This paper was accepted by Chemical Engineering
Communication on August 20, 2004 and it is in press.
_______________________________________________________________
ENVIRONMENTALLY CONSCIOUS HOIST SCHEDULING FOR
ELECTROPLATING FACILITIES
_______________________________________________________________________________________________________________________________________
ISIK KUNTAY
QIANG XU
KORKUT UYGUN
YINLUN HUANG
Department of Chemical Engineering and Materials Science
Wayne State University, Detroit, MI 48202
Hoist scheduling in electroplating operations has long been considered a key
factor for improving the production rate. It is, however, recently recognized that
hoist scheduling can also play an important role in waste minimization. In this
work, a new hoist scheduling method is introduced for simultaneously achieving
both the economic and environmental goals. A two-step dynamic optimization
algorithm is introduced in this paper for identifying an optimal hoist schedule that
can minimize the quantity and toxicity of wastewater streams from an
electroplating line without loss of production rate. To improve computational
efficiency, an engineering approach is adopted to reduce the number of binary
decision variables in the optimization problem. An application to an actual
electroplating process shows a significant reduction of both chemical and water
consumption, which equates to a simultaneous realization of wastewater
reduction and increase of profits.
Keywords –
Electroplating, hoist scheduling, pollution prevention
Address correspondence to Prof. Yinlun Huang, Department of Chemical Engineering and Materials
Science, Wayne State University, Detroit, MI 48202. Email: (yhuang@wayne.edu).
1
INTRODUCTION
Electroplating is performed in a typical hybrid system, where workpieces are processed
through a series of cleaning, rinsing, and plating units in a batch mode, while the rinsing
water flows through a number of rinse systems in a continuous mode. Electroplating is
commonly utilized to apply a thin layer of metal coating on workpieces. In the US, there are
over 10,000 electroplating plants, which provide numerous plated workpieces to the
electronics, automobile, aerospace and other industries. Electroplating processes typically
use various chemicals, and consume huge amounts of fresh water and energy in operation.
These plants generate tremendous amounts of hazardous or toxic waste daily in the form of
wastewater, sludge and spent solution, which includes over 100 toxic chemicals that are
regulated by the EPA (Duke, 1994). The environmental problems have made the
electroplating industry the second most regulated one in the nation. Over the past three
decades, the industry has developed and adopted a variety of pollution control and pollution
prevention (P2) technologies in their systems (Cushnie, 1994). The success in improving
environmental quality has been tremendous.
However, increasingly stringent environmental regulations and economic competition
have been forcing the industrial sectors to look for P2 methods that would not reduce their
profits. Lou and Huang (2000) introduced a new concept called profitable pollution
prevention (P3). This designates the simultaneous reduction of pollution and increase in
profit. In recent years, a number of P3 technologies have been developed for waste
minimization in electroplating industry.
The essential route to P3 is source reduction, which is the elimination of waste
before it has been generated. It is recognized that in most electroplating plants, the
consumption of the resources, such as chemicals, water, and energy, are much higher than
theoretically calculated necessary amounts. This has suggested great opportunities for P3,
as reduced consumption of these resources leads to simultaneous realization of waste
reduction (environmental impact) and cost reduction (economic incentive).
In the environmentally conscious hoist scheduling method to be introduced in this
work, environmental concerns are taken into account simultaneously with cost related
criteria. The method simultaneously determines the optimal operation times for the units in
an electroplating line, and an optimal hoist schedule for productivity and environmental
cleanness.
ELECTROPLATING AND SCHEDULING BASICS
Figure 1 depicts a general electroplating process. Workpieces, usually loaded in barrels or
on racks, are covered by grease, soil, dirt, rust, etc., on their surface. For simplicity, the
following text will use “barrel” only; the methodology is entirely applicable to both the barrel
based and the rack based processing. The surface of workpieces must be treated before
they can be plated, which is achieved by use of preprocessing units. For the process
displayed in Figure 1, the barrels are processed in soaking (Unit C1), followed by two-step
rinsing (Units R1, and R2), electrocleaning (Unit C2), followed by another two-step rinsing
(Units R3, and R4), and acid cleaning (Unit C3) succeeded by one-step rinsing (Unit R5).
After electroplating in Unit P, the workpieces need to be post-processed, typically in a couple
of more solutions in order to stabilize the plating or extend its features, such as in Units R 6
and R7, in this case.
The sequential operations are managed by a hoist or hoists. The barrels are usually
placed at one end of the plating line, which is a loading/unloading zone. Each barrel is then
moved by the hoist from one unit to the other, according to the pre-specified processing
procedure. The residence time of a barrel in a unit varies according to the type of processing
(i.e., cleaning, rinsing, and plating). There is a minimum amount of time a barrel must stay in
each unit for processing, and for most processes, there is also a maximum processing time
limit. The residence time is also limited by the scheduling, so that the production schedule is
2
feasible using a minimum number of hoists. Therefore, the workpiece processing time is
directly related to the production rate of the line. If the hoist can pick barrels from the loading
station more frequently, the production rate will be higher.
