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A goal approach to assembly line balancing
Computers Opns Res Vol. 17. NO. 5, pp. 509-521, PnntedI” GreatBntain.All nghts reserved
A GOAL
53.00 + 0.00 0305-0548/90 Copyright c, 1990 Pergamon Press plc
1990
APPROACH
TO ASSEMBLY
RICHARD F. DECKRO’**
and
SARANGAN
LINE
BALANCING
RANGACHARI”‘f
‘College of Commerce & Industry, The University of Wyoming, Laramie, WY 82071, U.S.A. and
‘Legato Systems, 260 Sheridan Avenue, Palo Alto, CA 94306, U.S.A. I Recewed Ju1.v 1986: rewed
December 1989)
Scope and Purpose-The assembly line balancing problem has been a task of interest to the operations
researcher for a number of years. Single objective models have traditionally concentrated on either the mmimizatlon of cycle time or the minimization of the number of workstations [l. Working Paper WPS-86-41. However, it has been shown that a number of potentially conflicting goals are present in the operational line balancing problem [2. J. Opns Mgmt 3, 209-221 (1983)]. The model developed here provides the tlexibihty to simultaneously consider variations in cycle time, workstations, zoning considerations, sequencing, idle time and cost. This increase in modeling flexibility, coupled with the versatility to consider multiple criteria, is particularly useful in the planning stages of plant and line layout. In addition, under certain circumstances, the zero-one goal model has a simple optimality condition which can improve computatlonal effectiveness [3. Computers Opns Res. 9, 279-285 (1982)]. Abstract-A zero-one goal programmmg model for the assembly line balancing problem is developed. The goal model provides Increased flexibility in constructing line balances by simultaneously considering varying operational requirements, such as zoning, sequencing, idle time, cycle time and costs. A sensitivity analysis of an example model highlights the usefulness of the approach in the planning stages of a line balance development.
INTRODUCTION
The assembly line balancing problem has attracted the attention of the operations researcher for a number of years. Salvenson [4]. in a 1955 article, was one of the lirst researchers to propose a mathematical programming based solution technique to the line balancing problem. Bowman [S] formalized the connection in 1960. Since that time, a number of models, formulations, heuristics and algorithms have been proposed for the solution of the assembly line balancing problem. Ghosh and Gagon [l], in a recent extensive review of line balancing approaches, cite 162 separate publications dealing with balancing assembly lines. Through the years, a number of authors have recognized the multiple criteria aspects of operational scheduling problems [2.3,6-91. However, very few reported studies have utilized a multiple criteria approach to the assembly line balancing problem. Zanakis and Gupta [lo], in a categorized bibliography of goal programming models, cite only one goal model for the line balancing problem, the model developed by Gunther, Johnson and Peterson [2]. In a survey by Romero [l l] of goal programming models from the period 1970-1982 no titles refer directly to the assembly line balancing problem. In addition to the Gunther et al. model, Rangachari [9] has proposed a multiple objective assembly line model, while Deckro [ 123 has suggested a single objective cost model which balances cycle time and workstations. In general, however, there has been very little in the available, published literature dealing with the multiple criteria aspects of the assembly line balancing problem. Even a cursory review of the literature reveals a number of varied, and potentially conflicting, objectives which have been used in assembly line balancing. A goal based approach would appear to be a natural modeling medium for the assembly line balancing problem. This paper presents a zero-one, goal programming model for the assembly line balancing problem. It utilizes the zero-one formulation given by Patterson and Albracht [13] to take advantage of *Richard F. Deckro is an Associate Professor of Industrral and Operations Management at The University of Wyoming. He holds a BSIE from The State Unrverslty of New York at Buffalo, an MBA and a DBA in decision sciences from Kent State University. His primary areas of research are in applied mathematical programming, multi-criteria decision models, and project management. Dr Deckro has previously published in Computers and Operations Research. Decision Sciences, The European Journal of Operational Research, IIE Transactions, OMEGA, The Journal of Operations Monugemenr and others. Dr Deckro is on the editorial advisory board of Computers and Operations Research. tsarangan Rangachari is a member of the technical staff of Legato Systems. He holds a BS in chemical engineering from The University of Madras (India) and a MS in industrial management from The University of Wyoming.
RICHARD
510
F. DECKRO and SARANGAN RANGACHARI
the variable limiting definitions used in their model. The model developed here can accommodate the goal formulations in the Gunther et al. model [2] and suggests some further useful goal constraints. The next section of the paper presents a programming model for goal assembly line balancing. This model incorporates a number of operational features beyond those used in single objective models. The third section highlights the proposed model through several illustrative examples. The paper concludes with a summary of the approach and some suggestion for future research. THE MODEL In this section the model proposed by Patterson various goals and criteria [2,3, 121. Each relation objective model of Patterson and Albracht. The multiple goal requirements. Adopting the notation utilized by Patterson and additions, we have:
and Albracht [13] is amended to accommodate will first be presented as it appears in the single constraints will then be adjusted to meet the Albracht [13, pp. 167-1681, with several minor
Constants : N number of tasks, M maximum number of possible workstations, subset of all tasks that must precede task i, pi subset of all tasks that must succeed task i, si Wj subset of all tasks that could be assigned to station j by virtue of task-precedence constraints, task time of activity i, Ti earliest workstation task i can be assigned to given sequencing requirements, Ei latest workstation task i can be assigned to given sequencing requirements, Li C,, lower bound on allowable cycle time, Cub upper bound on allowable cycle time. Variables : XL
c
a zero-one variable which equals 1 if task i is assigned to station j, zero otherwise, cycle time; a continuous variable where C,, < C < C,b (C may also be an integer variable, if such a restriction is desirable).
Patterson and Albracht have limited the number of variables required in the line balancing problem by utilizing the precedence relationships to preclude infeasible assignments. The cycle time, which is a fixed value in the Patterson-Albracht model, is used to calculate Ei and Li for each task. Following Deckro [12], we have allowed the cycle time to be a variable, with an upper and lower bound. This designation alters the Patterson-Albracht definitions slightly by using the upper bound in the calculation of Ei and Li instead of the constant value for cycle time.
I 1
E, =
if
(T+&
T,)=O i=l,2,...,N
[( T’z,
otherwise
‘j)/‘ub]’
M
if
(?;+z,
Li =
Tj)=O
i=1,2,...,N (M+
I-[(Ti+z,
Tj)/C”b]’
otherwise
where [cl+ is the smallest integer greater than or equal to u. If the cycle time is a constant, C,b would be replaced by the constant in the above formulas, reverting to the original PattersonAlbracht specification [ 13, p. 1683.
A goal approach to assemblyline balancing
511
Assignment constraints
The assignment constraints assure that each task is assigned to at most one workstation. Task splitting, if it results in a material difference in task completion time, is accommodated by representing each possible split as an individual task. If the split of some task k is allowed and does not materially increase task time, T,, one could model this condition by allowing X~j for all j to be continuous. X~j would be bounded on the interval from zero to one. In order for such an expression to be an accurate representation of the production setting, task k would have to be of a nature such that it could be represented as continuous. An example of such a task might be attaching exterior bolts or caps, where they can be added at any or all of the final workstations. In the single objective model, the assignment constraints are expressed as follows:
2 X,=1
i-l,2
,_..,
N.
(1)
j=E,
In general, the assignment constraints are hard constraints, not suitable for consideration as goals. A possible exception to this would be in a planning stage, where a task might be optional or where multiple task assignments are allowed. If the exclusion of a particular non-essential task would allow the satisfaction of some higher level goal, a goal assignment constraint would be expressed as follows:
j=E,
where d; = 1 would allow the exclusion of task i, while d+ > 0 would allow task i to be assigned to more than one workstation. d+ would not be necessary in equation (2) if multiple assignments were not allowed. By using a bounded variable solution approach, acceptable ranges of variations could be created for the deviational variables. Precedence
constraints
Patterson and Albracht [13, p. 1633 have developed a single objective precedence relation for two tasks a and b where a precedes b as follows:
for each precedence pairing of tasks a and b where L, 2 E,. Like the assignment constraints, the sequencing constraints are usualy hard constraints. In general, we are not likely to desire a change in the tasks’ sequence. In some specific environments, however, we may have a preferred sequence which need not be enforced for all tasks [3, p. 2823. This can be expressed by the addition of a deviational variable to constraint (3). $J j*(Xaj)I=&
2
k*(Xb,)-dl
k=Eb
~0,
(4)
where d,’ = 0 if task a precedes b; d,’ > 0 does not precede b. If it is desirable to have task a completed within p workstations of task b, this may be accomplished by changing the right-hand side of constraint (3) from zero to ~1.Alternatively, if the limit of p workstations is a strict requirement, ,u can be an upper bound for dl. Finally, if this were a desirable condition, but not a necessary condition, the preference could be expressed as a goal constraint. d; +d,
-d,+
=p,
(3
where d; < ,a indicates the assignment of task a to a workstation no more than fl workstations after the workstation to which task b is assigned; d: k 0 indicates task a was assigned more than p workstations after the workstation to which task b is assigned. Zoning constraints Z~com~atible
workstation
zoning. Incompatible zoning considers tasks whose combination at a given is undesirable. Patterson and Albracht [13, p. 1683 have modeled the incompatible
512
RICHARD F. DECKROand SARANGAN RANGACHARI
zoning constraint for the single objective model for some tasks a and b, where a succeeds b, as ig
i*(Xi,)-
5
i*(Xj,)
2
l.
