Graph-Assisted Cyclic Hoist Scheduling for Environmentally Benign Electroplating

Transcription

1 Ind. Eng. Chem. Res. 2004, 43, Graph-Assisted Cyclic Hoist Scheduling for Environmentally Benign Electroplating Qiang Xu and Yinlun Huang* Department of Chemical Engineering and Materials Science, Wayne State University, Detroit, Michigan Hoist scheduling is a class of production scheduling of batch systems in the manufacturing industries. Cyclic hoist scheduling (CHS) is a type of hoist scheduling that deals with the scheduling that involves one hoist and produces some type of product in a multistage production line. The only resource in the production is the hoist for which various operations compete with each other in every production cycle. Currently available CHS approaches, regardless of whether they are algorithmic or heuristic based, are almost all designed to maximize the production rate by optimizing hoist movements; environmental issues are not a concern during development of the hoist schedule. This paper introduces a methodology for solving a general CHS problem where the production rate and waste reduction are simultaneously taken into account. Using this methodology, all feasible hoist schedules for a given CHS problem can be rapidly identified using a graph-assisted search algorithm. The schedules are then evaluated in terms of waste minimization capability, using the corresponding operational strategies. In evaluation, the system dynamics associated with each selected feasible hoist schedule is examined. A final hoist schedule is the best for maximum production rate and minimum waste generation. The efficacy of the methodology is demonstrated by solving an industrial electroplating example. 1. Introduction In the manufacturing industries, material handling always involves the handling of various types of jobs in a multistage operational setting. Coating processes, such as electroplating and polymeric coating processes, are typical examples. The material handling is usually performed by hoist(s). Each hoist is automatically controlled to move the jobs among the processing units in a production line, according to a pre-set hoist schedule. Development of the hoist schedule, or hoist scheduling, is an integral part of the design and operation of a manufacturing process. An optimal hoist schedule can improve production efficiency and manufacturing costs significantly. According to Kumar, as high as 20% reduction in mean job waiting time and 50% improvement in standard deviation of cycle time can be reached in operation through the use of new scheduling policies. 1 Hoist scheduling for improving the production rate is known as a NP-hard problem. 2 Cyclic hoist scheduling (CHS) involves one hoist for processing a single type of product in a multistage production system. The available CHS approaches can be divided into two categories: algorithmic-based and heuristic-based. Phillips and Unger are among the earliest to use mixed integer programming (MIP) to schedule a process with one hoist in one-capacity processing units. 3 Later, Shapiro and Nuttle proposed a linear-programming (LP)-based branch and bound algorithm in solution identification. 4 Subsequently, Lei and Wang 5 and Armstrong et al. 6 improved their branch and bound methods. In parallel, Baptiste et al. addressed the CHS problem with constraint logic programming (CLP) as a modeling language. 7 Rodosek and Wallace then combined CLP * To whom correspondence should be addressed. Tel.: Fax: yhuang@wayne.edu. and MIP to generate a hybrid algorithm. 8 Chen et al. introduced a new branch and bound algorithm that was claimed to gain better production efficiency. 9 Although computationally efficient, their approach is intended for solving a class of relatively simple CLP problems where no environmental issues could be integrated. Dutilleul and Denat 10 presented their solution strategy for hoist scheduling with a P-time Petri nets tool. 11 It is worth noting that approaches based on mathematical programming generally cannot guarantee global optimality for large-scale hoist scheduling problems. In most cases, truly optimal solutions may not be identifiable. In addition, their computational loads are usually (very) heavy. In industry, practicing CHS approaches are almost all heuristic-based, because they rely on experience and solutions can be easily understood and modified. The algorithms by Yih and Yin, 12 Geiger et al., 13 Zhou and Li, 14 as well as a genetic algorithm by Lim, 15 are among the effective ones. The approaches are generally computationally efficient; however, solution optimality is a major issue. It has been noted that, in almost all the existing hoist scheduling approaches, environmental issues are not a concern. A misconception is that environmental problems only need to be handled at the management level, during process design, and through process control. Note that hoist scheduling focuses on the spatial and temporal aspects of a hoist. The spatial aspects involve the locations where the hoist stays, acts, and moves, whereas the temporal aspects involve the time sequence that a hoist needs to follow in taking action. Each hoist action time is directly related to the job processing time in each processing unit. The processing time is one of the key factors in determining product quality, processing cost, waste generation, and production efficiency. In an electroplating process, for example, if the rinsing time for a job in a rinse tank is longer than necessary, the /ie CCC: $ American Chemical Society Published on Web 11/19/2004