Fresh water
Parts out
Parts in
C1
R1
R2
C2
R3
R4
C3
R5
P
R6
R7
Wastewater
Legend:
Water flow
Parts flow
Figure 1. Sketch of a plating process.
Electroplating is an interesting type of hybrid process due to the added complexity of
the hoist scheduling. The task of hoist scheduling (HS) is to identify the sequence and the
time of hoist movements such that the production objective(s) can be reached. A typical
objective is to maximize production rate. Practically, all the operations, and therefore the
objectives, in electroplating are related with the hoist schedule.
Historically, hoist schedules were developed based on experience. The first reported
effort on computerized scheduling was made by Phillips and Unger (1976). The formulation
for scheduling is initiated with the information given on the system and processing, which
includes information, such as layout, chemistry of processes, hoist operation speeds,
workpiece processing sequence and processing time ranges.
Phillips and Unger (1976) investigated the scheduling problem for the simplest type
of electroplating process that involves a single type of products handled by a single hoist. In
this case, the hoist must perform the same sequence of operations on each barrel to
maintain uniform processing. This results in a repeating sequence of moves, which is
referred to as a cycle (Shapiro and Nuttle, 1988). This type of operation is named a Cyclic
Hoist Scheduling (CHS). In such a case, the hoist is controlled by a programmed logic
controller executing a loop of commands, such as lift and release actions. The instants when
the barrel is lifted from a unit and released into another unit are named the lift time and
release time, respectively. In CHS, each move starts with a lift action and ends with a
release action.
In addition to CHS, there are other kinds of HS, namely real-time hoist scheduling
(RHS), predictive hoist scheduling (PHS), and dynamic hoist scheduling (DHS) (Lamothe et
al., 1995). The RHS deals with the case when different types of workpieces are processed
in a process line with a given order of processing. The PHS covers the operations in which
the processing orders are to be determined. The DHS, on the other hand, is concerned with
the workpieces of different types arriving randomly at the loading station. Riera and YorkeSmith (2002) provide a comprehensive survey on the history and classification of hoist
scheduling problems, where a hybrid algorithm combined with constraint logic programming
and mixed integer programming is presented.
3
Figure 2 is an illustration of a “time-way diagram” (Shapiro and Nuttle, 1988), which
is a graphical representation of a cycle. In this example, four units are employed. In the
figure, each inclined line segment represents a hoist move. A hoist move can be either
loaded (called loaded move) or unloaded (called free move). A loaded move starts from a lift
action (symbolized by “↑” in the figure). The hoist then carries the barrel or rack to another
unit. It stops above that unit and then releases the barrel into it (symbolized by “↓” in the
figure). A free move is a hoist movement during which it does not carry any barrel. Thus, it
starts from “↓” and ends at “↑” in the figure. A hoist can also pause over some unit for some
period (indicated by “W” in the figure).
Time
(sec)
70
W
60
Loaded move
50
Cycle
Free move
40
30
Processing time
in Tank 1
20
10
0
5
4
3
2
1
Tank 4
Tank 3
Tank 2
Tank 1
Loading
Figure 2. Time-way diagram representation of a hoist cycle.
As shown in the figure, the cycle time for this case is 75 seconds. This means every
75 seconds of operation, one barrel will enter the process and one barrel of plated
workpieces will leave the process. Or equivalently, every 75 seconds, the hoist will appear
at the same location for the same operation of the process. If the lifting of a barrel in the
loading station is considered the starting point of the hoist operation, then the hoist will lift the
next barrel from the same station in 75 seconds. In each cycle, there are numerous
operation states. Each state can be characterized by: a) the location of the hoist, b) the
number and location of the barrels in the system, and c) the elapsed processing time of each
barrel in its current process (Shapiro and Nuttle, 1988).
A cycle is said to be feasible if the associated sequence of hoist movements is
executable and the associated processing time in each unit is within the allowed ranges.
The throughput of a cycle is defined as the rate at which barrels are processed through the
system. CHS is to determine a feasible, hopefully also optimal, cycle that allows a maximum
throughput for a given process (Shapiro and Nuttle, 1988).
MODELING FOR WASTE REDUCITON
In order to discover waste reduction opportunities, mechanisms of waste generation must be
precisely known. Luo and Huang (1997) identified two types of waste in plating lines:
unavoidable and avoidable. The unavoidable waste is generated from the removal of the dirt
4
on workpiece surface that is essential for ensuring cleaning quality. The avoidable waste, on
the other hand, is caused by over cleaning, and loss of chemicals through drag-in/drag-out.
Through drag-in/drag-out, a certain amount of chemical and plating solutions is carried over
to succeeding rinsing systems and finally enters the wastewater. This type of waste should
be avoided to the maximum possible extent.
