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A cost evaluation method for dynamic line balancing
JOURNAL
OF OPERATIONS
Vol. 4, No. I, November
MANAGEMENT
1983
A Cost Evaluation Method for Dynamic Line Balancing K.
ROSCOE
DEENA
DAVIS*
D. KUSHNER**
EXECUTIVE
SUMMARY
A procedure is presented for calculating stochastic costs, which include operator (labor) and inventory costs, associated with dynamic line balancing. Dynamic line balancing, unlike the traditional methods of assembly and production line balancing, assigns operators to one or more operations, where each operation has a predetermined processing time and is defined as a group of identical parallel stations. Operator costs and inventory costs are stochastic because they are functions ofthe assignment process employed in balancing the line, which may vary throughout the balancing period, and the required flow rate. Earlier studies focused on the calculation of the required number of stations and demonstrated why the initial and final inventories at the different operations are balanced. The cost minimization method developed in the article can be used to evaluate and compare the assignment of operators to stations for various assignment heuristics. Operator costs and inventory costs are the components of the cost function. The operator costs are based on the operations to which operators are assigned and are calculated for the entire work week regardless of whether an operator is given only a partial assignment which results in idle time. It is assumed that there is no variation in station speeds, no learning curve effect for operators’ performance times, and no limit on the number of operators available for assignment. The costs associated with work-in-process inventories are computed on a “value added” basis. There is no charge for finished goods inventory after the last operation or raw material before the first operation. The conditions which must be examined before using the cost evaluation method are yield, input requirements, operator requirements, scheduling requirements and output requirements. Yield reflects the output of good units at any operation. The input requirement accounts for units discarded or in need of reworking. The operator requirements define the calculation of operator-hours per hour, set the minimum number of operators at an operation, and require that the work is completed. The scheduling requirements ensure that operators are either working or idle at all times, and that no operator is assigned to more than one operation at any time. The calculation of the output reflects the yield, station speed, and work assignments at the last operation on the line. An application of the cost evaluation method is discussed in the final section of the article. Using a simple heuristic to assign operators, the conditions for yield, inputs, operators, scheduling, and output are satisfied. The costs are then calculated for operators and inventories. In conclusion, the cost evaluation method for dynamic balancing enables a manager to compare the costs of assigning operators to work stations. Using this method to calculate the operator and inventory costs, a number of different heuristics for assigning operators in dynamic balancing can be evaluated and compared for various configurations of the production line. The least cost solution procedure then can be applied to a real manufacturing situation with similar characteristics.
* University of Georgia, Athens, Georgia. ** Georgia State University, Atlanta, Georgia.
Journal of Operations
Management
41
INTRODUCTION Dynamic line balancing is unlike the traditional methods of assembly and production line balancing, which can be modeled using the general mathematical formulations proposed by Salveson [5] and Hunt [3]. The focus of dynamic line balancing is on the assignment of operators to one or more operations, where each operation has a predetermined processing time and is defined as a group of identical parallel stations. In dynamic balancing, inventories are balanced at each operation at the beginning and end of the balancing period, which can range in length from a few hours to a week. The typical manufacturing line which can be addressed with dynamic balancing is a series of operations separated by work-in-process or inventory banks, as shown in Figure 1. Davis and Taylor [l] have shown that the resource requirements for each operation in dynamic line balancing can be determined after establishing the amount of output required from the line and the work pace at each operation. In addition, Davis, Taylor, and Kushner [2] have shown how the required number of stations per operation can be calculated and have demonstrated why the initial and final inventories at the different operations are balanced. The actual assignment of operators to stations has not been addressed in prior research. Before a method of assigning operators can be developed, each of the cost factors associated with dynamic balancing must be identified and evaluated. The objective of this paper therefore is to analyze the cost component of dynamic balancing, and to develop a cost minimization method. The costing method can be used to evaluate and compare the assignment of operators to stations for various assignment heuristics. Following the definition of the cost components and mathematical terms, the conditions which must be met in order to apply the cost minimization method are presented. In the last section of the paper, an example of dynamic balancing is given and used to illustrate the application of the costing method. COST COMPONENTS As has been indicated, operator costs and inventory costs are the two components that make up the cost function for the dynamic balancing problem. All the details of the cost FIGURE Production
-+j
1
Line for Dynamic
...
Balancing
qJ+~”
s,(I-y,) Legend:
Pn
units of work-in-process
SI
station speed
Yi t%
42
S#- Y,, at operation
at operation
yield expressed
as good units/all
hours of work required
i at time t
i, expressed
as units/operator-hour
units input
at operation
i, expressed
as operator-hours/hour.