(6)
j=Eb
a
The incompatible zoning constraint can be expressed as a goal by the addition of deviational variables to obtain in
i*(Xi,)II
~
j*(Xjb)+dh
-d:
= 1.
(7)
j=Eb
If d; = 0, the incomptible tasks a and b will not be assigned to the same workstation. If it is desirable that task a and b be more than one workstation apart, the right-hand side of constraint (7) can be set at any desirable level. Alternatively, d,,+ can be maximized. Compatible zoning constraints. Compatible zoning is a simple variation of incompatible zoning [12, p. 1081. In compatible zoning, it is desirable to combine certain tasks at a given workstation. The single objective compatible zoning constraint is expressed as:
ig
i*(Xi,)0
This constraint variables.
$
i*(Xjb)=O.
(8)
j=Eb
type may be adjusted to consider multiple goals by the addition of deviational
5 i*(X,)-
i=E.
5 j*(Xjb) +d;
-d:
=O.
(9)
j=Eb
If d; and d: are minimized to zero, the compatible tasks a and b will be assigned to the same workstation. Through the use of differential weighting, a preference can be expressed for assigning task a or b to the station first, if they cannot be assigned to the same workstation. Such weights might be the additional cost required if a compatible assignment cannot be made. The same approaches of bounding and the creation of deviational constraints (5) used with the precedence constraints can also be used with the zoning constraints. For example, if it is desirable to have tasks a and b assigned in the same vicinity, but it is not necessary to assign the tasks to exactly the same workstation, some upper limit, v, can be set on d; and d:. The objective function could be used to attempt to place task a and b at the same workstation, but the upper bound would assure that the tasks would be no more than v stations apart. By the same logic, a lower bound on the variables d; and d: in constraint (7) would force tasks a and b to be separated by a minimum number of workstations. Cycle time constraints
For a fixed cycle time, Patterson and Albracht have given the constraint [13, p. 1683 iJ$ TiXij~C
j=1,2
,...,
M,
(10)
,
with C a constant for the single objective model. This constraint was amended to T,X,,-C
1
j=1,2
,...,
M,
(11)
ieWJ
with C a variable such that Clb < C < C,, in [12, p. 1081. For a fixed value of cycle time, the goal constraint would be C TiXij+dJ’
-di
=C
j = 1,2,. . . , M.
(12)
ioW,
The single deviational variable, d+ is used for all j constraints each workstation. If C were a variable, the goal would be expressed as 1 ieW,
7;,Xij+d~-C=O
j=1,2,...,M
to assure the same cycle time for
(13)
513
A goal approach to assembly line balancing
omitted. The cycle time goal constraints, with either fixed or variable cycle time, offer several goal possibilities. First, the minimization of the sum of the d,: variables will minimize the idle time at each workstation. If this is to be a primary goal, d+ should be omitted from constraint (12). Constraint (13), with variable cycle time, will tradeoff longer cycle time at each workstation for decreased idle time. A second modeling possibility arising from the cycle time constraints concerns limiting the allowable deviations between workstations, For constraint type (12), an upper bound on d’ will limit the possible increase in the cycle time bust as C,, does in constraint type (13)]. An upper bound on d,: for either type of cycle time constraint will limit the allowable idle time at any workstation. As with the previous constraints these bounds may themselves be expressed as goals. The final possibility offered for the expression of goals from the cycle time constraints considers the equalization of workload by time. While the other cycle time constraints indirectly deal with the equalization of workload by time (by decreasing idle time and/or limiting allowable deviation), the goal formulation allows the concept to be directly expressed. with d’
~jld,~-dj+li~~j
j=1,2,...,M-l,
04)
where & S
is the lower bound on the allowable variation in idle time by workstation, is the upper bound on the allowable variation in idle time by workstation.
Constraint types (14) will restrict the variation in total task time from workstation to workstation to a specified range. It should be noted, however, that such constraints should be used cautiously. The steps necessary to put the bounded absolute values into standard programming form will cause a rapid increase in the number of constraints in the problem. The option is available, however, should it prove to be of value in a particular modeling environment.
Tusk load constraints The task load constraints limit the number of individual tasks which can be assigned to a workstation k to some predetermined maximum, 0,. The most common situation is for 0, to be equal to the same value for all k, Shop work rules or employee contracts may limit the number of tasks which can be assigned to a particular workstation. C Xi,<@k
k=l,2,...,M.
(15)
iaWk
The task load constraints variables.
can be written as goal constraints C
Xik+d;-d:=t)k
k=l,2
,...,
with the addition of deviational M.
(16)
By minimizing d&T,the task count at each workstation will be forced to the. target level 0,. By setting upper bounds on the dl and d; variables, the actual variation from the target values ok can be limited, as has been done with previously discussed constraints. These bounds, in turn, can also be turned to goal constraints. Finally, the task load deviation from workstation to workstation can be limited in the same fashion as constraint type (14) limited the variation in time from workstation to workstation. Unfortunately, as with the cycle time deviations, the number of constraints can grow rapidly when ranges among workstations are established, but the option is available.
Workstation constraints A workstation constraint has been developed for the Patterson and Albracht model in 112, p. lOS]. Let Xj be a zero-one variable, where Xj equals one if workstation j is utilized and Xj is zero otherwise. The set ~j is the set of all tasks which can be assigned to workstation j, subject to the
514
RICHARD F. DECKRO and SARANGAN RANGACHARI
known precedence relations. Let (1Wjjl be the cardinality of the set Wj, then 1 Xij-IlWjllXj~O
j=1,2,...,M.
(17)
isW,
If any task is assigned to workstation j, Xj must be one. The addition of the workstation constraints allows for the formulation of a number of goal constraints. If we wished to specify a target number of workstations, S, we could add a workstation goal as follows: ~ Xj+d,
-d:
=S
j=l
(18)
in conjunction with the workstation constraints. If d: can be minimized to zero, the number of workstations utilized will be S or less. The maximization of d, will have the effect of minimizing the total number of workstations used. A variation on the workstation goal involves considering a budget for tooling. Let B be the total planned budget for establishing workstations, and bj be the individual cost of workstation j. Again, in conjunction with the workstation constraints, the following budget constraint could be established: ~ bjXj~B. j=l
(19)
At an aggregate planning stage, this budget constraint would be expressed as a goal relation: ~ bjXj_d,-d,+=B j=l
w-8
d, and d,’ would give the variation from the targeted budget. The budget constraint could be extended to individual lines, shops or work areas, as desired. A number of modeling possibilities have been outlined above. While the work above is by no means a totally exhaustive list of possible goals for an assembly line problem, the key options have been considered. Objectives
Goal programming is characterized by the attempt to establish a desired level of goal attainment. This attainment is measured as the underachievement or overachievement of some target or threshold level of satisfaction for specific goals [14, p.2821. Objectives are constructed which attempt to minimize the weighted deviation from desired goal levels. The weights selected in the goal programming objective dictate the preference structure for the specific goals. The weighting may be numerical, Archimedian weights, or preemptive weights may be selected. Archimedian weights assume that the goal attainments can be measured in the same metric, while preemptive weights allow the combination of incommensurate deviations to be considered. Lee [7] and Ignizio [15, 163, among others, provide detailed discussions of goal modeling and objective construction. By varying the goals and the weightings included in the objective, a wide variety of preferences can be expressed. Preemptive weighting has been utilized here for the goal assembly line model. Preemption assures that upper level goals will be satisfied before lower priority goals are considered. While numeric weighting may be used to differentiate between goals at the same priority level, the satisfaction of an upper level goal is considered to be infinitely more important than the satisfaction of any lower level goals. The weight s is defined to be much greater than Pk+, where Pk is the preemptive weight assigned to priority k. By varying the order of priority, the sensitivity to the preference structure can be explored [17, pp. 152-1541. A number of priority structures are possible. Gunther et al. [2], in a survey of automotive engineers, determined 11 goals of interest in automobile manufacturing. Table 1, from [Z], lists the Gunther et al. goals in order of preference. The majority of their goals are zoning type goals. Goals 5,8,9, 10, and 11 are compatible zoning type goals. They would be expressed using constraint
A goal approach to assembly Table number
1. Assembly
Goal
1
Mmtmize
Goal
2
Assure
Goal
3
Assure
the workload
Goal
4
Adhere
to plant
Goal
5
Combine
Goal
6
Avoid
Goal
7
Mamtain
Goal
8
Asslgn
tasks with
common
toohng
Goal
9
Asslgn
tasks with
common
parts
Goal
10
Asslgn
tasks requiring
Goal
11
Grouo
tasks together
that
of workmg
the
sum
actlvlty
time
at
for a stanon
12, p
2101
a
workstation
or employee
does
not
has not Increased
exceed
the
cycle
time
since the last balance
requirements
for asstgnment
several
precedence
hne goals from
areas and employees
the
assignment
layout
activities
asstgnmg
of
515
line balancing
to a worker
physlcally
demanding
that
are more
interestmg
tasks at a workstatlon
relations
smular which
to the same workstatlon to the same workstatton
crafts or skills to the same statlon requre
stmultaneous
motions
type (9) with d; and d: minimized at the appropriate priority weight. Goal 6 from Table 1 is an incompatible zoning goal. Goal 6 would be formulated using constraint type (7) and minimizing d; at the appropriate weight (P6 for the Gunther et al. example). Goals 4 and 7 from Table 1 would be modeled using the precedence goal constraint (4). If di can be forced to zero, the desired precedence is maintained, meeting layout and technological requirements. Goal 2 concerns staying within an established cycle time. Constraints type (12) will meet this goal. d+ would be minimized in the objective function at weight PZ. Goal 3, concerning workload balance from previous assignments, could be handled in several ways. Directly, constraint type (15), the task load constraint, could be used. 0, would be the previous task load at a given station. Minimizing d: at priority P3 would satisfy the goal. If the desire is to balance the task time at a workstation, a variation of constraint type (12) could be used. Rather than C as a right-hand side, ST,, the previous task time for the workstation, would be utilized. d’ would be replaced by df . As an objective, the minimization of d: to zero for all i would indicate no increase in task time at station j while forcing d,: and df both to zero would assure the same task time for workstation j. By combining some or all of the goals presented above, a number of operational settings can be modeled. Specific objectives should reflect the preferences and needs of the decision makers involved. By varying the preemptive weights, the differential weights and the deviations in the objective, a wide range of options can be considered. The Appendix contains the actual formulation of model one, below, with its specific objectives. The sensitivity of the formulation is discussed in the next section. ILLUSTRATIVE
EXAMPLES
To illustrate the goal model discussed here, several examples utilizing subsets of the goals given above have been solved. The examples are based upon a problem from Johnson and Montgomery [18, p. 395) and two problems from Tonge [19, p. 651. The conditions of each problem are summarized in Figs 1,2 and 3, while Table 2 gives a summary of the results of each solution. In each case, a maximum for the number of possible workstations has been specified. (If no maximum is known from the operational setting, M may be set equal to N, the number of tasks. It is, however, computationally advantageous to keep M as small as possible.) The actual number of zero-one variables required will depend upon M, N, and the nature of the precedence relations [13]. The number of deviational variables necessary will be a strict function of the goals utilized in modeling the situation. Each of the models were solved sequentially (by goal) on the VAX 8800 processor at the University of Wyoming using the LINDO software package. Times were calculated using the FORTRAN interface in LINDO with the VAX run time library routine LIBSSHOW _TIMER. LINDO utilizes a branch and bound approach to solve integer programs. A review of the figures and Table 2 demonstrates that as the goal conditions and problem dimensions vary, both the computational performance and the ultimate solution varies. For example, Model 2, with 11 tasks. actually took less time to solve than Model 1, with 9 tasks, when the final goal in Model 2.2 was to limit cycle time.