The focal point of an operational strategy is to identify the upper limit of allowable dirt
residue on parts and minimizing chemical use such that cleaning is done only up to this
limiting amount. Existing P3 methodologies try to achieve this goal by reducing the waste
generated in each processing unit, the waste transferal among units, chemicals, water and
energy, and meanwhile ensuring cleaning, rinsing and plating quality. The determination of
waste generation, or a minimum amount of chemical necessary, requires information about
process dynamics.
Traditional scheduling formulations are static, i.e., the dynamics in the units are
ignored and lumped into maximum and minimum bounds on processing times, which are
generally determined in a heuristic and not optimal way. Implementation of waste
minimization in HS requires dynamic information that can be acquired from system
simulations such that the waste generation can be determined precisely. Gong et al. (1997)
developed a set of dynamic models for cleaning, rinsing, and electroplating operations.
These models were later improved by Lou and Huang (2001) and Zhou et al. (2001).
The algorithm for calculating the needed water consumption of a fresh waterreceiving unit, developed by Gong et al. (1997), is depicted in Figure 3. Based on the
processing time of all rinsing units ( t p (1) , …, t p (r ) ), the algorithm is initiated with an
initial guess ( W
f ,g
) for water flow rate ( W
f
). Next, the initial contamination of Unit 1
( C i (1) ) in the system, which is the fresh-water receiving unit, is assigned an initial guess
value ( C i , g 1 ). Calculations that require the solution of coupled differential equations are
performed for Unit 1 to find the final contamination in this unit ( C f (1) ).
The initial
contamination of Unit 1 is set to the final contamination calculated and new calculations are
performed. This iterative process continues until the initial and final contaminations are
within some neighborhood (  ). Similarly, Unit 2 is subjected to the iterative calculations,
noticing that it now does not receive fresh water but water that has been contaminated in
Unit 1. Calculations are continued in the order the water flows through the system. Once
the final calculation is finished, the qualities of rinsing of all the units ( Ce (k ) ) are evaluated.
If any of them is not in the vicinity (  ) of the minimum requirement ( C e ,l ( k ) ), the iteration
will continue with a water flow rate a certain level (  ) higher or lower than the previous
guess. If all units meet the requirement, the water flow rate is decreased and calculations
are continued until one unit hits the limit of satisfactory rinsing.
ENVIRONMENTALLY CONSCIOUS HOIST SCHEDULING
The methodology introduced in this section is referred to as Environmentally Conscious
Cyclic Hoist Scheduling (ECCHS), as only cyclic processes are considered at this point. In
development of a feasible hoist schedule, details regarding waste minimization must be
considered to maximize the efficiency of hoist usage. Also, the issues concerning profit must
be taken into account for P3 goals to be realized. As such, a general discussion of the
problems in ECCHS will be presented first.
5
Obstacles and Strategy
A hoist schedule for an electroplating operation has to address three major issues. The
primary issue is production rate, as it reflects the production efficiency and profitability of a
plant. The secondary issue is product quality. A hoist schedule implements the scheduled
processing times, which is one of the most important factors of product quality assurance.
The relationship between processing time and waste generations creates the ternary issue of
P2. Figure 4 summarizes some key ideas of P2 through improved HS. It suggests that P2
can be enhanced through adjusting the processing times in cleaning and rinsing systems.
Needless to say, the processing time determination must be coordinated with the hoist
schedule.
Given k = 1, and
Δt p (1) ~ Δt p ( r )
W f  W f ,g
Ci ( k )  Ci , g ( k )
k=1
Ci ( k )  C f ( k )
No
If max Ce (k)  Ce,l (k)  0
Dynamic Model for
the k-th Rinsing Tank
C i (k )  C f (k )  
k 1,, r
Then W f  W f  
Yes
If max Ce,l (k )  Ce (k )  
k = k+1
k 1,, r
Then W f  W f  
Yes
k r
No
No
max Ce (k)  Ce,l (k)  
k 1,, r
Yes
Wopt  W f
Figure 3. Algorithm for the optimization of water flow rate into a rinsing system with r
units in series and one fresh water inlet (Gong et al., 1997).
It is easily realized that if the chemical concentration in a cleaning unit is reduced,
then the chemical loss through drag-out will be proportionally reduced, and thus the chemical
contents in the wastewater stream will be reduced by the same amount. Of course, this
chemical concentration reduction must not affect the cleaning quality adversely. Obviously,
at a lower chemical concentration, the cleaning time of each barrel in the unit should be
properly increased in order to maintain the cleaning quality. Likewise, if the flow rate of
rinsing water is decreased, the consumption of water and the volume of the wastewater
stream will also decrease. This again requires an adjustment of the processing time such
that the desired rinsing quality is achieved. An increased operation time, however, may lead
to a reduced production rate for the electroplating process. This creates the trade-off
6
between environmental quality and economic benefits that necessitates optimization to find
the best solution.
