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A bicriterion objective for levelling the schedule of a mixed-model, JIT assembly process
Mathl.
Pergamon
Comput. Modelling Vol. 20, No. 2, pp. 123-134, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/94 $7.00 + 0.00
0895-7177(94)00109-x
A Bicriterion Objective for Levelling the Schedule of a Mixed-Model, JIT Assembly Process G. STEINER Management Science and Information Systems Area Faculty of Business, McMaster University Hamilton, Ontario, L8S 4M4, Canada S. YEOMANS Management Science Area Faculty of Administrative Studies, York University North York, Ontario, M3J lP3, Canada Abstract-A
bicriterion objective for levelling the schedule of a mixed-model, Just-In-Time assembly process is examined. The two objectives considered are: minimizing the sum of deviations from a perfectly level schedule (min-sum), and minimizing the maximum deviation (min-max). This bicriterion approach combines the desirable properties of both criteria. It is shown that several Pareto optimal solutions may be efficiently determined. Furthermore, the in-core storage required to solve the bicriterion objective subject to a constraint on the maximum deviation is shown to be significantly lower than the storage required for the ‘standard’ min-sum problem, because there are fewer edges in the associated graph. Keywords-Just-In-Time objectives.
systems,
Smooth
schedules,
Matching
in bipartite
graphs,
Bicriterion
1. INTRODUCTION Mixed-model assembly processes with negligible switch over costs between products allow for diversified small-lot production; for example, the mixing of models (products) on a car assembly line. Just-In-Time (JIT) production methods can be used to control such assembly systems. The goal of these methods is to satisfy customers’ demands for a variety of products without holding large inventories or incurring large shortages. One problem that must be solved for the effective utilization of these systems is to determine the sequence in which different products are scheduled on the assembly process [l]. The most important goal of’s JIT production system is to keep the quantity of each part used by the assembly process as constant as possible per unit time [2]. Hall [3] refers to this as levelling or balancing the schedule. Miltenburg [4] provided a nonlinear integer programming formulation of the level scheduling problem for mixed-model assembly processes operating under JIT production methods which possessed a ‘sum of deviations’ (min-sum) objective function. Kubiak and Sethi [5,6] subsequently developed an optimization algorithm for this min-sum formulation that runs in polynomial time in the total number of products produced, i.e., the size of the input data. They This research was supported in part by the Natural Sciences and Engineering Research Council under Grant A1798. The authors would like to thank the Guest Editor and two anonymous referees for many useful suggestions which led to a clearer and more concise paper. Typeset 123
by A+@-TEX
G. STEINER AND S. YEOMANS
124
show that when the objective function is represented by penalties for deviation from the most even but unrealizable (i.e., fractional) distribution of demand and if these penalty functions are non-negative, symmetric and convex, then the problem can be reduced to an assignment problem. Steiner and Yeomans [7] d eveloped an efficient optimization procedure for an alternative, min-max objective of the level schedule problem. An extensive review of the state of the art in mixed-model, JIT, level scheduling problems can be found in [8] (see also [9]). In this paper, a new bicriterion model, which simultaneously incorporates both the min-sum and min-max objectives, is presented. Problem formulations with bicriterion objectives are not new. Gupta and Kyparisis [lo], in a survey of scheduling research, indicated that many scheduling problems of the future are likely to incorporate multiple objectives in their formulation. Examples of recent scheduling research which have used multiple objectives are [ll-161. Scheduling mixedmodel, JIT facilities with more than one objective, however, has not been considered previously. By exploiting certain desirable properties introduced in [7], it will be shown that, subject to a bound on the min-max objective, efficient computational procedures can be developed for determining several Pareto optimal solutions to this new bicriterion problem. A bicriterion solution to the level schedule problem is Pareto optimal if no other assembly sequence exists which has an objective value that is at least as good in both of the criteria and strictly better in at least one of them [17, p. 581. Q uit e surprisingly, the procedure for finding a Pareto optimal solution, under the constraint that the value of the min-max objective does not exceed the upper bound proved for it in [7], has an improved computational complexity in comparison to the algorithm for finding the min-sum solution in [5,6]. It will also be shown, for the first time, that a good upper bound exists on the optimal objective function value for two particular formulations of the minsum scheduling problem. If either of these commonly used min-sum objectives is subsequently implemented in conjunction with the bicriterion model presented, then useful bounds on the solutions to these bicriterion problems are readily established.
2. MATHEMATICAL MODEL AND REVIEW OF PREVIOUS RESULTS Assume that there are n products to be produced within the planning horizon with demands d,, for a total demand of D = ~~=, di units. If it is assumed that one time unit is di,dz,..., required to process each copy of each product, then an implied time horizon of D time units can be inferred; where one copy of product i, i = 1,2,. . , n will be produced in each time period. If ri = di/D then the level scheduling objective is to maintain the total production per time period as close to ri as possible, i.e., ideally kri units of product i should be produced in the first k time periods (k = 1,2,. . . D). LettingZik,i=l,2 ,..., n,k=l,2 ,..., D, be the total production of product i in time periods 1 through k, then the bicriterion model can be written as: minimize 2 = m,;x ]xik - kr,l , S =
9 2 F, (Xik k=l
i = 1,2 ,...,
72, k = 1,2,.
. , D,
kri) ,
i=l
= k,
subject to kz,
k=1,2
,...,
D,
(Pl)
i=l Zi(k-1)
5
zik,
i =
1.2 ,...,
zio = 0,
i=1,2
5%~
i =
=
4,
xik > 0 and integer,
n,
,...,
12,
1,2 ,...,
72,
i = 1,2 ,...,n,
,...,
D,
1,2 ,...,
D.
k=1,2
k=
Levelling JIT
Assembly
Process
125
convex function with minimum F,(O) = 0, i = 1,2,. . . , n, for deviafunctions that have been considered for 5’ in the literature [4-6,191
where Fi(.) is a unimodal, tion 0 [18]. Two min-sum are 5
2
k=l
i=l
FF(xik-kri)2.
and
1zilc - kril
k=l
z=l
The constraints in (Pl) ensure that exactly k units are scheduled in periods 1 through k and that for each product either one unit is scheduled in a given period or else is not scheduled at all in the period. The min-sum
objective
introduced
by Miltenburg
endeavours
to produce
a “smooth”
schedule
on the average by ensuring that the total deviation of a production sequence from a perfectly level schedule for all products over all time periods is low. It does not preclude, however, the possibility of relatively large deviations occurring for some products in certain time periods. In contrast, the min-max objective seeks a “smooth” schedule in every time period by requiring that all production be maintained as closely as possible to the desired level. This objective possesses an applicable physical interpretation since it provides the maximum overproduction or underproduction (the maximum inventory or shortage) from the desired level of production that occurs at any time during the schedule. By considering both objectives simultaneously, the production sequence created will be smooth for all products in general and the deviation in the worst case for any product will never be more than the value obtained for the min-max objective. A method is developed in [5,6] whereby the min-sum level schedule problem could be represented using an assignment problem formulation, and hence, solved to optimum in 0(D3) time. This method can be summarized in the following way. Let (i, j) refer to the jth copy of product i, i = l,..., n,j = l)..., d,. Define 2; = [kijl ([XT] is th e nearest integer not less than z, i.e., the ceiling of X) to be the ideal location in the final production sequence for (i,j), where kzj is the unique crossing point satisfying
That is, k,j represents the point of intersection between the function, function, Fi, for the preceding copy (i, j - l), j = 1,. . , di.
