Solving Flexible Job-shop Scheduling Problem with Transfer Batches, Setup Times and Multiple Resources in Apparel Industry Miguel Ortiz1, Dionicio Neira2, Genett Jiménez3, Hugo Hernández4 1
Department of Industrial Engineering, Universidad de la Costa CUC, Barranquilla, Colombia.
[email protected] 2 Department of Industrial Engineering, Universidad de la Costa CUC, Barranquilla, Colombia.
[email protected] 3 Department of Industrial Engineering, Institución Universitaria ITSA, Soledad, Colombia
[email protected] 4 Department of Business Management, Universidad Del Atlántico, Puerto Colombia, Colombia
[email protected]
Abstract. Apparel industry is characterized by the presence of flexible job-shop systems that have been structured to manufacture a wide range of customized products. However, Flexible Job-shop Scheduling is really challenging and even more complex when setup times, transfer batches and multiple resources are added. In this paper, we present an application of dispatching algorithm for the Flexible Job-shop Scheduling Problem (FJSP) presented in this industry. Days of delay, throughput, earlier date and monthly demand are used as rules of operation selection. A case study in apparel industry is shown to prove the validity of the proposed framework. Results evidence that this approach outperforms the company solution and other algorithms (PGDHS and HHS/LNS) upon reducing average tardiness by 61.1%, 2.63% and 1.77% respectively. The inclusion of throughput in the model resulted in low tardiness for orders with high speed to make money. Promising directions for future research are also proposed. Keywords: Flexible Job-shop Scheduling Problem, Dispatching Algorithm, Throughput, Apparel Industry, Transfer Batches, Setup Times.
1
Introduction
Empirical evidence indicates that customized goods offer bigger value than standardized goods [1]. In reply to this, some manufacturing enterprises offer a wide
range of customized products to satisfy different customer requirements. However, this involves creating flexible production systems whose scheduling is highly complex. One of these systems is known as flexible job shop, defined as an extension of the classical job shop system which lets an operation to be processed by any machine from a specified set. This system has gained importance because companies need to produce more customized goods, which requires smaller batches, and machines capable to perform different operations. Given the above, this arrangement can be found more often in apparel industry [1]. Nevertheless, the difficulty of addressing flexible job shop scheduling (FJSS) is a well-known and complex Non-Polynomial (NP) hard combinational optimization problem [3] - [4]. According to Rakesh [2], the decisions to be made in a FJSS include the selection among optional machines on which to perform an operation or the selection among flexible process plan of a part-type. This is complemented with the objective of minimizing the makespan, average tardiness or mean flow time of parts [2]. FJSS is a prominent topic of study for researchers because of its theoretical, computational, and empirical significance since it was introduced. Owing to the complexity of FJSS, different solution techniques such as various meta-heuristics approaches, (Like Nature based heuristics, genetic algorithms, simulated annealing, among others) and heuristic approaches have been developed. The fact that a large number of small to medium companies operate at flexible job shops environment gives importance to the search for an efficient method to solve FJSS. Optimizing FJSS helps companies to increase its production efficiency; reduce cost and improve product quality [2]. It is also of particular interest to reduce average tardiness in highthroughput orders to ensure meaningful returns on investment (ROI) for companies with flexible job-shop systems since bad schedules could make clients return these kinds of orders, which generates superior affectations than low-throughput orders. Particularly, this paper is organized as follows: Section 2 presents a literature review referred to some techniques that have been explored to solve FJSP. Section 3 describes dispatching algorithm with rules of operation selection, machine selection and robustness. Section 4 presents an application of this approach in apparel industry. As final point, in Section 5 conclusions are given.
