Sequencing Method for Products of Various Assembly Timeand Different Parts Prices at a Single Workstation
－Mixed-model Assembly Scheduling Problem in JIT Production System
OkayamaUniversity, Jiajun ZHANG, Shigeji MIYAZAKI and Yoshinari YANAGAWA
Abstract
Just-in-time production systems are intended not only to trim total inventory levels, but also to trim the total inventory value and to thereby increase profits.We have proposed a heuristic method, Price Based Goal Chasing Method(PBGC), for different parts compositions and different parts prices in a mixed-model assembly scheduling situation at a single work station. Themethod is based on price-based goal chasing method to decrease variations in purchase expenses of parts.
In this paper, we propose a new heuristic method,Assembly Time and Parts Prices Based Goal Chasing Method (TPBGC), considering the different assembly times of respective products.In addition, problems of neglecting the different assembly times of products are discussed through comparison of results obtained using PBGC and TPBGC.Numerous better results demonstrate the effectiveness of the new heuristic method.
Keywords: JIT, mixed-model,assembly time,part prices,the goal chasing method.
1. Introduction
Just-in-Time (JIT) production is intended to reduce all wastes that produce no value. A manufacturing process is accustomed to having sufficient inventories to allow continuation of production products. To reduce inventories, it is important to find wastes in a manufacturing process because inventories are not only buffers between related processes; they are also shelters of waste from the view of JIT production. Previous examinations of JIT have described how to reduceinventories; some have particularly investigated job sequencing in processes [1]–[3]. Goal chasing method [1] is the most popular job sequencing method in JIT because the method can obtain a feasible sequence and seems to be not so far off the mark. Goal chasing method, however, has some inherent problems, e.g., non-optimal sequencing, assumption of non-fluctuating or equal worktime, and so on. This study specifically examines the differences of parts that are assembled to make products using goal chasing method. In previous studies[3]–[5], all different parts were assumed to equally hide wastes of the process and are treated as having no value differences. Nevertheless, we consider that each parthas a different value and should therefore be treated differently. A new job sequencing method, price-based goal chasing method (PBGC)[6], based on quantities and purchasing prices of respective parts, was proposed to reduce the fluctuation of necessary funds that are used to purchase parts.
This paper presents a new heuristic method, Assembly Time and Parts Prices Based Goal Chasing Method (TPBGC), considering the difference of assembly time of each product into account.In addition, problems of neglecting the different assembly times of respective products are discussed through comparison of results obtained using the PBGC and TPBGC.Numerous better results obtained using the latter method demonstrate its effectiveness.
2.Scheduling Model
2.1 Manufacturing Model
The manufacturing model used in this study is shown in Fig.1.In a process, Itypes of products are assembled from J types of parts. In addition, bijmeans the number of consumed parts hjthat are consumed to assemble a productHi; pj represents the value of parthj. We suppose that the value equals the purchase
Fig.1 Manufacturing Model
price of hjin this study. Qi is the number of assembled product Hi in the planning horizon.Assembly time of each product Hi is Titimeunit.
2.2Preconditions
In this paper, we assume the manufacturing condition as follows:
1. I types of product are assembled.
2. J types of parts are required to assemble all products.
3. Assembly times of respective products are different.
4. Values of respective parts are different.
5. These products are assembled at a single work station.
6. It is not possible to assemble more than one product simultaneously.
7. The set-up time of each type of product is independent of the preceding job in the sequence.
8. Slack time between jobs is not allowed.
3. Mathematical Formation
3.1 Notation
Hi :Product i (i=1,2,...,I)
i: Number of types of products
hj : Parts j (j=1,2,...,J)
j : Number of types of parts
Qi: Number of assembled product Hi that are assembled in a planning horizon
K: Total of all products that are assembled in a planning horizon
(1)
bij: Total of parts hj that are used to assemble one product Hi
Nj: Total of parts hjthat are used to assemble all products in a planning horizon
(2)
pj: Purchase price of one part hj
Pj: Amount of purchase price of parts hj, which are used to assemble all products in a planning horizon
(3)
k : kth order in which a kth product is assembled in a process (k=1,2,...,K)
Eij : Purchase price of parts hj that are ideally used to assemble one unit of product Hi
Hi*:Product selected and allocated into the sequence
Sk : Set of products that are assigned from order 1 to order k
(4)
(5)
Ti: Assembly time of one unit of productHi
tk : Total time from job sequence 1 until job sequence k
(6)
(7)
tK : Total time from job sequence 1 until all products are completed
(8)
:Amount of purchase price of parts hj, which is used to assemble all products, belong to Sk
:Amount of purchase price of parts hj, which is used from order 1 to order k-1 and at order k to which product Hiis assigned
: Amount of purchase price of parts hj, which is ideally used from order 1 to order k
: Amount of purchase price of parts hj, which is ideally used from order 1 to order k to which product Hi is assigned
: Variation value when product Hiis assigned to order k. The value is square root of sum of the squared difference between and
3.2 Objective Function
According to goal chasing method [1], the objective of this problem is to determine the job sequence that minimized the distance between the ideal purchase price of a used part () and the actual purchase price of a used part (), mathematically, as
minimizing (9)
In which the actual purchase price of part hj required until job sequence k () is
(10)
(11)
Amount of purchase price of parts hj which is used from order 1 to order k-1 and at order k to whichproduct Hi is assigned () is
. (12)
According to the PBGC and the TPBGC, the ideal purchase price of used part hj required until job sequence k () is, however, defined differently. Considering these two different definitions, we discuss various evaluation measures in the next section.
3.3 Evaluation Measure
3.3.1Price Based Goal Chasing Method (PBGC)
Figure2 shows assembly orders of products and parts purchase price. In Fig. 2, the horizontal axisshows assembling orders of products in a process and the vertical axis shows the purchase price of parts hj. The diagonal line in Fig. 2 shows ideal amounts of the parts price (); actual amounts of parts price at order k, , are also shown in Fig. 2. The diagonal line in Fig. 2 is called the ideal line of the amount of parts purchase price, PPP hjin this study. This study shall attempt bring a line that is drawn by actual amount of PPP hjclose to an ideal line of PPPhj. The assembled products sequence in the process should minimize the difference between the actual amount line of PPP hjand the ideal amount line of PPP hj, which depend on demands of products, bills for materials, and parts prices.
The ideal purchase price of part hj required until job sequence k() is
.(13)
(14)
At each job sequence, the deviation of each product Hi is obtained according to
.(15)
Fig. 2 Relations among, and k Fig. 3 Relations among, and tk
As a result, the deviation of the parts purchase price at job sequence k resulting from each product Hi() is
.(16)
3.3.2 Assembly Time and Parts Prices Based Goal Chasing Method (TPBGC)
Comparison of PBGC and TPBGCconsiders the difference of the assembly time of each product, defined differently. Fig. 3 shows that we defined as the ratio of total time at job sequence k () to total time when all products are completed (). Mathematically, it is:
.(17)
At each job sequence, deviation of each product Hi is obtained according to
,(18)
where
.(19)
As a result, deviation of the parts purchase price at job sequence k resulting from each product Hi () is
. (20)
4. Algorithm of TPBGC
In this paragraph, i is a value from 1 to I, j ranges from 1 to J and k ranges from 1 to K.
[Step 0] Data preparation
Enter input data: I, J, bij, pj, Qi, Ti.
Set initial variables: S0=0, =0, t0=0, =0,k=1.
Calculate necessary data: K, Nj,, tK, Eij from eqs. (1),(2),(3), (8) and (19).
[Step 1] and at order k are calculated from eqs. (10) and (18) to find the variation value, , for each candidate product at order k from eq. (20).
[Step 2]A product that minimizes in Step 1 is determined to be Hi* and is fixed at order k.
[Step 3]Product Hi* has been included in Sk as an assembled product at order k. and are respectively set as and . If all products are assigned to all orders, then this algorithm is complete, otherwise, letk = k + 1 and go to Step 1.
The proposed method (TPBGC) should be compared with PBGC under many numerical experiments to elucidate features of the TPBGC in the following section 5.
5.Numerical Experiments
5.1Experiments Data
Let types of products = 10 and types of parts = 8 (Table 1).The BOM for numerical experiments isshown in Table 2. 8 sets demands that are randomly set for numerical experiments (Table 3).
