Sciences Shenyang, Shenyang, China.

Solving the Two-Objective Shop Scheduling Problem in MTO

Manufacturing Systems by a Novel Genetic Algorithm

Lili Yao

, Haibo Shi

, Chang Liu

, Zhonghua Han

Graduate University of the Chinese Academy of Sciences, Beijing, China; 2

Key Laboratory of Industrial Informatics, Shenyang Institute of Automation, Chinese Academy of Sciences Shenyang , Shenyang, China.

a

[email protected], [email protected], [email protected], [email protected]

Keywords: Make-To-Order; two-objective shop scheduling; Genetic Algorithm.

Abstract. In this paper, a novel genetic algorithm (GA) is proposed to solve the two-objective shop scheduling problem in make-to-order (MTO) manufacturing systems. This algorithm can ensure that all jobs meet their deadlines; simultaneously, it can satisfy another performance goal which the enterprise pursues. Referring to the principle of population updating with survival of the fittest in traditional genetic algorithm and taking advantage of the idea of two sub-modules, the novel algorithm is controlled by the two nested closed-loops, and the strategy that feasible solutions are preferred while infeasible solutions are remade is employed to make the search forward. Finally the novel algorithm and the traditional algorithm are used to solve the two-objective hybrid flow-shop scheduling problem (HFSP) in MTO manufacturing systems. The result shows that the novel algorithm has an obvious advantage and good feasibility compared with the traditional algorithm.

Introduction

As customer requirements become various and individual, make-to-order (MTO) production is accepted by a lot of manufactories. In a MTO enterprise, the planners organize production according to customer orders and sale contracts, and how to fulfill customer orders on time is crucial [1, 2]. In MTO manufacturing systems, the production data should be accurate, and reasonable plans such as balancing the production capacity, solving the bottleneck problem of constrains, maintaining equipments and instruments with reasonable arrangements, optimizing the production process, controlling the work shop jobs and so on are very important. But the most important thing is to ensure that all jobs meet their deadlines.

Scheduling is the key process in the computer integrated production system, which is the link between management and control. It determines the specific processing paths, work times, machines and operations for each processing object. Excellent production scheduling plays an important role in increasing economic efficiency and improving the production system. Certainly excellent production scheduling can fulfill customer orders on time, at the same time reasonable scheduling also can optimize some other performance goals, such as cost minimization and punishment minimization.

In MTO enterprises, scheduling software usually adopt the method of rule-based reverse scheduling. The method takes the order time as start time and arranges each process from back to front; finally we can get the latest start-time of each job. This method can better guarantee the due date. In addition to the delivery performance, MTO enterprise decision-makers usually hope to reduce the cost of production or improve resource utilization through reasonable scheduling, but reverse scheduling method can’t solve the multi-objective problem. Traditional multi-objective optimization methods maybe can optimize these objectives [3, 4, 5, 6, 7], but can’t ensure the performance that all jobs complete on time. In this paper, a novel genetic algorithm is proposed, which can ensure that all jobs meet their deadlines and can optimize another performance goal which the enterprise pursues.

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Features and Scheduling Performance Goals of MTO Enterprise

Features of MTO Enterprise. The features of MTO enterprise include multiple variety and small batch products, large fluctuations in demand, orders changing frequently, complicated processing techniques and so on. Automobile manufacturing industries mostly are MTO manufacturing systems. Since these features in MTO enterprise, it is difficult to deal with the scheduling on manual. Many enterprises which arrange the scheduling on manual have experienced the following problems: Incomplete plan results material storage and work broken frequently, and the production efficiency and the profit are engulfed seriously; different processing pace results that the “slack at the beginning and speed up towards the end” phenomenon arises in the assembly department. So implementation of automatic scheduling has a great significance in MTO enterprise.

Performance Goals of MTO Manufacturing Systems. The most important thing for the MTO enterprise is to ensure that all jobs meet their deadlines. It can be described as Eq.1:

i im i i im i i n i i D C if U D C if U U U > = ≤ = =

∑

Where i is the job number; n is the total number of jobs; m is the last stage; C_im is the time when the job i is finished at stage m; D_iis the due date of the job i. U_iis a variable, when C_im is earlier thanD_i, it shows that the job i meets the deadline, U_i =0; otherwise the job i delays the completion, U_i =1.