2 8308 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 Figure 1. Example of a time-way graph. rinse water usage will be low in efficiency. In turn, wastewater generated from the tank will be more than it should be. This analysis is valid as well for cleaning and electroplating operations in the electroplating line. Therefore, hoist scheduling can and should contribute to waste reduction. Very recently, Kuntay et al. introduced a mathematical-programming-based CHS methodology that can not only improve the production rate but also reduce wastewater in an electroplating line. 16 In their methodology, the two objectives are optimized in a sequential manner, i.e., derive an optimal schedule for maximizing the production rate first, and identify the optimal rinsing time in each rinse unit later. This optimization process is repeated until the final solution satisfies a predefined criterion. However, their methodology is incapable of providing more than one solution for screening. Moreover, the computational time for identifying an optimal solution (in fact, a local optimal solution) is very long (on the magnitude of hours). In this paper, a new environmentally conscious CHE methodology is introduced. This methodology, which targets simultaneous maximization of the production rate and minimization of waste, is designed to generate an optimal schedule in a computationally very efficient way. It is particularly effective for the scheduling of general coating operations. 2. Graph-Assisted Analysis The Gantt chart has been widely used to analyze CHS problems. 4,14 Although it can depict a hoist move sequence in a given production line, it is difficult to represent some critical information, such as the processing capacity of a processing unit where more than one job is processed in it simultaneously and the hoist waiting mode in certain production areas. In this regard, a hoist schedule representation, based on the time-way graph by Shapiro and Nuttle, 4 is presented below Hoist Schedule Representation Scheme. In an electroplating line, multistep cleaning, rinsing, and plating operations occur in different processing units. The processing time of each job in a specific unit is usually restricted to a predetermined range. A hoist is used to move each job from one unit to the other in the line. Note that there is always more than one job in the production line at any time, so that the production efficiency can be increased. Thus, hoist movements must be well-scheduled, so that any job-move request can be immediately taken by the hoist. Hoist movements can be divided into two types: the loaded move during which the hoist carries a job, and the free move during which it does not carry anything. Figure 1 shows a time-way diagram, where each loaded move, which is labeled by a circled number in the figure (e.g., 2), is always initiated by lifting a job from a unit (e.g., from unit 1, with symbol v ), moving it (represented by a solid line), and releasing it into another unit (e.g., into unit 2, with symbol V ). The time needed for a loaded move is listed above the related solid line (e.g., 10 s for loaded move 2). A free move (e.g., a dotted line between units 1 and 3) occurs usually between two consecutive loaded moves (e.g., between 1 and 4). Figure 1 provides a complete cyclic hoist schedule example for a process that contains six units. This figure, in fact, contains the following additional important information Hoist Cycle Time. A hoist cycle is the time span between two subsequent jobs entering a production line. In Figure 1, for example, the hoist cycle time is 105 s. This means that every 105 s, a new job enters the line, or equivalently, a job finishes the entire processing and is moved out from the line. Note that each loaded move is determined based on the job processing requirement. For example, a job after cleaning needs to be moved by the hoist for rinsing. Figure 1 shows seven loaded moves (symbolized as 1-7) that are needed for each job to go through the line. Also note that the total amount of time for all the required loaded moves in a cycle is fixed. However, the total amount of free moves varies, depending on a