APICS
factors will not be discussed since they are described in a previous study by Kushner [4], but an overview is necessary. Operator Costs
The direct costs in dynamic balancing are those associated with the workers. These costs are based on the operations to which operators are assigned. If an operator is assigned to only one operation and that operation has a low wage rate, the operator’s labor cost is accordingly low. If an operator can be assigned to more than one operation, then each operator’s labor cost is based on the highest wage rate for any operation in the group of operations at which the operator works. No operator is paid at a rate higher than that for any individual operation worked by that operator, regardless of the number of operations worked. The effect of this policy is to discourage combinations of operations which have widely disparate wage rates. There is no direct or indirect charge for transfers affecting either the wage rate or the amount of time spent working. The operators are paid for a full day of eight hours, regardless of the fractional parts of a day worked. Thus, an assignment heuristic which does not permit the full utilization of the operators’ time, sometimes results in higher labor costs. Under a few sets of circumstances the operators’ assignments determined by the heuristics may be revised. If at a particular operation there is less than one hour of work to be completed by the next operator, the hours are reassigned such that neither operator works at the operation for less than one hour. The first operator at the operation is granted some idle time so that the second operator, who is assigned to complete the operation, has one hour of work at the operation before moving to another operation. If the revision results in the first operator having less than one hour of work at the operation, the assignment is further revised to remove all work obligations for the first operator at the operation in question, leaving the second operator solely responsible for completing the required work. This restriction is imposed to make the assignments conform more closely to reality. For similar reasons, an operator with less than one hour of idle time will not be assigned to work a fraction of an hour at another operation. The remaining assumptions regarding labor, such as the deterministic nature of the model, learning curve effect, groupings of operations, and adjacency requirements, are made in order to avoid complications which might mask the efficiency of the heuristic used to assign the operators. It is assumed that there is no variation in station speeds, no learning curve effect for operators’ performance times, and no limit on the number of operators available for assignment. It is also assumed that in the actual assignment of operators to operations there are no prespecified groupings of operations, such as requiring that an operator working at one operation must use any extra time at a predetermined second operation. Similarly, it is assumed that the operations to which an operator is assigned have to be neither adjacent nor separated by one or more operations. Inventory Costs
Costs associated with work-in-process inventories are computed on a “value added” basis. This means that a unit of work-in-process at an operation near the start of the production process costs less than a unit at an operation further down the line. There is no charge for finished goods inventory after the last operation or raw material before the first operation. Because it is assumed that the operators always work at the expected station speed until the work is completed, regardless of whether they will be idle afterward, the
Journal of Operations Management
43
Yield merely states that there can not be more than at any operation, i. Input
100 percent
of good units
of output
Requirements -
I
T
T
Yi s 0
Pi+l,&
=
s0
for all i, i = 1, . . . , n.
Pitdt
For every operation i, i = 1, . . . , n, the work-in-process at the i + 1 operation integrated over the T hours in the balancing period consists only of the good units produced at the ith operation. To account for the units discarded or in need of reworking after operation i, the input to operation i + 1 is divided by the yield at operation i. Using equation (2) and the yields at each operation, the amount of input required at the ith operation can be computed, given that the units of output are specified. Operator
Requirements
for all i, i = i, . . . , n,
(3)
n
C Pi
=
r,
(4)
i=l T M-m o
C
s
uijtdt 2 0
foralli,i=
l,...,
n,
(5)
= 1, . . . . n.
(6)
j=l T M-m
TUi +
o s
C uijtdt = T/3i j=l
foralli,i
Equation (3) is used to calculate the number of operator hours per hour, pi, required atoperationi,i= l,..., n. The numerator is the work-in-process at operation i which, when divided by the T hours in the balancing period, becomes the average input speed to the operation, expressed as units per hour. When divided by the station speed, Si, expressed as units per operator hour, the result is the number of operator hours required at that operation for each hour of clock time in the balancing period. A similar equation for calculating the number of workers is given in Davis, Taylor, and Kushner [2]. The total number of operator hours required, I’, is computed using equation (4). Inequality (5) specifies that over the time period [0, T] there could be no fewer than the number of full-time operators at any operation i, i = 1, . . . , n. By setting the limit of the summation over operators, j, to only partial operators, M - m, the condition becomes a non-negativity constraint on the operators’ assignments. Equation (6) ensures that the operators’ assignments fulfill the work requirements at each operation. The number of full-time operators at operation i, Ui, is multiplied by the hours in the balancing period, T, and the number of partial operators is integrated over the interval [0, T]. The result is the total number of operator-hours worked during the balancing period.
Journal of Operations Management
45
Scheduling Requirements
s Uijt +
oT ,% (uijt + vijJdt = T
forallj,j=
l,...,
M-m,
(7)
1-l
1
Ll?jt I
for all i, j, and t, i = 1, . . . , n, j = 1, . . . , M - m, t E [0, T], where i’ # i,
Uijt +
Vijt =
(8) (9)
for all i, j, and t, i = 1, . . . , n, j = 1, . . . , M - m, t E [0, T].
1
Equation (7) ensures that operator j, j = 1, . . . , M - m, is either working or idle at one or more operations i, i = 1, . . . , n, during the entire time period of the length T. To prevent a partial operator j, 1, . . . , M - m, from working at any time t, t E [0, T] at more than one operation, scheduling requirement (8) states that for any pair of operations i and i’, where i’ # i, operator j can work no more than one operation at any time during the balancing period. Equation (9) ensures that partial operator j, j = 1, . . . , M - m, is either working or idle at some operation i, i = 1, . . . , n at any time t during the balancing period interval [0, T].
j=
Output Requirements Y”%[TU”
+ iT>!
u,tdt]
= TQ-
(10)
The output of the manufacturing process depends on the number of good units produced by the last operation on the line. To compute only the good units of output, the yield is multiplied by the station speed at the last operation, y&. The terms in the brackets, T M-m
TU, [
+ s
o
C j=l
u”j,dt
give the 12
number
of operator
hours in the balancing
period.
The
number of operators working at the last operation, u”jt, is integrated over the hours in the balancing period because it is a function of time. The number of full-time operators at the last operation, U, , are multiplied by the hours in the balancing period because the operators produce the output during the entire period of length T. COST
MINIMIZATION
To develop follows: R = ratio Wi = base Wij = wage ...)
