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Ind. Eng. Chem. Res. 2009, 48, 3234–3243
Development of a Scheduling Model for Polarizer Manufacturing Logistics Kyung Tae Park,† Jun-hyung Ryu,*,† Ho-Kyung Lee,‡ and In-Beum Lee† Department of Chemical Engineering, POSTECH, Pohang, Korea, and LG Chem, Ltd., Research Park, Daejeon, 305-380, South Korea
Polaroid films, or the so-called “polarizer”, is an optical device that transforms natural light into some form of polarized light. When the form of films, it is widely used in thin display equipment, such as liquid crystal displays (LCDs). The polarizer manufacturing industry is under constant competition to lower its price with the continuous price reduction in LCD markets. This paper is particularly concerned with its logistics problems, with a focus on minimizing the cost for purchasing cores used in moving films in the process. The core movement for polarizer logistics is formulated into a linear programming (LP) problem by employing an event-based pull approach that has been proposed earlier by Karimi et al. [Karaimi, I. A.; Sharafali, M.; Mahalingam, H. AIChE J., 2004, 51, 178-197]. Two numerical examples are presented to illustrate the applicability of the proposed model, and some remarks also are made. 1. Introduction Polaroid films (or the so-called “polarizer”) is an optical device that transforms natural light into some form of polarized light. When in the form of films, it is used in thin display equipment such as liquid crystal displays (LCDs). With the explosive increase in the market for thin-display devices, the manufacturing capacity of the polarizer has been also expanded continuously. Polarizer manufacturing processes generally consist of a front-end process with key operations and a back-end process that has relatively low value. As illustrated in Figure 1, the front-end process consists of treatment, stretching, and coating processes and the back-end process consists of lamination and aging and cutting processes. During the process, the intermediate products, in the form of films, are wrapped or unwrapped in a core as the process continues, as shown in Figure 2. Because front-end and backend processes are often located in geographically distinctive locations, intermediate films should be delivered from the front end to the back end by wrapping them in cores. If the process runs out of cores, then the entire process should be stopped. Therefore, a sufficiently large number of cores must be purchased to avoid that situation, because the prices of cores are thought to be inexpensive and it takes some time to actually facilitate the cores after ordering them. However, unnecessarily redundant cores just raise the costs, in terms of procurement and, particularly, inventory holding. Therefore, cores should be managed in terms of the necessary amount and proper allocation. The need to manage cores is also consistent with the business environment, which is under severe pressure to reduce prices to remain competitive. In the literature, few works have been reported on polarizers or polarizer scheduling. When the scope is expanded to the logistics problems of chemical products, only recently have some researchers begun to show interest.In 2002, Chen et al.1 presented a discrete event simulation study of logistics activities in chemical plants. In 2004, Jetlund and Karimi2 addressed the scheduling problems of multiparcel tankers that were engaged in the shipment of bulk liquid chemicals as a mixed-integer linear programming (MILP) formulation. In 2005, Karimi et * To whom correspondence should be addressed. E-mail address:
[email protected]. † Department of Chemical Engineering, POSTECH. ‡ LG Chemicals, Ltd.
al.3 modeled a linear programming (LP) formulation for transporting multiproduct chemical containers and expanded the formulation for several cases. They presented an event-based “pull” approach to compute the solutions of the formulated model. Motivated by the lack of explicit relevant research in the face of economic potential, this paper proposes a mathematical model in the context of core logistics problems of the polarizer manufacturing process. The remainder of this paper is organized as follows. Section 2 describes the problem for entire polarizer manufacturing movements and the associated assumptions. Section 3 constructs the mathematical formulation using an event-based pull approach. Two numerical examples will be then presented to illustrate the applicability of the proposed model, and some remarks will be given as well. 2. Problem Description Logistics is an integral part of manufacturing companies; it involves the procurement, storage, and transportation of goods and people. Some important strategic logistic decisions include concepts such as what (and how much) transportation equipment, loading and unloading facilities, and storage are needed to ensure that products are produced and shipped to customers on time.1 To address the logistics of the polarizer manufacturing process, we will start from the diagram of the process network. Table 1. Transport Times for One Core in Example 1 Transport Time (days) depot
Site S1
Site S2
Site S3
Site S4
D1 D2
3 3
4 4
2 2
30 30
Table 2. Transport Costs in Example 1 Transport Costs (US$ per core) depot
Site S1
Site S2
Site S3
Site S4
Depot-to-Site Transport Cost per Empty Core D1 D2
120 137
181 163
89 124
250 203
Depot-to-Site Transport Cost per Loaded Core D1 D2
180 201
264 246
10.1021/ie8009044 CCC: $40.75 2009 American Chemical Society Published on Web 02/24/2009
115 172
335 310
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3235
Figure 1. Schematic of the procedure of Polaroid film production.