3 Ind. Eng. Chem. Res., Vol. 43, No. 26, particular hoist schedule. Apparently, a longer time for free moves indicates a longer cycle time and, thus, a lower production rate. Therefore, one of the hoist scheduling objectives is to sequence the loaded moves in a way so that the total time for free moves is minimized in each hoist cycle Free Move Matrix. To analyze free moves, a representation scheme is needed. Note that the time for a free move is proportional to the distance between the two units where the hoist travels, assuming that the hoist moves at a constant speed. For a system that has M loaded moves, there are a total of M (M - 1) possible free moves (excluding the free move from a unit to itself) in each cycle. For example, Figure 1 contains seven loaded moves, and thus there are 42 options of free moves. The travel time of these free moves generates a free move matrix, F, where each entry, f i,j (i, j ) 1,..., 7; i * j), contains the time needed for a hoist free move from the unit where the hoist ends the ith loaded move to the unit where the hoist starts the jth loaded move. In Figure 1, for instance, loaded move 2 ends above unit 2, whereas loaded move 6 starts at unit 5. If the time needed for a hoist move from one unit to its adjacent unit is 2 s, then, the move from unit 2 to unit 5 needs to pass three units and, thus, the total amount of time for the free move will be 6 s. Therefore, the entry f 2,6 contains a value of 6. A detailed complete example of the free move matrix will be given in section 5, where an industrial example is fully studied Job Processing Time. Assume that every unit in a production line has a single capacity (i.e., is capable of accommodating only one job at a time). This means that, in any hoist cycle, each unit is loaded with a job only once and is unloaded only once. In Figure 1, therefore, there exists only one pair of arrows for each unit (i.e., one v and one V positioned vertically above each unit). The job processing time in a unit can be calculated using the following formulas: t p i ) { t l r i - t i (if t r i < t l i ) T - (t r i - t l i ) (if t r i > t l i ) (1) where t i p is the processing time of a job in the ith unit; t i l and t i r are, respectively, the time when a job is lifted from and the time when a job is released into the ith unit in the same cycle; T is the hoist cycle time. The formulas can be readily understood with the help of Figure 1, where two examples (with shaded vertical bars) are given. For a unit with multijob capacity (i.e., simultaneous processing of n (>1) jobs in a unit), the job processing time can be evaluated as follows: t p i ) { (n - 1)T + t l r i - t i (if t r i < t l i ) nt - (t r i - t l i ) (if t r i > t l i ) (2) In this case, a job that has started processing in the jth cycle will finish the processing in the line in the (n + j - 1)th or (n + j)th hoist cycle. Figure 2 gives an example of processing time evaluation, where n is set to Search Space Reduction. For a CHS problem that involved M loaded moves in each cycle, the solution space, S t (M), will be M!. Clearly, to generate all of them through an exhaustive search and then evaluate each Figure 2. Graphical analysis for job processing time in a two-job capacity unit. In panel a, t i r < t i l, where a job finishes processing in the (2 + j - 1)th cycle; in panel b, t i r > t i l, where a job finishes processing in the (2 + j)th cycle. of them is very time-consuming and impractical. Therefore, an effective solution search method is needed. Figure 3 shows loaded moves involving a singlecapacity unit i. In Figure 3a, the loaded move sequence is a-b. As indicated, the hoist must wait above unit i for the job to finish processing in it. Unless it is operationally required, in this case, a-b can be considered as a grouped loaded move, which will be introduced later; otherwise, the hoist should not wait there but instead leave for other activities first. Note that any unnecessary waiting time means an increment of the hoist cycle time and, thus, a reduction of production rate. Therefore, any hoist operational sequence containing unnecessary hoist waiting should be discarded from the solution space. For the example in Figure 1, the sequences involving any loaded-move pairs (1-2, 2-3, 3-4, 4-5, 5-6, 6-7, and 7-1) must be eliminated. The reduced solution space, S r (M), where M is the total number of processing units, can be calculated as S r (M) ) 1 M k)1 M-2 (-1) M-k M! (M - k)! (3) In the example of Figure 1, the feasible sequence space, S r (7), is only 265; this is reduced from S t (7) (i.e., 7!), a 94.7% reduction. Note that if a system contains a multicapacity unit(s), then S r (M) is the lower bound of the reduced solution space. The solution space S r (M) may be further reduced by eliminating those sequences that do not accommodate proper processing time(s) in certain processing units.

