rolled together through the workshop. Besides the machines and the assemblyline, the workshop also includes a number of workers. As mentioned earlier there are three types: shipbuilders, robot-controllers and hand-welders. The workers can work in three different shifts, at different wages. The number of workers is not fixed, but can be changed for each week, within some limits i.e. there is a minimum and a maximum number of workers, which can work at each station.. Constraints Because we are working with a real jobshop task, not all scheduling plans are possible: There are two different kinds of constraints or limitations in the solution-space: Geometrical constraints: Each package contains some blocks, which must stay inside the rectangle defining the stations. The possible packages are thus restricted. Time constraints: These are more weak constraints, which are introduced because the rest of the shipyard is dependent upon the delivery of the blocks! Therefore, for each block there is a starting constraint-date and a finishing constraint-date associated. These constraints are called weak, because there are no physical obstacles for breaking the constraints, but a good scheduling plan 4, which breaks these time-constraints, may cost a fortune for other parts of the shipyard. Finally it should be noted that naturally all blocks should be produced. This may seem trivial, but some of our genetic operators actually may violate this "constraint" and the resulting individuals have to be repaired.. The scheduling plan The macro scheduling plan defines the frame in which the scheduling plan for the workshop should be designed. For each block and for each type of work the number of working hours, which is to be performed, is given. This measure is quite precise. Further, the macro scheduling plan establishes the time constraint-dates i.e. defines when the production of a block can be started and should be finished. The scheduling plan for the workshop should be constructed within these requirements. The scheduling plan can be viewed as consisting of a number of parameters, which should be optimized. These parameters are: Definition of the packages. The order in which the packages are to be produced. The number of men assigned to each station for each week and each shift. The last point is the means with which it is possible to change the capacity of the workshop. The number of men assigned to a station has 4 Good in this perspective means locally good for the workshop, but maybe bad from a wider perspective. to be below a certain limit, in order to avoid the men from obstructing each other. Further, it should be noted that the scheduling plan should not reduce the production time of the blocks, but instead reduce the needed resources, yet deliver the blocks within the limits of the macro scheduling plan. Notice that all parameters depend upon each other making the problem epistatic and non-separable. The Genetic Algorithm In the following we present the different parts of the GA, which was constructed to produce the needed scheduling plans..1 The basic GA The skeleton of the GA is presented in figure. This is a so called steady state GA, where the population is gradually changed through generation of new individuals, replacing old individuals one by one. This kind of GA was chosen in order to avoid premature convergence and to make efficient use of the information supplied by the fitnessfunction. Further, this steady state is possible because we are working with a deterministic fitness-function [4]. The selection pressure is obtained through three different tournament-selection procedures. SelectForMutation performs a binary tournament i.e. randomly selecting two different individuals from the population and returning the fittest. This amounts to a selection pressure equal to a linear fitness ranking procedure, where the best individual has twice the probability to be selected, compared to the medium individual and the worst individual has zero probability. SelectForCrossover selects at random 4 different individuals from the population, returning the two fittest for crossover. The selection procedure SelectDead selects at random different individuals and returns the least fit individual. This automatically creates an elite of two. All the remaining functions: Initialize, Evaluate, ReplicateAnd- Mutate and Crossover are domain dependent and will be presented in the following subsections.. Constraints The constraints are a very important part of the problem and the strategy of how to solve the problem under the given constraints affects the rest of the algorithm. In general there are two different methods of using GA on constrained problems: Allow constraint-violating solutions and ensure that the penalty given through the fitness-function makes those regions of search space unattractive to the GA. Forbid constraint-violating solutions and repair chromosomes whenever genetic operators create new individuals, which break the constraints. The problem with the time-constraints cannot be easily solved using the forbidding approach, since it is very unclear how one should change an individual, where one or more blocks break the timeconstraints. On the other hand, the geometrical constraints can be checked rather simply and the removal of blocks from too big packages constitutes a simple way of repairing these packages. For these reasons we choose to use the fitness approach on the time-constraints and forbid packages breaking geometrical constraints. Planning, Scheduling, and Reasoning about Actions 40 Stidsen, Kragelund and Mateescu