FUNCTION
the cost minimization
function,
additional
terms are required.
of inventory to operator costs wage rate at operation i, i = 1, . . . , n, rate for operator j at operation i; wij 2 Wi; j = 1, . . . , M - m, i = 1, n.
The cost function includes only the cost of partial operators. Therefore, the total operator cost associated with all operators who do not work operation. Mathematically this is n M-m X = 2 2 Wij.
46
These are as
X is defined as full-time at an
(11)
APES
The M - m term is the number of operators who work at more than one operation during the entire balancing period. The m workers who do work full-time at a single operation during the balancing period are excluded from the cost calculations because they are not assigned using heuristics. The costs associated with these m operators are the same regardless of how each operator is assigned. The derivation of the inventory cost of all work-in-process inventories, Y, is more complex than that of the labor cost, X. The total cost of work-in-process inventory is defined by
s T
Y=R
”
2 Pi,( & Wc)dt, i’ < i. i’=] O i-1
(12)
The environmental effect of the ratio of inventory to operator costs, is included in the calculation of Y by treating R as a multiplier. The inventory term, Pi,, is summed over alli,i= l,..., n, to compute the costs associated with each operation’s work-in-process. Because the amount of work-in-process at each operation changes with the number of operators working at the preceding and succeeding operations, Pi, is integrated over the time in the balancing period, t E [0, T]. Finally, the inventory at each operation at a given point in time is multiplied by the sum of the base wage rates, WC, i’ < i, for all operations preceding the operation of interest, i. Then x+y is minimized where X is the total operator cost associated with all operators work full-time at one operation and Y is the inventory cost of all work-in-process
who do not inventories.
EXAMPLE The cost of assigning section. Let n ; w, s, YI
R
operators
on an example
production
line is developed
in this
ZZ 2 operations = 8 hours = 400 units/hour = $5.00 and WZ = $4.00 = 100 and S2 = 150 units/operator/hour = 0.98 and y2 = 0.95 good units output/units = 0.001.
input
Note that condition (1) is satisfied because yi and y2 are both less than 1.00. The average hourly input requirement for each operation is calculated using equation (2) beginning with the second operation, where P3,, = Q = 400 units/hour. P 2,i = P3,,/y2 = 40010.95 P 1.1 = P2,,/y, = 42110.98
= 421 = 430.
On the average, during each hour of the eight hour balancing period, 430 units should be in the WIP bank preceding operation 1 and 421 units should be in the bank between operations 1 and 2, if the average hourly output is to satisfy the requirement of 400 units.
Journal of Operations
Management
47
The number of operator hours required at operations 1 and 2 during balancing period can be calculated using definition (3) as follows: pr = P&S, = 430/100 f12 = P2,,/S2 = 421/150 By equation
each hour of the
= 4.300 = 2.807
(4) r = ~ pi = 4.300 + 2.807 = 7.107
which means on the during the period, 7.107 hours must completed for hour of time. The requirements for operation can computed using (5) and The integer of p, & yield and 2 respectively, required operations 1 2 for entire balancing The fractional when multiplied the eight in the period give hours of per period a partial at each Using the
s 8 s
T M-m
o
C
uijtdt = TPi - TUi,
J=l
the calculation
for the first operation is M-m urjtdt = 8 X (4.3 - 4.0) = 2.4, 0
and for the second
= j=l
operation, 8 M-m s0
uytdt = 8 X (2.807 - 2.000) = 6.456. = j=l
A total of 8.856 operator hours are required at the two operations during the balancing period. Because more than eight operator hours are required, one operator will be insufficient to complete the work. Consequently, two partial operators are used. After the operator requirements are calculated, they are filled by assigning the appropriate number of operators to work the length of time required at each operation. The least complex assignment method that can be used, for purposes of explanation, is to assign four full-time operators and one partial operator to the first operation and two full-time operators and one partial operator to the second operation. Each full-time operator would work eight hours while the partial operators would work 2.4 and 6.456 hours, respectively, spending the remaining 5.6 and 1.544 hours idle at their respective operations. This satisfies the scheduling condition (7). Because neither partial operator j, j = 1, 2, works at more than one operation, ui5t is undefined while ui,r,t and ui,Z,t are 0 or 1, satisfying condition (8). Finally, condition (9) is satisfied because the partial operators are not working and idle at the same time. To compute the cost efficiency of the assignment of the operators in the example, only costs associated with the partial operators are included. The work-in-process at the operations is calculated during the eight hours of the balancing period and then the inventory costs
48
APES
and labor costs are summed. The wage rates for the partial operators, w,,r and w,,~, are equal to the base wage rate at the operation to which each is assigned because neither partial operator works at more than one operation. Accordingly, WI.1 = W, = $5.00 per hour w2,2 = W2 = $4.00 per hour and by equation
(11) the total hourly X = i i i-1
wage rate for partial
operators
is
wij = 5.00 + 4.00 = 9.00.