4 8310 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 Figure 3. Infeasible sequence. Panel a shows the sequence a-b, which requires the hoist to be waiting above unit i. Panel b shows sequence a-c-b, where the processing time in unit i is out of the limits. Figure 4. Graphical representation of the processing time when the hoist waiting time is considered: (a) case I, single capacity; (b) case I, multi-capacity; (c) case II, single capacity; and (d) case II, multi-capacity. In Figure 3b, if the processing time of a job in unit i (see the shaded vertical bar) is beyond the permissible processing time range, then the sequence a-c-b should be discarded. In this way, more unacceptable sequences can be identified by checking the relevant processing times. If all the unacceptable sequences has the space of S p (M), then the final feasible solution space (S f (M)) will be S f (M)) S t (M) - S p (M) (4) Conceivably, when a scheduling problem is large, S f - (M) will be increased correspondingly Additional Hoist Operational Features. For general applications, the following concerns should be taken into account Grouped (Loaded) Move. In various handling processes for coating materials, certain processing tasks must be performed one by one by the hoist without any interruption, during which the hoist will not be allowed to have other activities. Job dunking is an example, in which the hoist carries a job, dips it into a unit for one or more times without any interruption, and then moves it to another unit for further processing. In a graph, a grouped move is depicted by several connected solid line segments (an example will be shown in Figure 7 later) (Hoist) Home Station. In operation, a hoist may be required to start a new cycle at a specified location (above a specific processing unit). This location is called the home station. In Figure 1, for example, if unit 2 is designated as the home station, then the first hoist move in each cycle is from unit 2 (to unit 1). At the end of the cycle (after loading a job into unit 5), the hoist must move back to unit 2 (the home station). In scheduling, the introduction of a home station intro-

5 Ind. Eng. Chem. Res., Vol. 43, No. 26, Figure 5. Sketch of a flow rinse unit. Table 1. Determination of the Hoist Waiting Time Limit actual status of t i p t i p < t i min t i min e t i p e t i max t i p g t i max cases of Figure 4a (single capacity) infeasible t i,wait min ) 0; t i,wait max ) τ infeasible case of Figure 4b (unit capacity of n g 2) case of Figure 4c, 4d (unit capacity of n g 1) t i,wait min ) t i min - t p i n - 1 t i,wait max ) min{ t i max - t p i n - 1 t i,wait min ) t i min - t i n t i,wait max ) min{ t i max - t p i n p, τ}, τ} t i,wait min ) 0; t i,wait max ) min{ t i max - t i n - 1 t i,wait min ) 0; t i,wait max ) min{ t i max - t i n p p, τ}, τ} infeasible infeasible duces two additional free moves, which means that after a hoist operational sequence is determined, the time for the home-station-related hoist free moves should also be determined. If the home station is included, the free move matrix should be augmented by adding one row and one column; this will be discussed in section 5 with an example (Hoist) Waiting Time. In some material handling process, a hoist is set to wait at the home station for some time before starting a new cycle. The waiting time is determined based on job processing in certain units. According to Figure 4, an introduction of the waiting time (represented by shaded horizontal bars) increases the cycle time and, thus, the processing times in some units, except for a special case depicted in Figure 4a. Therefore, some infeasible hoist schedule, which is caused by insufficient processing time in some units, can become feasible when an appropriate waiting time is introduced. An optimal waiting time is usually difficult to determine, because different units have different waiting time requirements. Here, a method for identifying the minimum waiting time suitable for the entire system is introduced. Suppose the processing time in unit