In addition to the delivery performance, most MTO enterprises try their best to reduce the production cost. Eq.2 is the mathematical description. For shop scheduling, the production cost is always named as direct production cost including three parts: machine working cost, machine waiting cost and job storage cost. They are described as Eq.3-Eq.5.

}

min{ oststorage

wait ost work ost C C C + + . (2) 1 1 1 j M m n work v

ost ijk j k ijk

j i k C Y F− P = = =      = _ __ ⋅ __      

∑ ∑ ∑

C is the machine working cost, and it is mainly constituted by depreciation expenses of fixed assets, loss cost of cooling fluid and machining tools, the wages of production unit managers and so on. In Eq.3, M_jis the total number of machines at the stage j; F_jv−_kis processing rate of the machine k

at stage j; Y_ijkis a variable, when the job i is processed on machine k at stage j, Yijk =1, otherwise 0

=

ijk

Y ; P_ijk is the processing time when the job i is processed on machine k at stage j. Eq.3 shows that the machine working cost is equal to the product sum of machine processing rate and machine processing time. 1 1 j M m wait S s ost j k j k j k C T− F− = =   = _ ⋅ _  

∑ ∑

C is the machine waiting cost, and it mainly contains machine idling depreciation and machine maintenance cost. In Eq. 4, T_js−_kis the waiting time before machine k starts to work at stage j; Fjs−k is

the waiting rate of the machine k at stage j. The machine waiting cost is equal to the product sum of machine waiting rate and waiting time.

(

)

∑ ∑

storage ost

C is the job storage cost. For many enterprises, there are some quench times between two consecutive processes. Some enterprises put jobs into storehouse during these quench times; others have the online storage management, and they put jobs near the current machine or the next machine which will start to process them. Anyway, storage will result the maintenance cost in order to ensure the product quality, and this cost is named as storage cost in shop scheduling. In Eq.5, S_ijis the time when the job i starts to be processed at stage j; w

i

F is the storage rate of the job i; C_i,j−1 is the time when the job i is finished at stage j-1.Thestorage cost is equal to the product sum of quench time and storage rate.

There are some other performance goals in MTO enterprises such as the improvement of resource utilization; however they do not have to be mentioned here.

Two-Objective Genetic Algorithm based on Sub-Module

For MTO enterprises, it looks like traditional two-objective optimization problem to achieve the order deadline goal and optimize another enterprise own performance goal. Actually it is different from the traditional optimization problem. The goal that must ensure all productions meet their deadlines is an absolute goal in MTO enterprise but not an optimization goal. Therefore the traditional multi-objective optimization algorithm can’t solve the two-objective optimization in MTO enterprise. In this paper, a novel genetic algorithm is proposed to solve the two-objective shop scheduling problem in MTO manufacturing systems.

The Idea of the New Algorithm. The two sub-module idea is applied into the search process in the new algorithm. It is controlled by two nested closed-loops, the inner of which is controlled by order delivery fitness and the outer one is controlled by optimization goal fitness. In the search process, the inner loop works with the manner that populations change by comparison. There are two solutions in the inner loop: feasible solution and infeasible solution. The search will jump out of the inner loop when the feasible solution arises; otherwise the unfeasible solution will be changed by crossover and mutation operations until the feasible solution arises. The outer loop is controlled by the traditional GA search strategy. The algorithm is controlled by two nested closed-loops and goes forward.

The stopping criterion of the outer loop is as same as the traditional GA’s. It is either the number of evolution generations or threshold value. Similarly, the termination condition of the inner loop should be given in order to prevent the order deadline from being unreasonable.

Operations of the New Algorithm. The new algorithm is described as follows:

Step1 Initialize mutation factor Pm, crossover factor Pc, the population size Np, the number of the

inner loop evolution generations, the number of the outer loop evolution generations. Let K=0, and randomly initialize population P(0).

Step2 Determine whether the outer loop stopping criterion is met, if the outer loop stopping criterion is met, jump out of the search and output the best solution; otherwise go to step3.

Step3 Let the individual number m=0.

Step4 Let the inner loop current evolution generation i=0.

