A Hierarchical Production Planning Approachfor Multiprocessor Flow Shops

Daniel Quadt

Department of Production, Logistics and Operations Management, CatholicUniversity of Eichstatt-Ingolstadt, Auf der Schanz 49, 85049 Ingolstadt, Germanydaniel.quadt@ku-eichstaett.de

Summary. We consider the lot-sizing and scheduling problem for multiprocessorow shops. In this environment, lot-sizing and scheduling are interdependent. Ahierarchical planning approach is presented together with an overview of the devel-oped solution procedures. The objective is to minimizes setup, inventory holding andback-order costs as well as the mean ow time. Computational results are illustrated.

1 Introduction

A multiprocessor ow shop is a ow shop with several parallel machines onsome or all production stages. Figure 1 shows an example. Multiprocessor owshops are ubiquitously found in various industries, for example in the auto-motive, the chemical, the electronics, the packaging and the textile industries(see e.g. [1, 2, 15, 18]).

We consider the short-term production planning problem for multiproces-sor ow shops, which comprises the lot-sizing and the scheduling problem. The

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Fig. 1. Example of a multiprocessor owshop

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lot-sizing problem is to calculate production orders (lot-sizes) for an upcom-ing, short-term planning horizon, which typically covers one to several weeks.The production orders have to meet certain demand volumes for a number ofproducts. If a demand cannot be met, the volume is back-ordered and has tobe produced after its due date. The scheduling problem is to assign one of theparallel machines on each stage and to dene precise production start- andend-times for all jobs and thus the job sequence on each machine.

Several jobs have to be scheduled for a number of products. Setup timesand costs are incurred when changing a machine from one product to another.For this reason, a batching procedure is usually embedded in the schedulingphase. The batching procedure determines how many units of a product toconsecutively produce on a specic machine. In the industrial practice, prod-ucts can often be aggregated to (logistical) product families. A solution pro-cedure should exploit this fact, as typically, setup times and costs are lowerwhen changing a machine between products of the same family.

The machines of a stage are assumed to be identical, meaning that allmachines of a stage can produce all products, and their process times arethe same. However, the process times may vary for dierent products andproduction stages. Hence, to reach similar production rates among stages, thenumber of machines per stage may be dierent as well. A solution procedurehas to consider the parallel machines individually, as typically, a setup hasto be performed on all machines that produce a product. In contrast, usingaggregated machines and production volumes usually implies that these setupscannot be modeled accurately.

Lot-sizing and scheduling decisions are interdependent: It is not possible todetermine a schedule without knowing the lot-sizes that have to be produced.On the other hand, one cannot calculate optimal lot-sizes without knowingthe machine assignments and the product sequences. These interdependenciesmake it impossible to eectively solve the problems separately [10].

This paper is organized as follows: Sect. 2 presents a short literature review.An overview of a new, integrative solution procedure is illustrated in Sect. 3.Computational results are presented in Sect. 4. We give a summary and drawconclusions in Sect. 5.

2 Literature Review

So far, the lot-sizing and the scheduling problem for multiprocessor ow shopshave mainly been considered separately. Such dichotomic approaches cannotcoordinate the interdependencies of the two problems. Nevertheless, both thelot-sizing and the scheduling problem are known to be NP-hard, even whensolved separately.

There is little research on the lot-sizing problem for multiprocessor owshops. Some authors consider similar systems, such as parallel machines orjob shops with alternative routings (e.g. [4, 5, 8, 9]).

Production Planning for Multiprocessor Flow Shops 11

There are numerous studies covering the stand-alone scheduling compo-nent of the problem, e.g. [3, 7, 12, 16, 17]. A recent survey on optimal solutionprocedures is given in [6], a survey focussing on heuristics in [14].

There is a lack of solution procedures that consider the combined lot-sizing and scheduling problem. Such approaches are needed to incorporatethe interdependencies between the two problems.

3 An Integrative Solution Approach

We illustrate an integrative lot-sizing and scheduling approach for multipro-cessor ow shops. The objective is to minimize setup, inventory holding andback-order costs as well as the mean ow time through the ow shop.

The solution approach consists of the phases Bottleneck planning, Sched-ule roll-out and Product-to-slot assignment. Figure 2 shows an overview anddepicts the developed procedures. Although the rst phase is bottleneck ori-ented, a special model formulation ensures that the other stages are takeninto account implicitly also in this phase.

The rst phase of the solution approach solves a single stage lot-sizingand scheduling problem on the bottleneck stage. Products are aggregated toproduct families. A novelty of the procedure is that it is based on a newheuristic model formulation that uses integer variables in contrast to binaryvariables as employed by standard formulations. This makes it possible tosolve the model to optimality or near-optimality using standard algorithmsthat are embedded in commercial software packages (e.g. CPLEX). However,the heuristic model cannot capture all capacity restrictions as imposed by theoriginal problem. For this reason, it is embedded in a period-by-period heuris-tic, which iteratively solves instances of the model using standard algorithms(CPLEX).