i is restricted to the interval t i min, t i max. If a possible hoist schedule is selected, the initial processing time ( t i p ) will be determined according to eq 1 or 2. The minimum waiting time for unit i ( t i,wait min ) then can be determined by t i p and t i min, and the maximum waiting time for unit i ( t i,wait max ) can be determined by t i p and t i max. However, the waiting time range, [ t i,wait min, t i,wait max ], may not exist in some cases. Table 1 gives the formulas for calculating the upper and lower bounds of the waiting time. The parameter τ in the table is the maximum permissible waiting time at the home station. If there is one unit i for which the waiting time range does not exist, then the corresponding hoist schedule candidate is invalid and it should be eliminated from the list of solution candidates. If every unit has a feasible waiting time range ([ t i,wait min, t i,wait max ), then these ranges will be used to determine an optimal waiting time ( t wait ) at the home station. This can be derived using the max-min rules below. t max min ) max{ t i,wait min /i ) 1, 2,..., Nu} (5) t min max ) min{ t i,wait max /i ) 1, 2,..., Nu} (6) t wait ) { t max min (if t max min e t min max ) N/A (if t max min > t min max ) (7) where Nu is the total number of processing units. Note that if t max min is larger than t min max, then t wait does not exist and the hoist schedule candidate is not feasible. 3. Dynamic Optimization of Water Re-use Network As stated early, wastewater can be reduced through an optimizing water allocation network (WAN) in a coating process, where a feasible hoist scheduling must be associated. Figure 5 depicts the mass flow in a general rinse unit, for which Zhou et al. developed the following dynamic model. 17 (a) Rinse water cleanliness dynamics. dc i dt ) W in in i C i + D in i C din i (H(t - t din is ) - H(t - t din ie )) - V i V i [W out i + D out i (H(t - t dout is ) - H(t - t dout ie ))]C i (8) V i (b) Mass balance at the inlet of the unit. J W in i ) W f in i + W ji j)1 (9) (c) Contaminant mass balance at the inlet of the unit.

6 8312 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 (d) Mass balance for the entire unit. (e) Average contaminant concentration at the outlet of the unit. where W f i is the freshwater flow rate into the ith unit; W in i and W out i are the total water flow rate into and out of the ith unit, respectively; D in i and D out i are the dragin and drag-out flow rates of the ith unit, respectively; C din i is the drag-in contaminant concentration of the ith unit; V i is the capacity of the ith unit; t din i,s and t din i,e are, respectively, the job entering time and the drag-in ending time in the ith unit; t dout i,s and t dout i,e are the job leaving time and the drag-out ending time in the ith unit, respectively; and H(t - a) is the step function. For a given WAN that contains Nr rinsing units, the rinsing time in each unit is obviously a key factor for minimizing wastewater. Note that the reduction of wastewater generation is equivalent to the reduction of freshwater consumption. A dynamic optimization model for minimizing the total freshwater, W f, is formulated below: s.t. where dc(t) dt J W in i C in i ) W f i C f + W in in ji C ji j)1 W i in + D i in ) W i out + D i out dc(t) dt (10) (11) C out T C i ) i 0 dt (12) T J( t p ) ) min W f (13) )RC in (t) + βc din (t) - γc(t) (14) C(t e dout ) e C lim (15) W f g0 (16) C(0) ) C 0 (17) 0 et et (18) t p ) ( t p 1,..., t Nr ) T (19) W f ) (W f f 1,..., W Nr ) T (20) ) ( dc 1 (t),..., dc Nr (t) dt dt ) T (21) C in (t) ) (C in 1 (t),..., C in Nr (t)) T (22) C din (t) ) (C din 1 (t),..., C din Nr (t)) T (23) C(t) ) (C 1 (t),..., C Nr (t)) T (24) C 0 ) {C 1,0,..., C Nr,0 } (25) R)diag{ W in 1,..., W in V 1 V Nr} (26) β ) diag{ D in 1 (H(t - t din 1s ) - H(t - t din 1e )),..., V 1 D in Nr (H(t - t din Nr,s ) - H(t - t din Nr,e )) V Nr } (27) γ ) diag{ W out 1 + D out 1 (H(t - t dout 1s ) - H(t - t dout 1e )),..., V 1 W out Nr + D out Nr (H(t - t dout Nr,s ) - H(t - t dout Nr,e )) V Nr } (28) C(t dout