RICHARD F. DECKROand SARANGAN RANGACHARI
516 6
M=5
Cub= 18
ctb = 10
Zero-One Variables = 31 Goals Problem 1.1
Problem 1.2
Goal 1
Limit Tasks at Each Workstation to Thne
Goal 2
Compatible zoning of Tasks 2 and 5; Incompatible Zoning of Tasks 3 and 6
Goal 3
Minimize Cycle Time
Goal 1
Limit Time Cycle to 15
Goal 2
Limit Tasks at Each Workstation to Three
Goal 3
Compatible Zoning of Tasks 2 and 5; Incompatible Zoning of Tasks 3 and 6
Fig. 1. Model 1 [18, p. 3951 precedence relations with task times.
Sensitivity analysis
An important aspect of any solution procedure centers upon post optimality analysis. Once an optimal solution has been established, it is necessary to test the sensitivity of the solution to changes in the operational parameters. As Lee points out [20, p. 1621, if the solution is extremely sensitive to variations in specific parameters, more care must be maintained in developing these critical values. Conversely, if the solution is relatively insensitive to parameter variation, resources may be applied elsewhere in the analysis procedure. In addition, by establishing specific ranges on the parameter values, the analyst can be confident of the stability of the solution during the daily operational variations. Ignizio [ 16, p. 11151 points out that a given linear goal programming model may be subjected to a post optimality analysis in much the same fashion as a continuous linear program. He goes on to reference both his own text [lS] and Lee’s text [20] as sources of how this analysis may be conducted. More recently, Steuer [14, p. 298-J points out that in preemptive goal programming, analyst often conduct selectivity analysis by interchanging priorities. This rotation of priorities allows the analyst to review the effects of the ordering of the preemptive weights. Steuer [ 14, p. 2981 further suggests this analyst be carried out by resolving the goal program with the new priority specification. The sensitivity analysis of the goal assembly line model is further restricted by the integer nature of the decision variables. Nauss [21] has provided one of the early studies dedicated to sensitivity analysis in integer programs. Nauss suggests parameterizing the integer model, and looking for efficient ways to resolve the parameterized model. Combining all of these works [15, 16,20-223 appears to be the most effective way to pursue a sensitivity analysis of the zero/one goal programming assembly line balancing model. Actual results will be highly dependent upon the input values of the parameters and the structure of the given operational model, however. To illustrate the suggested approach, an analysis of the objective of Model 1 has been made, as well as the parameterization of one of the constraints of the model for a given objective. In rotating the priorities of a preemptive goal program, there will be p! different rotations for a
517
A goal approach to assembly line balancing 2
5
6
2
wo&tations 4 GOdS Problem 2.1
Problem 2.2
Goal 1
Limit Wotitations to Three
Goal 2
Limit Tasks per Workstation to Four
Goal 3
Limit Tie
Goal 1
Limit Workstations to Three
Goal 2
Compatible Zoning of Tasks 5 and 7; Incompatible Zoning of Tasks 2 and 3
Goal 3
Limit Time Cycle to Eighteen
Cycle to Sixteen
Fig. 2. Model 2 [19, p. 631 precedence relations with task times.
model with p priorities [14, p. 2983. Such an analysis has been carried out on the example model in the Appendix. Table 3 summarizes the results of this analysis. The rotation of the priorities for Model 1.3 highlights the inter-relation between the cycle time and the number of workstations. While the task limit goal (Goal C) is satisfied in all rotations, it is not possible to simultaneously satisfy the workstation limit (Goal A) and the cycle time limit (Goal B). The various rotations offer the decision maker a spectrum of solutions to choose from. The final selection will hinge upon the decision makers preference (or abhorrence) of a longer cycle time or more workstations. To further investigate these choices, a sensitivity analysis was undertaken on the right-hand side of the workstation goal using Model 1.3, Problem 1. Table 4 summarizes the results of this analysis. Three possible workstation limits were reviewed. At a limit of two workstations, it was not possible to satisfy either the workstation limit or the cycle time limit. When the workstation limit was increased to three, that goal was met, but the cycle time goal was exceeded. Finally, at a workstation limit of four, all goals can be met. The analyst, in conjunction with the decision maker, might wish to extend the sensitivity analysis to other constraints. The actual direction of the analysis would be dictated by the operational setting. While sensitivity analysis on a zero-one goal programming model is more cumbersome than on a linear goal program or a linear program, it can provide the analyst with a wealth of critical information. This goal approach does suffer from the weaknesses of all general zero-one approaches. As the problem increases in size, the computational time will also grow. Unless specialized structures can be exploited, the problem will eventually become intractable. The goal model does, however, provide the analyst with a method to model varied operational settings. This modeling flexibility comes at a potential computational cost, however. Clearly, this cost should be balanced against any modeling gain. It would appear that the goal approach should be particularly useful in the planning stages of a line balance development.
RICHARD F. DECKROand SARANGAN RANGACHARI
528
M=f
Zero-One Variables: Assignment 83 workstations 5 GO& Problem 3.1
Problem 33
Goal 1
Limil Number of Workstations to Four
Goal 2
Minimiu: Task Splitting of T&s 8 and 12
Goat5
Limit Number of Tasks at Each Wwkstation to Five
Gull
Limit Number of Tasks at FBiehW~~tati~
Goat2
Limit Numberof Workstations to Four
Goa 3
M~~rni~ Workstatior~Expense
to Five
Fig. 3. Model 3 [H, p. 651 precedence relations with task times. Table 2. Exampk problems results GtXP
Problem
status
CPU? time (sccl
Final non-zero vanables
C
519
A goal approachto assemblyline balancing Table 3. Scnsitivicy analysis on
Problem No.