Scheduling
Minimum
Cycle Time
Increased
Production
Rate
Increased
Processing Times
Cleaning Tanks:
Less Chemical
Concentration
Less Chemical Loss
From Drag-out
Rinsing Tanks:
Less Water Flow
Rate
Less Fresh Water
Consumption
Figure 4. Scheduling strategy analysis.
The observations above suggest that hoist schedule could be designed to assist in
waste reduction during electroplating operation, by allowing an increase in the workpiece
processing times in units. Nevertheless, this is a rather difficult task with the following
conceivable technical barriers to be removed.
a) The processing time in a cleaning or rinsing unit cannot be increased without
bounds because of the requirements on production rate.
b) The environmental benefit may not be noticeable when the processing time is
longer than some threshold. Note that this threshold is somewhat difficult to determine, as it
requires an evaluation of system dynamics.
c) Since the rinsing time in one unit is related to the operation of other units as well
as the source of rinse water, the determination of the optimal rinse times is a system issue.
d) When an environmental issue is integrated into a hoist schedule development
process, the decision-making will be a multiple objective optimization problem: to minimize
waste and to maximize (or retain) production rate.
e) The optimization involves evaluating the dynamic performance of cleaning and
rinsing operations to ensure product quality. Thus, the optimization will be computationally
expensive. Since hoist schedule development in production is always an on-site issue, the
computational time needed must be well acceptable for real applications.
Algorithm
To realize the dual objectives of ECCHS, a two-step optimization procedure is introduced.
The basic idea of this stepwise algorithm is to optimize for production rate first and to
minimize waste subsequently, which is detailed in Figure 5. As shown, the first step is to
identify the minimum achievable hoist cycle time, tc ,min ; note that the production rate is
the reciprocal of the hoist cycle time. In this optimization, no environmental concerns are
taken into account, and the problem is essentially equivalent to solving a traditional CHS
7
problem (Phillips and Unger, 1976; Shapiro and Nuttle, 1988). Note that it is possible that
the production rate is sometimes determined based on market needs rather than process
capacity.
a) Hoist Scheduling for Maximum
Production Rate
t csp  Min t c
g (x)  0
h(x) = 0
or
b) User Specified Cycle Time
t csp
t csp
Hoist Scheduling for Maximum Waste
Reduction
Minimize Waste
tc  tcsp
s.t.
g (x)  0
h(x) = 0
Figure 5. Suggested HS scheme for meeting multi-objectives.
In the second optimization step, the objective is to minimize chemical and water
consumption while maintaining the optimized or desired production rate. Note that reducing
the amount of chemicals and water used will reduce both the operational cost and the total
waste generation; hence the environmental and economic goals coincide in this stage. The
sp
target cycle time ( t c ) identified in the first step is set as a constraint in this optimization
problem. Note that if optimization for minimizing waste generation were to be performed
without any constraint on the cycle time, the solution identified would probably be very
satisfactory in terms of environmental issues but would be deficient in terms of production
rate.
In order to evaluate water and chemical consumption in production, the second step
must utilize process dynamic models, and therefore it is a dynamic optimization. In each
iteration during optimization, the process must be simulated. This requires performing the
rinsing unit simulation iterations as given by Gong et al. (1997). The iterative nature of the
simulations introduces a significant difficulty in solving the second-level problem, particularly
in evaluating the objective function. For clarity, we will first discuss the constraints involved
along with the notation, and then proceed to the discussion of the objective. After that, a bilevel algorithm developed for solving the optimization problem will be introduced.
Constraints. The necessary constraints associated with the optimization problem
are listed below. Note that indices (i, j, and k) are all used to designate the units in the
electroplating line.
Processing time limits. The range of permissible processing time for the parts in
each operational unit is given below.
8
t p , min (i )  t p (i )  t p , max (i )
(1)
where t p (i ) is the processing time in Unit i. The processing time is allowed to be very
large, as long as the rinsing quality is satisfactory for rinsing units.
Loaded move. A relationship between the lift time from the i-th unit ( t l (i ) ) and the
release time into the succeeding j-th unit in the electroplating procedure ( t r ( j ) ) is given
below.
(2)
tr ( j )  tl (i)  tL  td  tm (i, j )  tR
Note that the hoist-related operations for each loaded move include lifting, draining, loaded
moving, and releasing. The time needed for moving the hoist from Unit i to Unit j ( t m (i, j ) )
is usually independent of whether the hoist is loaded or not. The time for lifting ( t L ) and
the time for releasing ( t R ) are constant for all units, and the drainage time ( t d ) is
generally 3~5 seconds.