e ) ) (C(t dout 1,e ),..., C(t dout Nr,e )) T (29) C lim ) (C lim 1,..., C lim Nr ) T (30) In the model, inequality constraint 15 is the concentration limit when a job is removed from each rinsing unit. Note that the optimization objective is to minimize freshwater consumption, which is a function of all the rinsing times. Also note that, in this optimization step, water consumption has no upper limit, because the rinse quality is of utmost importance. Therefore, there always exists a solution for water use such that the rinse quality is satisfied. With this dynamic optimization model, every feasible hoist schedule can be evaluated for optimal freshwater consumption. The best hoist schedule should be the one with a minimum cycle time and minimum freshwater consumption. 4. Graph-Assisted Search The basic elements for the CHS described previously form the basis of a graph-assisted search algorithm for optimal hoist schedule identification. In this work, the minimization of cycle time and that of freshwater consumption are the two objectives in optimization. If the two objectives are combined into a single objective function, the optimization will become very complex because the cycle-time-based hoist scheduling is already a difficult task. Furthermore, the computational load will be incredibly heavy. Thus, a two-stage optimization strategy is proposed for this task. In the first stage, the cycle time is the sole objective (steps 1-5 in the algorithm below), because the production rate is a major concern of a plating system. In this stage, the candidate schedules with the minimum cycle time will be identified. In the second stage (steps 6 and 7 in the algorithm below), the minimum water consumption for each candidate schedule will be calculated. Among them, the schedule with the smallest water consumption will be identified as the optimal solution. Note that this sequential optimization cannot guarantee global optimality, but it does greatly facilitate optimal solution identification and is proven to be very efficient for the real industrial application. The stepwise algorithm is shown below. Step 1. Define grouped loaded moves and form a list of complete loaded moves if there are any operational requirements. Step 2. Generate a hoist free move matrix, according to Section 2.1. Step 3. Determine the reduced search space, S f (M), based on eq 4. Step 4. If S f (M) is not empty, select a sequence, s j f, from it and go to Step 5; otherwise, go directly to Step 6.

7 Ind. Eng. Chem. Res., Vol. 43, No. 26, Figure 6. Flowsheet of the known electroplating line. Table 2. Process Specification for the Industrial Example operation type unit number unit processing capacity processing time limit (s) soak cleaning 2 1 g445 electrocleaning 6 1 g460 acid cleaning electroplating 9 8 g3905 rinsing 3, 5, 7, 11 1 g30 Between Two Adjacent Tanks From Unit 8 to Unit 9 From Unit 1 to Unit 16 free move time 2 s 7 s 7 s Step 5. Calculate the cycle time and the processing time of each unit (if waiting time is needed, identify t wait and then update the cycle time). If the processing time of any unit is not within the limits (or t wait does j not exist), delete s f from S f (M); otherwise, record the calculation results of s j f and add s j f to the solution set S 0 (M). Then, delete s j f from S f (M) and return to Step 4. Step 6. Evaluate the freshwater consumption with the dynamic model of eqs for each schedule in the solution set S 0 (M). Step 7. Select the most desirable schedule in S 0 (M). 