Priority 1
Priority 2
Priority 3
1
A
B
C
2
B
A
C
3
B
C
A
4
A
C
B
5
C
B
A
6
C
A
B
rotationof pnoritin-Model
Non-zero dccislon variables ~,,,x,,x,..~,,x53~64.~?5~~ss~~,,x,.~5 ~,,.~,,.~,,.~4,xs1Js5.~~s~s5~9s~l.~3.~4~5 X,*.XI~.Xs,.XI3.X52.~6~.~75~ss.~9~.~1.~3.~4.~s ~,,,~,,.~,,~l,.~,,.~64.~,,,x,,sr,,x,x,,~s ~,,.~,,.~,,sc4,~s,x,,,x,,sr,,,x,,x,.~95~~.~3~4~~s ~**r~11~~34.~41r~s3.~64~~15~ss.~95~~z.~~.~s
1.3 Unsatisfied priority/ goal
Cycle time
Workstations
2/B
16
3
2/A
15
4
3/A
I5
4
3/e
16
3
3/A
15
4
318
16
3
Goal A: limit number of workstations 10 no more than 3 Goal B: hmit cycle time to no more than IS. Goal C limit task per workstatIon lo no more than 3
Table 4. Sensitivity analysis on RHS of workstation goal Model 1.3 (pnority A, B, C), WorkstatIon hrmt 2
Non-zero decision variables X~,.X2,.X~r,X.,,Xs3.~s~.x,s.~*s.~~s.~,.~..~5
3
X,,.Xll.X3I.XII.X53.~64.~75~~5~95.~1.~4~5
4
~,,.~,2.~,,~.,.~5,.~6~~75.~~5~95.~2~3~4~5
Problem 1
Unsatisfied goals
Cycle time
A, B
16
3
16
3
15
4
B -
Workstatmns
Goal A: lirmt number of workstatIons. Goal B: hnnt cycle time to no more than IS. Goal C: hmit tasks per workstatIon lo no more than 3
CONCLUSION The approach discussed above for the goal modeling of the assembly line balancing problem allows the modeler and/or decision maker greater latitude in creating a line balance. The goal approach allows the modeler to consider multiple criteria in selecting a preferred schedule. In addition, through the use of weights and/or the altering of preemptive priorities, one can entertain various trade-offs among the criterion. Such flexibility should prove to be particularly useful in the planning stages of a line balancing. The use of the Patterson-Albracht [ 133 variable definition had aided in reducing the total number of variables and constraints required to model the problem. The Patterson-Albracht model can accommodate all of the goals outlined by Gunther et al. [2]. Even with these reductions, however, this approach has some of the same limitations that previous zero-one approaches to the line balance problem have had. The models can get very large very quickly. The goal model will have all the zero-one variables required by the single objective model and the additional deviational variables required to model the goals. However, if a goal can be satisfied, it is not necessary to spend computational time on establishing optimality. Once the deviational variables at a goal are driven to zero, no further computation is required at that goal [3]. In this paper, we have provided a goal based approach to model line balancing problems. Such a formulation allows the decision maker greater flexibility in considering alternatives. While the existence of an optimality condition should make large formulations more computationally tractable, the approach does suffer from the weaknesses of all general zero-one programs. The goal approach developed here for a generalized model may be used as a basis for future research which exploits the network structure. Acknowledgements-The authors wish to thank the editor and the reviewers of this paper for their useful insights and patience. We also wish to thank Mr Per H. Grlmsrud for his assistance in running these examples.
REFERENCES 1. S. Ghosh and R. Gagon, A comprehenslve
literature review and hierarchical taxonomy for the design and balancing ofassembly lines. Working Paper WPS-86-4, College of Administrative Science, The Ohio State University. Jan. (1986). 2. R. E. Gunther, G. D. Johnson and R. S. Peterson, Currently practiced formulations for the assembly line balance problem. J. OpnsMgmt 3.209-221 (1983).
RICHARDF. DECKROand SARANGANRANGACHARI
520
3. R. F. Deckro, J. E. Hebert and E. P. Winkofsky, Multiple criteria job-shop scheduling. Computers Opns Res. 9,279-285 (1982). 4. M. E. Salveson, The assembly line balancing problem. J. lndust. Engng 6, 18-25 (1955). 5. E. H. Bowman, Assembly line balancing by linear programming. 0pn.r Res. 8, 385-389 (1960). 6. K. D. Lawrence and J. S. Burbidge, A multiple goal linear programming approach for coordinated production and logistic planning. Inc. J. Product. Res. 14, 215-222 (1976). 7. S. M. Lee, E. R. Clayton and B. W. Taylor, A goal programming approach to multi-period production line scheduling. Computers Opns Res. 5, 205-211 (1978). 8. L. J. Moore and B. W. Taylor, Analysis of a tram-shipment problem with multiple conflicting objectives. Computers Opns Res. 5, 39-46 (1978).
9. S. Rangachari, Assembly line balancing using multiple objective zero-one integer programming. Plan B Paper for the MS in Industrial Management, The University of Wyoming, Dec. (1984). 10. S. H. Zanakis and S. K. Gupta, A categorized bibliographic survey of goal programming. Omega 13,21 l-222 (1985). 11. C. Romero, A survey of generalized goal programming (1970-1982). Eur. J. Opt Res. 25, 183-191 (1986). 12. R. F. Deckro, Balancing cycle time and workstations. f I E Trans. 21, 106-l 11 (1989). 13. J. II. Patterson and J. J. Albracht, Assembly-line balancing: zero-one programming with Fibonacci search. Opns Res. 23, 166-172 (1975). 14. R. E. Steuer, Multiple Criteria Optimization: Theory. Computation and Apphcation. Wiley, New York (1986). IS. J, P. Ignizto, Goof ~rogrummi~g and Extensions. Heath, Lexington. MA (1976). 16. J. P. Ignizio, A review ofgoal programming: a tool for multiobjective analysis. J. Opl Res. Sot. 29.1109-i 119 (1978). 17. R. F. Deckro and J. E. Hebert, Polynomial goal programming: a procedure for modeling preference tradeoffs. J. Opns Mgmt 7, 149-164 (1988). 18. L. A. Johnson and D. C. Montgomery, Operations Research in Production Planning, Scheduling and Inventory Control. Wiley, New York (1974). 19. F. M. Tonge, A Heuristic Program fir Assembly Line Balancing. Prentice-Hall, Englewood Cliffs, NJ (1961). 20. S. M. Lee, Goal Programming for Decision Analysis. Auerback, Philadelphia, PA (1972). 21. R. M. Nauss, Parametric Integer Programming. University of Missouri Press, Columbia, MO (1979). 22. C. A. Markowski and J. P. Ignizio, Duality and transformations in multiphase and sequential linear goal programming. Computers Opns Res. 10, 321-333 (1983).
23. R. B. Chase, Survey of paced assembly lines. Indust. Engng 14-18 Febr. (1974).
APPENDIX Model 1.3. Problem 1 Variables
xii = 1 if task i is assigned to workstation j 0 otherwise xj = 1 if workstation j is used 0 otherwise PW= overachievement of workstation limit NW= underachievement of workstation limit PC = overachievement of cycle time limit NCI = underachievement ofcycle time limit for workstation j PTj = overachievement of task limit for workstation j NT, = underachievement of task limit for workstation j Maximum cycle time is 20. Goafs
Priority 1: limit number of workstations to no more than 3. Priority 2: limit cycle time to no more than 15. Priority 3: limit task per workstation to no more than 3. Modes Mm
P,(PW), P,(PC), PJPT,
+ PT, + PT, + PT., $ PT,)
s.t. Assignment constraints: x11 +x,2+xl3
=
x,,+x22+~23+x24 x3l
+x32
+x33
+x34
x41
f-%2
+x43
+-x44
x5,+X32+X53+X54 x32
= =
+x63
+
xb4
1
=: =
=
1 1 1
x,,+x,,+x,, -1 x,,+xg*+xg> = 1 xg3 + xpq + xgs = 1. Precedence constraints: lx,, +2x,, + 3x,3 - lx,, -2x,, - 3x,, - 4x54 lx,, +2x,, + 3x,a - lx,, -2x4r3x,3 -4x,, lx,, + 2x,, + 3X,, +4x 1,,,+2rnt3~,,+4r::~::::~~~~~I:~~:-~~~~~~
A goal approach to assembly line balancing Ix,, + 2x42 + 3x,, + lx,, + 2x52 + 3x33 + lx,, + 2x,2 + 3x,, + 2x,, + 3X63+4x,, 3x,, +4x,, + 5x,, 3x,, +4x,, + 5x,, -
4x,, 4x,, 4x,, 3x,, 3x,3 3x9,
- 2x,, - 2x,, - 3x,, -4x,, -4x,, -4x,,
- 3x,, -4x,, - 3x,3 -4x,, -4% - 5x,, - 5x,, - 5x,, - 5x9,
< 0
<0 <0 SO GO SO.
Workstation: ~1,+~,,+~31+~4,+~,,-5~, x,2 +x21 +x32 + x42 +x52 - 5x, x,3 +x23 +x33 +x43 +x63 + x,3 +x ,,+x,,-9x,% x24 + x,g + x54 + X64+ x,4 + xgq + .xg‘$- lx, x,5 + xg5 + xgs -3x,
GO
GO SO.
Workstation goal: x, +x,+x,+.u,+x,+NW-PW=3. Cycle time goals. 4x,,+5x,,+6x3,+7x,,+5x,,+NCI-PC=15 4x1, + 5x2, + 6x3, + 7x4, + 5x5, + 10x,, + NC2 - PC = 15 4x,, + 5x13 + 6x33 + 7x,, + 5x,3 + 10x,, + 6x,, + ix,, +4x,, + NC3 - PC= 15 5x,, + 6x,, + 7x,, + 5x,, + lox,, +6x,, + lx,, +4x,, + NC4 - PC = 15 6x,, + 1x8,+4x,,+ NC5-PC= 15. Task goals: “~,+~~~+x~~+x~,+x~~+NTl-PTl=3 ~12 + ~22 + x32 + xd2 + xS2 + xs2 + NT2 - PT2 = 15 -xl3 + XZJ+ x33 + ~4~ + xs3 + xe3 + x,~ + xs3 + xg3 + NT3 - PT3 = 3 x,,+x,,+x,,+x,,+x,,+x,,+x,,+x,,+NT4-PT4=3 x,~ + xss + xgs + NT5 - PT5 = 3.