Step5 Evaluate whether the individual m meet the order deadline, if the individual solution meet the deadline, jump to Step8; otherwise go to Step6。

Step6 Let i=i+1, and determine whether the inner loop stopping criteria is met, if the inner loop stopping criteria is met, jump out of the search and output the warning information; otherwise go to step 7.

Step7 Randomly select another individual to perform crossover and mutation operations with the old individual, and produce a new individual. Then return to Step5.

Step8 Let m=m+1, and evaluate whether m is greater than Np, if m is greater than Np, go to Step9;

otherwise return to Step4.

Step9 Evaluate the optimization objective values of all individuals, and pick up the best individual to put aside.

Step10 Perform crossover and mutation operations for the original population to produce a new population.

Step11 Let K=K+1, and return to Step3.

Simulation and Comparison

The new two-objective algorithm based on sub-module is simulated by visual studio 2008 software, and is compared with the traditional two-objective optimization algorithm based on weight-set. They are used to solve the two-objective hybrid flow-shop scheduling problem (HFSP) in MTO manufacturing systems, The HFSP can be described as follows [8, 9, 10]: there are n jobs to be processed, and each job must experiences m stages in the same direction; there is at least one machine at each stage and not less than one stage existing multiple machines; each work piece should complete one process at each stage; each process can work on any machine at the same stage. The HSFP model is shown in Fig. 1.

m m M−

1

m−M m−M2

Fig. 1 Hybrid flow-shop scheduling problem detailed description

In order to verify the feasibility and effectiveness of the new algorithm, we select 4 sets of different scheduling schemes to make comparison and analysis. Table 1 lists some parameters of the 4 sets, some parameters are produced randomly, such as the number of machines in each process, the processing time, order due date, machine working cost rate, machine waiting cost rate and job storage rate, and others are computed by Eq.1-Eq.5. The two objectives are delivery performance and cost optimization objective.

Table 1 Some parameter sets of scheduling schemes Parameter sets The number of jobs The number of stages Gen (outer loop) Np Pc Pm gen (inner loop) The range of machine number The weight of cost objective The weight of delivery objective 1 4 3 500 30 0.9 0.3 2000 random1-2 0.15 0.85 2 7 3 1000 30 0.9 0.1 2000 random2-4 0.1 0.9 3 8 6 200 30 0.7 0.5 2000 random2-5 0.1 0.9 4 10 8 500 20 0.9 0.2 2000 random1-6 0.5 0.5

After simulated, the results are shown in Table 2 and Fig.2 – Fig.5. Table 2 shows the two objective values of the two algorithms. The four figures show the compared iterative curves of the two algorithms.

Table 2 Objective values

Objective values 1 2 3 4

The new algorithm

The number of the

tardy jobs 0 0 0 0

Production cost 284 507 1200 2062

The old algorithm

The number of the

tardy jobs 0 1 0 1

Fig. 2 Compared iterative curves(1) Fig. 3 Compared iterative curves(3)

Fig. 4 Compared iterative curves(2) Fig. 5 Compared iterative curves(4)

In Fig. 2, Fig. 3, Fig.4, Fig. 5, red curve indicates the iterative process of the new algorithm based on sub-module, and the green curve indicates the iterative process of the traditional algorithm based on weight-set. These four figures show that the new algorithm has the faster convergent speed in search process of optimization objective. Table 1 shows that the new algorithm can ensure all jobs meet their deadline, but the traditional algorithm can’t do it. Therefore it can say that the novel algorithm has an obvious advantage and good feasibility compared with traditional algorithm.

Conclusions

For the scheduling of MTO manufacturing systems, the most important thing is to ensure all jobs meet their deadlines; at the same time, many MTO enterprises hope to make their own interests maximize through production scheduling. Both of the rule-based reverse scheduling method and the traditional multi-objective optimization algorithm can’t solve the two-objective problem well. In this paper, a novel genetic algorithm is proposed which can ensure that all jobs are completed on time. In addition, it also can better optimize another performance which the enterprise pursues. In a word, the novel algorithm has the better feasibility, practicability and maneuverability to solve two-objective shop scheduling problem in MTO manufacturing systems.

Acknowledgements

This work is financially supported by the National High Technology Research and Development Program “863”-Program Foundation of China (2011ZX02601-005, 2007AA040702-3), the Natural Science Foundation of China (60904047).

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