Parts price data in Table 4 are given as 10sets, where pj means the range of parts prices and sd(pj) means the range of standard deviation of parts price data. Assembly times(Ti) are shown in Table 5. The average of the times data in each set is set to 5 and the ranges of the times data of each set respectively as 5–6, 4–7, 3–8, 2–9, and 1–10. One hundred patterns are generated randomly for each set of times data. The one hundred patterns are combined with eightsets of demands of products in Table 3and 10sets of parts price data in Table 4 for numerical experiments. In all, 40,000 numerical experiments were tested.
Table 1 Types of products and parts
types of products(I) / 10types of parts (J) / 8
Table 2 BOM for each product
partP
R
O
D
U
C
T
/ 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8
1 / 1 / 0 / 1 / 0 / 0 / 2 / 0 / 1
2 / 1 / 0 / 1 / 0 / 0 / 0 / 2 / 1
3 / 1 / 0 / 0 / 1 / 0 / 0 / 2 / 1
4 / 1 / 0 / 0 / 1 / 0 / 2 / 0 / 0
5 / 1 / 0 / 0 / 0 / 1 / 0 / 2 / 0
6 / 0 / 1 / 1 / 0 / 0 / 2 / 0 / 1
7 / 0 / 1 / 1 / 0 / 0 / 0 / 2 / 1
8 / 0 / 1 / 0 / 1 / 0 / 0 / 2 / 1
9 / 0 / 1 / 0 / 1 / 0 / 2 / 0 / 0
10 / 0 / 1 / 0 / 0 / 1 / 0 / 2 / 0
Table 3 Eight kinds of demand data for each product
Table 4 Price of every part of the 10 kinds
Table 5 Five kinds of assembly timeranges
Range of Ti(min) / 5~6, 4~7, 3~8, 2~9, 1~10
5.2 Results
Results of numerical experiments are shown in Table 6. The values of Aver(TPBGC) and Aver(PBGC) in the table respectively denote the average of objective function value based on TPBGC and PBGC. Worse in the table means the ratio of cases in which TPBGC is worse than PBGC about the value. Better in the table means the ratio of cases in which TPGBC is better than PBGC about the value. Draw in the table means the ratio of cases in which TPGBC and PBGC have the same value.
5.3Analysis
Generally speaking, TPBGC is more effectivethan PBGC from Table 6because the assembly time is morefluctuant. The difference of sd(Qi) in Table 6 doesnot affect the advantage of TPBGC. According to the above result, for decreasing the inventory value, it is necessary to considerthe partspurchasing price and product assembly time simultaneously.
6. Conclusion
In this paper, considering the difference of assembly times, we proposed a sequencing method for mix-model assembly scheduling in a single work station problem: Assembly Time and Parts Prices Based Goal Chasing Method (TPBGC). Numerous experimental results lead us to conclude that:
- Our proposed methods become more effective than PBGC when the range of assembly times becomes larger.
- Different assembly times must be considered in mixed-model assembly scheduling problem.
Table 6 Results of numerical experiments
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References
[1] Monden,Y., 1998, Toyota Production System: An Integrated Approach to Just-in-time, third edition, Engineering and Management Press, pp.253-264.
[2] Kubiak,W., 1993, Minimizing Variation of Production Rates in Just-in-Time Systems: A Survey, European J. Operational Res.,66, pp.259-271.
[3] Miltenburg, J., 1989, Level Schedules for Mixed-model Assembly Line in Just-in-time Production System, Management Science, 35, pp.192-207.
[4]Wangwiwatsin, P.,Yanagawa, Y., Miyazaki, S.,1995, Sequencing Method for Various-assembly-timed Products in Single Work Station, Production Scheduling Symposium’95,in Japanese, pp.205-210.
[5]Wangwiwatsin, P.,Yanagawa, Y., Miyazaki, S.,1997,Mixed-model Assembly Scheduling by Considering Assembly Times and Part Consumption Ratio in Single Work Station,Journal of The JapanSociety of Mechanical Engineers,（C）, in Japanese,Vol.63, No.610, pp.2181-2188.
[6] Zhang, J.,Yanagawa, Y., Miyazaki, S., 2004,Sequencing method for products in consideration of a different parts price in a single work station,Journal ofJapan Industrial Management Association,in Japanese, Vol.55, No.5, pp.227-233.
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