j-1
The inventory costs are based on the operators’ work schedules. It is assumed that the operators work at the expected pace of the station speed for the required length of time before becoming idle for the remainder of the eight hours in the balancing period. This assumption separates the inventory calculations into three time periods: (1) both partial operators are working, (2) one partial operator is idle while the other continues to work, and (3) both are idle. The first time period is from 0.0 to 2.4 hours; the second is from 2.4 to 6.456 hours; and the third is from 6.456 to 8.0 hours. To use equation ( 12), the units of work-in-process must be known. This can be computed by employing simulation. First, the initial work-in-process levels at the operations are set at 9900 units, a level high enough to prevent the first operation from being starved or blocked and the second operation from being starved. As the calculations for the first simulation show, Table 1, the initial inventory levels are preset at 9900 and the ending levels reflect the production rate required to meet a 400 unit per hour requirement. The fluctuations reflect the flow rate of units into or out of the WIP banks. The work-in-process at the first operation initially exhibits a negative flow rate of 500 units an hour because four full-time operators and one partial operator are each working at a station speed of 100 units. As long as the partial operator is not transferred, the five operators take 500 units from the first WIP bank each hour they work. After 2.4 hours the partial operator stops working at the first operation. Consequently the flow rate rises to -400 units for each of the remaining 5.6 hours in the balancing period. The inventory levels reached during the first simulation are used to compute the initial WIP required for the second simulation in which inventory costs are calculated. To calculate the “Adjusted Initial WIP” in the second simulation shown in Table 1, the difference between the highest and lowest work-in-process levels for the heuristic in the first simulation is multiplied by 105 percent. At the first operation the low “Ending WIP” of 6460 is subtracted from the high “Initial WIP” of 9900 yielding 3440, which, when multiplied by 105 percent, gives an “Adjusted Initial WIP” of 36 12. The flow rates at each operation are the same in the second simulation as they were in the first. To calculate the inventory costs at the first operation, the WIP levels are computed at half-hour intervals starting with the adjusted initial WIP shown in Table 1. Using the flow rate of -500 units an hour, 250 units are subtracted from 3612 to give 3362 units after half an hour of production. Multiplying by the ratio of inventory to operator costs, R = 0.001, at time 0.5 the inventory costs $3.362. The standard half hour interval for computing inventory costs is supplemented by the times when operators change assignments. In Table 2 changes occur after 2.4 and 6.456 hours.
Journal of Operations Management
49
TABLE 1 Inventory Fluctuation for the Example Operation Number
Operation Number
First Simulation Initial WIP Flow rate (units/ hour) X hours worked (-500 X 2.4) Flow rate X hours worked (-400 X 5.6) Ending
WIP
Initial WIP Flow rate X hours worked (((.98 X 500) - 450) X 2.4 = 40 X 2.4) WIP at 2.4 hours Flow rate X hours worked (((.98 X 400) - 450) X 4.056 = -58 X 4.056) WIP at 6.456 hours Flow rate X hours worked (((.98 X 400) - 300) x 1.544 = 92 x 1.544) Ending WIP
Second
Simulation
9900
-1200
_2240
Adjusted Initial WIP (1.05 x (99006460)) Flow WIP at 2.4 hours Flow Ending
3612
-1200 2412 -2240
WIP
172
6460 9900
96 9996
Adjusted Initial WIP Flow WIP at 2.4 hours FIOW WIP at 6.456 hours Flow Ending WIP
247 96 343
-235.248 107.752 142.048 249.8
~235.248 9760.752
142.048 9902.8
The average total cost is calculated by summing the operator shown in equation (13). For the example problem this is
and inventory
costs, as
$9.00 + 43.34 = $52.34. The total cost from the example problem can be computed to the same problem, as in the study by Kushner [4].
for other heuristic
solutions
SUMMARY The cost evaluation method for dynamic balancing enables a manager to compare the costs of assigning operators to work stations. Using this method, the operator and inventory costs can be calculated prior to actually assigning operators to the stations. Consequently, a number of different heuristics for assigning operators in dynamic balancing can be evaluated
50
APES
Calculation Time Elapsed 0.0 0.5 1.0 1.5 2.0 2.4 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.456 6.5 7.0 7.5 8.0
WIP Bank
of Inventory
TABLE 2 Costs for the Example Problem
1
3612 3362 3112 2862 2612 2412 2372 2172 1972 1771 1572 1372 1172 972 772 572 372 172
Total
cost
WIP Bank 2
-O3.362 3.112 2.862 2.612 2.412 2.372 2.172 1.972 1.772 I .572 1.372 1.172 0.972 0.772 0.572 0.372 0.172
247 267 287 307 327 343 337.2 308.2 279.2 250.2 221.2 192.2 163.2 134.2 107.752 111.8 157.8 203.8 249.8
29.62
cost -O0.9345 1.0045 1.0745 1.1445 1.0290 0.5058 1.0787 0.9772 0.8757 0.7742 0.6727 0.5712 0.4697 0.3534 0.1364 0.5523 0.7 133 0.8743 13.72
and compared for various configurations of the production line. The least cost solution procedure then can be applied to a real manufacturing situation with similar characteristics. REFERENCES 1. Davis, K. R. and B. W. Taylor, III, “A Heuristic Procedure for Determining In-Process Inventories,” Decision Sciences, Vol. 9, No. 3, (July 1978), pp. 452-466. 2. Davis, K. R., D. D. Kushner, and B. W. Taylor, III, “Production Line Balancing: A Dynamic Systems Approach,” Decision Sciences, submitted (1983). 3. Hunt, G. C., “Sequential
Journal of Operations
Arrays of Waiting
Management
Lines,”
Operations Research, Vol. 4, No. 6 (December 1956) pp. 674-683. 4. Kushner, D. D., “Dynamic Line Balancing: A Cost Analysis of Four Operator Assignment Heuristics,” Ph.D. dissertation, University of Georgia, September, 1980. 5. Salveson, M. E., “The Assembly-Line Balancing Problem,” Transactions of the ASME, Vol. 77, No. 6, (August 1955) pp. 939-948.