Processing time calculation. The processing time of a barrel in Unit i ( t p (i ) ) is
controlled by a hoist schedule. It is the time elapsed between the release and lift time of the
barrel from Unit i. In the following equation, the binary variable (y(i)) is an intermediate
variable necessary for the algebraic formulation of the logical OR statement, The
introduction of the binary variable ensures that the processing time is always less than the
cycle time and greater than 0.
t p (i )  t l (i ) - t r (i )  t c y (i )
(3)
where
y (i )  0
y (i )  1
if
tl(i)  tr(i)
(4)
if
tl(i) < tr(i)
(5)
Single hoist. Equation (6) is the elaboration of the feasibility condition, which states
that a sufficient time is needed for the hoist to travel from where it was last used to where it is
next needed.
k
t l (k )  (t r (i)  t m (i, k )) z (i, k )
(6)
where tl(k) is the time at which a barrel is lifted from and processing is ended in Unit k; tr(i)
the time at which a barrel is released and processes is started in Unit i; z(i,k) the binary
variable which equals 1 if there is a free hoist movement from Unit i to Unit k, or 0 otherwise.
Simple cycle. Equation (7) is the constraint that prevents the hoist to visit Unit k
more than once for lifting a barrel. Equation (8) is about the hoist being destined to only a
single unit after it has finished one job (i.e., after releasing a barrel). Equation (9) prevents
the hoist from visiting the unit it has just left.
N
 z (i, k )  1
k
(7)
 z (i, k )  1
i
(8)
z (i, i )  0
i
i 1
N
k 1
(9)
Complete cycle. A desired solution of a scheduling process is a cycle that starts and
ends at the same unit. Also, the hoist should visit all the units in the line, which means the
cycle should contain all the units in the unit index superset, i. Without special constraints to
force it so, it is possible for a solution to be composed of two different and completely
uncoupled cycles. These two cycles can satisfy all the other constraints, but the solution will
not be acceptable since two distinct cycles running simultaneously will require two hoists in
the line on different tracks.
9
To formulate the problem mathematically, let us split the hoist cycle into two distinct
sets, E and F, and they are arbitrarily extracted from I such that:
E  F=I
(10)
E  F=
(11)
|E| > 1 (i.e., E has more than one element)
(12)
|F| > 1
(i.e., F has more than one element)
(13)
For a solution not to be composed of two distinct cycles, there must exist at least one
movement between the units indexed in E and F that connects these two sets. If e and f are
the elements of E and F, respectively, this can be expressed as:
z (e, f )  1
(e  E and f  F)
(14)

e
f
Note that, E and F were arbitrarily chosen subsets that satisfy equations (10) through
(13). Let each combination of E and F that satisfy these conditions be represented by the
index s S. The constraint (14) must be applied to every single one of these combinations,
which means it should be restated as:
z (e( s), f ( s))  1
(15)
s , e(s)  E(s) and f(s)  F(s)

e( s ) f ( s )
The number of different combinations of sets E and F can be calculated as follows:
S 
n
I!
 p!( I
p2
(16)
- p )!
where n is the largest integer smaller than or equal to I /2; S is the set of the index s; S is
the number of elements in the set S (i.e., the size of set S).
Equation (16) is the summation of the number of possible combinations obtainable
from a grand set of I with p elements. Note that the summation index starts from 2 as there
is no need to include sets E with single elements in Equation (15), since it is impossible to
have a cycle with a single element. The reason for terminating the summation at n instead of
( I -1) is to exclude the combinations already accounted for.
Cycle start and end constraints. The time of lifting a barrel from the loading station is
to be taken as the reference point in time (see equation (17)). The cycle length has to be
greater than the ending time of the last move plus the free movement time from the final
location of the hoist (after completing the last move) to the starting point of the cycle (the
loading station). This is expressed mathematically by equation (18). Also note that the last
move is the one for which z(i, 1) is equal to 1, as the definition of cycle requires. Equation
(18) provides a generalization of this constraint so that whenever the term z(i, 1) is active for
the i-th tank, the cycle time has to be greater than the addition of the release time of a barrel
into Unit i and the free movement time from Unit i to the loading station (Unit 1).
(17)
t l (1)  0
tc  (tr (i)  tm (i,1)) z (i,1)
i
(18)
Objective. The waste reduction oriented second optimization step involves the
following decision variables: the sequence of the hoist moves, the lift and release times of
barrels from units during the cycle, and the processing times of all units. A basic form of the
objective function is as follows:
Min J(tp) = Jc(tp) + Jf(tp)
(19)
where
tp = (tp(1) tp(2) … tp(I))T
(20)
In equation (19), J(tp) is the combined total chemical and fresh water costs ($/yr), Jc(tp)
the chemical consumption cost ($/yr), and Jf(tp) the fresh water consumption cost ($/yr).
The cost function includes those related to the processing times of workpieces in all cleaning
and rinsing units. Note that the processing times directly determine the consumption of
10
chemicals and water, which can be calculated with iterations as outlined above (Gong et al.,
1997). Therefore, Jc(tp) and Jf(tp) can be derived according to following equations.