Limit on cycle time: PC65 x,,=o or 1 xj=o or 1 all other variables are non-negative.
521
A GOAL
53.00 + 0.00 0305-0548/90 Copyright c, 1990 Pergamon Press plc
1990
APPROACH
TO ASSEMBLY
RICHARD F. DECKRO’**
and
SARANGAN
LINE
BALANCING
RANGACHARI”‘f
‘College of Commerce & Industry, The University of Wyoming, Laramie, WY 82071, U.S.A. and
‘Legato Systems, 260 Sheridan Avenue, Palo Alto, CA 94306, U.S.A. I Recewed Ju1.v 1986: rewed
December 1989)
Scope and Purpose-The assembly line balancing problem has been a task of interest to the operations
researcher for a number of years. Single objective models have traditionally concentrated on either the mmimizatlon of cycle time or the minimization of the number of workstations [l. Working Paper WPS-86-41. However, it has been shown that a number of potentially conflicting goals are present in the operational line balancing problem [2. J. Opns Mgmt 3, 209-221 (1983)]. The model developed here provides the tlexibihty to simultaneously consider variations in cycle time, workstations, zoning considerations, sequencing, idle time and cost. This increase in modeling flexibility, coupled with the versatility to consider multiple criteria, is particularly useful in the planning stages of plant and line layout. In addition, under certain circumstances, the zero-one goal model has a simple optimality condition which can improve computatlonal effectiveness [3. Computers Opns Res. 9, 279-285 (1982)]. Abstract-A zero-one goal programmmg model for the assembly line balancing problem is developed. The goal model provides Increased flexibility in constructing line balances by simultaneously considering varying operational requirements, such as zoning, sequencing, idle time, cycle time and costs. A sensitivity analysis of an example model highlights the usefulness of the approach in the planning stages of a line balance development.
INTRODUCTION
The assembly line balancing problem has attracted the attention of the operations researcher for a number of years. Salvenson [4]. in a 1955 article, was one of the lirst researchers to propose a mathematical programming based solution technique to the line balancing problem. Bowman [S] formalized the connection in 1960. Since that time, a number of models, formulations, heuristics and algorithms have been proposed for the solution of the assembly line balancing problem. Ghosh and Gagon [l], in a recent extensive review of line balancing approaches, cite 162 separate publications dealing with balancing assembly lines. Through the years, a number of authors have recognized the multiple criteria aspects of operational scheduling problems [2.3,6-91. However, very few reported studies have utilized a multiple criteria approach to the assembly line balancing problem. Zanakis and Gupta [lo], in a categorized bibliography of goal programming models, cite only one goal model for the line balancing problem, the model developed by Gunther, Johnson and Peterson [2]. In a survey by Romero [l l] of goal programming models from the period 1970-1982 no titles refer directly to the assembly line balancing problem. In addition to the Gunther et al. model, Rangachari [9] has proposed a multiple objective assembly line model, while Deckro [ 123 has suggested a single objective cost model which balances cycle time and workstations. In general, however, there has been very little in the available, published literature dealing with the multiple criteria aspects of the assembly line balancing problem. Even a cursory review of the literature reveals a number of varied, and potentially conflicting, objectives which have been used in assembly line balancing. A goal based approach would appear to be a natural modeling medium for the assembly line balancing problem. This paper presents a zero-one, goal programming model for the assembly line balancing problem. It utilizes the zero-one formulation given by Patterson and Albracht [13] to take advantage of *Richard F. Deckro is an Associate Professor of Industrral and Operations Management at The University of Wyoming. He holds a BSIE from The State Unrverslty of New York at Buffalo, an MBA and a DBA in decision sciences from Kent State University. His primary areas of research are in applied mathematical programming, multi-criteria decision models, and project management. Dr Deckro has previously published in Computers and Operations Research. Decision Sciences, The European Journal of Operational Research, IIE Transactions, OMEGA, The Journal of Operations Monugemenr and others. Dr Deckro is on the editorial advisory board of Computers and Operations Research. tsarangan Rangachari is a member of the technical staff of Legato Systems. He holds a BS in chemical engineering from The University of Madras (India) and a MS in industrial management from The University of Wyoming.
RICHARD
510
F. DECKRO and SARANGAN RANGACHARI
the variable limiting definitions used in their model. The model developed here can accommodate the goal formulations in the Gunther et al. model [2] and suggests some further useful goal constraints. The next section of the paper presents a programming model for goal assembly line balancing. This model incorporates a number of operational features beyond those used in single objective models. The third section highlights the proposed model through several illustrative examples. The paper concludes with a summary of the approach and some suggestion for future research. THE MODEL In this section the model proposed by Patterson various goals and criteria [2,3, 121. Each relation objective model of Patterson and Albracht. The multiple goal requirements. Adopting the notation utilized by Patterson and additions, we have:
and Albracht [13] is amended to accommodate will first be presented as it appears in the single constraints will then be adjusted to meet the Albracht [13, pp. 167-1681, with several minor
Constants : N number of tasks, M maximum number of possible workstations, subset of all tasks that must precede task i, pi subset of all tasks that must succeed task i, si Wj subset of all tasks that could be assigned to station j by virtue of task-precedence constraints, task time of activity i, Ti earliest workstation task i can be assigned to given sequencing requirements, Ei latest workstation task i can be assigned to given sequencing requirements, Li C,, lower bound on allowable cycle time, Cub upper bound on allowable cycle time. Variables : XL
c
a zero-one variable which equals 1 if task i is assigned to station j, zero otherwise, cycle time; a continuous variable where C,, < C < C,b (C may also be an integer variable, if such a restriction is desirable).
Patterson and Albracht have limited the number of variables required in the line balancing problem by utilizing the precedence relationships to preclude infeasible assignments. The cycle time, which is a fixed value in the Patterson-Albracht model, is used to calculate Ei and Li for each task. Following Deckro [12], we have allowed the cycle time to be a variable, with an upper and lower bound. This designation alters the Patterson-Albracht definitions slightly by using the upper bound in the calculation of Ei and Li instead of the constant value for cycle time.
I 1
E, =
if
(T+&
T,)=O i=l,2,...,N
[( T’z,
otherwise
‘j)/‘ub]’
M
if
(?;+z,
Li =
Tj)=O
i=1,2,...,N (M+
I-[(Ti+z,
Tj)/C”b]’
otherwise
where [cl+ is the smallest integer greater than or equal to u. If the cycle time is a constant, C,b would be replaced by the constant in the above formulas, reverting to the original PattersonAlbracht specification [ 13, p. 1683.
A goal approach to assemblyline balancing
511
Assignment constraints
The assignment constraints assure that each task is assigned to at most one workstation. Task splitting, if it results in a material difference in task completion time, is accommodated by representing each possible split as an individual task. If the split of some task k is allowed and does not materially increase task time, T,, one could model this condition by allowing X~j for all j to be continuous. X~j would be bounded on the interval from zero to one. In order for such an expression to be an accurate representation of the production setting, task k would have to be of a nature such that it could be represented as continuous. An example of such a task might be attaching exterior bolts or caps, where they can be added at any or all of the final workstations. In the single objective model, the assignment constraints are expressed as follows:
2 X,=1
i-l,2
,_..,
N.
(1)
j=E,
In general, the assignment constraints are hard constraints, not suitable for consideration as goals. A possible exception to this would be in a planning stage, where a task might be optional or where multiple task assignments are allowed. If the exclusion of a particular non-essential task would allow the satisfaction of some higher level goal, a goal assignment constraint would be expressed as follows:
j=E,
where d; = 1 would allow the exclusion of task i, while d+ > 0 would allow task i to be assigned to more than one workstation. d+ would not be necessary in equation (2) if multiple assignments were not allowed. By using a bounded variable solution approach, acceptable ranges of variations could be created for the deviational variables. Precedence
constraints
Patterson and Albracht [13, p. 1633 have developed a single objective precedence relation for two tasks a and b where a precedes b as follows:
for each precedence pairing of tasks a and b where L, 2 E,. Like the assignment constraints, the sequencing constraints are usualy hard constraints. In general, we are not likely to desire a change in the tasks’ sequence. In some specific environments, however, we may have a preferred sequence which need not be enforced for all tasks [3, p. 2823. This can be expressed by the addition of a deviational variable to constraint (3). $J j*(Xaj)I=&
2
k*(Xb,)-dl
k=Eb
~0,
(4)
where d,’ = 0 if task a precedes b; d,’ > 0 does not precede b. If it is desirable to have task a completed within p workstations of task b, this may be accomplished by changing the right-hand side of constraint (3) from zero to ~1.Alternatively, if the limit of p workstations is a strict requirement, ,u can be an upper bound for dl. Finally, if this were a desirable condition, but not a necessary condition, the preference could be expressed as a goal constraint. d; +d,
-d,+
=p,
(3
where d; < ,a indicates the assignment of task a to a workstation no more than fl workstations after the workstation to which task b is assigned; d: k 0 indicates task a was assigned more than p workstations after the workstation to which task b is assigned. Zoning constraints Z~com~atible
workstation
zoning. Incompatible zoning considers tasks whose combination at a given is undesirable. Patterson and Albracht [13, p. 1683 have modeled the incompatible
512
RICHARD F. DECKROand SARANGAN RANGACHARI
zoning constraint for the single objective model for some tasks a and b, where a succeeds b, as ig
i*(Xi,)-
5
i*(Xj,)
2
l.