51
OF OPERATIONS
Vol. 4, No. I, November
MANAGEMENT
1983
A Cost Evaluation Method for Dynamic Line Balancing K.
ROSCOE
DEENA
DAVIS*
D. KUSHNER**
EXECUTIVE
SUMMARY
A procedure is presented for calculating stochastic costs, which include operator (labor) and inventory costs, associated with dynamic line balancing. Dynamic line balancing, unlike the traditional methods of assembly and production line balancing, assigns operators to one or more operations, where each operation has a predetermined processing time and is defined as a group of identical parallel stations. Operator costs and inventory costs are stochastic because they are functions ofthe assignment process employed in balancing the line, which may vary throughout the balancing period, and the required flow rate. Earlier studies focused on the calculation of the required number of stations and demonstrated why the initial and final inventories at the different operations are balanced. The cost minimization method developed in the article can be used to evaluate and compare the assignment of operators to stations for various assignment heuristics. Operator costs and inventory costs are the components of the cost function. The operator costs are based on the operations to which operators are assigned and are calculated for the entire work week regardless of whether an operator is given only a partial assignment which results in idle time. It is assumed that there is no variation in station speeds, no learning curve effect for operators’ performance times, and no limit on the number of operators available for assignment. The costs associated with work-in-process inventories are computed on a “value added” basis. There is no charge for finished goods inventory after the last operation or raw material before the first operation. The conditions which must be examined before using the cost evaluation method are yield, input requirements, operator requirements, scheduling requirements and output requirements. Yield reflects the output of good units at any operation. The input requirement accounts for units discarded or in need of reworking. The operator requirements define the calculation of operator-hours per hour, set the minimum number of operators at an operation, and require that the work is completed. The scheduling requirements ensure that operators are either working or idle at all times, and that no operator is assigned to more than one operation at any time. The calculation of the output reflects the yield, station speed, and work assignments at the last operation on the line. An application of the cost evaluation method is discussed in the final section of the article. Using a simple heuristic to assign operators, the conditions for yield, inputs, operators, scheduling, and output are satisfied. The costs are then calculated for operators and inventories. In conclusion, the cost evaluation method for dynamic balancing enables a manager to compare the costs of assigning operators to work stations. Using this method to calculate the operator and inventory costs, a number of different heuristics for assigning operators in dynamic balancing can be evaluated and compared for various configurations of the production line. The least cost solution procedure then can be applied to a real manufacturing situation with similar characteristics.
* University of Georgia, Athens, Georgia. ** Georgia State University, Atlanta, Georgia.
Journal of Operations
Management
41
INTRODUCTION Dynamic line balancing is unlike the traditional methods of assembly and production line balancing, which can be modeled using the general mathematical formulations proposed by Salveson [5] and Hunt [3]. The focus of dynamic line balancing is on the assignment of operators to one or more operations, where each operation has a predetermined processing time and is defined as a group of identical parallel stations. In dynamic balancing, inventories are balanced at each operation at the beginning and end of the balancing period, which can range in length from a few hours to a week. The typical manufacturing line which can be addressed with dynamic balancing is a series of operations separated by work-in-process or inventory banks, as shown in Figure 1. Davis and Taylor [l] have shown that the resource requirements for each operation in dynamic line balancing can be determined after establishing the amount of output required from the line and the work pace at each operation. In addition, Davis, Taylor, and Kushner [2] have shown how the required number of stations per operation can be calculated and have demonstrated why the initial and final inventories at the different operations are balanced. The actual assignment of operators to stations has not been addressed in prior research. Before a method of assigning operators can be developed, each of the cost factors associated with dynamic balancing must be identified and evaluated. The objective of this paper therefore is to analyze the cost component of dynamic balancing, and to develop a cost minimization method. The costing method can be used to evaluate and compare the assignment of operators to stations for various assignment heuristics. Following the definition of the cost components and mathematical terms, the conditions which must be met in order to apply the cost minimization method are presented. In the last section of the paper, an example of dynamic balancing is given and used to illustrate the application of the costing method. COST COMPONENTS As has been indicated, operator costs and inventory costs are the two components that make up the cost function for the dynamic balancing problem. All the details of the cost FIGURE Production
-+j
1
Line for Dynamic
...
Balancing
qJ+~”
s,(I-y,) Legend:
Pn
units of work-in-process
SI
station speed
Yi t%
42
S#- Y,, at operation
at operation
yield expressed
as good units/all
hours of work required
i at time t
i, expressed
as units/operator-hour
units input
at operation
i, expressed
as operator-hours/hour.