2
Flexible Job Shop Problem (FJSP): A Literature Review
The main difference between the classical Job Shop Problem (JSP) and the Flexible Job Shop Problem (FJSP) lies in the fact that in JSP, each operation is processed on a predefined machine; on the other hand, each operation in the FJSP can be processed on one out of several machines[5] - [6]. According to Demir Y. [6] and Zhang G. [5], the FJSP can be split into: “the routing sub-problem that assigns each operation to a machine selected out of a set of capable machines, the scheduling sub-problem that consists of sequencing the assigned operations on all machines in order to obtain a feasible schedule to minimize the predefined objective function”.
Based on the survey of Sobeyko and Mönch [8], Brucker and Schlie [7] addressed the FJSP for the first time. They considered a job shop consisting of different multipurpose machines. They proposed a polynomial algorithm for the special case of two jobs. Furthermore, Hurink et al. [9] used a tabu search approach to optimize the makespan (Cmax). A hierarchical algorithm for the flexible job shop scheduling problem was presented by Brandimarte [10] in which the operations were initially assigned to specific machines and a job shop scheduling problem was solved after this decision. In this research, the minimization of makespan and total weighted tardiness were considered as primary objectives. In addition to the formulation problem, another difficulty emerged with the problem solution when increased its size by including more machines, alternative machines, more jobs, and other considerations that turn out to be complex when have to be represented in accordance with the real production system. The need to solve this kind of formulations in a fast way, forces the use of metaheuristics. Nevertheless, according to Calleja and Pastor [11] the advantage of having very capable machines might disappear. It also important to indicate that flexibility has a positive effect on the manufacturing system performance. The processes should be flexible enough to overcome adversities related with operations, machines, and raw materials that potentially would alter the process performance and divert it from its goal (minimizing makespan, for example). According to this principle, Kim et al. [12] pointed that there are three types of flexibility to generate flexible process plans: operation flexibility, sequencing flexibility and processing flexibility. The inclusion of these flexibilities would make a process plan be better prepared to come through disturbances occurring at the shop floor. The main advantage of a flexible process plan relies on the fact that it helps to maximize flexibility preservation because the scheduler can choose an alternative way at any time during the production process. This allows the decision maker to react fast to any problem on the shop floor. The incorporation of flexibility in job-shop scheduling, increases dramatically the complexity of the problem that is already considered very hard and costly to solve. It is also important to highlight that these production schedules could be made many times in a week, reason by which it is important to find tools with quick solutions. According to Wu [13], several researchers have studied the effects and relationships of flexibility on system performance. Different heuristic procedures such as dispatching rules, local search and meta-heuristics procedures have been applied to solve FJSS and find near optimal schedules. Some works are mentioned here. For example, Tanev et al. [14] proposed a hybrid evolutionary algorithm involving priority dispatching rules with a GA to generate a schedule for a flexible job-shop system (FJS) in a plastic injection process. One of the most utilized tools for solving the FJSS is genetic algorithms. Many authors have implemented genetic algorithms with different variations for this purpose. One of these approaches is presented by Pezzella et al. [15] where different strategies are integrated to select an initial population and individuals for reproduction are selected. The results evidenced that the integration of more strategies in a genetic approach leads to a better performance. On the other hand, Gao et al [16] proposed a hybrid variable neighborhood descent genetic algorithm with the following objectives: makespan, maximal machine
workload, and total workload. A combined adaptive randomized decomposition large neighborhood search scheme was also proposed by Pacino and Van Hentenryck [17] with the makespan minimization as the objective. Many other authors addressed the FJSP with the use of dispatching rules, and heuristics like shifting bottleneck. Finally, it is important to point that Calleja and Pastor [11] developed a dispatching rule-based algorithm to minimize average tardiness for FJSP having transfer batches. This heuristic is used in this case study with the inclusion of throughput as second rule of operation selection [18] – [19] since this measure has not been considered inside these algorithms. In conclusion, literature review reveals that the effect of the flexible process plans on average tardiness and other performance measures in job shop scheduling, has not been studied deeply, reason by which this paper represents a relevant contribution to the development of this research field.