Fi, for copy
(i, J’) and the
Let Cijk be the cost attributable to placing (i, j) in the kth position of the final production sequence. If (i,j) is sequenced in its ideal location (i.e., k = Zz>) then Cijk = 0. If (i,j) is placed too soon in the production sequence (i.e., k < Zz’,) then inventory costs $+ are incurred in the positions from p = k to p = 2%; - 1. If, however, (i,j) is produced too late (i.e., k > Z&) then shortage costs $ijp are incurred from p = Zzj to p = k - 1. If these inventory and shortage costs are calculated as tiijp =
IF,(j -
Pri)
- F,(j - 1 -
Prt)I
i
1
=
j=
l,...,n, l,...,d,,
p=l,...,D, then the penalties for each copy (i,j) k=l,...,Darecalculatedas
to be assigned
z,*J -
c
I
p=k
ci.yk
=
to any location
1
hip,
k < Z;, k = Z&
0,
k-l C
P=z,*l
hip,
k
>
ZZ;.
in the production
sequence
k,
G. STEINERAND S.
126
Defining an indicator variable as 1, if
YEOMANS
i,j
is sequenced in location k,
0, else, then the assignment problem is
k=l
22
i=l
yjk
j=l
=
1,
k=1,2
,...,
i=1,2
,.‘.,
D,
subject to
i=l j=l 2, Y& = 1,
P4
72, j=l,...,di.
k=l
It is proved in [5] that an optimal sequence for the level schedule problem can be constructed from any optimal solution to the above assignment problem in O(D) time. By induction, it is shown that the implied ordering for the copies of each product will not be violated in at least one production sequence which is optimal for the assignment problem (i.e., the (j + l)st copy of product i will not appear before the jth copy in an optimal sequence). The following theorem regarding the ordering of the copies has important consequences for the bicriterion problem. THEOREM
1. [5] There is always a solution for (P2) which satisfies the ordering constraints given
in (Pl). In [7], a target, T, is set for the value of the min-msx objective and a test is performed to determine if a feasible production sequence (i.e., a sequence in which the min-max objective value does not exceed the target) can be constructed for the given T. If this sequence exists, then call T feasible. As in [5,6], each product is separated into its individual copies. It is shown that the value of T restricts the positions in the production schedule where each copy of product may be sequenced. For the jth copy of product i, E(i, J’) and L(i, J’) are defined to be the earliest and latest starting times possible in the final production sequence such that its “attributable” deviation does not exceed the target. E(i,j) and L(i,j) are calculated as the unique integers in the respective intervals
0; [j -T]
and
0;
1<
E(i,j) <
[(j - 1) +Tl - 1 < L(i,j)
5
(1)
(+) z [(j -
1) +T].
(2)
For a given T, early and late starting times can be calculated for each copy of each product in a one pass procedure and, hence, determined in O(D) time. This process can be visualized from Figure 1 by moving along the target line, T = 1, from left-to-right and finding where this line intersects the deviation curves of each copy. Appropriate integer adjustments, consistent with (1) and (2), are then made to these intersection points. The early and late starting times and the feasible starting intervals [E(i, j), L(i, j)] for each (i,j) are shown at the bottom of the figure. The test for the feasibility of T can be represented as a matching problem in a bipartite graph G(T) = (Vl,V2,E). Let VI = {O,l,. . . , D - 1) represent the starting times and let Vz correspond to the copies of each product. Construct an edge between k E VI and (i, j) E V, if (i,j) may start at time k. A bipartite graph is defined to be VI-corwez if together (i, 4) E E and (k, q) E E with i, k E VI and i < k implies that (p, q) E E for i 5 p < k [20]. Since each (i,j) E Vz is incident exactly to the points in [E(i,j), L(i,j)] c VI, the bipartite graph constructed above
Levelling JIT Assembly Process
127
PRODUCT I (i L II 4 = 7,r, = 7f20
T=l
k
0 W.1)
L(l.l) I E(l2) Early and Late Starting Times for Each Copy
Figure 1. Feasible starting times for the individual copies of product target value of T = 1.
i = 1 with a
A matching in a graph is a subset M of edges such that no two edges in M are incident to the same node [21]. A matching incident to every vertex is a perfect matching. Finding a feasible sequence in the level schedule problem is analogous to finding a perfect matching in the bipartite graph G(T), with the additional property that lower numbered copies of a product are always matched to earlier starting times than higher numbered copies. Such a matching will be referred to as order preserving. If an order preserving perfect matching can be found in the convex bipartite graph then the corresponding target value is feasible. Steiner and Yeomans [7] prove that if a perfect matching exists in G(T), then an order preserving perfect matching also exists and show how this matching problem can be solved in O(D) time. is Vi-convex.
THEOREM
2.
[7] Lower and upper bounds on the optimal
value of the min-max
objective
function
are 1 - rmax and 1, where rmaX = maxi {ri}.
Of particular significance is the upper bound, which implies that a feasible schedule always exists such that the actual production at no time deviates from the desired level of production for any product by more than one unit. By searching for feasible targets within the bounds, Steiner and Yeomans [7] prove that the optimal production schedule can be constructed in no more than 0( D log D) time. COROLLARY
1. Upper bounds on the optimal min-sum objective
and squared functions
respectively.
function
values for the absolute
are,
F
9
k=l
i=l
lxik - lcril 5 nD
and
F
T
k=l
z=l
(xik - kr,)2 < nD,
STEINERANDS. YEOMANS
G.
128
PROOF.
By Theorem
objective
is less than
the min-max
objective
summations Hence,
schedule
always exists
Use this sequence
of this assembly
sequence
for the first time,
an upper bound
which have been considered
is identical
is no greater
previously
each product
of these penalty
is separated
Let CE(i,j) and latest deviation and (2). that
calculations
strictly
of the assignment
could be considerable
to the storage
of both objectives
completion
times
“attributable” By simply
the min-max
I used min-
the value of this
for
of starting
If an order preserving
perfect
matching
times,
for T has a significant bipartite
contains
such that
matching
problem
can be constructed impact
graphs generated
substantially
the
from (1)
can be found in G(T),
must be the case that 2 5 T. The value selected G(1)
sequence
an equivalent
instead
Clearly
are the earliest
values can be easily calculated
times
T = 1 and T = 2, respectively.
memory
burden.
(i, j) in the final production
2 and 3 show the convex
of in-core
and CL(i,j)
edges in G(T).
Figures
values would have to be
It can be shown that a concurrent
this storage
. . , D},
number
size. For instance,
portions
Then CE(i,j)
VI to VI = {1,2,
using completion
problem.
values.
decreases
= L(i,j)+l.
The total
of a practical
Significant
to it does not exceed T. These
renumbering
above,
of these penalty
values (the Czjk’s)
algorithm.
then l,OOO,OOOpenalty
algorithm.
substantially
possible
for problems
problem,
and CL(i,j)
= E(i,j)+l
described
Surprisingly,
copies in [5,6], D2 penalty
into its distinct
prior to the start of the assignment
implementation
must hold. for two commonly
in [4-6,191.
Since
in the above
MEASURES
prior to the implementation
if R = 10 and D = 1000 in a particular must be devoted
objectives.
1, every term
for both functions.
must be calculated
calculated
the value of the min-max
than
has been established
3. BICRITERION Because
such that
for the two min-sum
must be less than or equal to 1 and so both inequalities
sum objectives bound
2, a feasible or equal to 1.
to for
then
it
on the number
of
by the target
fewer edges than
values
G(2).
Producti 3
Times-
The demand for each product is;
lJ,=
7.d
2
‘h.d7=‘l.d
Each copy of a product is labeled by the copy it represents.
J
=z.d
s
=I
An edge joins a
copy vertex to a time vertex if that copy may feasibly start at that time.
Figure 2. Bipartite graph of feasible starting times for 5 products induced by a target value of T = 1. An alternate problem
representation
in a complete
of the min-sum
bipartite
graph.
This
assignment complete
problem graph
is as a weighted
would
have CE(i,
j)
matching =
1 and
Levelling JIT Assembly Process
129
Product i 3
Starting Times
2
The demand
for each product
Each copy of a product
d,=7,d
is;
is labeled
2
=6.d
3
4
=2,d5=1
An edge joins a
by the copy it represents.
copy vertex to a time vertex if that copy may feasibly
=4,d
start at that time.