APICS
factors will not be discussed since they are described in a previous study by Kushner [4], but an overview is necessary. Operator Costs
The direct costs in dynamic balancing are those associated with the workers. These costs are based on the operations to which operators are assigned. If an operator is assigned to only one operation and that operation has a low wage rate, the operator’s labor cost is accordingly low. If an operator can be assigned to more than one operation, then each operator’s labor cost is based on the highest wage rate for any operation in the group of operations at which the operator works. No operator is paid at a rate higher than that for any individual operation worked by that operator, regardless of the number of operations worked. The effect of this policy is to discourage combinations of operations which have widely disparate wage rates. There is no direct or indirect charge for transfers affecting either the wage rate or the amount of time spent working. The operators are paid for a full day of eight hours, regardless of the fractional parts of a day worked. Thus, an assignment heuristic which does not permit the full utilization of the operators’ time, sometimes results in higher labor costs. Under a few sets of circumstances the operators’ assignments determined by the heuristics may be revised. If at a particular operation there is less than one hour of work to be completed by the next operator, the hours are reassigned such that neither operator works at the operation for less than one hour. The first operator at the operation is granted some idle time so that the second operator, who is assigned to complete the operation, has one hour of work at the operation before moving to another operation. If the revision results in the first operator having less than one hour of work at the operation, the assignment is further revised to remove all work obligations for the first operator at the operation in question, leaving the second operator solely responsible for completing the required work. This restriction is imposed to make the assignments conform more closely to reality. For similar reasons, an operator with less than one hour of idle time will not be assigned to work a fraction of an hour at another operation. The remaining assumptions regarding labor, such as the deterministic nature of the model, learning curve effect, groupings of operations, and adjacency requirements, are made in order to avoid complications which might mask the efficiency of the heuristic used to assign the operators. It is assumed that there is no variation in station speeds, no learning curve effect for operators’ performance times, and no limit on the number of operators available for assignment. It is also assumed that in the actual assignment of operators to operations there are no prespecified groupings of operations, such as requiring that an operator working at one operation must use any extra time at a predetermined second operation. Similarly, it is assumed that the operations to which an operator is assigned have to be neither adjacent nor separated by one or more operations. Inventory Costs
Costs associated with work-in-process inventories are computed on a “value added” basis. This means that a unit of work-in-process at an operation near the start of the production process costs less than a unit at an operation further down the line. There is no charge for finished goods inventory after the last operation or raw material before the first operation. Because it is assumed that the operators always work at the expected station speed until the work is completed, regardless of whether they will be idle afterward, the
Journal of Operations Management
43
Yield merely states that there can not be more than at any operation, i. Input
100 percent
of good units
of output
Requirements -
I
T
T
Yi s 0
Pi+l,&
=
s0
for all i, i = 1, . . . , n.
Pitdt
For every operation i, i = 1, . . . , n, the work-in-process at the i + 1 operation integrated over the T hours in the balancing period consists only of the good units produced at the ith operation. To account for the units discarded or in need of reworking after operation i, the input to operation i + 1 is divided by the yield at operation i. Using equation (2) and the yields at each operation, the amount of input required at the ith operation can be computed, given that the units of output are specified. Operator
Requirements
for all i, i = i, . . . , n,
(3)
n
C Pi
=
r,
(4)
i=l T M-m o
C
s
uijtdt 2 0
foralli,i=
l,...,
n,
(5)
= 1, . . . . n.
(6)
j=l T M-m
TUi +
o s
C uijtdt = T/3i j=l
foralli,i
Equation (3) is used to calculate the number of operator hours per hour, pi, required atoperationi,i= l,..., n. The numerator is the work-in-process at operation i which, when divided by the T hours in the balancing period, becomes the average input speed to the operation, expressed as units per hour. When divided by the station speed, Si, expressed as units per operator hour, the result is the number of operator hours required at that operation for each hour of clock time in the balancing period. A similar equation for calculating the number of workers is given in Davis, Taylor, and Kushner [2]. The total number of operator hours required, I’, is computed using equation (4). Inequality (5) specifies that over the time period [0, T] there could be no fewer than the number of full-time operators at any operation i, i = 1, . . . , n. By setting the limit of the summation over operators, j, to only partial operators, M - m, the condition becomes a non-negativity constraint on the operators’ assignments. Equation (6) ensures that the operators’ assignments fulfill the work requirements at each operation. The number of full-time operators at operation i, Ui, is multiplied by the hours in the balancing period, T, and the number of partial operators is integrated over the interval [0, T]. The result is the total number of operator-hours worked during the balancing period.
Journal of Operations Management
45
Scheduling Requirements
s Uijt +
oT ,% (uijt + vijJdt = T
forallj,j=
l,...,
M-m,
(7)
1-l
1
Ll?jt I
for all i, j, and t, i = 1, . . . , n, j = 1, . . . , M - m, t E [0, T], where i’ # i,
Uijt +
Vijt =
(8) (9)
for all i, j, and t, i = 1, . . . , n, j = 1, . . . , M - m, t E [0, T].
1
Equation (7) ensures that operator j, j = 1, . . . , M - m, is either working or idle at one or more operations i, i = 1, . . . , n, during the entire time period of the length T. To prevent a partial operator j, 1, . . . , M - m, from working at any time t, t E [0, T] at more than one operation, scheduling requirement (8) states that for any pair of operations i and i’, where i’ # i, operator j can work no more than one operation at any time during the balancing period. Equation (9) ensures that partial operator j, j = 1, . . . , M - m, is either working or idle at some operation i, i = 1, . . . , n at any time t during the balancing period interval [0, T].
j=
Output Requirements Y”%[TU”
+ iT>!
u,tdt]
= TQ-
(10)
The output of the manufacturing process depends on the number of good units produced by the last operation on the line. To compute only the good units of output, the yield is multiplied by the station speed at the last operation, y&. The terms in the brackets, T M-m
TU, [
+ s
o
C j=l
u”j,dt
give the 12
number
of operator
hours in the balancing
period.
The
number of operators working at the last operation, u”jt, is integrated over the hours in the balancing period because it is a function of time. The number of full-time operators at the last operation, U, , are multiplied by the hours in the balancing period because the operators produce the output during the entire period of length T. COST
MINIMIZATION
To develop follows: R = ratio Wi = base Wij = wage ...)