(6)
j=Eb
a
The incompatible zoning constraint can be expressed as a goal by the addition of deviational variables to obtain in
i*(Xi,)II
~
j*(Xjb)+dh
-d:
= 1.
(7)
j=Eb
If d; = 0, the incomptible tasks a and b will not be assigned to the same workstation. If it is desirable that task a and b be more than one workstation apart, the right-hand side of constraint (7) can be set at any desirable level. Alternatively, d,,+ can be maximized. Compatible zoning constraints. Compatible zoning is a simple variation of incompatible zoning [12, p. 1081. In compatible zoning, it is desirable to combine certain tasks at a given workstation. The single objective compatible zoning constraint is expressed as:
ig
i*(Xi,)0
This constraint variables.
$
i*(Xjb)=O.
(8)
j=Eb
type may be adjusted to consider multiple goals by the addition of deviational
5 i*(X,)-
i=E.
5 j*(Xjb) +d;
-d:
=O.
(9)
j=Eb
If d; and d: are minimized to zero, the compatible tasks a and b will be assigned to the same workstation. Through the use of differential weighting, a preference can be expressed for assigning task a or b to the station first, if they cannot be assigned to the same workstation. Such weights might be the additional cost required if a compatible assignment cannot be made. The same approaches of bounding and the creation of deviational constraints (5) used with the precedence constraints can also be used with the zoning constraints. For example, if it is desirable to have tasks a and b assigned in the same vicinity, but it is not necessary to assign the tasks to exactly the same workstation, some upper limit, v, can be set on d; and d:. The objective function could be used to attempt to place task a and b at the same workstation, but the upper bound would assure that the tasks would be no more than v stations apart. By the same logic, a lower bound on the variables d; and d: in constraint (7) would force tasks a and b to be separated by a minimum number of workstations. Cycle time constraints
For a fixed cycle time, Patterson and Albracht have given the constraint [13, p. 1683 iJ$ TiXij~C
j=1,2
,...,
M,
(10)
,
with C a constant for the single objective model. This constraint was amended to T,X,,-C
1
j=1,2
,...,
M,
(11)
ieWJ
with C a variable such that Clb < C < C,, in [12, p. 1081. For a fixed value of cycle time, the goal constraint would be C TiXij+dJ’
-di
=C
j = 1,2,. . . , M.
(12)
ioW,
The single deviational variable, d+ is used for all j constraints each workstation. If C were a variable, the goal would be expressed as 1 ieW,
7;,Xij+d~-C=O
j=1,2,...,M
to assure the same cycle time for
(13)
513
A goal approach to assembly line balancing
omitted. The cycle time goal constraints, with either fixed or variable cycle time, offer several goal possibilities. First, the minimization of the sum of the d,: variables will minimize the idle time at each workstation. If this is to be a primary goal, d+ should be omitted from constraint (12). Constraint (13), with variable cycle time, will tradeoff longer cycle time at each workstation for decreased idle time. A second modeling possibility arising from the cycle time constraints concerns limiting the allowable deviations between workstations, For constraint type (12), an upper bound on d’ will limit the possible increase in the cycle time bust as C,, does in constraint type (13)]. An upper bound on d,: for either type of cycle time constraint will limit the allowable idle time at any workstation. As with the previous constraints these bounds may themselves be expressed as goals. The final possibility offered for the expression of goals from the cycle time constraints considers the equalization of workload by time. While the other cycle time constraints indirectly deal with the equalization of workload by time (by decreasing idle time and/or limiting allowable deviation), the goal formulation allows the concept to be directly expressed. with d’
~jld,~-dj+li~~j
j=1,2,...,M-l,
04)
where & S
is the lower bound on the allowable variation in idle time by workstation, is the upper bound on the allowable variation in idle time by workstation.
Constraint types (14) will restrict the variation in total task time from workstation to workstation to a specified range. It should be noted, however, that such constraints should be used cautiously. The steps necessary to put the bounded absolute values into standard programming form will cause a rapid increase in the number of constraints in the problem. The option is available, however, should it prove to be of value in a particular modeling environment.
Tusk load constraints The task load constraints limit the number of individual tasks which can be assigned to a workstation k to some predetermined maximum, 0,. The most common situation is for 0, to be equal to the same value for all k, Shop work rules or employee contracts may limit the number of tasks which can be assigned to a particular workstation. C Xi,<@k
k=l,2,...,M.
(15)
iaWk
The task load constraints variables.
can be written as goal constraints C
Xik+d;-d:=t)k
k=l,2
,...,
with the addition of deviational M.
(16)
By minimizing d&T,the task count at each workstation will be forced to the. target level 0,. By setting upper bounds on the dl and d; variables, the actual variation from the target values ok can be limited, as has been done with previously discussed constraints. These bounds, in turn, can also be turned to goal constraints. Finally, the task load deviation from workstation to workstation can be limited in the same fashion as constraint type (14) limited the variation in time from workstation to workstation. Unfortunately, as with the cycle time deviations, the number of constraints can grow rapidly when ranges among workstations are established, but the option is available.
Workstation constraints A workstation constraint has been developed for the Patterson and Albracht model in 112, p. lOS]. Let Xj be a zero-one variable, where Xj equals one if workstation j is utilized and Xj is zero otherwise. The set ~j is the set of all tasks which can be assigned to workstation j, subject to the
514
RICHARD F. DECKRO and SARANGAN RANGACHARI
known precedence relations. Let (1Wjjl be the cardinality of the set Wj, then 1 Xij-IlWjllXj~O
j=1,2,...,M.
(17)
isW,
If any task is assigned to workstation j, Xj must be one. The addition of the workstation constraints allows for the formulation of a number of goal constraints. If we wished to specify a target number of workstations, S, we could add a workstation goal as follows: ~ Xj+d,
-d:
=S
j=l
(18)
in conjunction with the workstation constraints. If d: can be minimized to zero, the number of workstations utilized will be S or less. The maximization of d, will have the effect of minimizing the total number of workstations used. A variation on the workstation goal involves considering a budget for tooling. Let B be the total planned budget for establishing workstations, and bj be the individual cost of workstation j. Again, in conjunction with the workstation constraints, the following budget constraint could be established: ~ bjXj~B. j=l
(19)
At an aggregate planning stage, this budget constraint would be expressed as a goal relation: ~ bjXj_d,-d,+=B j=l
w-8
d, and d,’ would give the variation from the targeted budget. The budget constraint could be extended to individual lines, shops or work areas, as desired. A number of modeling possibilities have been outlined above. While the work above is by no means a totally exhaustive list of possible goals for an assembly line problem, the key options have been considered. Objectives
Goal programming is characterized by the attempt to establish a desired level of goal attainment. This attainment is measured as the underachievement or overachievement of some target or threshold level of satisfaction for specific goals [14, p.2821. Objectives are constructed which attempt to minimize the weighted deviation from desired goal levels. The weights selected in the goal programming objective dictate the preference structure for the specific goals. The weighting may be numerical, Archimedian weights, or preemptive weights may be selected. Archimedian weights assume that the goal attainments can be measured in the same metric, while preemptive weights allow the combination of incommensurate deviations to be considered. Lee [7] and Ignizio [15, 163, among others, provide detailed discussions of goal modeling and objective construction. By varying the goals and the weightings included in the objective, a wide variety of preferences can be expressed. Preemptive weighting has been utilized here for the goal assembly line model. Preemption assures that upper level goals will be satisfied before lower priority goals are considered. While numeric weighting may be used to differentiate between goals at the same priority level, the satisfaction of an upper level goal is considered to be infinitely more important than the satisfaction of any lower level goals. The weight s is defined to be much greater than Pk+, where Pk is the preemptive weight assigned to priority k. By varying the order of priority, the sensitivity to the preference structure can be explored [17, pp. 152-1541. A number of priority structures are possible. Gunther et al. [2], in a survey of automotive engineers, determined 11 goals of interest in automobile manufacturing. Table 1, from [Z], lists the Gunther et al. goals in order of preference. The majority of their goals are zoning type goals. Goals 5,8,9, 10, and 11 are compatible zoning type goals. They would be expressed using constraint
A goal approach to assembly Table number
1. Assembly
Goal
1
Mmtmize
Goal
2
Assure
Goal
3
Assure
the workload
Goal
4
Adhere
to plant
Goal
5
Combine
Goal
6
Avoid
Goal
7
Mamtain
Goal
8
Asslgn
tasks with
common
toohng
Goal
9
Asslgn
tasks with
common
parts
Goal
10
Asslgn
tasks requiring
Goal
11
Grouo
tasks together
that
of workmg
the
sum
actlvlty
time
at
for a stanon
12, p
2101
a
workstation
or employee
does
not
has not Increased
exceed
the
cycle
time
since the last balance
requirements
for asstgnment
several
precedence
hne goals from
areas and employees
the
assignment
layout
activities
asstgnmg
of
515
line balancing
to a worker
physlcally
demanding
that
are more
interestmg
tasks at a workstatlon
relations
smular which
to the same workstatlon to the same workstatton
crafts or skills to the same statlon requre
stmultaneous
motions
type (9) with d; and d: minimized at the appropriate priority weight. Goal 6 from Table 1 is an incompatible zoning goal. Goal 6 would be formulated using constraint type (7) and minimizing d; at the appropriate weight (P6 for the Gunther et al. example). Goals 4 and 7 from Table 1 would be modeled using the precedence goal constraint (4). If di can be forced to zero, the desired precedence is maintained, meeting layout and technological requirements. Goal 2 concerns staying within an established cycle time. Constraints type (12) will meet this goal. d+ would be minimized in the objective function at weight PZ. Goal 3, concerning workload balance from previous assignments, could be handled in several ways. Directly, constraint type (15), the task load constraint, could be used. 0, would be the previous task load at a given station. Minimizing d: at priority P3 would satisfy the goal. If the desire is to balance the task time at a workstation, a variation of constraint type (12) could be used. Rather than C as a right-hand side, ST,, the previous task time for the workstation, would be utilized. d’ would be replaced by df . As an objective, the minimization of d: to zero for all i would indicate no increase in task time at station j while forcing d,: and df both to zero would assure the same task time for workstation j. By combining some or all of the goals presented above, a number of operational settings can be modeled. Specific objectives should reflect the preferences and needs of the decision makers involved. By varying the preemptive weights, the differential weights and the deviations in the objective, a wide range of options can be considered. The Appendix contains the actual formulation of model one, below, with its specific objectives. The sensitivity of the formulation is discussed in the next section. ILLUSTRATIVE
EXAMPLES
To illustrate the goal model discussed here, several examples utilizing subsets of the goals given above have been solved. The examples are based upon a problem from Johnson and Montgomery [18, p. 395) and two problems from Tonge [19, p. 651. The conditions of each problem are summarized in Figs 1,2 and 3, while Table 2 gives a summary of the results of each solution. In each case, a maximum for the number of possible workstations has been specified. (If no maximum is known from the operational setting, M may be set equal to N, the number of tasks. It is, however, computationally advantageous to keep M as small as possible.) The actual number of zero-one variables required will depend upon M, N, and the nature of the precedence relations [13]. The number of deviational variables necessary will be a strict function of the goals utilized in modeling the situation. Each of the models were solved sequentially (by goal) on the VAX 8800 processor at the University of Wyoming using the LINDO software package. Times were calculated using the FORTRAN interface in LINDO with the VAX run time library routine LIBSSHOW _TIMER. LINDO utilizes a branch and bound approach to solve integer programs. A review of the figures and Table 2 demonstrates that as the goal conditions and problem dimensions vary, both the computational performance and the ultimate solution varies. For example, Model 2, with 11 tasks. actually took less time to solve than Model 1, with 9 tasks, when the final goal in Model 2.2 was to limit cycle time.