Figure 3. Bipartite graph of feasible starting times for 5 products induced by a target value of T = 2.
CL(i,j) weight
= D for every calculated
matching
(i,j).
Each
penalty
of the assignment
for an edge in the graph.
A minimum
weight
in which the sum of the weights of the matched
of the edge weights
of any other
perfect
matching.
problem
would correspond
matching
to a
would be a perfect
edges is less than or equal to the sum
Trivially,
such a complete
bipartite
graph
would also be convex. In some G(T),
consider the consequences
choice of T, the number of calculated in the complete
If an order preserving
graph. matching
This matching
would have the property
max objective
is 2 5 T.
perfect
for the corresponding Since,
this bicriterion
for incomplete
bipartite
procedure graphs
matching
is addressed
function
an appropriate
exists in G(T), then the minimum graph could also be determined. value, S, such that the min-
of T, denote
S is a function
would provide a solution
With
reduced from the total required
that it provides the min-sum
in this case,
Hence, such an approach
CL(i,j)].
weighted bipartite
with a value of (S(T), Z 5 T) for the objective Before
only those edge weights (i.e., the penal-
weights can be significantly
weight perfect
as S(T).
of calculating
at each k E VI such that k E [CE(i,j),
ties, CQ~) for each (i,j)
to the bicriterion
this min-sum scheduling
value
problem
in (Pl).
in further
detail, a useful corollary
to Theorem
1
will be shown.
COROLLARY 2. If in some minimum weight perfect matching in G(T) two copies of a product are interchanged in order to restore their correct ordering in the corresponding production sequence, then the value of S(T) does not increase. PROOF.
production
Assume that the implied ordering sequence
set by the matching.
of the two copies (i, j) and (i, j + 1) is violated
CL(i,j
+ l)] # 4. H ence, it must be possible to interchange sequence. The weights on the edges in G(T) are identical assignment represent
formulation. possible
assignments
value of the objective identical
minimum
Thus,
function
both the initial in (P2).
By Theorem
for the assignment
weight perfect
solution
matching,
+ I),
(i, j) and (i, j + 1) in the production to those
calculated
and the solution
1, this
objective
in the
[CE(i, j), CL(i, j)] n [CE(i,j
This implies that
interchange
function.
S(T), will not increase.
after
for the min-sum the interchange
does not increase
Therefore,
the
the value of the I
G.
130
STEINERAND S. YEOMANS
On account of Corollary 2, it is possible, without any loss of optimality, to restrict the class of solutions for the minimum weight matching problem to those that are order preserving matchings (i.e., to those solutions which are feasible for the level schedule problem). The computational complexity issues of the bicriterion matching procedure proposed above can be addressed in the following manner. The check for a perfect matching in G(T) requires O(D) time as mentioned previously. For a given graph G = (V, E), Papadimitriou and Steiglitz [22, p. 2671 note that the min-sum assignment method (to find a minimum weight perfect matching) for weighted (not necessarily complete) bipartite graphs can be implemented in 0 (IV/ /El log [VI) time. By necessity, for a perfect matching to exist in G(T), it must be the case that ]E] > D. Since /VI = 20 in G(T), a minimum weight perfect matching in the graph could be determined in 0 ( IEl D log D) > 0(D2 log D) time. The correct ordering for any assignment could easily be restored in O(D) time by successive swapping operations. Hence, the time complexity of the bicriterion procedure depends strictly upon the time complexity of the minimum weight perfect matching algorithm and therefore upon (El. Because ]E] is determined by the value of T, the time complexity of the bicriterion algorithm must also depend upon T. It will be shown that for values of T 5 1, ]E] is significantly smaller than the edge set of a complete graph, thereby allowing for a redzlction in the time complexity of the bicriterion problem in comparison to that of the min-sum procedure. The solutions to these bicriterion problems will also possess many highly desirable properties which significantly add to the efficiency of the procedure. LEMMA 1. The number of edges created in G(T),
with T 5 1, is (E] 1. nD + 20.
PROOF. Trivially, the number of edges in a graph G(T) with T < 1 is fewer than in the graph G(1). Hence, consider the case in which T = 1. The number of edges incident to a given (i,j) is CL(i,j) - CE(i,j) + 1. Using T = 1 in (1) and (2), and with the definitions of CE(i, j) and CL(i, j), + 0 2
+ 1 > L(i, j) - E(i, j) = [CL(i, j) - l] - [CE(i, j) - l] )
(3)
and therefore, (
;
z>
+ 1 r: CL(i,j)
- CE(i,j).
(4
From (4), for the total number of edges in the convex, bipartite graph G(T)
IEI = fg =gd,
~C~~~~_d - CE(Gj) + 116 g [(g)
+2]
[ (9)
with T = I
+ 21
=nD+2D.
(5) I
Consider, again, the example in which n = 10 and D = 1000. With T < 1, only 12,000 penalties would need to be calculated for the weighted matching problem in G(T) which is substantially fewer than the l,OOO,OOO required in the complete assignment formulation. Hence, in-core memory requirements for this instance of the problem are reduced considerably. LEMMA 2. The min-sum assignment solution for the convex can be determined in O(nD2 log D) time.
bipartite graph, G(T)
with T 5 1,
PROOF. By Lemma 1, the number of edges in the graph is JE] < nD + 20. Calculating the edge weights requires O((E]) = O(nD) time. The min-sum assignment algorithm for weighted bipartite graphs requires O(lVl [El log IV]) t ime. Since JV( = 20, this optimal assignment can therefore be determined in 0(nD2 log D) time. I
LevellingJIT AssemblyProcess LEMMA 3. A bicriterion JIT level schedule with a solution (S(l),
131
2 5 T = 1) can be found in
0(nD2 log D) time. PROOF. Set T = 1. By Theorem 2, the production sequence could be determined By Lemma 2, this schedule can be found corresponding to this particular min-sum
bicriterion
objective
value of this sequence
graph G(1) contains a feasible schedule. A min-sum for G(1) which would have the objective value, S(1). in O(nD2 log D) time. The min-max objective value solution to G(1) is necessarily 2 5 T = 1. Hence, the is (S(l),
2 5 T = 1).
I
Using T 5 1 ensures that the bicriterion solution possesses several useful properties. Since Z 5 T = 1, the production sequence created in Lemma 3 retains the highly desirable property that the production of any product never deviates from its ideal production level by more than one unit. Furthermore this sequence is the optimal assignment, in the min-sum sense, of all sequences with this property. Thus, the level schedule created will be good on the average and also in the worst case. However, while the sequence found above possesses these useful properties, it can be observed that S* I S(1) and Z* < T = 1; where 5” and Z* are the optimal values for the min-sum and min-max objectives, respectively. Hence, the sequence may not be optimal with respect to either measure. With some adjustments to the method of Lemma 3, a Pareto optimal solution can easily be determined using a similarly efficient approach. THEOREM 3.
The Pareto optimal solution (S(Z*), Z*) can be determined in O(nD2 log D) time.
algorithm in O(Dlog D) time to determine Z”. Then by Theorem 2 and Lemma 2 with T = Z*, the perfect matching which provides the min-sum objective value S(Z*) could be constructed in at most 0(nD2 log D) time. Because the min-max objective is Z*, this production sequence must be Pareto optimal. Therefore, the overall complexity for determining this Pareto optimal solution is O(nD2 log D) time. I PROOF. Run the min-max
Thus, a Pareto optimal solution for the bicriterion JIT level scheduling problem can be found in O(nD2 log D) time. This solution procedure runs, in general, far faster than O(D3) since, complexity for in practice, it is generally the case that 7~ < D. Hence, the computational determining a Pareto optimal solution is lower than that for determining an optimal min-sum solution alone. Furthermore, with only minor modifications to this approach, all of the Pareto optimal solutions with Z 2 1 can be efficiently generated. The following algorithm can be used to determine all such Pareto optimal solutions with this property. Algorithm:
T +- 1 - T,,, . Generate the edge set for the convex bipartite graph corresponding to T. Denote this edge set by E(T). Determine if a perfect matching exists in this bipartite graph. If there is no perfect matching, then let T e T-t l/D and go to 2. If a perfect matching exists, then determine a minimum weight order preserving perfect matching in G(T). The corresponding production sequence will be Pareto optimal with Z = Z* and S = S(Z’). Let E = E(T), Smin = S(Z*) and go to 4. Let T + T + l/D. If T > 1, then stop-all required Pareto optimal solutions with Z 5 1 have been determined. Otherwise, go to 5. Generate the edge set, E(T), for the convex bipartite graph corresponding to T. If E(T) = E, then go to 4. Otherwise set E = E(T) and go to 6.