5. Application The developed graph-assisted CHS methodology has been used to investigate optimal hoist scheduling for many material-handling systems. Figure 6 gives one example of an industrial electroplating system. In this production line, three types of unit operationsscleaning, rinsing, and platingsare performed in 16 tanks (i.e., processing units). The WAN associated with 7 rinse tanks is also depicted, for which the total freshwater consumption (or wastewater generation) is gal/ min. In this line, the home station is designated as the rinse 5 tank. The maximum waiting time, τ, is 8 s. Other process parameters and processing time requirements are given in Table 2. By following the graph-assisted search algorithm, all the solutions can be derived in a systematical way as follows. Step 1. Nine (hoist) loaded moves, numbered from 1 to 9, are defined (see Figure 7). Among those, moves 7, 8, and 9 are grouped loaded moves. Step 2. A (hoist) free move matrix is generated (see Table 3). Table 3. Free Move Matrix for the Industrial Example group number home station home station Step 3. Infeasible sequential loaded moves are identified. In this case, the sequential loaded move pairs 1-2, 2-3, 3-4, 4-5, 5-6, 6-7, 8-9, and 9-1 are infeasible (see Section 2.1); the hoist schedule candidates that contain any of them should be excluded from the solution space S t (9). This will reduce the solution space from 9! to (i.e., S f (9)), according to eq 4. Steps 4 and 5. In S f (9), a total of 314 feasible schedules are identified and form solution space S 0 (9). These schedules have the cycle times ranging from 507 s to 544 s, which indicates a 7.3% difference of the production rate among them. In these solutions, only 25 schedules have the same minimum cycle time (507 s). Note that these 25 solutions may have different wastewater generation levels. Steps 6 and 7. The water consumptions in the WAN of all 25 solutions are evaluated. Because the rinsing time in each of units 8/10, 14, and 15 is fixed, according to the grouped moves, 7, 8, and 9, respectively, the water flow to these units will not affect the selection of an optimal hoist schedule. Thus, only the water flow rate through units 7 and 3 (i.e., W f 7 ) and that through f units 11 and 5 (i.e., W 11 ) should be determined (see Figure 8). The optimal values of these two variables are obtained by solving the dynamic optimization problem, using eqs The minimum freshwater flow rates, f f W 7 and W 11, for each of the 25 schedules are listed in Table 4. Solution 1 in the table is the most preferred, whose time-way graph is plotted in Figure 7. Figure 9 demonstrates the dynamics of the four relevant rinse units of the optimal schedule, where the shaded area

8 8314 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 Figure 7. Optimal schedule for the industrial example. Figure 8. Subsystem of the water re-use network for the industrial example. represents the rinse mode of a specific unit. It is shown that the rinse quality in each tank satisfies the rinse standard. As a comparison, the original schedule with water consumption is also listed in the same table, whose cycle time is 538 s, which is 31 s longer that the optimal one. Thus, the production rate using the optimal schedule can be increased by 6%. The given schedule requires the water consumption at gal/min. However, the optimal solution consumes gal/min, 7.8% less than the original schedule. Note that the first six schedules in Table 4 have a very small difference in terms of freshwater consumption, and they all have the same cycle time. Thus, any

9 Ind. Eng. Chem. Res., Vol. 43, No. 26, Figure 9. Rinsing subsystem dynamics for solution 1 in Table 4: (a) unit 7, (b) unit 3, (c) unit 11, and (d) unit 5. Table 4. Solution Comparison for the Industrial Example solution number Chemical loaded/grouped move Processing Time (s) sequence soak electrocleaning acid plating waiting time (s) Rinsing Time (s) Water Consumption unit (gal/min) 11 W f 11 W f 7 W f sub Org unit 3 unit 7 unit 5 of these solutions is acceptable as long as the small difference of freshwater consumption (or, equivalently, wastewater generation) is not a concern. The computation for solving this example is very efficient. The computer processing unit (CPU) time needed to identify all 314 feasible solutions from a total of solution candidates is only 28 s on a Dell Workstation PWS 450. This evidently shows that the developed graph-assisted solution identification algorithm is fast enough for on-line applications. 6. Concluding Remarks Material handling in general, and coating process operation in particular, always requires optimal hoist scheduling to obtain a maximum production rate. A