FUNCTION
the cost minimization
function,
additional
terms are required.
of inventory to operator costs wage rate at operation i, i = 1, . . . , n, rate for operator j at operation i; wij 2 Wi; j = 1, . . . , M - m, i = 1, n.
The cost function includes only the cost of partial operators. Therefore, the total operator cost associated with all operators who do not work operation. Mathematically this is n M-m X = 2 2 Wij.
46
These are as
X is defined as full-time at an
(11)
APES
The M - m term is the number of operators who work at more than one operation during the entire balancing period. The m workers who do work full-time at a single operation during the balancing period are excluded from the cost calculations because they are not assigned using heuristics. The costs associated with these m operators are the same regardless of how each operator is assigned. The derivation of the inventory cost of all work-in-process inventories, Y, is more complex than that of the labor cost, X. The total cost of work-in-process inventory is defined by
s T
Y=R
”
2 Pi,( & Wc)dt, i’ < i. i’=] O i-1
(12)
The environmental effect of the ratio of inventory to operator costs, is included in the calculation of Y by treating R as a multiplier. The inventory term, Pi,, is summed over alli,i= l,..., n, to compute the costs associated with each operation’s work-in-process. Because the amount of work-in-process at each operation changes with the number of operators working at the preceding and succeeding operations, Pi, is integrated over the time in the balancing period, t E [0, T]. Finally, the inventory at each operation at a given point in time is multiplied by the sum of the base wage rates, WC, i’ < i, for all operations preceding the operation of interest, i. Then x+y is minimized where X is the total operator cost associated with all operators work full-time at one operation and Y is the inventory cost of all work-in-process
who do not inventories.
EXAMPLE The cost of assigning section. Let n ; w, s, YI
R
operators
on an example
production
line is developed
in this
ZZ 2 operations = 8 hours = 400 units/hour = $5.00 and WZ = $4.00 = 100 and S2 = 150 units/operator/hour = 0.98 and y2 = 0.95 good units output/units = 0.001.
input
Note that condition (1) is satisfied because yi and y2 are both less than 1.00. The average hourly input requirement for each operation is calculated using equation (2) beginning with the second operation, where P3,, = Q = 400 units/hour. P 2,i = P3,,/y2 = 40010.95 P 1.1 = P2,,/y, = 42110.98
= 421 = 430.
On the average, during each hour of the eight hour balancing period, 430 units should be in the WIP bank preceding operation 1 and 421 units should be in the bank between operations 1 and 2, if the average hourly output is to satisfy the requirement of 400 units.
Journal of Operations
Management
47
The number of operator hours required at operations 1 and 2 during balancing period can be calculated using definition (3) as follows: pr = P&S, = 430/100 f12 = P2,,/S2 = 421/150 By equation
each hour of the
= 4.300 = 2.807
(4) r = ~ pi = 4.300 + 2.807 = 7.107
which means on the during the period, 7.107 hours must completed for hour of time. The requirements for operation can computed using (5) and The integer of p, & yield and 2 respectively, required operations 1 2 for entire balancing The fractional when multiplied the eight in the period give hours of per period a partial at each Using the
s 8 s
T M-m
o
C
uijtdt = TPi - TUi,
J=l
the calculation
for the first operation is M-m urjtdt = 8 X (4.3 - 4.0) = 2.4, 0
and for the second
= j=l
operation, 8 M-m s0
uytdt = 8 X (2.807 - 2.000) = 6.456. = j=l
A total of 8.856 operator hours are required at the two operations during the balancing period. Because more than eight operator hours are required, one operator will be insufficient to complete the work. Consequently, two partial operators are used. After the operator requirements are calculated, they are filled by assigning the appropriate number of operators to work the length of time required at each operation. The least complex assignment method that can be used, for purposes of explanation, is to assign four full-time operators and one partial operator to the first operation and two full-time operators and one partial operator to the second operation. Each full-time operator would work eight hours while the partial operators would work 2.4 and 6.456 hours, respectively, spending the remaining 5.6 and 1.544 hours idle at their respective operations. This satisfies the scheduling condition (7). Because neither partial operator j, j = 1, 2, works at more than one operation, ui5t is undefined while ui,r,t and ui,Z,t are 0 or 1, satisfying condition (8). Finally, condition (9) is satisfied because the partial operators are not working and idle at the same time. To compute the cost efficiency of the assignment of the operators in the example, only costs associated with the partial operators are included. The work-in-process at the operations is calculated during the eight hours of the balancing period and then the inventory costs
48
APES
and labor costs are summed. The wage rates for the partial operators, w,,r and w,,~, are equal to the base wage rate at the operation to which each is assigned because neither partial operator works at more than one operation. Accordingly, WI.1 = W, = $5.00 per hour w2,2 = W2 = $4.00 per hour and by equation
(11) the total hourly X = i i i-1
wage rate for partial
operators
is
wij = 5.00 + 4.00 = 9.00.