Figure 2. Schematic of the cores: (a) empty core and (b) loaded core.
Figure 4. Schematic diagram of the generation of events for core movement.
Figure 3. Schematic diagram of core flow.
Look at a polarizer core transportation network that consists of front-end and back-end manufacturing processes, a core market, a depot, and a site, as graphically illustrated in Figure 3. A depot denotes storage where cores are stored for the operation in the front-end process. It sends cores wrapped with the polarizer to sites and receives empty cores from them. A site denotes a place where cores are stored for the back-end process. A core market sells empty cores for each depot. The market may be a core manufacturing supplier or a core trade agent. An order (o) that provides information for the destination site (s) requires that the core quantities (De) and the due date (Dd) generally be known from the upper level production planning module. In this paper, we are going to develop a core scheduling framework from the polarizer manufacturing schedule that is already assumed to be given. Other assumptions are as follows: (1) transportation between sites and between depots is not allowed; (2) the core market has an unlimited capacity for empty cores; (3) parameters such as costs and times for orders are known; (4) a just-in-time (JIT) mode is given; (5) the front-end and back-end processes have constant process times, regardless of core quantities; (6) the intermediate film length per core is 900 m; (7) all cores return to depots at the end of the horizon,
Figure 5. Schematic representation of Example 1.
to prepare for the next operation; and (8) the transportation cost from a depot to a site is proportional to the number of cores. That is, there is no restriction on any minimum order quantity for delivery. 3. Model Formulation The proposed approach consists of two stages. The concept of events and event times are introduced. The corresponding mathematical formulation then is presented.
3236 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 Table 3. Event Times for Example 1 Event Times (days) Target
Result
t2
t3
t4
t6
order
destination site
demanded core
order
delivery core
depot
t1 (due date)
(D1)
(D2)
(D1)
(D2)
(D1)
(D2)
t5
(D1)
(D2)
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12 O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24 O25 O26 O27 O28
1 2 3 1 2 4 3 1 2 3 1 2 4 3 1 2 3 1 2 4 3 1 2 3 1 2 4 3
83 39 28 50 50 45 39 75 37 23 47 53 31 38 72 28 50 56 48 47 36 86 47 31 43 41 46 40
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 O11 O12 O13 O14 O15 O16 O17 O18 O19 O20 O21 O22 O23 O24 O25 O26 O27 O28
83 39 28 50 50 45 39 75 37 23 47 53 31 38 72 28 50 56 48 47 36 86 47 31 43 41 46 40
D1 D2 D1 D1 D2 D2 D1 D1 D2 D1 D1 D2 D2 D1 D1 D2 D1 D1 D2 D2 D1 D1 D2 D1 D1 D2 D2 D1
31 32 33 33 34 34 35 38 39 40 40 41 41 42 45 46 47 47 48 48 49 52 53 54 54 55 55 56
28 28 31 30 30 4 33 35 35 38 37 37 11 40 42 42 45 44 44 18 47 49 49 52 51 51 25 54
28 28 31 30 30 4 33 35 35 38 37 37 11 40 42 42 45 44 44 18 47 49 49 52 51 51 25 54
18 18 21 20 20 -6 23 25 25 28 27 27 1 30 32 32 35 34 34 8 37 39 39 42 41 41 15 44
18 18 21 20 20 -6 23 25 25 28 27 27 1 30 32 32 35 34 34 8 37 39 39 42 41 41 15 44
8 8 11 10 10 -16 13 15 15 18 17 17 -9 20 22 22 25 24 24 -2 27 29 29 32 31 31 5 34
8 8 11 10 10 -16 13 15 15 18 17 17 -9 20 22 22 25 24 24 -2 27 29 29 32 31 31 5 34
41 42 43 43 44 44 45 48 49 50 50 51 51 52 55 56 57 57 58 58 59 62 63 64 64 65 65 66
44 46 45 46 48 74 47 51 53 52 53 55 81 54 58 60 59 60 62 88 61 65 67 66 67 69 95 70
44 46 45 46 48 74 47 51 53 52 53 55 81 54 58 60 59 60 62 88 61 65 67 66 67 69 95 70
Table 4. Transport Times from Depots to Sites for One Core in Example 2 Transport Time (days) depot