RICHARD F. DECKROand SARANGAN RANGACHARI
516 6
M=5
Cub= 18
ctb = 10
Zero-One Variables = 31 Goals Problem 1.1
Problem 1.2
Goal 1
Limit Tasks at Each Workstation to Thne
Goal 2
Compatible zoning of Tasks 2 and 5; Incompatible Zoning of Tasks 3 and 6
Goal 3
Minimize Cycle Time
Goal 1
Limit Time Cycle to 15
Goal 2
Limit Tasks at Each Workstation to Three
Goal 3
Compatible Zoning of Tasks 2 and 5; Incompatible Zoning of Tasks 3 and 6
Fig. 1. Model 1 [18, p. 3951 precedence relations with task times.
Sensitivity analysis
An important aspect of any solution procedure centers upon post optimality analysis. Once an optimal solution has been established, it is necessary to test the sensitivity of the solution to changes in the operational parameters. As Lee points out [20, p. 1621, if the solution is extremely sensitive to variations in specific parameters, more care must be maintained in developing these critical values. Conversely, if the solution is relatively insensitive to parameter variation, resources may be applied elsewhere in the analysis procedure. In addition, by establishing specific ranges on the parameter values, the analyst can be confident of the stability of the solution during the daily operational variations. Ignizio [ 16, p. 11151 points out that a given linear goal programming model may be subjected to a post optimality analysis in much the same fashion as a continuous linear program. He goes on to reference both his own text [lS] and Lee’s text [20] as sources of how this analysis may be conducted. More recently, Steuer [14, p. 298-J points out that in preemptive goal programming, analyst often conduct selectivity analysis by interchanging priorities. This rotation of priorities allows the analyst to review the effects of the ordering of the preemptive weights. Steuer [ 14, p. 2981 further suggests this analyst be carried out by resolving the goal program with the new priority specification. The sensitivity analysis of the goal assembly line model is further restricted by the integer nature of the decision variables. Nauss [21] has provided one of the early studies dedicated to sensitivity analysis in integer programs. Nauss suggests parameterizing the integer model, and looking for efficient ways to resolve the parameterized model. Combining all of these works [15, 16,20-223 appears to be the most effective way to pursue a sensitivity analysis of the zero/one goal programming assembly line balancing model. Actual results will be highly dependent upon the input values of the parameters and the structure of the given operational model, however. To illustrate the suggested approach, an analysis of the objective of Model 1 has been made, as well as the parameterization of one of the constraints of the model for a given objective. In rotating the priorities of a preemptive goal program, there will be p! different rotations for a
517
A goal approach to assembly line balancing 2
5
6
2
wo&tations 4 GOdS Problem 2.1
Problem 2.2
Goal 1
Limit Wotitations to Three
Goal 2
Limit Tasks per Workstation to Four
Goal 3
Limit Tie
Goal 1
Limit Workstations to Three
Goal 2
Compatible Zoning of Tasks 5 and 7; Incompatible Zoning of Tasks 2 and 3
Goal 3
Limit Time Cycle to Eighteen
Cycle to Sixteen
Fig. 2. Model 2 [19, p. 631 precedence relations with task times.
model with p priorities [14, p. 2983. Such an analysis has been carried out on the example model in the Appendix. Table 3 summarizes the results of this analysis. The rotation of the priorities for Model 1.3 highlights the inter-relation between the cycle time and the number of workstations. While the task limit goal (Goal C) is satisfied in all rotations, it is not possible to simultaneously satisfy the workstation limit (Goal A) and the cycle time limit (Goal B). The various rotations offer the decision maker a spectrum of solutions to choose from. The final selection will hinge upon the decision makers preference (or abhorrence) of a longer cycle time or more workstations. To further investigate these choices, a sensitivity analysis was undertaken on the right-hand side of the workstation goal using Model 1.3, Problem 1. Table 4 summarizes the results of this analysis. Three possible workstation limits were reviewed. At a limit of two workstations, it was not possible to satisfy either the workstation limit or the cycle time limit. When the workstation limit was increased to three, that goal was met, but the cycle time goal was exceeded. Finally, at a workstation limit of four, all goals can be met. The analyst, in conjunction with the decision maker, might wish to extend the sensitivity analysis to other constraints. The actual direction of the analysis would be dictated by the operational setting. While sensitivity analysis on a zero-one goal programming model is more cumbersome than on a linear goal program or a linear program, it can provide the analyst with a wealth of critical information. This goal approach does suffer from the weaknesses of all general zero-one approaches. As the problem increases in size, the computational time will also grow. Unless specialized structures can be exploited, the problem will eventually become intractable. The goal model does, however, provide the analyst with a method to model varied operational settings. This modeling flexibility comes at a potential computational cost, however. Clearly, this cost should be balanced against any modeling gain. It would appear that the goal approach should be particularly useful in the planning stages of a line balance development.
RICHARD F. DECKROand SARANGAN RANGACHARI
528
M=f
Zero-One Variables: Assignment 83 workstations 5 GO& Problem 3.1
Problem 33
Goal 1
Limil Number of Workstations to Four
Goal 2
Minimiu: Task Splitting of T&s 8 and 12
Goat5
Limit Number of Tasks at Each Wwkstation to Five
Gull
Limit Number of Tasks at FBiehW~~tati~
Goat2
Limit Numberof Workstations to Four
Goa 3
M~~rni~ Workstatior~Expense
to Five
Fig. 3. Model 3 [H, p. 651 precedence relations with task times. Table 2. Exampk problems results GtXP
Problem
status
CPU? time (sccl
Final non-zero vanables
C
519
A goal approachto assemblyline balancing Table 3. Scnsitivicy analysis on
Problem No.