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Fig. 2. Summary of the solution approach and the developed procedures

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The second phase generates a product family plan on all production stages.Thus, it keeps the aggregation level of product families, but explicitly includesthe non-bottlenecks stages. It assigns a production time and a machine toeach product family unit, which means that it solves a multi-stage schedulingand batching problem. However, the number of machines per product familyand the batch-sizes are pre-determined by the rst phase. The objective is tominimize the mean ow time. This implies a minimization of waiting timesbetween stages, and hence of in-process inventory. As a side eect, the usage ofintermediate buers is also minimized. The basic idea of the solution procedureis to establish the same product family throughput on all stages. In this way,each preceding stage supplies the intermediate products just in time for thesubsequent stage. Thus, the products do not have to wait between stages andthe ow time as well as the inventory volumes of intermediate products areminimized. The result of Phase II are so-called machine/time slots, signallingat what time and on which machine a product family unit is to be produced.

The third phase considers the individual products and calculates the nalschedule, which consists of exact production times and machine assignmentsfor every product unit on all stages. The problem is to determine how manyand which machines to set up and when to produce the individual productunits. Hence, it is a scheduling and batching problem for all stages on productlevel. The solution is calculated using the machine/time slots from the secondphase: We assign an individual product unit of the respective family to eachof the slots under the objective of minimizing intra-family setup costs, whilekeeping the low ow time of the second phase. There is a trade-o betweenthe two objectives. Two nested Genetic Algorithms are employed to solvethe problem. A novelty of the approach is the representation scheme: Anouter Genetic Algorithm sets a target number of setups and a target batch-size for each product of the current family. With these target values, theproblem resembles a two-dimensional packing problem: On a Gantt-Chart,the available slots for the product family form a pallet and the products formrectangular boxes that have to be placed on the pallet. An inner GeneticAlgorithm tries to nd an optimal packing pattern. The packing pattern mayin turn be interpreted as a schedule.

A detailed description of the phases and the developed solution procedurescan be found in [10], a more detailed overview in [13].

4 Computational Results

Several variants of the algorithm for the rst phase have been implementedand compared with a direct implementation of a mixed integer programmingmodel of the problem in CPLEX. For 960 small test instances with 5 productsfamilies, 4 periods and 4 machines each, the best heuristic leads to solutionsthat are 23% worse than the optimal solutions computed by CPLEX. Theaverage computation time of the heuristic is 11 seconds, compared with 45

Production Planning for Multiprocessor Flow Shops 13

seconds required by CPLEX to prove optimality. In addition, 288 large testinstances have been generated with 20 product families, 6 periods and 10machines each. Using the direct implementation, CPLEX is able to solve only97 of these instances within a time-limit of 1 hour. None of these solutions isproven to be optimal. The best heuristic is able to solve 285 of the instances,with an average computation time of approximately 25 minutes. On average,the heuristic solutions are approximately 24% better than the time-truncatedCPLEX solutions [11].

The heuristics of the second and the third phase have been evaluatedjointly [10]. 384 instances have been generated covering 3 production stages,20 product families, up to 10 products per family and up to 80 machines ona stage. The ndings show thatbesides the number of products and thenumber of machinesalso the demand pattern has a signicant inuence onthe number of setups: The number of setups is lower if there are high andlow volume products within a family compared with an evenly distributedvolume among the products of a family. The computation time depends onthe problem size. On average, it is 11 minutes. For large problems with 10products per family, it is approximately 21 minutes.

5 Summary and Conclusions

We have presented an integrative solution approach for the combined lot-sizingand scheduling problem for multiprocessor ow shops. Computational testshave shown that the procedures are able to solve practically sized problems in amoderate amount of time. Thus, they can be used in real business applications.The procedures presented for the second and third phase are embedded in ascheduling system for a semiconductor assembly facility. In a comparativestudy with the performance of the real shop oor that has been plannedmanually, the system has led to a ow time reduction of 55%, while at thesame time, only 80% of the capacity on the bottleneck stage has been utilized.The system has also been used for a capacity planning study [10].

There are some aspects open for further research: Alternative solutionapproaches should be developed to compare the heuristic. Especially optimalprocedures would be benecial. At the same time, the presented planningapproach is exible in the sense that it does not depend on the heuristicsdeveloped for each of the phases. Alternative procedures could be developedfor one or more phases to take dierent scenarios into account, such as non-stable bottlenecks or dierent objectives.

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