Site S1
Site S2
Site S3
Site S4
Site S5
Site S6
Site S7
Site S8
Site S9
Site S10
D1 D2 D3
7 5 9
8 10 15
12 15 10
5 8 5
2 3 7
9 10 10
4 8 11
15 20 13
6 10 8
10 8 9
Table 5. Transport Cost from Depots to Sites for One Core in Example 2 Transport Cost per Core (US$) depot
Site S1
Site S2
D1 D2 D3
130 124 115
D1 D2 D3
141 139 129
Site S3
Site S4
Site S5
Site S6
Site S7
Site S8
Site S9
Site S10
115 127 162
From Depots to Sites for Empty Cores 160 113 98 155 127 184 145 142 98 163 124 168 134 148 127 155 132 177
134 154 145
154 148 162
From Depots to Sites for Loaded Cores 127 171 120 107 164 134 194 136 153 149 108 171 138 194 172 141 160 135 164 141 189
150 167 159
164 163 168
A. Events and Event Times. At first, an event denotes something that happens and is mainly generated by an order. The event time is when the event happens. To address the transportation process, six events (which are denoted as event1, event2, event3, event4, event5, and event6) and the corresponding six event times (which are denoted as t1, t2, t3, t4, t5, and t6) are used. The event generation procedure is as follows. The event that is designated as “event1” represents the arrival of loaded cores at the destination site and is generated by an order o. The event time t1 represents the due date for an order. The event that is designated as “event2” represents the arrival of loaded cores at the origin depot and is generated by event1. Note that the event time t2 is t1 minus DST(t1,o,d,s), where DST(t1,o,d,s) denotes the transport time from the origin depot to the destination site for an order. The event that is designated
as “event3” is the arrival of empty cores at the origin depot and is generated by event2. Between event2 and event3, the loaded cores are operated in the front-end process. The event time t3 is given as t2 minus the front-end process time (TL). The event that is designated as “event4” is the purchase of the core from the core-market and is generated by event3. The event time t4 is given as t3 minus QDT(t3,o,q,d), where QDT(t3,o,q,d) denotes the transport time from the core market (q) to the origin depot for an order. The event that is designated as “event5” represents the arrival of empty cores at the destination site and is generated by event1. Between event1 and event5, the Polaroid films are processed in the back-end process. The event time t5 is given as t1 plus the back-end process time (TP). The event that is designated as “event6” denotes the arrival at depots for empty cores and is generated by event5. The event time t6 is given as t5 plus SET(t5,o,s,e), where SET(t5,o,s,e) denotes the transport time from the destination site to another depot e for an order. The sequence of the event times becomes t4, t3, t2, t1, t5, and t6. These event times are arranged in the increasing order of actual occurrence, while discarding negative event times and eliminating the overlapping event times. Figure 4 graphically describes the generation of events for the core transportation. B. LP Model Formulation. The core transportation movement model is presented here. At first, the objective of the scheduling framework is to minimize the core management cost. The objective function can be formulated as follows:
∑ ∑ ∑ ∑ x XC + ∑ ∑ ∑ ∑ y YC + ∑ ∑ h SI (t - t ) + ∑ ∑ ∑ ∑ w WC + ∑ ∑ L FC
C)
dskj
d
s
k
sekj
s
e
k
dskj
jgk
sekj
jgk
s
s
qdkj
q
d
k
jgk
sk k+1
k
k 1) (3)
e
k
jgk
+
∑ ∑ h SI s
s
∑ ∑ ∑ ∑w
sk(tk+1
k 1) (2)
SIsk ) SIs(k-1) +
Empty cores may remain at sites after their usage in the back end. However, it is assumed that all cores must return to the depots after they are used in sites until the last event time, because they will be used for the next operation.
