Genetic optimization of order scheduling with multiple uncertainties
Introduction
Facing with ever increasing global market competition, today’s manufacturers have to continuously improve their production performance so as to be more competitive in the market. Effective production scheduling plays a significant role in maximizing the resource utilization and shortening the production lead time.
A wide literature base has been published on production scheduling, focussing mostly on the scheduling for various types of production systems at the shop floor or assembly-line level, such as job shop scheduling (Adam et al., 1993, Fayad and Petrovic, 2005, Guo et al., 2006, Guo et al., 2006, Kondakci and Gupta, 1991), flow shop scheduling (Ishibuchi et al., 1994, Iyer and Saxena, 2004, Morita and Shio, 2005, Nagar et al., 1996), machine scheduling (Baek and Yoon, 2002, Dimopoulos and Zalzala, 2001, Fowler et al., 2003, Liu and Tang, 1999), assembly line scheduling (Guo et al., 2006, Guo et al., 2006, Kaufman, 1974, Vargas et al., 1992, Zhang et al., 2000), etc. Ashby and Uzsoy (1995) have presented a set of scheduling heuristic to solve the order release and order sequencing problem in a single-stage production system. Axsater (2005) has discussed the order release problem in a multi-stage assembly network by an approximate decomposition technique. Their studies only focused on determining the starting times for different processes of each production order (also called order), where the process should be performed has not been considered. Chen and Pundoor (2006) have considered the order assignment and scheduling in the supply chain level, they focused on assigning orders to different factories and finding a schedule for processing the assigned orders at each factory. However, multiple shop floors and multiple assembly lines are setup in most factories. The order scheduling problem at the factory level, where the production process of each order scheduled to the appropriate assembly line, has not been reported so far.
The great majority of previous studies on production scheduling are based on the deterministic estimation of the processing time of each production process and the arrival time of each order. In real-life production environment, various uncertainties often occur, such as uncertain customer orders and uncertain estimation of processing time. The deterministic estimation is not in accordance with the industrial practice and will lead to an unsatisfactory scheduling solution. Moreover, without considering the uncertain factors, it is difficult for an optimized production schedule to be realized in real-life production environment owing to various uncertainties. This optimized solution cannot be helpful to achieve the optimal performance in the real production environment. For example, if a schedule is generated without considering possible orders in the future, new rush orders may interrupt those already scheduled, causing serious violation of their desired due dates.
This paper will investigate the order scheduling problem at the factory level, in which each production process is corresponding to a unique shop floor comprising one or multiple assembly lines. The objectives are firstly to maximize the total satisfaction level of orders’ actual competition times, and also minimize these orders’ total throughput time by determining which assembly line and when the production process of each order will be processed. In a make-to-order manufacturing environment, it is very important to predict whether the due date can be satisfied before receiving a new order from the customer and to schedule the production of each process in different assembly lines. A typical example is apparel manufacturing.
Some possible uncertain factors on order scheduling will also be investigated in this paper. We consider the uncertain processing time as a continuous random variable, and uncertain orders as well as arrival times as discrete random variables. On the basis of the stochastic processing time, the stochastic beginning time and completion time of processes are derived using the probability theory approach. The genetic algorithm (GA) will be adopted to solve the order scheduling problem, in which a novel process order-based representation with variable length of sub-chromosome is presented.
The rest of this paper is organized as follows. Section 2 defines the notations which are used in this paper. A detailed problem formulation for the order scheduling problem is presented firstly in Section 3. In Section 4, how to calculate the stochastic beginning time and completion time are explained. The proposed GA is then presented to solve the addressed order scheduling problem in Section 5. In Section 6 experiments are conducted to validate the effectiveness of the proposed methodology using the real production data from an apparel manufacturing factory. Lastly, concluding remarks are presented and further research are suggested in Section 7.
Section snippets
Nomenclature
The following notations are used in developing the mathematical model of the order scheduling addressed in this paper.
- Ai
arrival time of order Pi
- Bij
beginning time of process Rij
- Ci
completion time of order Pi
- Cij
completion time of process Rij
- Di
due date of order Pi
- ETij
transportation time between assembly lines processing process Rij and its following process
- Lkl
lth assembly line of shop floor Sk
- Pi
ith production order (1 ⩽ i ⩽ m)
- Rij
jth production process of order Pi
- Sk
kth shop floor
- SALij
set of assembly
Problem formulation
This section explains the formulation of the order scheduling problem in an order-based manufacturing factory. Production processes of each order should be performed in different types of shop floors, respectively. Each type of shop floor comprises one or more assembly lines. According to a predetermined production flow, production processes involved in each order must be completed on an assembly line of the corresponding shop floors. For simplicity, assume that there is no WIP in each shop
Completion time of production process
The completion time Cij of process Rij is determined by its beginning time and processing time. Since the beginning time and the processing time are independent, the probability density function of Cij is equal to the convolution of probability density functions of its beginning time and processing time according to the theory of probability.
Beginning time of production process
Since the processing time and completion time of process Rij are uncertain, the beginning time of Ri,j+1, and the subsequent process of Rij, are also
Genetic algorithm for order scheduling problem
The addressed order scheduling problem is categorized as the combinational optimization problem of NP-hard type (Ross & Corne, 2005) and the number of its possible solutions grows exponentially with the number of assembly line, orders and processes. It is very difficult for the classical technique to solve the type of problem. Since the GA has been proven to be very powerful and efficient in finding heuristic solutions from a wide variety of applications (Goldberg, 1989), it is adopted in this
Experimental results and discussion
To evaluate the performance of the proposed algorithm for the order scheduling problem, a series of experiments were conducted. The experimental data were collected from a make-to-order apparel manufacturing factory producing outerwear and sportswear. This section highlights three out of these experiments in detail. Each example includes several cases. In each case, the order scheduling result generated by the proposed method is compared with that of the practical method from industrial
Conclusions
This paper dealt with a multi-objective order scheduling problem at the factory level, where uncertainties are described as continuous or discrete random variables. The objectives considered were to maximize the total satisfaction level of all orders and minimize their total throughput time, which are particularly helpful to meet the due dates of orders and reduce the WIP in each shop floor.
Based on the uncertain processing time of production process, uncertain completion time and beginning
Acknowledgements
The authors would like to thank Innovation and Technology Commission of the government of the Hong Kong SAR and Genexy Company Limited for the financial support in this research project (Project No. UIT/62).
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