1. Initialize
2. 3.
4. 5.
Pareto
G. STEINERAND
132
S. YEOMANS
6. Determine a minimum weight order preserving perfect matching in G(T). If S(T) < Smin, then this sequence will be Pareto optimal with 2 = T and S = S(T), set Smin = S(T) and go to 4. Otherwise the sequence is not Pareto optimal. Go to 4. THEOREM 4. Algorithm Pareto determines the production solutions (S, Z), with 2 5 1, in O(nd,,,D’ log D) time.
sequences for all Pareto optimal
By Theorem 2, 1 - Tmax 5 Z* < 1. Since Z* = ]zik - Icri] for some i and Ic, DZ* must be an integer in the interval [D - d,,,, D]. By the way in which T is incremented, every integer in the preceding interval (and so every possible value for 2) is considered and there are only dmax iterations of the algorithm. Therefore, starting with the initial target of T = 1 - rmaX and PROOF.
by Theorem 2, the minimum weight perfect matching in G(T) determined by step 3 must have 2 = 2’. Hence, steps 1 to 3 must determine one Pareto optimal solution. Steps 4 to 6 determine if any other Pareto optimal solutions with 2’ < 2 5 1 exist. G(Z*) has a perfect matching and since G(Z) contains G(Z*), G(Z) also has a perfect matching. Because T increases monotonically, if the edge set of a graph has not changed from that of a previously generated graph, then its min-sum solution cannot have decreased and, therefore, the solution cannot be Pareto optimal. Conversely, if the edge set has changed, then a new Pareto optimal solution could exist only if the value of the min-sum objective, S(T), is less than the min-sum value of any previously considered solution, S’min. Step 5 determines whether G(T) contains more edges than any previously considered graph. Step 6 determines if G(T) contains a new Pareto optimal solution. The test for the matching feasibility in stages 1 to 3 and the generation of each edge set can be performed in O(D) time. A check requiring constant [i.e., O(l)] t ime quickly determines whether the edge set of a graph contains more edges than the previously generated graph. Calculating the min-sum weights for a given graph requires O(]E]) 5 O(nD) time, and therefore, a minsum solution can be determined in O(nD’log D) time. The complete algorithm requires d,,, iterations. Hence, the time required for determining all Pareto optimal solutions with the property that 2 < 1 is, O(nd,,D210gD). I This algorithm demonstrates that many Pareto optimal solutions can be efficiently found in O(nd,,, D2 log D) < O(nD3 log D) time. Determining all of these production sequences requires the addition of a factor of less than n log D to the time complexity of the min-sum, assignment procedure in [5,6]. Upper bounds have been shown for two commonly used min-sum objectives; the sum of squared and sum of absolute deviations. Since 2 5 1 in the Pareto algorithm, S(Z)
_< nD for either of these two formulations. Hence, upper bounds are readily established on
the objective objectives.
function values of the bicriterion solution using either of these common min-sum
4.
CONCLUSIONS
AND
EXTENSIONS
The level scheduling problem of an assembly process in a mixed-model, JIT environment has been examined. A bicriterion, min-max, min-sum objective, not considered previously, is introduced to control this system. By exploiting certain properties from the previously considered min-max problem, many efficiencies are incorporated into this new bicriterion problem. Of particular importance is the reduction of the in-core storage requirements of the bicriterion problem in comparison to those of the min-sum problem. An algorithm is presented for determining several Pareto optimal solutions to this problem. The min-sum criteria ensures that the level schedules created by this algorithm are good in general, while the min-max criteria ensures that these schedules are good in the worst case. The absolute function has been considered for 2, since its value provides a meaningful, physical interpretation and bounds on its optimal value have been established previously. This function
Levelling JIT Assembly Process
allowed for the calculation of the time and space requirements of the bicriterion also for the efficient determination of several Pareto optimal solutions.
133
procedure and
It would be possible to consider more general functions (pi’s) for 2 and to use a similar approach to that described in this paper to determine bicriterion solutions. However, such general functions would not necessarily possess such an applicable physical interpretation as the absolute function. Changing the function would necessitate that complexity results specific to the appropriate function be determined. A given bound, T, on a general min-max function could still be used to restrict the positions in the production schedule where each copy of product could be sequenced. The convex bipartite graph G(T) could still be constructed and the check for a perfect matching (i.e., for the feasibility of T) could be performed in O(D) time. The correct ordering for any assignment could, also, still be restored in O(D) time. However, since (E( 2 D and (VI = 20 for any perfect matching in G(T), determining the minimum weight perfect matching (and, hence, a bicriterion solution) would require O(IEjD log D) > 0(D2 log D) time and O(lEl) space. The value of [El would necessarily depend upon the specific min-max function selected, and thus, the time and space requirements would also depend upon the function chosen. Furthermore, this bicriterion solution would most probably not be Pareto optimal. Moreover, it is highly unlikely that Theorems 2-4 and Lemmas l-3 would continue to hold for a general function. Therefore, new bounds on the optimal value of the min-max objective function would have to be established. Complexity results for calculating an optimal solution to the min-max formulation alone would have to be determined (i.e., the time required to find the optimal min-max solution could conceivably require more than the 0( IE( D log D) time necessary for determining a bicriterion solution). Thus, any procedure for determining Pareto optimal solutions with a different min-max criteria would necessarily depend upon the specific function selected. These types of bicriterion problems will be the subject of future study.
REFERENCES A Heuristic algorithm for the assembly line model-mix sequencing problem to minimize the risk of stopping the conveyor, Int. J. Prod. Res. 1’7, 233-247, (1979). Y. Monden, Toyota Production Systems, Industrial Engineering & Management Press, Norcross, GA, (1983). R. Hall, Zero Inventories, Dow Jones-Irwin, Homewood, IL, (1983). J. Miltenburg, Level schedules for mixed-model assembly lines in Just-In-Time production systems, Mgmt. Sci. 35 (2), 192-207, (1989). W. Kubiak and S. Sethi, Optimal level schedules for flexible assembly lines in JIT production systems, working paper, Faculty of Management, University of Toronto, (1993). W. Kubiak and S. Sethi, A note on “Level schedules for mixed-model assembly lines in Just-In-Time production systems”, Mgmt. Sci. 37 (l), 121-122, (1991). G. Steiner and S. Yeomans, Level schedules for mixed-model, Just-In-Time assembly processes, Mgmt. Sci. 39 (6), 728-735, (1993). W. Kubiak, Minimizing variation of production rates in Just-In-Time systems: A survey, Eur. J. Opnl. Res. 66, 259-271, (1993). S. Yeomans, Optimal level schedules for mixed-model, Just-In-Time assembly systems, Ph.D. Dissertation, Dept. of Management Science and Information Systems, McMaster University, Hamilton, Ontario, (1992). K. Gupta and J. Kyparisis, Single machine scheduling research, Omega 15 (2), 181-203, (1987). L. Van Wassenhove and L. Gelders, Solving a bicriterion scheduling problem, Eur. J. Opnl. Res. 4, 42-48, (1980). R. Nelson, R. Sarin and R. Davies, Scheduling with multiple performance measures: The one machine case, Mgmt. Sci. 32 (4), 464-479, (1986). T. Sen, F. Raiszadeh and P. Dileepan, A branch and bound approach to the bicriterion scheduling problem involving total flow time and range of lateness, Mgmt. Sci. 34 (2), 254-260, (1988). U. Bagchi, Simultaneous minimization of mean and variation of Row time and waiting time in single machine systems, Opns. Res. 37 (l), 118-125, (1989). R. Daniels and R. Sarin, Single machine scheduling with controllable processing times and number of jobs tardy, Opns. Res. 37 (6), 981-984, (1989). R. Daniels, A multi-objective approach to resource allocation in single machine scheduling, Eur. J. Opnl. Res. 48 (2), 226-241, (1990). S. French, Sequencing and Scheduling, Ellis-Horwood, Chichester, West Sussex, England, (1982). M. Minoux, Mathematical Programming. Theory and Algorithms, John Wiley & Sons, New York, NY, (1986).