3 Population P; Initialize(P); Evaluate(P); while (not Finished()) do if (SelectOperator() = mutation) then parent := SelectForMutation(P); child := ReplicateAndMutate(parent); else ** SelectOperator() = crossover ** (father, mother) := SelectForCrossover(P); child := Crossover(father, mother); endif Evaluate(child); dead := SelectDead(P); P[dead] := child; endwhile result := BestOf(P); An individual P1 P P P4 Shift 1: 8 Shift : Shift : 4 8 Workers, week 1 Workers, week Station 4 Station 5 Station Station 8 Station Station Station 11 Station 1 Station 4 Station 5 Station Station 8 Station Station Station 11 Station 1 Block-info, block 1 Block-info, block Block-info, block Block-info, block 4 Block-info, block 5 Block-info, block Block-info, block Block-info, block 8 Block-info, block Block-info, block Shift 1: Shift : 4 Shift : 0 Figure. The tournament based steady state Genetic Algorithm Figure. An individual constituted of 4 packages produced in weeks.. Representation The choice of representation is very important. The choice affects most of the other parts of the GA and a bad representation may seriously harm the mutation operators and crossover operators etc. Two different things have to be represented in the chromosomes: The order of the packages to be produced together with the blocks contained, and the workers to be assigned to the different stations in the workshop for each week and each shift. There are several different possibilities, but we have chosen a direct representation. An example of an individual is shown in figure. The chromosome is divided into two different parts (in figure, upper and lower part). The upper describes the order of the packages and the blocks, which they contain. Each block is represented in the package by an integer, which addresses further information about the block. It should be noted, that each package fulfills the geometrical constraint. The lower part assigns an amount of workers to each station and each shift for all weeks in the production period. This representation facilitates some interesting characteristics: Packages can be swapped without violation of the geometrical constraints, if they are already satisfied. A crossovermade between packages will satisfy the geometrical constraints. The fact that the rolling times for all the rolls in the workshop are not represented in the chromosomes, reflects that the fitness-function is a greedy algorithm, which propagates the packages through the workshop as quickly as possible, see subsection.. Hence the rolling times would be redundant information..4 Initialization Initialization is normally not considered important and for most GA s, when working with binary chromosomes, these are just initialized at random. In this jobshop assignment the initialization task is complicated by the constraints. The fact that we want to create individuals in the population, which do not break the geometrical constraints, enforces a more intelligent initialization. Further the time-constraints enforce some order in the blocks such that an initialization, which creates a loose ordering of the blocks, will narrow down the search space. Because we do not want to break the time-constraints, the GA focuses on searching the interesting part of the search space. On the other hand it is very important to create a diverse population initially, otherwise the GA is hampered from the start and will just perform local hillclimbing. We have constructed an advanced initialization algorithm, but it is quite complex and we will not discuss it in further detail..5 Mutations The standard argument for mutations is, that they ensure the possibility that all parts of the search space may be visited. To accomplish this, we have constructed several mutations. To change the contents of the packages, we have designed the following mutation types: Swap packages: This mutation chooses a random point between two packages and swaps the packages. Move block: This mutation chooses a random block in a random package and attempts to move this block to one of the neighboring packages. Swap blocks: This mutation chooses two neighboring packages and attempts to swap a pair of blocks between these, if possible. To change the worker scheme, we have designed the following mutation types: Add men: This mutation chooses a random week, a random station and a random shift and adds a small random number of men, if possible. Remove men: This mutation chooses a random week, a random station and a random shift and removes a small random number of men, if possible. Planning, Scheduling, and Reasoning about Actions 41 Stidsen, Kragelund and Mateescu