s.t.
d
k
jgk
+
- tk) +
∑ ∑L
dkFCdk
d
k
3238 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
DIdk ) DId(k-1) +
∑ ∑y
sdjk
s
-
jek
∑ ∑x
dskj
s
+
jgk
∑ ∑w
qdjk
q
(DIdk e SIsk ) SIs(k-1) +
∑ ∑x
dsjk
d
-
jek
dK
d
)
d0
d
+
sekj
jgk
∑ ∑ ∑ ∑w
SIsU, k
qdkj
q
d
dsjk
d
k
Figure 7. Schematic representation of Example 2.
jgk
) Psk ) Desk
jek
∑ ∑x
dskj
∑ ∑y e
∑ DI
> 1)
∑ ∑x s
(SIsk e
∑ DI
jek DIUd , k
> 1)
) Ldk
jgk
Two numerical examples are presented to illustrate the applicability of the proposed models. Example 1. Consider a polarizer film manufacturing company that has two depots (D1 and D2), four sites (S1-S4), and 28 orders (O1-O28). Figure 5 shows the regular order data for sites, and
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3239
Figure 8. Results of Example 2: inventory level in (a) depot D1, (b) depot D2, and (c) depot D3.
Table 1 lists the transport time for cores from depots to sites. Table 2 lists the transport costs per unit core. In the beginning of event time t0, the sites have no core, while depot d1 has 100 cores and depot d2 has 200 cores. Other information is as follows: (1) The transport time from a core market to the depots is 10 days, (2) The purchasing price of one core is $50, (3) The front-end manufacturing process cost is $120 for D1 and $135 for depot D2, (4) The holding cost at sites for one core is $10, and (5) The front-end process time and back-end process time are each 10 days. The proposed model is used to compute the solution of this problem. At first, events and events times are computed as
summarized in Table 3. For example, the event generation procedure of order 19 at site 2 is as follows. The event times are t1 ) 48, t2(1) ) 44, t2(2) ) 44, t3(1) ) 34, t3(2) ) 34, t4(1) ) 24, t4(2) ) 24, t5 ) 58, t6(1) ) 62, t6(2) ) 62, where t( · ) denotes event time t at a depot or site. When negative event times and the overlapping event times are modified, the total event time becomes 63 days. Refer to Table 3 for the list of all unmodified event times. The results according to the proposed model are graphically illustrated in Figure 6. The final depot level is D1 ) 757 and D2 ) 512. At event time T15, the number of empty cores in depot D1 increases, because 657 empty cores are purchased to meet the order. At event time T18, the empty core level in depot D2 increased, because 312 empty cores are purchased to meet
3240 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 9. Results of Example 2: core transport (a) from the sites to depot D1, (b) from the sites to depot D2, and (c) from the sites to depot D3.
the orders. In depot D2, there are not many cores between T41 and T45. In fact, it denotes that cores from the sites are immediately used to send films to the sites. With regard to the computational statistics, the number of equations is 1164 and the number of variables is 1051. The solution of the problem is computed using CPLEX 9.5 within GAMS on an Intel Core 2 2.39 GHz computer system with 2GB of RAM. The solution time is 0.016 s. Example 2. Consider another example with three depots (D1-D3) and 10 sites (S1-S10) to handle 102 orders (O1-O102). Table 4 lists the core transport times from the
depots to the sites, and Table 5 lists the transport costs per core. Figure 7 illustrates the problem structure with the corresponding order data. Although sites have no cores in the beginning, D1 has 1200 cores, D2 has 750 cores, and D3 has 600 cores at event time t0. The transport time from the core market to the depots for cores is known to be 10 days. The price of one core is $50; the front-end process cost is $120 for depots D1 and D3 and $135 for depot D2. The proposed “pull” approach model is used to obtain the solution of the problem. The total event time becomes 97 days. To prevent empty cores from remaining in sites for the sake of
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3241
Figure 10. Results of Example 2: core transport from (a) depot D1 to the sites, (b) depot D2 to the sites, and (c) depot D3 to the sites.