3
Dispatching Algorithm for FJSP
This section describes the FJSP problem and the dispatching algorithm proposed by Calleja and Pastor [11] with the inclusion of throughput as second rule of operation selection to solve this problem in apparel industry. Throughput is included to ensure a minimum tardiness for orders with high speed to make money.
3.1
Problem Description
The Flexible Job-shop Problem may be defined as follows: Let be a set of n orders, which can be performed by m machines . Each order consists of a predefined sequence of operations . Each operation has to be manufactured on a machine chosen from a given set of available machines. The allocation of the operation to the machine involves the occupation of the last one at a time defined as and has been defined as availability time of operation . On the other hand, has been denoted as the earlier time to start a new operation on the machine (if no queue, infinite value is assigned) In this paper, the objective is to minimize average tardiness of orders and get the lower tardiness scores in those with high throughput. The preliminary conditions considered in this system have been summarized as follows: ─ Each machine can manufacture one operation at a time. ─ Setup times and transfer batches are considered. ─ Each operation can be performed without interruption on one of the available machines of the set.
─ Priorities are assigned to the orders according to the next criteria and order of priority: days of delay, throughput, earlier date and monthly demand. ─ All orders are released at time 0 and all machines are available at this time. ─ Breakdowns are not taken into account. ─ The sequence of operations for each order is predefined. 3.2
Steps of Dispatching Algorithm
3.2.1 Start ─ Put the first operations of the parts with their respective of candidate operations ─ For each machine , estimate value. ─ Calculate and its respective machine
values in the subset
3.2.2 Machine Selection ─ If , all operations have been scheduled. ─ Otherwise, select a machine according to (Rule 1: Order with more days of delay). In case of a tie, select one according to Rules 2 (Order with the highest throughput), 3 (Order with earlier delivery date) and 4 (Order with the highest monthly demand) of operation selection. ─ When selecting the machine, create the subset of candidate operations that is composed by the eligible operations that the machine is able to perform . 3.2.3 Operation Selection ─ If there is only one candidate operation, this must be scheduled. ─ Otherwise, apply the priority rules for operation selection to choose the operation to program. 3.2.4 Update ─ Schedule the selected operation setting its initial time (See Equation (1) and Equation (2)).
and final
(1)
(2) Where represents the number of ordered pieces and of operation of the order in the machine
is the unit processing time
─ Place the eligible operation in the subset of schedule operations with its start and final times. ─ If it is not the final operation of order , move its next operation from (Unscheduled and unavailable operations) to subset. Transfer batches
Taking into account that is the machine associated to the next operation of order , is the transfer batch size and is the necessary time to move a transfer lot size , release date can be calculated as described in Table 1: Table 1. Formulas to calculate release dates with transfer batches
Relation between k and k´
Relation between p(j, i, k) and p(j+1, i, k) All possible relations between and
r(j + 1, i, k)
─ Update used
values of the machine . If the machine has not already . If any operation in machine has been scheduled, then, i.e., the machine will have an availability time that is equal to final time of the last scheduled operation in the machine . ─ Calculate according to Equation (3): (
─ Calculate
)
(3)
according to Equation (4) (
)
(4)
─ Return to step 1. 3.2.5 Objective Function Calculate average tardiness according to Equation (5): ∑
.
(5)
, represents the completion time of the order and
4
is the delivery date.
A Case Study in Apparel Industry
A case study in a company from apparel industry is shown in this paper to prove the validity of the proposed approach. This company presents a production system with 6 processes as shown in Figure 1.
