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1、CHINESE JOURNAL OF MECHANICAL ENGINEERING Vol. 20, No. 4, 2007117CHEN Kejia Jl PingDepartment of Industrial andSystems Engineering,The Hong Kong Polytechnic University,Hong Kong, ChinaDYNAMIC ADVANCED PLANNING AND SCHEDULING WITH FROZEN INTERVAL FOR NEW ORDERS*Abstract: A dynamic advanced planning a
2、nd scheduling (DAPS) problem is addressed where new orders arrive on a continuous basis. A periodic policy with frozen interval is adopted to increase stability on the shop floor. A genetic algorithm is developed to find a schedule at each rescheduling point for both original orders and new orders t
3、hat both production idle time and penalties on tardiness and earliness of orders are minimized. The proposed methodology is tested on a small example to illustrate the effect of the frozen interval. The results indicate that the suggested approach can improve the schedule stability while retaining e
4、fficiency. Key words: Dynamic advanced planning and scheduling Genetic algorithm Frozen interval0 INTRODUCTIONAdvanced planning and scheduling (APS) deals witi? effectively allocating productior resource.; to complete the multi-level products so that production constraints are satisfied and producti
5、on objectives are mrt (l2. Much of the research in APS is based on the assumption that the manufacturing environment is static, which rarely holds in real situations. In practice, some unexpected events, such as the arrival of new orders, may arise and disrupt the manufacturing system, which leads t
6、o the study of dynamic advanced planning and scheduling (DAPS).In recent years, more studies have considered planning and scheduling problems in the dynamic environment, as reviewed by Refs. 3-4, BEAN, et al51, provided a heuristic method by reconstructing part of the schedule, when a disruption occ
7、urs, to match up with the pre-schedule at some future time. Also, match-up approaches with schedule changes minimization were adopted for responding to disturbances including insertion of new orders in Ref. 6. BIERWIRTH, et al7, described genetic algorithms for job shop scheduling and rescheduling,
8、and demonstrated that their approaches produce far better results than priority-rule based methods. When jobs arrive at the job shop on a continuous basis, RANGSARITRATSAMEE, et al11, proposed a genetic local search methodology that simultaneously considers efficiency and stability through a multi-o
9、bjective measure. Unfortunately, most of the research efforts have concentrated on one machine, flow shop, and job shop situations, assuming operations are performed in series. There appears to be scant research on introducing dynamic mechanism into APS.This paper addresses a DAPS problem where new
10、orders arrive on a continuous basis. A periodic policy with frozen interval is adopted to increase stability on the shop floor. A genetic algorithm is developed to find a schedule at each rescheduling point for both original orders and new orders that both production idle time and penalties on tardi
11、ness and earliness of orders are minimized. The proposed methodology is tested on a small example to illustrate the effect of the frozen interval. The results indicate that the suggested approach can improve the schedule stability while retaining efficiency.This paper is organized as follows: In sec
12、tion 2, a periodic policy with frozen interval is introduced, and a GA-based method to achieve the production objective is presented. A small example This project is supported by the Hong Kong Polytechnic University.China (No.G-RGF9). Received August 7, 2006; received in revised form November 14, 20
13、06; accepted December 7, 2006to illustrate the methodology is show i.i section 3. Finally, section 4 concludes the paper.I PROPOSED METHODOLOGY1.1PolicyThis research addresses a DAPS problem where new orders arrive on a continuous basis. In such a situation, if we construct a new schedule every time
14、 a new order arrives, the system may be in a permanent state of replanning and rescheduling, and instability will be induced on the shop floor. Meanwhile, it is observed that the stability of the production system will decrease more when changes are made closer to the current period 18l Therefore, t
15、his paper adopts a periodic policy with frozen interval. In other words, the schedule is revised periodically at the rescheduling point but not every time a new order arrives. Moreover, operations near the current time and within the frozen interval are fixed. Those operations outside the frozen int
16、erval and the newly arrived orders are available for building a new schedule. This policy provides a framework for balancing efficiency and stability.1.2Objective functionAt each rescheduling point, our problem is to find a schedule for all the orders including original orders and new orders that bo
17、th production idle time and penalties on tardiness and earliness of orders are minimized. Minimizing production idle time is equivalent to minimizing flow time or maximizing machine utilization. In addition, production idle time is chosen as the objective to be reduced because it is able to reflect
18、two focuses in shops: manufacturing lead time and work-in-process (WIP) inventory level. Another objective of the problem is to find a schedule with all orders completed as closed to their due dates as possible, which is on the fact that either early or late delivery of an order results in an increa
19、se in the costs.1.3GA-based approachGenetic algorithms (GAs), invented by Holland and his associates in the 1960s, have been successfully implemented to find good solutions to a wide variety of dynamic scheduling problems 78l In this section, a GA-based approach for solving the DAPS problem is discu
20、ssed.Encoding. Our encoding scheme is based on the concept of random keys suggested by BEANI0. This scheme encodes a solution with a string of random numbers. Each available item in both original orders and new orders has one random number generated from the range 0, 1. These random numbers denote t
21、he priorities of the items, while the smaller value represents the 1994-2007 China A demic Journal Elec ronic Publishing House. A1 rights reserv d. http/.n nt* 118 CHEN Kejia, et al: Dynamic advanced planning and scheduling with frozen interval for new ordershigher priority. The random key encoding
22、has the advantage that it eliminates the offspring feasibility problem and is robust to problem structures01. On the basis of this idea, a feasible schedule can be derived from the product structure and the string of random numbers. From the random numbers, the priorities of items are obtained first
23、. An item with the highest priority is selected among the available items which have no child items or whose child items have all been scheduled. The starting time of the selected item is determined considering both the finish time of its child item and the available time of its processing machine.