j-1
The inventory costs are based on the operators’ work schedules. It is assumed that the operators work at the expected pace of the station speed for the required length of time before becoming idle for the remainder of the eight hours in the balancing period. This assumption separates the inventory calculations into three time periods: (1) both partial operators are working, (2) one partial operator is idle while the other continues to work, and (3) both are idle. The first time period is from 0.0 to 2.4 hours; the second is from 2.4 to 6.456 hours; and the third is from 6.456 to 8.0 hours. To use equation ( 12), the units of work-in-process must be known. This can be computed by employing simulation. First, the initial work-in-process levels at the operations are set at 9900 units, a level high enough to prevent the first operation from being starved or blocked and the second operation from being starved. As the calculations for the first simulation show, Table 1, the initial inventory levels are preset at 9900 and the ending levels reflect the production rate required to meet a 400 unit per hour requirement. The fluctuations reflect the flow rate of units into or out of the WIP banks. The work-in-process at the first operation initially exhibits a negative flow rate of 500 units an hour because four full-time operators and one partial operator are each working at a station speed of 100 units. As long as the partial operator is not transferred, the five operators take 500 units from the first WIP bank each hour they work. After 2.4 hours the partial operator stops working at the first operation. Consequently the flow rate rises to -400 units for each of the remaining 5.6 hours in the balancing period. The inventory levels reached during the first simulation are used to compute the initial WIP required for the second simulation in which inventory costs are calculated. To calculate the “Adjusted Initial WIP” in the second simulation shown in Table 1, the difference between the highest and lowest work-in-process levels for the heuristic in the first simulation is multiplied by 105 percent. At the first operation the low “Ending WIP” of 6460 is subtracted from the high “Initial WIP” of 9900 yielding 3440, which, when multiplied by 105 percent, gives an “Adjusted Initial WIP” of 36 12. The flow rates at each operation are the same in the second simulation as they were in the first. To calculate the inventory costs at the first operation, the WIP levels are computed at half-hour intervals starting with the adjusted initial WIP shown in Table 1. Using the flow rate of -500 units an hour, 250 units are subtracted from 3612 to give 3362 units after half an hour of production. Multiplying by the ratio of inventory to operator costs, R = 0.001, at time 0.5 the inventory costs $3.362. The standard half hour interval for computing inventory costs is supplemented by the times when operators change assignments. In Table 2 changes occur after 2.4 and 6.456 hours.
Journal of Operations Management
49
TABLE 1 Inventory Fluctuation for the Example Operation Number
Operation Number
First Simulation Initial WIP Flow rate (units/ hour) X hours worked (-500 X 2.4) Flow rate X hours worked (-400 X 5.6) Ending
WIP
Initial WIP Flow rate X hours worked (((.98 X 500) - 450) X 2.4 = 40 X 2.4) WIP at 2.4 hours Flow rate X hours worked (((.98 X 400) - 450) X 4.056 = -58 X 4.056) WIP at 6.456 hours Flow rate X hours worked (((.98 X 400) - 300) x 1.544 = 92 x 1.544) Ending WIP
Second
Simulation
9900
-1200
_2240
Adjusted Initial WIP (1.05 x (99006460)) Flow WIP at 2.4 hours Flow Ending
3612
-1200 2412 -2240
WIP
172
6460 9900
96 9996
Adjusted Initial WIP Flow WIP at 2.4 hours FIOW WIP at 6.456 hours Flow Ending WIP
247 96 343
-235.248 107.752 142.048 249.8
~235.248 9760.752
142.048 9902.8
The average total cost is calculated by summing the operator shown in equation (13). For the example problem this is
and inventory
costs, as
$9.00 + 43.34 = $52.34. The total cost from the example problem can be computed to the same problem, as in the study by Kushner [4].
for other heuristic
solutions
SUMMARY The cost evaluation method for dynamic balancing enables a manager to compare the costs of assigning operators to work stations. Using this method, the operator and inventory costs can be calculated prior to actually assigning operators to the stations. Consequently, a number of different heuristics for assigning operators in dynamic balancing can be evaluated
50
APES
Calculation Time Elapsed 0.0 0.5 1.0 1.5 2.0 2.4 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.456 6.5 7.0 7.5 8.0
WIP Bank
of Inventory
TABLE 2 Costs for the Example Problem
1
3612 3362 3112 2862 2612 2412 2372 2172 1972 1771 1572 1372 1172 972 772 572 372 172
Total
cost
WIP Bank 2
-O3.362 3.112 2.862 2.612 2.412 2.372 2.172 1.972 1.772 I .572 1.372 1.172 0.972 0.772 0.572 0.372 0.172
247 267 287 307 327 343 337.2 308.2 279.2 250.2 221.2 192.2 163.2 134.2 107.752 111.8 157.8 203.8 249.8
29.62
cost -O0.9345 1.0045 1.0745 1.1445 1.0290 0.5058 1.0787 0.9772 0.8757 0.7742 0.6727 0.5712 0.4697 0.3534 0.1364 0.5523 0.7 133 0.8743 13.72
and compared for various configurations of the production line. The least cost solution procedure then can be applied to a real manufacturing situation with similar characteristics. REFERENCES 1. Davis, K. R. and B. W. Taylor, III, “A Heuristic Procedure for Determining In-Process Inventories,” Decision Sciences, Vol. 9, No. 3, (July 1978), pp. 452-466. 2. Davis, K. R., D. D. Kushner, and B. W. Taylor, III, “Production Line Balancing: A Dynamic Systems Approach,” Decision Sciences, submitted (1983). 3. Hunt, G. C., “Sequential
Journal of Operations
Arrays of Waiting
Management
Lines,”
Operations Research, Vol. 4, No. 6 (December 1956) pp. 674-683. 4. Kushner, D. D., “Dynamic Line Balancing: A Cost Analysis of Four Operator Assignment Heuristics,” Ph.D. dissertation, University of Georgia, September, 1980. 5. Salveson, M. E., “The Assembly-Line Balancing Problem,” Transactions of the ASME, Vol. 77, No. 6, (August 1955) pp. 939-948.
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