Fig. 1. Layout of flexible job-shop studied in apparel industry – Case study
In this application, 35 orders were considered with units and min. PRINTING process has 1 machine (H), LONGITUDINAL SECTION is composed by 2 worker-machines (CO1, CO2), SIDE SEAM is integrated by 4 worker-machines (CL1, CL2, CL3, CL4), CROSS CUT has 2 worker-machines (CT1, CT2), HEAD SEWING is composed by 7 worker-machines (CC1, CC2, CC3, CC4, CC5, CC6, CC7) and CLEANING has 4 worker-machines (L1, L2, L3, L4). Each product has its own production sequence as shown in Table 2. Black cells indicate that the product does not need to be processed in that stage. Table 3 shows the orders received by the company in each product reference. Each cell indicates the order size and the delivery date of the order where “A” means August, “S” September and “O” October. In this way, “1A” indicates August 1st. If the cell contains a number in brackets, this number denotes the days of the delay corresponding to the order. Table 2. Processing times in min for each product reference in each process PRODUCT REFERENCE
30 X 50 35 X 60
PRINTING
LONGITUDINAL SECTION
SIDE SEAM
CROSS CUT
HEAD SEWING
CLEANING
0.02 0.02
0.43 0.43
0.06 0.06
0.75 0.75
0.67 0.14
50 X 90 35 X 60 R 33 X 56 P 70 X 140 60 X 120 30 X 30 30 X 20 P 60 X 120 P 60 X 120 G 35 X 60 G 30 X 30 G 60 X 120 H 70 X 140 H
0.08
0.04 0.19
70 X 140 HG
0.02 0.02 0.02
0.75 0.43 0.43
0.01 0.01
0.38 0.38
0.02 0.01 0.04 0.045 0.045
0.43 0.38 1 1.2 1.2
0.07 0.06 0.06 0.1 0.083 0.04 0.04 0.083 0.083 0.06 0.04 0.083 0.1 0.1
1 0.75 0.75 1.2 1.2 0.9 0.9 1.2 1.2 0.75 0.9 1.2 1.2 1.2
0.42 0.14 0.14 0.67 0.67 0.13 0.13 0.67 0.67 0.14 0.13 0.67 0.67 0.67
Table 3. Orders per product reference and delivery dates PRODUCT REFERENCE
ORDER 1
ORDER 2
ORDER 3
30 X 50
12000 – 24S
12000 – 4O
4797 – 14S
35 X 60
7800 – 1A (9)
2000 – 1A (2)
6000 – 9A
50 X 90
24000 – 1A (5)
8000 – 28S
6000 – 28S
35 X 60 R
24000 – 1A (8)
13405 – 16S
8500 – 16S
33 X 56 P
20400 – 28S
70 X 140
2400 – 1A (2)
60 X 120
2400 – 1A (4) 2000 – 22S
30 X 30 30 X 20 P 60 X 120 P 60 X 120 G 35 X 60 G 30 X 30 G 60 X 120 H 70 X 140 H
17000 – 4O
5000 – 26S
12600 – 4O
4400 – 4O
5351 – 30S
15000 – 2O
406 – 12S
1600 – 15S
70 X 140 HG
100 – 12S
ORDER 4
ORDER 5
ORDER 6
2000 – 12S
10000 -2O
7800 – 26S
5000 – 23S
1600 – 15S 1600 – 15S 200 – 5S
150 – 9S
100 – 5S 200 – 12O
On the other hand, Table 4 defines the throughput and monthly demand of each product reference to be considered at the moment of scheduling. It is noticed that
product references 60 x 120 and 70 x 140 present the highest throughput rates which has to be taken into account by the algorithm to ensure low tardiness scores. Products with high throughputs represent better returns on investment (ROI) for the company. Table 4. Throughput and monthly demand of each product reference
PRODUCT REFERENCE
Throughput ($/min)
30 X 50 35 X 60 50 X 90 35 X 60 R 33 X 56 P 70 X 140 60 X 120 30 X 30 30 X 20 P 60 X 120 P 60 X 120 G 35 X 60 G 30 X 30 G 60 X 120 H 70 X 140 H 70 X 140 HG
13530 18942 21250 20951 16669 71400 39827 20062 5986 39827 88560 25830 12177 110208 120540 150005
Monthly demand (units/month) 1360 10991 26145 2229 16669 1998 5044 1407 2436 1387 274 1302 37 140 179 86
Furthermore, Table 5 shows the first seven operations scheduled by dispatching algorithm. It can be noted that LONGITUDINAL SECTION operation for the product reference 35 x 60 (Order 1) has been selected as the first scheduled operation since this order presents 9 days of delay. At this time, both first operations and machines are available. In total, 167 operations were programmed under this algorithm and priority rules. Table 5. First 7 operations scheduled by dispatching algorithm with the inclusion of throughput Product Reference