24、This procedure continues until all items are allocated. Finally, since our objective is to minimize both production idle time and penalty including earliness and tardiness costs, unforced idle time may be introduced into the schedule. To achieve this, for the early orders in the obtained schedule, t
25、he final products are moved to their latest possible time, based on the fact that only the completion time of the final product affects the earliness or tardiness of an order. If the overall objective improves, the new schedule updates the former one; Otherwise, the former one remains.Initialization
26、. The initialization of the population of chromosomes can be done by generating the chromosomes randomly as much as the desired population size. The genetic operations are then performed on the chromosomes, the random keys, not on the schedule, which always leads to a feasible solution.Evaluation. T
27、he schedule represented by each chromosome is evaluated by using the fitness function given in the following equation, which aggregates production idle time, earliness and tardiness penalty. Let F(Xh) be the fitness function for chromosome A* in the scheduling problem, thenFxky.CiX/ + i(CtxL;+Cex/)w
28、heren-Q-1-C,-Number of orders-Cost of idle time per hour-Total production idle time-Cost of tardy orders per day per job-Cost of early orders per day per job-Number of tardy days (integer) for order 0,ENumber of early days (integer) for order OfSelection. The well-known roulette wheel approach is em
29、ployed for selecting some chromosomes to conduct genetic operations. Based on this approach, the probability of selecting a chromosome is determined by its fitness. Chromosomes having larger fitness values are more likely being selected. Although the roulette wheel selection mechanism chooses chromo
30、somes probabilistically, not deterministically, it is certain that on average a chromosome will be selected with the probability proportional to its fitness.Genetic operations. Genetic operations such as reproduction, crossover and mutation are executed to produce a new set of chromosomes called off
31、spring. There are many variations of genetic operations that could be used in GAs. Since random keys encoding preserves to generate feasible solutions, there is no need to design specialized operations. The genetic operations employed here are elitist reproduction, parameterized uniform crossover an
32、d immigration, which have proved very robust in computational tests ft.The algorithm. The overall procedure of the proposed GA approach is listed as follows.Step 1: Set the GA parameters, including the maximum generation, the population size, the number of reproduction, the number of crossover, the
33、crossover probability, and the number of mutation.Step 2: Generate the population size initial chromosomes according to the encoding strategy.Step 3: Evaluate the fitness value F(Xh) for all chromosomesin the population according to the evaluation strategy.Step 4: Perform the genetic operations.Step
34、 5: Repeat steps 34 until the maximum generation is reached.2 NUMERICAL STUDYDue to space limitation, only a small example is presented to verify the proposed methodology. The three-level product structure is shown in Fig. 1. Two machines, with 8 h available per day, are eligible to process the item
35、s (Table 1). The penalty rates are as follows: cost of idle time at $50 per hour, cost of tardiness at $250 per day per order, and cost of earliness at $50 per day per order. At the beginning of the planning horizon (r = 0), the ready time of Mi and M2 are Hour 5 and Hour 3, respectively. There are
36、two orders, one requiring 10 product F)S with due date Day 3 and the other requiring 25 product Sis with due date Day 4. For this original problem, an optimal schedule could be obtained by solving a mixed integer programming (MIP) l2. It is assumed that the rescheduling interval is 1 d, that u, 8 h.
37、 Then, at the rescheduling point (t = Day 1), two new orders are collected: 5 product F)S with due date Day 7 and 20 product C2s with due date Day 3.Fig. I Three-level product structureTable I Machine processing time for the itemsItemMachine numberProcessing time l/hF,Mi0.7s,Mi0.5c,M20.5c2M20.1c,M20
38、.2To solve this dynamic problem, our proposed methodology was coded in C language, and run on a personal computer with Pentium 2.4 GHz CPU and 1 GB RAM. Frozen interval = 2 h was used, and the genetic parameters were set to maximum generation = 200, population size = 100, number of reproduction = 10
39、, number of crossover = 80, crossover probability = 0.7, number of mutation = 10. The best solution with the total costs of 1 200 was obtained in 0.68 s. The best operation sequences are graphically represented in the Gantt chart as shown in Fig. 2. For convenience, item/? of order O, is denoted Oip
40、. From the Gantt chart, it is easy to find that order 1 is finished after its due date, while order 3 is early.In order to compare the results and examine the effect of the frozen interval, we also adopted GA and MIP without frozen interval to tackle the dynamic problem. As in Ref. 8, stability is m
41、easured by two components. One is the starting time deviation for all operations between the new schedule and the original one, and the other is the total deviation from the current time. Table 2 lists the testing results, and it can be seen that the suggested approach can improve the schedule stabi
42、lity while retaining efficiency.994- 07 hnCHINESE JOURNAL OF MECHANICAL ENGINEERING 119 M20102SISi010102020303CM(ilC2C3C2C3C2C3C262402030103MlSISIFtFl0103M2Cli 124483240Fig. 2 Output in the form of Grantt chartTable 2 Results from different methodologiesMethodologyStabilityObjective functionGA with
43、frozen interval GA without frozen interval MIP without frozen interval33.57 38.72 38.721200 1 200 12003 CONCLUSIONSThe issue of DAPS is studied to allow for ths airival of new orders. In order to trade oft efficiency &ni stability, a periodic policy with frozen intervy.i is suggested. We also develo
44、p a GA-based method for the dynamic production system, with the objective of minimizing cost of both production idle time and earliness-tardiness penalty for all orders including both original orders and new orders. The numerical results confirm that the proposed methodology can improve the schedule
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