1. K. Okamura and H. Yamashita,
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. MCM
Z&Z-J
134
G. STEINER AND S. YEOMANS
19. R. Inman and R. Bulfin, Sequencing JIT mixed-model assemblly lines, Management Science 37 (i’), 901-904, (1991). 20. U. Derigs, 0. Goecke and R. Schrader, Bisimplicial edges, Gaussian elimination and matchings in bipartite graphs, In Graph Theoretic Concepts in Computer Science Proceedings West Germany ‘84, (Edited by U. Pape), pp. 79-87, Trauner Verlag, Lina, (1984). 21. C. Berge, Graphs, 2 nd ed., North-Holland Mathematical Library, Elsevier Science Publishers B.V., Amsterdam, (1985). 22. C. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, (1982).
Pergamon
Comput. Modelling Vol. 20, No. 2, pp. 123-134, 1994 Copyright@1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0895-7177/94 $7.00 + 0.00
0895-7177(94)00109-x
A Bicriterion Objective for Levelling the Schedule of a Mixed-Model, JIT Assembly Process G. STEINER Management Science and Information Systems Area Faculty of Business, McMaster University Hamilton, Ontario, L8S 4M4, Canada S. YEOMANS Management Science Area Faculty of Administrative Studies, York University North York, Ontario, M3J lP3, Canada Abstract-A
bicriterion objective for levelling the schedule of a mixed-model, Just-In-Time assembly process is examined. The two objectives considered are: minimizing the sum of deviations from a perfectly level schedule (min-sum), and minimizing the maximum deviation (min-max). This bicriterion approach combines the desirable properties of both criteria. It is shown that several Pareto optimal solutions may be efficiently determined. Furthermore, the in-core storage required to solve the bicriterion objective subject to a constraint on the maximum deviation is shown to be significantly lower than the storage required for the ‘standard’ min-sum problem, because there are fewer edges in the associated graph. Keywords-Just-In-Time objectives.
systems,
Smooth
schedules,
Matching
in bipartite
graphs,
Bicriterion
1. INTRODUCTION Mixed-model assembly processes with negligible switch over costs between products allow for diversified small-lot production; for example, the mixing of models (products) on a car assembly line. Just-In-Time (JIT) production methods can be used to control such assembly systems. The goal of these methods is to satisfy customers’ demands for a variety of products without holding large inventories or incurring large shortages. One problem that must be solved for the effective utilization of these systems is to determine the sequence in which different products are scheduled on the assembly process [l]. The most important goal of’s JIT production system is to keep the quantity of each part used by the assembly process as constant as possible per unit time [2]. Hall [3] refers to this as levelling or balancing the schedule. Miltenburg [4] provided a nonlinear integer programming formulation of the level scheduling problem for mixed-model assembly processes operating under JIT production methods which possessed a ‘sum of deviations’ (min-sum) objective function. Kubiak and Sethi [5,6] subsequently developed an optimization algorithm for this min-sum formulation that runs in polynomial time in the total number of products produced, i.e., the size of the input data. They This research was supported in part by the Natural Sciences and Engineering Research Council under Grant A1798. The authors would like to thank the Guest Editor and two anonymous referees for many useful suggestions which led to a clearer and more concise paper. Typeset 123
by A+@-TEX
G. STEINER AND S. YEOMANS
124
show that when the objective function is represented by penalties for deviation from the most even but unrealizable (i.e., fractional) distribution of demand and if these penalty functions are non-negative, symmetric and convex, then the problem can be reduced to an assignment problem. Steiner and Yeomans [7] d eveloped an efficient optimization procedure for an alternative, min-max objective of the level schedule problem. An extensive review of the state of the art in mixed-model, JIT, level scheduling problems can be found in [8] (see also [9]). In this paper, a new bicriterion model, which simultaneously incorporates both the min-sum and min-max objectives, is presented. Problem formulations with bicriterion objectives are not new. Gupta and Kyparisis [lo], in a survey of scheduling research, indicated that many scheduling problems of the future are likely to incorporate multiple objectives in their formulation. Examples of recent scheduling research which have used multiple objectives are [ll-161. Scheduling mixedmodel, JIT facilities with more than one objective, however, has not been considered previously. By exploiting certain desirable properties introduced in [7], it will be shown that, subject to a bound on the min-max objective, efficient computational procedures can be developed for determining several Pareto optimal solutions to this new bicriterion problem. A bicriterion solution to the level schedule problem is Pareto optimal if no other assembly sequence exists which has an objective value that is at least as good in both of the criteria and strictly better in at least one of them [17, p. 581. Q uit e surprisingly, the procedure for finding a Pareto optimal solution, under the constraint that the value of the min-max objective does not exceed the upper bound proved for it in [7], has an improved computational complexity in comparison to the algorithm for finding the min-sum solution in [5,6]. It will also be shown, for the first time, that a good upper bound exists on the optimal objective function value for two particular formulations of the minsum scheduling problem. If either of these commonly used min-sum objectives is subsequently implemented in conjunction with the bicriterion model presented, then useful bounds on the solutions to these bicriterion problems are readily established.
2. MATHEMATICAL MODEL AND REVIEW OF PREVIOUS RESULTS Assume that there are n products to be produced within the planning horizon with demands d,, for a total demand of D = ~~=, di units. If it is assumed that one time unit is di,dz,..., required to process each copy of each product, then an implied time horizon of D time units can be inferred; where one copy of product i, i = 1,2,. . , n will be produced in each time period. If ri = di/D then the level scheduling objective is to maintain the total production per time period as close to ri as possible, i.e., ideally kri units of product i should be produced in the first k time periods (k = 1,2,. . . D). LettingZik,i=l,2 ,..., n,k=l,2 ,..., D, be the total production of product i in time periods 1 through k, then the bicriterion model can be written as: minimize 2 = m,;x ]xik - kr,l , S =
9 2 F, (Xik k=l
i = 1,2 ,...,
72, k = 1,2,.