Operation
Order Number
Possible resources
Selected resource
tstart (h)
tfinal (h)
1
CO1, CO2
CO1
0
2,68
1
CO2
CO2
0
8,08
H CL1, CL2, CL3, CL4
H
0
17,94
CL1
0,01
55,94
35 x 60 R
LONGITUDINAL SECTION LONGITUDINAL SECTION
60 x 120 P
PRINTING
1
35 x 60
SIDE SEAM
1
35 x 60
SIDE SEAM
1
CL2, CL3, CL4
CL2
0,01
172,04
50 X 90
LONGITUDINAL SECTION
1
CO1
CO1
2,68
10,76
50 X 90
SIDE SEAM
1
CL3, CL4
CL3
2,69
302,72
35 x 60 R
Figure 2 shows a comparative graph among proposed algorithm; company solution, Pareto-based grouping discrete harmony search algorithm (PGDHS) [20] and the integrated HHS/LNS approach [21]. It is shown that the proposed algorithm obtains the best results for tardiness performance measures since they are equal to or less than provided by company solution and the other algorithms. Average tardiness was improved by proposed algorithm with 3.17 days; while company solution, HHS/LNS approach and Pareto-based grouping discrete harmony search algorithm offered a schedule with 8.14, 3.26 and 3.23 days respectively. This represents an improvement percentage of 61.1% which is meaningful for the company. Deviation standard of tardiness was reduced from 13.88 days to 11.2 days (Improvement percentage: -19.2%). Compared to PGDHS and HHS/LNS, average tardiness was reduced by 2.63% and 1.77% correspondingly.
Fig. 2. Comparison of proposed algorithm, company solution and other algorithms
Upon evaluating the number of late orders (although it is not the primary aim of this algorithm), dispatching algorithm presented 6 late orders (17.1%); while results from company solution showed 25 (71.4%). Finally, tardiness of orders with high throughput was also assessed in the study through a test for differences between means. With T = -2.07 and P-value of 0.03 (α = 0.05), it can be said that average tardiness of high-throughput orders of proposed algorithm (0.36 days) is statistically less than provided by company solution (1.82 days). A comparative graph (Figure 3)
for this evaluation between proposed method (PM) and company method (CM) is shown to confirm the statistical results.
Fig. 3. Comparison between proposed algorithm and company solution for high-throughput orders
5
Conclusions
In this study, a dispatching algorithm [11] with the inclusion of throughput as second rule of operation selection has been proposed to solve FJSP in apparel industry. Application results prove the validity of the proposed approach with 61.1% improvement in average tardiness, 19.2% reduction in standard deviation of tardiness, 54.3% reduction in late orders and a minor average tardiness for high-throughput orders in a company from apparel industry. It also offers better performance of tardiness measures than provided by PGDHS and HHS/LNS with 2.63% and 1.77% reduction correspondingly. This signifies a cost reduction for the company since some penalties for non-compliance are avoided. In addition, customer satisfaction is increased because of the reduction in average tardiness of orders. Lower average tardiness in high-throughput orders are less likely to be returned by clients; so that high return on investments can be ensured. For future research, it is recommended to explore scheduling changes upon varying transfer lot size; moreover, breakdowns can be included in order to evaluate strategies that allow reducing non-compliance, possible complaints and cost overruns.
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