avoiding transport cost from sites to depots at the final event time, the constraint that forces all cores to be sent back to sites is added in the model. The issue of which depot to select then is determined from the optimization of the overall model. The corresponding results, according to the proposed approach, are graphically summarized in Figures 8, 9, 10, and 11. Figure 8 graphically describes the inventory profile for each depot. At event time T11, the empty core level in depot D1 is increased by purchasing 964 empty cores to meet the order. For the time period between T44 and T51 for D1, the zero level
denotes that all of the cores from sites are used to send films to the sites. This situation also occurs for D2 for the time period between T41 and T51 and for D3 for the time period between T48 and T50. On the other hand, the result that the level is empty for D3 for the time period between T34 and T36 indicates that cores are not transported. Its level increases only after event time T37, after 529 cores are purchased from the core market. With regard to computational statistics, the objective value is $2,026,953, the number of equations is 3589, and the number of variables is 3377. The solution of the problem is computed using
3242 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009
Figure 11. Results of Example 2: number of cores transported from the depots to the front-end processes. Table 6. Comparison between Our LP Model and the Heuristic-Based LP Model That Describes the Real Factory Value parameter
our LP model
LP model describing the real factory
number of equations number of variables objective value CPU time
3589 3377 $1,983,380 0.031 s
3589 3377 $2,024,264 0.032 s
CPLEX 9.0 within GAMS on an Intel Core2 2.39 GHZ computer system with 2GB of RAM. The solution time is 0.031 s. It was computed that sites S2, S4, and S9 send cores to depot D1 and sites S5, S7, S8, and S10 send cores into depot D2. Sites S1, S3, and S6 send cores into depot D3, as shown in Figure 9. The inventory level for depot D1 is relatively low at the last event time, because the number of cores transported from sites S2, S4, and S9 to depot D1 is small; in contrast, the
amount for depot D2 is large, because the number of transported cores is large. Figure 10 illustrates the actual movement of loaded cores from individual depots to individual sites. Figure 11 graphically shows the supply of empty cores for the front-end process for orders. Before conducting the proposed modeling approach, we have studied the industrial practices on core management. Generally speaking, in industry, the connection between depots and sites are fixed. That is to say, cores are transported from a depot to the fixed sites. The empty cores at sites are also sent back to the original depot from which they had come. This type of heuristics has been used because the core management has been done manually by very limited personnel who are also involved with other types of jobs. Therefore, by applying the proposed modeling approach and establishing the corresponding system, companies are expected
Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3243
to automate core management and improve the core distribution efficiency, in addition to the advantage of saving labor costs. To compare the difference between the industry practice and the proposed modeling approach, we solved Example 2 again, with the assumption that the core is transferred from a fixed depot to fixed sites in the LP model (representing practice). Specifically, it is assumed that (i) depot d1 sends cores to sites s2, s4, s5, and s7; (ii) depot d2 sends cores to sites s1, s9, and s10; and (iii) depot d3 sends cores to sites s3, s6, and s8. The resulting cost, according to this heuristic approach, is $2,024,264, whereas the cost using the proposed event based approach is $1,983,380, as shown in Table 6. Using the proposed LP model, the costs could be reduced by as much as $40,884. That represents the potential contribution of the proposed approach. Remarks. There are some issues that are worthy of comment. First, we can reduce the number of cores by using this model rather than not using it. The quantity of cores saved is measured by calculating the number of cores transported from sites to depots until empty cores are finally transported from depots to front-end processes. The result is that depot D1 saves 718 cores, depot D2 saves 67 cores, and depot D3 saves 50 cores. If this model runs for long times, more cores can be saved. Second, there are similarities and differences between the tankcontainer management problem that is described by Karimi et al.3 and the presented polarizer manufacturing