4 Move men: Chooses two random weeks, two random stations and two random shifts and moves a random number of men from the one spot to the other, if possible. Together, the previously mentioned mutations enable the possibility of reaching all parts of the search space. It might though be argued, that none of these mutations, except Swap packages, are big mutations. But several small mutations may accomplish the same effect, and this is possible even if each of the small mutations reduces the fitness, because we only exert a rather weak selection pressure on the population. Besides, it is rather easy to enlarge the mutations and for instance, distribute the size of the mutation according to the gaussian distribution. All the previous mutations are so called blind mutations, which alter the chromosomes with an equal probability on all parts. Because the problem of fulfilling the time-constraints proved to be quite hard, we added some special mutations, which increase the probability of improving the worst parts of the chromosomes. F1 F F F4 F1 F 8 F F4 8 M 1 M M M 4 M M 1 M M M 4 M Split package: This mutation chooses a package, which consists of blocks, which have very different finishing constraint-dates. This package is then divided into two more "sensible" packages. Sensible here means packages, where the finishing constraint-dates are more alike. Swap package: This mutation chooses two neighboring packages, where the latest finishing constraint-dates are ordered in the wrong way and swaps the two packages. Figure 4. C1 C C C4 C5 4 8 The one point crossover of the package part of the chromosome. We now have a total of 8 different mutation types, which have to be selected according to a certain probability compared to the other genetic operators. We have not attempted to fine tune these parameters, but an obvious solution would be to use the adaptive probabilitiestechnique as suggested in [4], chapter.. Crossover The crossover is needed to ensure transfer of high quality parts of the chromosome to other individuals. Unfortunately, it is quite difficult to construct a crossover, which does not create offspring breaking the constraints. We have chosen to make two different crossover operators, one-point and two-point, which produce legal offspring, but at the expense of more packages. See figure 4, where the one-point crossover, performed on the package part of the chromosome, is sketched. The danger with this kind of crossover is, that it may create many new packages, since the parents disagree on the order/contents of the packages. Hence the crossover-operation creates new packages to the disputed blocks. We have tested this on the data and found, that on average for each crossover-operation, the offspring was expanded with half a package, which should be compared with the fact, that we work with individuals consisting of 0 to 5 packages. We have constructed a two-point crossover, which is very similar to one-point crossover, but we will not describe it in further detail.. Fitness function In order to achieve good results with the GA it is necessaryto give the GA a fairly accurate measure of the quality of each of the scheduling plans, i.e. each individual in the population. To achieve this, we have constructed a fitness-function, which simulates the production of the blocks, according to a given scheduling plan, in a rather crude fashion. The idea is quite simple and relates to the fact, that we want to produce as fast as possible with as few men as possible, without breaking the constraints. For each block, types of work must be performed: shipbuilder work, robot-controller work and handwelder work. All of this work has to be performed in the workshop, at the corresponding stations. Given a legal scheduling, the simulation of the production is done step by step, as prescribed in the following: 1. Check the package just outside the workshop. For each block in the package, check the starting constraint-dates and check the time to see whether it is legal 5 to start production of this package now. If it is legal, roll the entire assemblyline and insert the new package at station 4. If it is not legal, roll the entire assemblyline making an empty station 4.. Given a number of packages in the workshop: For each of the stations: Station 8, the last shipbuilder station, station, the robot station, and station 1, the last handwelder station, calculate when these stations with the given crew will have finished their jobs. The latest of these dates is the next rolling time.. Return to 1. A few things should be noted about this kind of simulation: All the work is finished, i.e. a sufficient amount of working hours is assigned to all the blocks. We do not break any starting constraint-dates. We do not optimize any "obvious" drawbacks in the scheduling plan, i.e. we do not perform any local optimization. When the simulation of a schedulingplan is finished,the quality of the scheduling plan according to the simulation of it has to be estimated. In this case the fitness-function consists of three parts: 5 Here legal means that all packages have a fulfilled starting constraint. Planning, Scheduling, and Reasoning about Actions 4 Stidsen, Kragelund and Mateescu