Jc(tp) = Uc Wc(tp)
(21)
Jf(tp) = Uf Wf(tp)
(22)Where
Uc is the unit cost of chemicals ($/gal); Uf the unit cost of water
consumption ($/gal); Wc (tp) the total chemical consumption (gal/min); Wf (tp) the total
water consumption (gal/min);  the conversion factor (i.e.,  = 432,000 for the conversion
from min-1 to yr-1, assuming 300 manufacturing days per year).
Since it is nearly impossible to express J(tp) analytically, an evaluation of the
objective function requires a numerical simulation of the system, which introduces a second
level into the optimization problem; this should not be confused with the two-step procedure
used in the overall solution algorithm. Figure 6 presents a simplified overview of the ECCHS
algorithm.
a) Hoist Scheduling for
Maximum Production Rate
t csp  Min t c
Step1
b) User Specified Cycle Time
t csp
t p
Hoist Scheduling for
Maximum Waste Reduction
Minimize Cost (J)
s.t.
t c  t csp
Step2
t p ,0
Rinsing and Cleaning
Unit Simulations
J = f( t p )
Determine J = f( t p )
Level 1
Level 2
Figure 6. Schematic overview of the ECCHS algorithm.
There are two methods for tackling such a bi-level problem. One is solving the
simulation in each step of iteration in optimization, which is a straightforward, but brute-force
approach. The drawback is that since a large number of simulations are necessary, this
makes the computational time infeasible for any realistic process. An alternative way was
developed by Uygun and Huang (2002). In this method, rather than performing a simulation
within the optimization algorithm, linear approximations to the solution of the simulation are
constructed.
This methodology, which is a dynamic analogy of sequential linear
programming approach, is briefly delineated below:
1)
Simulations are performed with an initial hoist scheduling design.
2)
Linear approximations to the objective function and constraints as functions
of decision variables are constructed (in this case, approximations for waste water
production and chemical consumption as functions of processing times).
3)
The approximate optimization problem, formed by the linear approximations,
is solved with an optimization algorithm to obtain a new design.
11
4)
If the convergence criterion is met, then an optimum design is obtained and
iterations are terminated. Otherwise, Steps 1 to 3 are repeated based on a new design.
The above algorithm can significantly reduce the number of simulations and render
the solution of the scheduling problem practical. A generalized iteration form for linear
approximations to the chemical and water consumption in this problem is as follows:



 

J t pNS  J t pNS  1  t pNS  t pNS  1 SNS

T
 
(23)

 UW t pNS  1  t pNS  t pNS  1 SNS
T
where J  is the total chemical or water cost corresponding to initial processing times;
the processing time vector of the NS-th iteration (sec);
t pNS
t pNS 1 the initial processing time of
NS
the (NS-1)-th iteration (sec); S  the sensitivity of the NS-th iteration.
Equation (23) is a first-order Taylor series expansion of the cost function around the
initial processing times. This representation of the total cost function is exact within the neighborhood of the previous points, but declines in accuracy when evaluated far from the
linearization point. Sensitivities are first-order derivatives of the cost function with respect to
the variables (i.e., the processing times). They are essential information for constructing
first-order Taylor series expansion, and are evaluated using the following general finite
difference formula.
NS
S







 J  t pNS (1)  J  t pNS 1 (1)
J  t pNS ( I )  J  t pNS 1 ( I )


, ,

t pNS (1) - t pNS 1 (1)
t pNS ( I ) - t pNS 1 ( I )


T


(24)
APPLICATION
The capacity of the ECCHS algorithm is demonstrated via an application on a typical, largesize electroplating plant performing barrel plating. The process involves 10 processing units
as depicted in Figure 7. Each unit has a special functionality designated as cleaning (C),
rinsing (R), plating (P), or post processing (S), and each has its own operational
requirements. The units need to be operated within certain limits of temperature, processing
time and chemical composition.
P
9
R
8
R R
10 5
C
4
R
7
R
3
C
6
C S
2 11
L
1
Figure 7. Water flow pattern and unit layout of the plating line.
Process Specification
The electroplating unit (Unit 9) holds a zinc-containing solution from which the zinc plating is
deposited on the workpiece surface. The plating thickness is specified depending on
customer request. The amount of electric current passing through the unit is fixed while the
workpieces are allowed to stay in this unit for a specific period of time (67.7 minutes). These
constraints are crucial for achieving the correct plating thickness. Unit 9 has an 8-barrel
capacity. It is partitioned into 8 slots, each reserved for one barrel. Preserving the
12
conventions for multi-capacity units (Shappiro and Nuttle, 1988), the processing time of each
slot is p(9)
c.
In this process, Unit 2 is a primary treatment unit, involving a soak cleaning agent,
used for removing dust and grease off metal surfaces. Unit 4 also involves a soak cleaning
agent, but much stronger than that in Unit 2. Unit 6 is an electro-cleaning unit, which
employs an electric current to remove metal scales and impurities. Unit 11 is a postprocessing unit. It contains a solution that stabilizes the plated surface of the workpieces.