logistics problem. With regard to similarities, there are reverse logistics that can reuse logistics equipment (such as the core and container) after use. Regarding the difference, in the tank-container management problem, the container can be borrowed, whereas in the polarizer manufacturing logistics problem, the core should be purchased. Also, the load times are different: The load time of the container is short, because workers only put materials into the container, whereas the load time of the core is relatively long, because there are various processes that change empty cores to loaded cores. Third, the proposed model is based on the deterministic data. Work is currently underway to consider the presence of uncertainty in the model. For example, it is assumed that a fixed length of polarizer is wrapped in each core. In practice, the lengths of the individual polarizers vary. Therefore, the daily necessary number of cores also can vary. The variation should be considered in the operation and the corresponding model. Because the delivery time of cores is ∼10 days from the core market to depots, empty cores for at least 10 days of operation should be reserved in the depot. The issue would then be how to reduce the safety stock level via efficient core scheduling. Fourth, there is no restriction on the minimum order quantity for delivery. In practice, the intermediate cores are delivered after being shipped in a container. The container can deliver a certain number of cores at a time or per day. This type of constraint may be added in the model. Finally, the film widths vary. The size of the core changes depending on the width of films. For this example, the film width is 1 m and the size of the core must be longer than 1 m, because the films are wrapped around the core. Real factories generally have various film widths and core sizes. 4. Conclusions The logistics of polarizer manufacturing is addressed by optimizing the number of cores that are used to deliver intermediate films from the front-end process to the back-end process. Therefore, a scheduling framework that minimizes redundant cores is presented, using the event-based pull approach. This work is an example showing that there exist many
potential opportunities for the expertise of process systems engineering to contribute to improving industrial practices. Acknowledgment This work is the outcome of a Manpower Development Program for Energy & Resources supported by the Ministry of Knowledge and Economy (MKE). Notation Indices d ) origin depot that sends cores to sites e ) another depot that receive cores from sites s ) destination site that receive cores from origin depot j, k ) event time for an order o ) order q ) core market K ) the last event time Parameters t1, t2, t3, t4, t5, t6 ) event time for an order tk ) kth time of event times arranged in increasing order DST(t1, o, d, s) ) transport time from origin depot d to destination site s for an order o QDT(t3, o, q, d) ) transport time from the core-market q to origin depot d for an order o SET(t5, o, s, e) ) transport time from origin site s to another depot e for an order o XCdskj ) transport cost from depot d at time tk to site s at time tj YCsekj ) transport cost from site s at time tk to depot d at time tj WCqk ) cost of purchasing one core from the core market q at time tk FCdk ) front-end process cost for depot d at time tk Desk ) required core quantities in site s at time tk DIdU ) capacity restriction of cores in depot d SIsU ) capacity restriction of cores in site s hs ) staying cost in site s Variables DIdk ) number of cores in depot d at time tk SIsk ) number of cores in site s at time tk xdsjk ) number of loaded cores transported from depot d at tj to site s at tk ysejk ) number of empty cores transported from site s at tj to depot e at tk wqdjk ) number of empty cores transported from the core market q at tj to depot d at tk Ldk ) number of cores transferred from depot d to the front-end process at tk Psk ) number of cores transferred from site s to the back-end process at tk C ) total cost for polarizer manufacturing transportation
Literature Cited (1) Chen, E. J.; Lee, Y. M.; Selikson, P. L. A simulation study of logistics activities in a chemical plant. Simul. Model. Pract. Theory 2002, 10, 235– 245. (2) Jetlund, A. S.; Karimi, I. A. Improving the logistics of multicompartment chemical tankers. Comput. Chem. Eng. 2004, 28, 1267–1283. (3) Karimi, I. A.; Sharafali, M.; Mahalingam, H. Scheduling Tank Container Movements for Chemical Logistics. AIChE J. 2005, 51 (1), 178– 197.
ReceiVed for reView June 9, 2008 ReVised manuscript receiVed December 10, 2008 Accepted February 17, 2009 IE8009044