. , D,
kri) ,
i=l
= k,
subject to kz,
k=1,2
,...,
D,
(Pl)
i=l Zi(k-1)
5
zik,
i =
1.2 ,...,
zio = 0,
i=1,2
5%~
i =
=
4,
xik > 0 and integer,
n,
,...,
12,
1,2 ,...,
72,
i = 1,2 ,...,n,
,...,
D,
1,2 ,...,
D.
k=1,2
k=
Levelling JIT
Assembly
Process
125
convex function with minimum F,(O) = 0, i = 1,2,. . . , n, for deviafunctions that have been considered for 5’ in the literature [4-6,191
where Fi(.) is a unimodal, tion 0 [18]. Two min-sum are 5
2
k=l
i=l
FF(xik-kri)2.
and
1zilc - kril
k=l
z=l
The constraints in (Pl) ensure that exactly k units are scheduled in periods 1 through k and that for each product either one unit is scheduled in a given period or else is not scheduled at all in the period. The min-sum
objective
introduced
by Miltenburg
endeavours
to produce
a “smooth”
schedule
on the average by ensuring that the total deviation of a production sequence from a perfectly level schedule for all products over all time periods is low. It does not preclude, however, the possibility of relatively large deviations occurring for some products in certain time periods. In contrast, the min-max objective seeks a “smooth” schedule in every time period by requiring that all production be maintained as closely as possible to the desired level. This objective possesses an applicable physical interpretation since it provides the maximum overproduction or underproduction (the maximum inventory or shortage) from the desired level of production that occurs at any time during the schedule. By considering both objectives simultaneously, the production sequence created will be smooth for all products in general and the deviation in the worst case for any product will never be more than the value obtained for the min-max objective. A method is developed in [5,6] whereby the min-sum level schedule problem could be represented using an assignment problem formulation, and hence, solved to optimum in 0(D3) time. This method can be summarized in the following way. Let (i, j) refer to the jth copy of product i, i = l,..., n,j = l)..., d,. Define 2; = [kijl ([XT] is th e nearest integer not less than z, i.e., the ceiling of X) to be the ideal location in the final production sequence for (i,j), where kzj is the unique crossing point satisfying
That is, k,j represents the point of intersection between the function, function, Fi, for the preceding copy (i, j - l), j = 1,. . , di.
Fi, for copy
(i, J’) and the
Let Cijk be the cost attributable to placing (i, j) in the kth position of the final production sequence. If (i,j) is sequenced in its ideal location (i.e., k = Zz>) then Cijk = 0. If (i,j) is placed too soon in the production sequence (i.e., k < Zz’,) then inventory costs $+ are incurred in the positions from p = k to p = 2%; - 1. If, however, (i,j) is produced too late (i.e., k > Z&) then shortage costs $ijp are incurred from p = Zzj to p = k - 1. If these inventory and shortage costs are calculated as tiijp =
IF,(j -
Pri)
- F,(j - 1 -
Prt)I
i
1
=
j=
l,...,n, l,...,d,,
p=l,...,D, then the penalties for each copy (i,j) k=l,...,Darecalculatedas
to be assigned
z,*J -
c
I
p=k
ci.yk
=
to any location
1
hip,
k < Z;, k = Z&
0,
k-l C
P=z,*l
hip,
k
>
ZZ;.
in the production
sequence
k,
G. STEINERAND S.
126
Defining an indicator variable as 1, if
YEOMANS
i,j
is sequenced in location k,
0, else, then the assignment problem is
k=l
22
i=l
yjk
j=l
=
1,
k=1,2
,...,
i=1,2
,.‘.,
D,
subject to
i=l j=l 2, Y& = 1,
P4
72, j=l,...,di.
k=l
It is proved in [5] that an optimal sequence for the level schedule problem can be constructed from any optimal solution to the above assignment problem in O(D) time. By induction, it is shown that the implied ordering for the copies of each product will not be violated in at least one production sequence which is optimal for the assignment problem (i.e., the (j + l)st copy of product i will not appear before the jth copy in an optimal sequence). The following theorem regarding the ordering of the copies has important consequences for the bicriterion problem. THEOREM
1. [5] There is always a solution for (P2) which satisfies the ordering constraints given
in (Pl). In [7], a target, T, is set for the value of the min-msx objective and a test is performed to determine if a feasible production sequence (i.e., a sequence in which the min-max objective value does not exceed the target) can be constructed for the given T. If this sequence exists, then call T feasible. As in [5,6], each product is separated into its individual copies. It is shown that the value of T restricts the positions in the production schedule where each copy of product may be sequenced. For the jth copy of product i, E(i, J’) and L(i, J’) are defined to be the earliest and latest starting times possible in the final production sequence such that its “attributable” deviation does not exceed the target. E(i,j) and L(i,j) are calculated as the unique integers in the respective intervals
0; [j -T]
and
0;
1<
E(i,j) <
[(j - 1) +Tl - 1 < L(i,j)
5
(1)
(+) z [(j -
1) +T].
(2)
For a given T, early and late starting times can be calculated for each copy of each product in a one pass procedure and, hence, determined in O(D) time. This process can be visualized from Figure 1 by moving along the target line, T = 1, from left-to-right and finding where this line intersects the deviation curves of each copy. Appropriate integer adjustments, consistent with (1) and (2), are then made to these intersection points. The early and late starting times and the feasible starting intervals [E(i, j), L(i, j)] for each (i,j) are shown at the bottom of the figure. The test for the feasibility of T can be represented as a matching problem in a bipartite graph G(T) = (Vl,V2,E). Let VI = {O,l,. . . , D - 1) represent the starting times and let Vz correspond to the copies of each product. Construct an edge between k E VI and (i, j) E V, if (i,j) may start at time k. A bipartite graph is defined to be VI-corwez if together (i, 4) E E and (k, q) E E with i, k E VI and i < k implies that (p, q) E E for i 5 p < k [20]. Since each (i,j) E Vz is incident exactly to the points in [E(i,j), L(i,j)] c VI, the bipartite graph constructed above
Levelling JIT Assembly Process
127
PRODUCT I (i L II 4 = 7,r, = 7f20
T=l
k
0 W.1)
L(l.l) I E(l2) Early and Late Starting Times for Each Copy
Figure 1. Feasible starting times for the individual copies of product target value of T = 1.
i = 1 with a
A matching in a graph is a subset M of edges such that no two edges in M are incident to the same node [21]. A matching incident to every vertex is a perfect matching. Finding a feasible sequence in the level schedule problem is analogous to finding a perfect matching in the bipartite graph G(T), with the additional property that lower numbered copies of a product are always matched to earlier starting times than higher numbered copies. Such a matching will be referred to as order preserving. If an order preserving perfect matching can be found in the convex bipartite graph then the corresponding target value is feasible. Steiner and Yeomans [7] prove that if a perfect matching exists in G(T), then an order preserving perfect matching also exists and show how this matching problem can be solved in O(D) time. is Vi-convex.
THEOREM
2.
[7] Lower and upper bounds on the optimal
value of the min-max
objective
function
are 1 - rmax and 1, where rmaX = maxi {ri}.
Of particular significance is the upper bound, which implies that a feasible schedule always exists such that the actual production at no time deviates from the desired level of production for any product by more than one unit. By searching for feasible targets within the bounds, Steiner and Yeomans [7] prove that the optimal production schedule can be constructed in no more than 0( D log D) time. COROLLARY
1. Upper bounds on the optimal min-sum objective
and squared functions
respectively.
function
values for the absolute
are,
F
9
k=l
i=l
lxik - lcril 5 nD
and
F
T
k=l
z=l
(xik - kr,)2 < nD,
STEINERANDS. YEOMANS
G.
128
PROOF.
By Theorem
objective
is less than
the min-max
objective
summations Hence,
schedule
always exists
Use this sequence
of this assembly
sequence
for the first time,
an upper bound
which have been considered
is identical
is no greater
previously
each product
of these penalty
is separated
Let CE(i,j) and latest deviation and (2). that
calculations
strictly
of the assignment
could be considerable
to the storage
of both objectives
completion
times
“attributable” By simply
the min-max
I used min-
the value of this
for
of starting
If an order preserving
perfect
matching
times,
for T has a significant bipartite
contains
such that
matching
problem
can be constructed impact
graphs generated
substantially
the
from (1)
can be found in G(T),
must be the case that 2 5 T. The value selected G(1)
sequence
an equivalent
instead
Clearly
are the earliest
values can be easily calculated
times
T = 1 and T = 2, respectively.
memory
burden.
(i, j) in the final production
2 and 3 show the convex
of in-core
and CL(i,j)
edges in G(T).
Figures
values would have to be
It can be shown that a concurrent
this storage
. . , D},
number
size. For instance,
portions
Then CE(i,j)
VI to VI = {1,2,
using completion
problem.
values.
decreases
= L(i,j)+l.