This essential operation requires a fixed processing time (2.9 minutes).
Once a barrel comes out of a chemical solution, the workpiece surface has to be
rinsed before entering a different solution. Units 3, 5, 7, 8 and 10 should provide sufficient
rinsing. However, Unit 8 operates in a different manner than the others. This unit is a dragin/out unit. The workpieces are almost immediately lifted after being released into this unit.
Figures 8 and 9 show the processing sequence and corresponding hoist schedule for
the workpieces used in the plant. Note that the slope of the inclined lines between Units 8
and 9 is larger than the slope of the inclined lines anywhere else because the space between
these units in the figure represents a larger distance compared with the distance between
other units.
Parts Out
P
9
R
8
R R
10 5
C
4
R
7
R
3
C
6
C
2
S
11
L
1
Parts In
Figure 8. Workpiece processing procedure.
It is observed from Figure 9 that though the hoist is free after it has released a barrel
in Unit 11, it does not leave the unit. This is because of the restriction on the hoist that it
should not make any move while a barrel is being processed in this unit.
The cycle time of this process has been pre-specified as 520 seconds (8.67
minutes). This is the minimum cycle time to achieve the 67.7 minutes of electroplating time
in Unit 9. In Figure 9, the processing time of Unit 9 appears to be 7.0 minutes, which would
be the processing time if unit 9 was a single-capacity unit. This is 1.67 minutes shorter than
the cycle time. During this 1.67 minute period, the hoist lifts a barrel from Unit 9, dunks it in
Unit 8, places it in Unit 10, lifts a barrel from Unit 7 and places it in Unit 9. For the slot of Unit
9 farthest from Unit 8, the 1.67 minutes is the shortest possible time to replace a barrel lifted
from Unit 9. For other slots, there are waiting times of the hoist between Units 8 and 9.
Without the waiting times, the processing time of the slot nearest to Unit 8 would have been
68.0 minutes, which is a violation of the limit.
The process time constraints are presented in Table 1. It is observed that Unit 8
does not have any processing time because it is a drag-in/out unit. Some units do not have
upper limits, which means their processing times can be increased as desired, while some
units have very strict limits (i.e., Units 9 and 11). The processing times in the original hoist
schedule in Figure 9 are also displayed in Table 1. These are used as an initial point for the
iterations. It can be seen that the processing time of Unit 4 is over the limit in the original
schedule.
13
t (sec)
2
500
7
8
9
3
5
6
300
W
4
11
200
W
100
10
1
0
P
9
R
8
R
10
R
5
C
4
R
7
R
3
C
6
C
2
S
11
L
1
Figure 9. Original hoist schedule in the plating line.
Table 1. Processing Time Specification
Unit
2
3
4
5
6
7
8
9
10
11
Lower Limit
(min)
6.67
2.78
5.00
0.42
7.50
0.54
0 (Dunk)
67.7
2.00
1.29
Upper Limit
(min)
None
None
6.00
None
None
None
0 (Dunk)
67.7
None
1.29
Original
(min)
6.67
6.43
6.57
1.63
7.50
0.57
0 (Dunk)
67.7
2.23
1.29
14
Engineering Simplifications
It is possible to represent a schedule by means of ordering, instead of a time-way diagram.
The ordering concept, introduced by Shapiro and Nuttle (1988), is used to elucidate the order
in which the moves are performed by the hoist in the cycle. The moves are numbered
according to the unit they start from. For example, Move 4 is a combination of the actions of
lifting a barrel from Unit 4, waiting on the Unit for 5 seconds for drainage, carrying the barrel
to Unit 5 and releasing it to Unit 5.
The binary variables show whether a move is followed by another move or not. For
instance, z(2,3) has a value of 1 (or 0) means that Move 2 is followed (or not followed) by
Move 3. In this context, the hoist schedule in Figure 9 may be expressed by listing the
binary variables that are non-zero as: z(1,10), z(10,11), z(11,4), z(4,6), z(6,5), z(5,3), z(3,9),
z(9,8), z(8,7), z(7,2), and z(2,1). The total number of binary variables z(i, j) is equal to 11 
11 = 121. An application of equation (15) to these variables requires the generation of sets
E(s) and F(s). The number of pairs of these sets is calculated using equation (16) as follows:
5
S 
p 2
11!
11! 11! 11! 11!




 1012
p!(11 - p)! 2! 9! 3! 8! 4!7! 5! 6!
(25)
This is a huge number of constraints. However, heuristic simplifications can be used to
reduce the number of binary variables so as to render the problem solvable.
A primary simplification follows from the processing time requirement of Unit 9.
Since the cycle time is pre-specified, the bulk of hoist moves during which one slot of Unit 9
is empty should last exactly 1.67 minutes. This means that the moves between the time the
hoist lifts a barrel from Unit 9 and the time it releases another barrel into the same slot in the
original schedule must be preserved, which is equivalent to fixing the value of z(9,8) and
z(8,7) to 1. Once this is done, the number of free binaries decreases to 9  9 = 81.