Priority 1
Priority 2
Priority 3
1
A
B
C
2
B
A
C
3
B
C
A
4
A
C
B
5
C
B
A
6
C
A
B
rotationof pnoritin-Model
Non-zero dccislon variables ~,,,x,,x,..~,,x53~64.~?5~~ss~~,,x,.~5 ~,,.~,,.~,,.~4,xs1Js5.~~s~s5~9s~l.~3.~4~5 X,*.XI~.Xs,.XI3.X52.~6~.~75~ss.~9~.~1.~3.~4.~s ~,,,~,,.~,,~l,.~,,.~64.~,,,x,,sr,,x,x,,~s ~,,.~,,.~,,sc4,~s,x,,,x,,sr,,,x,,x,.~95~~.~3~4~~s ~**r~11~~34.~41r~s3.~64~~15~ss.~95~~z.~~.~s
1.3 Unsatisfied priority/ goal
Cycle time
Workstations
2/B
16
3
2/A
15
4
3/A
I5
4
3/e
16
3
3/A
15
4
318
16
3
Goal A: limit number of workstations 10 no more than 3 Goal B: hmit cycle time to no more than IS. Goal C limit task per workstatIon lo no more than 3
Table 4. Sensitivity analysis on RHS of workstation goal Model 1.3 (pnority A, B, C), WorkstatIon hrmt 2
Non-zero decision variables X~,.X2,.X~r,X.,,Xs3.~s~.x,s.~*s.~~s.~,.~..~5
3
X,,.Xll.X3I.XII.X53.~64.~75~~5~95.~1.~4~5
4
~,,.~,2.~,,~.,.~5,.~6~~75.~~5~95.~2~3~4~5
Problem 1
Unsatisfied goals
Cycle time
A, B
16
3
16
3
15
4
B -
Workstatmns
Goal A: lirmt number of workstatIons. Goal B: hnnt cycle time to no more than IS. Goal C: hmit tasks per workstatIon lo no more than 3
CONCLUSION The approach discussed above for the goal modeling of the assembly line balancing problem allows the modeler and/or decision maker greater latitude in creating a line balance. The goal approach allows the modeler to consider multiple criteria in selecting a preferred schedule. In addition, through the use of weights and/or the altering of preemptive priorities, one can entertain various trade-offs among the criterion. Such flexibility should prove to be particularly useful in the planning stages of a line balancing. The use of the Patterson-Albracht [ 133 variable definition had aided in reducing the total number of variables and constraints required to model the problem. The Patterson-Albracht model can accommodate all of the goals outlined by Gunther et al. [2]. Even with these reductions, however, this approach has some of the same limitations that previous zero-one approaches to the line balance problem have had. The models can get very large very quickly. The goal model will have all the zero-one variables required by the single objective model and the additional deviational variables required to model the goals. However, if a goal can be satisfied, it is not necessary to spend computational time on establishing optimality. Once the deviational variables at a goal are driven to zero, no further computation is required at that goal [3]. In this paper, we have provided a goal based approach to model line balancing problems. Such a formulation allows the decision maker greater flexibility in considering alternatives. While the existence of an optimality condition should make large formulations more computationally tractable, the approach does suffer from the weaknesses of all general zero-one programs. The goal approach developed here for a generalized model may be used as a basis for future research which exploits the network structure. Acknowledgements-The authors wish to thank the editor and the reviewers of this paper for their useful insights and patience. We also wish to thank Mr Per H. Grlmsrud for his assistance in running these examples.
REFERENCES 1. S. Ghosh and R. Gagon, A comprehenslve
literature review and hierarchical taxonomy for the design and balancing ofassembly lines. Working Paper WPS-86-4, College of Administrative Science, The Ohio State University. Jan. (1986). 2. R. E. Gunther, G. D. Johnson and R. S. Peterson, Currently practiced formulations for the assembly line balance problem. J. OpnsMgmt 3.209-221 (1983).
RICHARDF. DECKROand SARANGANRANGACHARI
520
3. R. F. Deckro, J. E. Hebert and E. P. Winkofsky, Multiple criteria job-shop scheduling. Computers Opns Res. 9,279-285 (1982). 4. M. E. Salveson, The assembly line balancing problem. J. lndust. Engng 6, 18-25 (1955). 5. E. H. Bowman, Assembly line balancing by linear programming. 0pn.r Res. 8, 385-389 (1960). 6. K. D. Lawrence and J. S. Burbidge, A multiple goal linear programming approach for coordinated production and logistic planning. Inc. J. Product. Res. 14, 215-222 (1976). 7. S. M. Lee, E. R. Clayton and B. W. Taylor, A goal programming approach to multi-period production line scheduling. Computers Opns Res. 5, 205-211 (1978). 8. L. J. Moore and B. W. Taylor, Analysis of a tram-shipment problem with multiple conflicting objectives. Computers Opns Res. 5, 39-46 (1978).
9. S. Rangachari, Assembly line balancing using multiple objective zero-one integer programming. Plan B Paper for the MS in Industrial Management, The University of Wyoming, Dec. (1984). 10. S. H. Zanakis and S. K. Gupta, A categorized bibliographic survey of goal programming. Omega 13,21 l-222 (1985). 11. C. Romero, A survey of generalized goal programming (1970-1982). Eur. J. Opt Res. 25, 183-191 (1986). 12. R. F. Deckro, Balancing cycle time and workstations. f I E Trans. 21, 106-l 11 (1989). 13. J. II. Patterson and J. J. Albracht, Assembly-line balancing: zero-one programming with Fibonacci search. Opns Res. 23, 166-172 (1975). 14. R. E. Steuer, Multiple Criteria Optimization: Theory. Computation and Apphcation. Wiley, New York (1986). IS. J, P. Ignizto, Goof ~rogrummi~g and Extensions. Heath, Lexington. MA (1976). 16. J. P. Ignizio, A review ofgoal programming: a tool for multiobjective analysis. J. Opl Res. Sot. 29.1109-i 119 (1978). 17. R. F. Deckro and J. E. Hebert, Polynomial goal programming: a procedure for modeling preference tradeoffs. J. Opns Mgmt 7, 149-164 (1988). 18. L. A. Johnson and D. C. Montgomery, Operations Research in Production Planning, Scheduling and Inventory Control. Wiley, New York (1974). 19. F. M. Tonge, A Heuristic Program fir Assembly Line Balancing. Prentice-Hall, Englewood Cliffs, NJ (1961). 20. S. M. Lee, Goal Programming for Decision Analysis. Auerback, Philadelphia, PA (1972). 21. R. M. Nauss, Parametric Integer Programming. University of Missouri Press, Columbia, MO (1979). 22. C. A. Markowski and J. P. Ignizio, Duality and transformations in multiphase and sequential linear goal programming. Computers Opns Res. 10, 321-333 (1983).
23. R. B. Chase, Survey of paced assembly lines. Indust. Engng 14-18 Febr. (1974).
APPENDIX Model 1.3. Problem 1 Variables
xii = 1 if task i is assigned to workstation j 0 otherwise xj = 1 if workstation j is used 0 otherwise PW= overachievement of workstation limit NW= underachievement of workstation limit PC = overachievement of cycle time limit NCI = underachievement ofcycle time limit for workstation j PTj = overachievement of task limit for workstation j NT, = underachievement of task limit for workstation j Maximum cycle time is 20. Goafs
Priority 1: limit number of workstations to no more than 3. Priority 2: limit cycle time to no more than 15. Priority 3: limit task per workstation to no more than 3. Modes Mm
P,(PW), P,(PC), PJPT,
+ PT, + PT, + PT., $ PT,)
s.t. Assignment constraints: x11 +x,2+xl3
=
x,,+x22+~23+x24 x3l
+x32
+x33
+x34
x41
f-%2
+x43
+-x44
x5,+X32+X53+X54 x32
= =
+x63
+
xb4
1
=: =
=
1 1 1
x,,+x,,+x,, -1 x,,+xg*+xg> = 1 xg3 + xpq + xgs = 1. Precedence constraints: lx,, +2x,, + 3x,3 - lx,, -2x,, - 3x,, - 4x54 lx,, +2x,, + 3x,a - lx,, -2x4r3x,3 -4x,, lx,, + 2x,, + 3X,, +4x 1,,,+2rnt3~,,+4r::~::::~~~~~I:~~:-~~~~~~
A goal approach to assembly line balancing Ix,, + 2x42 + 3x,, + lx,, + 2x52 + 3x33 + lx,, + 2x,2 + 3x,, + 2x,, + 3X63+4x,, 3x,, +4x,, + 5x,, 3x,, +4x,, + 5x,, -
4x,, 4x,, 4x,, 3x,, 3x,3 3x9,
- 2x,, - 2x,, - 3x,, -4x,, -4x,, -4x,,
- 3x,, -4x,, - 3x,3 -4x,, -4% - 5x,, - 5x,, - 5x,, - 5x9,
< 0
<0 <0 SO GO SO.
Workstation: ~1,+~,,+~31+~4,+~,,-5~, x,2 +x21 +x32 + x42 +x52 - 5x, x,3 +x23 +x33 +x43 +x63 + x,3 +x ,,+x,,-9x,% x24 + x,g + x54 + X64+ x,4 + xgq + .xg‘$- lx, x,5 + xg5 + xgs -3x,
GO
GO SO.
Workstation goal: x, +x,+x,+.u,+x,+NW-PW=3. Cycle time goals. 4x,,+5x,,+6x3,+7x,,+5x,,+NCI-PC=15 4x1, + 5x2, + 6x3, + 7x4, + 5x5, + 10x,, + NC2 - PC = 15 4x,, + 5x13 + 6x33 + 7x,, + 5x,3 + 10x,, + 6x,, + ix,, +4x,, + NC3 - PC= 15 5x,, + 6x,, + 7x,, + 5x,, + lox,, +6x,, + lx,, +4x,, + NC4 - PC = 15 6x,, + 1x8,+4x,,+ NC5-PC= 15. Task goals: “~,+~~~+x~~+x~,+x~~+NTl-PTl=3 ~12 + ~22 + x32 + xd2 + xS2 + xs2 + NT2 - PT2 = 15 -xl3 + XZJ+ x33 + ~4~ + xs3 + xe3 + x,~ + xs3 + xg3 + NT3 - PT3 = 3 x,,+x,,+x,,+x,,+x,,+x,,+x,,+x,,+NT4-PT4=3 x,~ + xss + xgs + NT5 - PT5 = 3.
Limit on cycle time: PC65 x,,=o or 1 xj=o or 1 all other variables are non-negative.
521