The total
of a practical
Significant
to it does not exceed T. These
renumbering
above,
of these penalty
values (the Czjk’s)
algorithm.
then l,OOO,OOOpenalty
algorithm.
substantially
possible
for problems
problem,
and CL(i,j)
= E(i,j)+l
described
Surprisingly,
copies in [5,6], D2 penalty
into its distinct
prior to the start of the assignment
implementation
must hold. for two commonly
in [4-6,191.
Since
in the above
MEASURES
prior to the implementation
if R = 10 and D = 1000 in a particular must be devoted
objectives.
1, every term
for both functions.
must be calculated
calculated
the value of the min-max
than
has been established
3. BICRITERION Because
such that
for the two min-sum
must be less than or equal to 1 and so both inequalities
sum objectives bound
2, a feasible or equal to 1.
to for
then
it
on the number
of
by the target
fewer edges than
values
G(2).
Producti 3
Times-
The demand for each product is;
lJ,=
7.d
2
‘h.d7=‘l.d
Each copy of a product is labeled by the copy it represents.
J
=z.d
s
=I
An edge joins a
copy vertex to a time vertex if that copy may feasibly start at that time.
Figure 2. Bipartite graph of feasible starting times for 5 products induced by a target value of T = 1. An alternate problem
representation
in a complete
of the min-sum
bipartite
graph.
This
assignment complete
problem graph
is as a weighted
would
have CE(i,
j)
matching =
1 and
Levelling JIT Assembly Process
129
Product i 3
Starting Times
2
The demand
for each product
Each copy of a product
d,=7,d
is;
is labeled
2
=6.d
3
4
=2,d5=1
An edge joins a
by the copy it represents.
copy vertex to a time vertex if that copy may feasibly
=4,d
start at that time.
Figure 3. Bipartite graph of feasible starting times for 5 products induced by a target value of T = 2.
CL(i,j) weight
= D for every calculated
matching
(i,j).
Each
penalty
of the assignment
for an edge in the graph.
A minimum
weight
in which the sum of the weights of the matched
of the edge weights
of any other
perfect
matching.
problem
would correspond
matching
to a
would be a perfect
edges is less than or equal to the sum
Trivially,
such a complete
bipartite
graph
would also be convex. In some G(T),
consider the consequences
choice of T, the number of calculated in the complete
If an order preserving
graph. matching
This matching
would have the property
max objective
is 2 5 T.
perfect
for the corresponding Since,
this bicriterion
for incomplete
bipartite
procedure graphs
matching
is addressed
function
an appropriate
exists in G(T), then the minimum graph could also be determined. value, S, such that the min-
of T, denote
S is a function
would provide a solution
With
reduced from the total required
that it provides the min-sum
in this case,
Hence, such an approach
CL(i,j)].
weighted bipartite
with a value of (S(T), Z 5 T) for the objective Before
only those edge weights (i.e., the penal-
weights can be significantly
weight perfect
as S(T).
of calculating
at each k E VI such that k E [CE(i,j),
ties, CQ~) for each (i,j)
to the bicriterion
this min-sum scheduling
value
problem
in (Pl).
in further
detail, a useful corollary
to Theorem
1
will be shown.
COROLLARY 2. If in some minimum weight perfect matching in G(T) two copies of a product are interchanged in order to restore their correct ordering in the corresponding production sequence, then the value of S(T) does not increase. PROOF.
production
Assume that the implied ordering sequence
set by the matching.
of the two copies (i, j) and (i, j + 1) is violated
CL(i,j
+ l)] # 4. H ence, it must be possible to interchange sequence. The weights on the edges in G(T) are identical assignment represent
formulation. possible
assignments
value of the objective identical
minimum
Thus,
function
both the initial in (P2).
By Theorem
for the assignment
weight perfect
solution
matching,
+ I),
(i, j) and (i, j + 1) in the production to those
calculated
and the solution
1, this
objective
in the
[CE(i, j), CL(i, j)] n [CE(i,j
This implies that
interchange
function.
S(T), will not increase.
after
for the min-sum the interchange
does not increase
Therefore,
the
the value of the I
G.
130
STEINERAND S. YEOMANS
On account of Corollary 2, it is possible, without any loss of optimality, to restrict the class of solutions for the minimum weight matching problem to those that are order preserving matchings (i.e., to those solutions which are feasible for the level schedule problem). The computational complexity issues of the bicriterion matching procedure proposed above can be addressed in the following manner. The check for a perfect matching in G(T) requires O(D) time as mentioned previously. For a given graph G = (V, E), Papadimitriou and Steiglitz [22, p. 2671 note that the min-sum assignment method (to find a minimum weight perfect matching) for weighted (not necessarily complete) bipartite graphs can be implemented in 0 (IV/ /El log [VI) time. By necessity, for a perfect matching to exist in G(T), it must be the case that ]E] > D. Since /VI = 20 in G(T), a minimum weight perfect matching in the graph could be determined in 0 ( IEl D log D) > 0(D2 log D) time. The correct ordering for any assignment could easily be restored in O(D) time by successive swapping operations. Hence, the time complexity of the bicriterion procedure depends strictly upon the time complexity of the minimum weight perfect matching algorithm and therefore upon (El. Because ]E] is determined by the value of T, the time complexity of the bicriterion algorithm must also depend upon T. It will be shown that for values of T 5 1, ]E] is significantly smaller than the edge set of a complete graph, thereby allowing for a redzlction in the time complexity of the bicriterion problem in comparison to that of the min-sum procedure. The solutions to these bicriterion problems will also possess many highly desirable properties which significantly add to the efficiency of the procedure. LEMMA 1. The number of edges created in G(T),
with T 5 1, is (E] 1. nD + 20.
PROOF. Trivially, the number of edges in a graph G(T) with T < 1 is fewer than in the graph G(1). Hence, consider the case in which T = 1. The number of edges incident to a given (i,j) is CL(i,j) - CE(i,j) + 1. Using T = 1 in (1) and (2), and with the definitions of CE(i, j) and CL(i, j), + 0 2
+ 1 > L(i, j) - E(i, j) = [CL(i, j) - l] - [CE(i, j) - l] )
(3)
and therefore, (
;
z>
+ 1 r: CL(i,j)
- CE(i,j).
(4
From (4), for the total number of edges in the convex, bipartite graph G(T)
IEI = fg =gd,
~C~~~~_d - CE(Gj) + 116 g [(g)
+2]
[ (9)
with T = I
+ 21
=nD+2D.
(5) I
Consider, again, the example in which n = 10 and D = 1000. With T < 1, only 12,000 penalties would need to be calculated for the weighted matching problem in G(T) which is substantially fewer than the l,OOO,OOO required in the complete assignment formulation. Hence, in-core memory requirements for this instance of the problem are reduced considerably. LEMMA 2. The min-sum assignment solution for the convex can be determined in O(nD2 log D) time.
bipartite graph, G(T)
with T 5 1,
PROOF. By Lemma 1, the number of edges in the graph is JE] < nD + 20. Calculating the edge weights requires O((E]) = O(nD) time. The min-sum assignment algorithm for weighted bipartite graphs requires O(lVl [El log IV]) t ime. Since JV( = 20, this optimal assignment can therefore be determined in 0(nD2 log D) time. I
LevellingJIT AssemblyProcess LEMMA 3. A bicriterion JIT level schedule with a solution (S(l),
131
2 5 T = 1) can be found in
0(nD2 log D) time. PROOF. Set T = 1. By Theorem 2, the production sequence could be determined By Lemma 2, this schedule can be found corresponding to this particular min-sum
bicriterion
objective
value of this sequence
graph G(1) contains a feasible schedule. A min-sum for G(1) which would have the objective value, S(1). in O(nD2 log D) time. The min-max objective value solution to G(1) is necessarily 2 5 T = 1. Hence, the is (S(l),
2 5 T = 1).