In this scheduling problem, Units 2 and 6 are cleaning units. Their processing times
do not have an upper bound and can be as long as possible in order have satisfactory
plating quality. To maintain maximum processing times for these two units, z(2,1) and z(6,5)
are also set to unity. This decreases the number of binaries to 7  7 = 49.
Finally, because the hoist is not allowed to leave Unit 11 during a process in that unit,
z(10,11) also needs to be fixed to 1. This final simplification reduces the number of binaries
to 6  6 = 36, and the number of constraints to 35.
Hoist Schedule Development
When the bi-level optimization algorithm is run, starting from the original schedule as the
initial point, it yields repeating solutions after the fifth iteration. Each step of iteration uses a
new approximation and a new optimal solution is derived based on that particular
approximation. Table 2 lists the iteration points.
An economical analysis of the five schedule candidates shows that the fifth schedule
is the most desirable. Note that the economical analysis only includes Units 3, 4, 5, 7 and
10, since these are the only units with variable processing times. All the other units have the
same processing times and the same cost as in the original case in all iterations.
In performing economical analysis, the unit cost of water (Uf) is set to $0.0195/gal,
based on the prices obtained from a water municipality. This is higher than the cost of only
fresh water because the municipality sews the industry for wastewater discharge when they
purchase fresh water. In addition, the more the volumetric flow rate of water, and
consequently the wastewater, the higher the expense of wastewater treatment.
Unit 4 uses a soak-cleaning chemical at the cost of $12.5/gal. Once the amount of
chemicals lost through drag-out is determined, the amount of unnecessary chemical
consumption can be identified by multiplying it with the chemical concentration.
15
The calculations are based on 300 days of continuous production throughout the
year. The total consumption cost in the original case is $80,609/yr. The optimal hoist
schedule generated in iteration 5 results in $65,794/yr of total consumption cost due to the
reduction in Units 3, 4, 5, 7, and 10. The optimal hoist schedule is displayed in Figure 10.
Table 2. Water and Chemical Consumptions in the
Original Case and Each of the Iterations
Fresh Water
Chemical
Water
Chemical
Total of
Consumption, Consumption, Consumption Consumption Consumption
Wf (gal/min) Wc (gal/min) Cost, Jf ($/min) Cost, Jc ($/min) Costs ($/yr)
Original
8.2
0.0021
69,077
11,533
80,609
1
6
0.0029
50,544
15,725
66,269
2
7.9
0.0027
66,571
14,774
81,346
3
6.1
0.0027
51,408
14,472
65,880
4
7.9
0.0029
66,571
15,466
82,037
5
6.3
0.0024
unit price of water = 0.0195 $ / gal
unit proce of chemicals = 12.5 $ / gal
53,093
12,701
65,794
Iteration #
The case study has shown the potential savings realizable using an optimal hoist
scheduling. The water consumption has been reduced by 23%. The chemical consumption,
however, is increased slightly (10 %).
Nevertheless, this was inevitable since the
processing time of Unit 4 had to be brought within the feasible limits as specified in Table 1.
Overall, the total savings are approximately $15,000/yr with a negligible capital investment.
CONCLUDING REMARKS
Environmentally conscious hoist scheduling has been achieved by a two-step optimization
algorithm developed in this work. In this algorithm, the first optimization step involves the
maximization of production rate, while the second step focuses on the environmental
objectives and operating costs. The algorithm, therefore, enables simultaneous cost
reduction and waste minimization. The industrial case study demonstrates that significant
savings are achievable with the method introduced here.
A realization of wastewater minimization requires detailed information about the
dynamics of the process. In this study, the dynamic information had to be incorporated into
the otherwise static hoist-scheduling problem, which results in a rather tedious dynamic
optimization problem. The introduced method enables solving the problem with a practically
acceptable computational time, and identifying the optimum schedule in terms of both
environmental and economic goals.
Another useful technique in the methodology is the utilization of engineering
heuristics for simplifying the optimization problem. Practical cases in general involve many
limitations that do not appear in ideal cases. Instead of introducing these limitations as
constraints to the optimization, they have been used to simplify the optimization. A method
for grouping of variables applied here has drastically reduced number of constraints.
16
The introduced hoist-scheduling algorithm allows incorporating the dynamic information
into the scheduling problem with a minimum amount of additional computational load.
Therefore, it is possible to handle industrial problems with ease, as it is demonstrated in
the case study.
t (sec)
500
2
5
6
3
400
7
8
9
300
4
W
11
200
W
100
10
1
0
P
9
R
8
R
10
R
5
C
4
R
7
R
3
C
6
C
2
S
11
L
1
Figure 10. Optimal hoist schedule.
ACKNOWLEDGMENT
This work is in part supported by NSF (DMI-0225844).
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18