5 1. The entire expenditure associated with the worker wages in the production period.. Penalty terms associated with breaking the finishing constraintdates are added for each block, which has not been completed. We use the following formula: F = X 8blocks:t actual >t constraint k1 e k (t actual?t constraint ) (1) The constants in equation 1 are assigned the following values: k1 = 00, k = 1:4. Further t actual is the actual finishing date and t constraint is the finishing constraint-date.. A penalty term corresponding to the blocks finished before the finishing constraint-dates: F = X 8blocks:t actual t constraint k (tconstraint? tactual) () In equation, k = 500. This last term should model the expenses associated with the storing of the completed blocks. This mainly relates to the space these blocks occupy and the rust, which may harm the finished blocks during the storing time before the block is needed elsewhere at the shipyard. The exactness of this fitness-function may certainly be debated. The expenses associated with the workers are probably quite precise, but the two remaining terms are more dubious, because of the less concrete nature of these expenses. The rational is, that the expenses associated with a delayed block may affect a large part of the shipyard, leading to huge expenses and this should thus be avoided at any cost, whereas the storing expenses are of a more moderate nature. Finally it should be noted, that since we are only looking at the relative fitness values, the absolute values, which may be huge because of the exponential functions, are irrelevant. 4 Results To evaluate the algorithm we have performed tests on different datasets provided by MD. The datasets contain between 45 and 88 blocks. We experimented with populationsizes between 500 individuals and 1500 individuals and performed 5.0 million genetic operations for each run. Each of these runs took about 1.5 hours on a Silicon Graphics Indy computer. These tests showed that a populationsize of 500 yielded a higher degree of convergence, thereby obtaining better performance. On the other hand, there was a higher risk of finding sub-optimal solutions. But for all runs, no time-constraints were violated. It is not possible to compare the efficiency of the manual planning for the workshop with the GA planning for the workshop. This is due to different reasons: We have only access to results for the manual planning concerning the running-in period for the workshop. Our simulation of the production is quite simple, compared to the real production. Hence the resource expenditure is theorethical. For these reasons, we will not attempt to compare the GA results with the manual planning. The aim of this project was not to automate the manual planning. It was to investigate the possibility of creating a tool, which could act The running-in period refers to the time, when the robots were introduced in the workshop. This introduction caused planning problems, thus making the workshop inefficient. as a support for the manual planners. In other words, the GA should suggest initial plans, which the planners could then refine. The results show, that the GA is able to create reasonable plans. Hence, there is a basis for a development of a tool supporting the manual planners. 5 Conclusion and future prospects With this project we hope to have demonstrated, that it is possible to create an actual working GA, which solves a non-standard problem of great interest in a shipyard. We have attempted a pragmatic approach, where we have used a non-standard representation and found, that this representation do not hinder the GA from acquiring good results, if the genetic operators are constructed with care. Further we hope we have demonstrated: GA s are flexible i.e. they are quite easy to adjust to new problems. It is possible to implement domain knowledge through special mutations, without requiring total knowledge about the problem. It may be needed to create special initializations, in order to reduce the number of conflicts from the given constraints. Weak constraints, which cannot be solved by initialization methods or repair methods,can be solved by the GA by using penalty-terms. Quite an amount of work has to be performed, before a usable tool for the planning for the workshop is possible. Whether this is done, remains yet an open question. A lot of practical questions arise and have to be evaluated, before such a development is decided: Can a system consisting of a GA be trusted by the people, who use the system, if they do not understand how the system gets to solutions? - Is it necessary to educate the planning people to understand the GA? Will the system arrive to competitive solutions, given the computer resources and time available? Is the system flexible, i.e. can it adjust to new demands like changes in the workshop, different blocks to be produced etc.? These are non-technical considerations, which may seem irrelevant, but which are of great importance to the users of this system. ACKNOWLEDGEMENTS We would like to thank the entire project group at OSS: Helle Jensen (MD) who initiated this project, Ole Haastrup (MD), Christian Voigt (MD), Esben T. Horup (OSS) for letting us base a part of this project on his master s thesis and Claus Risager (OSS) for guidance throughout the project. Finally we want to thank our supervisor Professor Brian Mayoh for making this project possible. REFERENCES [1] D. Whitley, T. Starkweather and D. Shaner, The Traveling Salesman and Sequence Scheduling: Quality Solutions Using Genetic Edge Recombination, Handbook of Genetic Algorithms, chapter, Van Nostrand, New York, 11. [] Y. Uckun, S. Bagchi, K. Kawamura and Y. Miyabe, ManagingGenetic Search in Job Shop Scheduling, IEEE Expert, 8(Oct.1) [] B. Filipic, A Genetic Algorithm Applied to Resource Management in Production Systems, Evolutionary Algorithms in Management Applications, Springer-Verlag, Berlin Heidelberg, 15. [4] L. Davis, Handbook of Genetic Algorithms, Van Nostrand, New York, 11. Planning, Scheduling, and Reasoning about Actions 4 Stidsen, Kragelund and Mateescu