I
Using T 5 1 ensures that the bicriterion solution possesses several useful properties. Since Z 5 T = 1, the production sequence created in Lemma 3 retains the highly desirable property that the production of any product never deviates from its ideal production level by more than one unit. Furthermore this sequence is the optimal assignment, in the min-sum sense, of all sequences with this property. Thus, the level schedule created will be good on the average and also in the worst case. However, while the sequence found above possesses these useful properties, it can be observed that S* I S(1) and Z* < T = 1; where 5” and Z* are the optimal values for the min-sum and min-max objectives, respectively. Hence, the sequence may not be optimal with respect to either measure. With some adjustments to the method of Lemma 3, a Pareto optimal solution can easily be determined using a similarly efficient approach. THEOREM 3.
The Pareto optimal solution (S(Z*), Z*) can be determined in O(nD2 log D) time.
algorithm in O(Dlog D) time to determine Z”. Then by Theorem 2 and Lemma 2 with T = Z*, the perfect matching which provides the min-sum objective value S(Z*) could be constructed in at most 0(nD2 log D) time. Because the min-max objective is Z*, this production sequence must be Pareto optimal. Therefore, the overall complexity for determining this Pareto optimal solution is O(nD2 log D) time. I PROOF. Run the min-max
Thus, a Pareto optimal solution for the bicriterion JIT level scheduling problem can be found in O(nD2 log D) time. This solution procedure runs, in general, far faster than O(D3) since, complexity for in practice, it is generally the case that 7~ < D. Hence, the computational determining a Pareto optimal solution is lower than that for determining an optimal min-sum solution alone. Furthermore, with only minor modifications to this approach, all of the Pareto optimal solutions with Z 2 1 can be efficiently generated. The following algorithm can be used to determine all such Pareto optimal solutions with this property. Algorithm:
T +- 1 - T,,, . Generate the edge set for the convex bipartite graph corresponding to T. Denote this edge set by E(T). Determine if a perfect matching exists in this bipartite graph. If there is no perfect matching, then let T e T-t l/D and go to 2. If a perfect matching exists, then determine a minimum weight order preserving perfect matching in G(T). The corresponding production sequence will be Pareto optimal with Z = Z* and S = S(Z’). Let E = E(T), Smin = S(Z*) and go to 4. Let T + T + l/D. If T > 1, then stop-all required Pareto optimal solutions with Z 5 1 have been determined. Otherwise, go to 5. Generate the edge set, E(T), for the convex bipartite graph corresponding to T. If E(T) = E, then go to 4. Otherwise set E = E(T) and go to 6.
1. Initialize
2. 3.
4. 5.
Pareto
G. STEINERAND
132
S. YEOMANS
6. Determine a minimum weight order preserving perfect matching in G(T). If S(T) < Smin, then this sequence will be Pareto optimal with 2 = T and S = S(T), set Smin = S(T) and go to 4. Otherwise the sequence is not Pareto optimal. Go to 4. THEOREM 4. Algorithm Pareto determines the production solutions (S, Z), with 2 5 1, in O(nd,,,D’ log D) time.
sequences for all Pareto optimal
By Theorem 2, 1 - Tmax 5 Z* < 1. Since Z* = ]zik - Icri] for some i and Ic, DZ* must be an integer in the interval [D - d,,,, D]. By the way in which T is incremented, every integer in the preceding interval (and so every possible value for 2) is considered and there are only dmax iterations of the algorithm. Therefore, starting with the initial target of T = 1 - rmaX and PROOF.
by Theorem 2, the minimum weight perfect matching in G(T) determined by step 3 must have 2 = 2’. Hence, steps 1 to 3 must determine one Pareto optimal solution. Steps 4 to 6 determine if any other Pareto optimal solutions with 2’ < 2 5 1 exist. G(Z*) has a perfect matching and since G(Z) contains G(Z*), G(Z) also has a perfect matching. Because T increases monotonically, if the edge set of a graph has not changed from that of a previously generated graph, then its min-sum solution cannot have decreased and, therefore, the solution cannot be Pareto optimal. Conversely, if the edge set has changed, then a new Pareto optimal solution could exist only if the value of the min-sum objective, S(T), is less than the min-sum value of any previously considered solution, S’min. Step 5 determines whether G(T) contains more edges than any previously considered graph. Step 6 determines if G(T) contains a new Pareto optimal solution. The test for the matching feasibility in stages 1 to 3 and the generation of each edge set can be performed in O(D) time. A check requiring constant [i.e., O(l)] t ime quickly determines whether the edge set of a graph contains more edges than the previously generated graph. Calculating the min-sum weights for a given graph requires O(]E]) 5 O(nD) time, and therefore, a minsum solution can be determined in O(nD’log D) time. The complete algorithm requires d,,, iterations. Hence, the time required for determining all Pareto optimal solutions with the property that 2 < 1 is, O(nd,,D210gD). I This algorithm demonstrates that many Pareto optimal solutions can be efficiently found in O(nd,,, D2 log D) < O(nD3 log D) time. Determining all of these production sequences requires the addition of a factor of less than n log D to the time complexity of the min-sum, assignment procedure in [5,6]. Upper bounds have been shown for two commonly used min-sum objectives; the sum of squared and sum of absolute deviations. Since 2 5 1 in the Pareto algorithm, S(Z)
_< nD for either of these two formulations. Hence, upper bounds are readily established on
the objective objectives.
function values of the bicriterion solution using either of these common min-sum
4.
CONCLUSIONS
AND
EXTENSIONS
The level scheduling problem of an assembly process in a mixed-model, JIT environment has been examined. A bicriterion, min-max, min-sum objective, not considered previously, is introduced to control this system. By exploiting certain properties from the previously considered min-max problem, many efficiencies are incorporated into this new bicriterion problem. Of particular importance is the reduction of the in-core storage requirements of the bicriterion problem in comparison to those of the min-sum problem. An algorithm is presented for determining several Pareto optimal solutions to this problem. The min-sum criteria ensures that the level schedules created by this algorithm are good in general, while the min-max criteria ensures that these schedules are good in the worst case. The absolute function has been considered for 2, since its value provides a meaningful, physical interpretation and bounds on its optimal value have been established previously. This function
Levelling JIT Assembly Process
allowed for the calculation of the time and space requirements of the bicriterion also for the efficient determination of several Pareto optimal solutions.
133
procedure and
It would be possible to consider more general functions (pi’s) for 2 and to use a similar approach to that described in this paper to determine bicriterion solutions. However, such general functions would not necessarily possess such an applicable physical interpretation as the absolute function. Changing the function would necessitate that complexity results specific to the appropriate function be determined. A given bound, T, on a general min-max function could still be used to restrict the positions in the production schedule where each copy of product could be sequenced. The convex bipartite graph G(T) could still be constructed and the check for a perfect matching (i.e., for the feasibility of T) could be performed in O(D) time. The correct ordering for any assignment could, also, still be restored in O(D) time. However, since (E( 2 D and (VI = 20 for any perfect matching in G(T), determining the minimum weight perfect matching (and, hence, a bicriterion solution) would require O(IEjD log D) > 0(D2 log D) time and O(lEl) space. The value of [El would necessarily depend upon the specific min-max function selected, and thus, the time and space requirements would also depend upon the function chosen. Furthermore, this bicriterion solution would most probably not be Pareto optimal. Moreover, it is highly unlikely that Theorems 2-4 and Lemmas l-3 would continue to hold for a general function. Therefore, new bounds on the optimal value of the min-max objective function would have to be established. Complexity results for calculating an optimal solution to the min-max formulation alone would have to be determined (i.e., the time required to find the optimal min-max solution could conceivably require more than the 0( IE( D log D) time necessary for determining a bicriterion solution). Thus, any procedure for determining Pareto optimal solutions with a different min-max criteria would necessarily depend upon the specific function selected. These types of bicriterion problems will be the subject of future study.
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