10 8316 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 major difficulty involved in the optimal hoist schedule development is the computation time. In addition, in all available hoist scheduling approaches, waste reduction is not integrated. In this paper, a graph-assisted cyclic hoist scheduling (CHS) methodology is introduced that can quickly develop an optimal hoist schedule for a single-type-product multistage process system, where maximum production rate and minimum waste generation can be reached simultaneously. The electroplating process example has demonstrated the attractiveness of the methodology in industrial applications. The environmentally bearing CHS methodology is applicable to the process where multijob capacity in the processing units is permissible. This has also advanced the existing CHS approaches in application capacity. A future effort will be needed to extend this new methodology to the system where multiple types of products are processed in a production line. Acknowledgment This work was supported, in part, by the National Science Foundation (under CTS and DMI ) and the Institute of Manufacturing Research at Wayne State University. Literature Cited (1) Kumar, P. R. Scheduling Semiconductor Manufacturing Plants. IEEE Trans. Control Syst. Technol. 1994, 14, (2) Lei, L.; Wang, T. J. A Proof: The Cyclic Hoist Scheduling Problem is NP-complete. Working Paper No , Rutgers University, Piscataway, NJ, August (3) Phillips, L. W.; Unger, P. S. Mathematical Programming Solution of a Hoist Scheduling Program. AIIE Trans. 1976, 8, (4) Shapiro, G. W.; Nuttle, H. W. Hoist Scheduling for a PCB Electroplating Facility. IIE Trans. 1988, 20, (5) Lei, L.; Wang, T. J. Determining Optimal Cyclic Hoist Schedules on a Single-Hoist Electroplating Line. IIE Trans. 1994, 26, (6) Armstrong, R.; Lei, L.; Gu, S. A Bounding Scheme for Deriving the Minimal Cycle Time of a Single-Transporter N-Stage Process with Time Window Constraints. Eur. J. Oper. Res. 1994, 78, (7) Baptiste, P.; Legeard, B.; Varnier, C. Hoist Scheduling Problem: An Approach Based on Constraints Logic Programming. Proc. IEEE Conf. Rob. Automation 1992, 2, (8) Rodosek, R.; Wallace, M. G. A Generic Model and Hybrid Algorithm for Hoist Scheduling Problems. In Proceedings of the 4th International Conference on Principles and Practice of Constant Programming; 1998; pp (9) Chen, H.; Chu, C.; Proth, J. M. Cyclic Hoist Scheduling Based on Graph Theory. In Proceedings of the 1995 INRIA/IEEE Symposium on Emerging Technologies and Factory Automation; IEEE Computer Society Press: Los Alamitos, CA, 1995: Vol. 1, pp (10) Dutilleul, S. C.; Denat, J. P. P-time Petri Nets and the Hoist Scheduling Problem. In IEEE International Conference on Systems, Man, and Cybernetics (SMC 98), San Diego, CA, 1998, pp (11) Khansa, W.; Denat, J. P.; Dutilleul, S. C. P-time Petri Nets for Manufacturing Systems. In Proceedings of the International Workshop on Discrete Event System, Edimburg, Scotland, 1996, pp (12) Yih, Y.; Yin, N. C. Crane Scheduling in a Flexible Electroplating Line: A Tolerance-Based Approach. J. Electron. Manuf. 1992, 2, (13) Geiger, C. D.; Kempf, K. G.; Uzsoy, R. A Tabu Search Approach To Scheduling an Automated Wet Etch Station. J. Manuf. Syst. 1997, 16, (14) Zhou, Z.; Li, H. A Heuristic Method for Single Hoist Dynamic Scheduling. J. Syst. Simul. 2000, 12, (in Chin.). (15) Lim, J. A. Genetic Algorithm for a Single Hoist Scheduling in the Printed-Circuit-Board Electroplating Line. Comput. Ind. Eng. 1997, 33, (16) Kuntay, I.; Lou, H. H.; Huang, Y. L. Environmentally Conscious Hoist Scheduling in Electroplating. Presented at the 23rd AESF/EPA Conference on Environmental and Process Excellence, Daytona, FL, February 2-6, (17) Zhou, Q.; Lou, H. H.; Huang, Y. L. Design of a Switchable Water Allocation Network Based on Process Dynamics. Ind. Eng. Chem. Res. 2001, 40, Received for review February 9, 2004 Revised manuscript received September 24, 2004 Accepted September 27, 2004 IE