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Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering
production economics Int. J. Production
ELSEVIER
Economics
34 (1994) 1-15
Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering Harvey
H. MillaP*,
Minzhu
Yangb
“Finance and Management Science Department, Saint Mary’s University. Halifax, Nova Scotia, Canada, B3H 3C3 bDepartment of Management Engineering, Xian Jiao-tong University. Xian, China (Received
17 August
1992; accepted
12 August
1993)
Abstract This paper presents two algorithms for solving a network-based formulation of the capacitated multi-item lot-sizing problem with backordering. We employ Lagrangian decomposition and Lagrangian relaxation techniques which exploit the underlying network structure of the problem. In both approaches, we exploit a transportation subproblem which guarantees a primal feasible solution at every iteration of the procedure. Further, we use a primal partitioning scheme to produce additional primal feasible solutions. Valid lower bounds are obtained at each iteration of the algorithms, thereby providing a readily available ex post measure of the quality of the primal solutions. Computational analysis shows that both algorithms are quite effective, particularly when item setup and unit backorder costs are high. We also provide a means of evaluating the potential impact of permitting backorders under various problem characteristics.
1. Introduction
The mathematical model for the multi-item lot-sizing problem with (CMLSP-B) can be stated as follows:
capacitated backorders
Xirk
G
dik Yit,
Xitk
2
0,
yit{O,l)
ieN,tET,keT,
iEN,tET,kET ieN,tET,
(4) (5) (6)
P,:
up= Min CCC hitkxitk+ CC sitYit i
‘.”
f
FT
k
at&k
i
< Ct,
Exit, > diky i f
* Corresponding
t
t E
E
T,
N, k E T,
(3)
author
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)E0076-8
where Xitk is the quantity of product i in units produced in period t for use in period k, hitk is the carrying/backorder cost of one unit of item i produced in period t for use in period k, where t > k implies backordering, and t < k implies a carrying cost, sit is the setup cost for item i in period t, N is the set of items as well as its cardinality, ai is the capacity absorption rate in time per unit for item i, c, is the capacity available in period t in time units, T is the set of periods as well as its
2
H.H. Miller, M. Yung,iht.
cardinality, di, is the demand in period k and
in units
J. Production
Economics
34 11994)
l-15
for item i (
C(l)
1 if xxi,, Yir =
i
d(l.1)
> 0,
W,l)
f
1 0 otherwise. The objective function requires the minimization of carrying costs, backorder costs, and setup costs. We assume that the unit production cost is timeinvariant. As a result, we ignore the production costs. Production costs, however, can easily be incorporated in hitk. Constraints (2) imply that the capacity in each period cannot be exceeded, (3) imply that the demand must be met, and (4) are logical constraints which force a setup to be incurred if production of item i takes place in period t. The model assumes that d, reflects net demand for item i in period k. That is, initial inventory for item i is used to reduce corresponding demands until all the initial inventory is consumed. Figure 1 shows a well-defined underlying network structure for the lot-sizing problem. The first column of nodes represents the capacities available in each period. The capacity in a given period is used to supply demand for products in that period, or to satisfy demand in future periods (incurring inventory), or satisfy demands in earlier periods (incurring backorders). The quantity C(t) represents the available capacity in period t. The quantity u(t) represents the amount of the capacity C(t) that is allocated to backorders and inventory. An upper bound on u(t) is easily incorporated in the network. The amount v(i, t) is the total amount of u(t) allocated for backorder or inventory for item i in period t. As in the case of u(t), an upper bound on the amount v(i, t) is easily incorporated in the network. Finally, the quantity x(i, t, k) is the amount of u(i, t) that is used to satisfy demand in period k. If k > t, then x(i, t, k) is inventory, if k < t, then x(i, t, k) is a backorder quantity. Once again, the network structure easily facilitates upper bounds on x(i, t, k). The quantity d(4,3), referred to as PM, is used to illustrate how a required shutdown (for preventive maintenance for example) can be modelled. We treat the shutdown as a demand for product N + 1,
d(3,1)
Aggregate Inventory/ Backorders
N12)
C(2)
NW Item
Inventory/
KW
Backordeis
d(L3)
C(3)
d(2.3) d(3.3) d(4.3) (PM)
Fig. 1. The underlying
network
B&order structure
Arcs
of the lotsizing problem.
which in our example is item 4. This item would have zero carrying cost and a large setup cost. The large setup cost is used to force a single setup (shutdown) for PM. By using this approach, we can strategically place PM so as to minimize the total cost of the production schedule.
2. The literature The CMLSP-B, unlike the no-backorder case (which will be referred to as CMLSP), has received very little attention. There are at least two possible reasons: (1) the complexity of the problem makes it difficult to produce effective simple heuristics, and (2) assigning unit backorder costs. Though the use of backorders is not uncommon, assigning a unit backorder cost is situation-specific. It is influenced by the impact on customer goodwill if the product is unavailable, the risk of obsolescence, the impact on future operations if the item is needed in an
H.H. Millur. M. Yangjlnt. J. Production Economics 34 (1994) I-15
assembly process, the cost of the part (backordering cheap parts may not be sensible), the physical characteristics of the part (large voluminous parts may have to always be backordered due to physical restrictions), and the desired customer service policy. Inspite of these issues, careful analysis can produce estimates of the cost impact of backordering. However, we note that it is generally felt that the unit backordering cost for an item is more expensive than the corresponding unit carrying cost (i.e. it is better to have the part in stock than to be without it). Given the fact that backorder situations do arise (e.g. in MRP systems), there is a need for effective approaches to solve production lot-sizing problems in which backorders are permitted. Dixon and Silver [l] has shown that the CMLSP is NP-hard. It follows that CMLSP-B is also NP-hard, since CMLSP is a restriction of CMLSP-B. Further, every solution to CMLSP is also a solution to CMLSP-B. Several of the myopic heuristics and math programming approaches developed for the CMLSP (see surveys by Bahl et al. [2] and Maes and van Wassenhove [3]) may or may not easily be modified for the CMLSP-B. Maes and van Wassenhove [3] illustrated the inability of the Dixon-Silver [l] and the Lambrecht-Vanderveken [4] algorithms to handle adequately shelf-life restrictions for example. In many cases, the approaches do not guarantee of finding a feasible solution even if one exists. Similarly, the algorithm proposed by Thizy and van Wassenhove [S] would have to be modified. We note that as it currently stands, it does not even guarantee a feasible solution to the CMLSP. There is a need for approaches that specifically take into account backordering, shelf-life restrictions, shutdowns, etc. These features are easily handled by our formulation which has been shown to be quite effective [6]. Pochet and Wolsey [7] report the only significant piece of work we found on the CMLSP-B. In fact, Pochet and Wolsey [7] made no reference to other approaches to the CMLSP-B. The authors addressed the CMLSP-B using two formulations. One approach used a shortest-path reformulation similar to the variable reformulation approach of
3
Eppen and Martin [S] and Barany et al. [9] for fixed charge network-flow problems. This formulation is solved by standard IP codes. The second approach used a plant location reformulation for which a cutting plane algorithm is developed. Both approaches produce near-optimal solutions to large problems. However, the computational effort is quite significant. It is in this light that we believe that the algorithm presented in this paper is a significant contribution to “faster” approaches to the CMLSP-B. In this paper we present two Lagrangian-based algorithms to solve the CMLSP-B. One algorithm is based on Lagrangian decomposition, while the other is based on Lagrangian relaxation. The solution schemes produce feasible primal solutions at every iteration of the corresponding Lagrangian procedure, and guarantee finding a feasible solution if one exists. Further, valid lower bounds are available at each iteration thereby providing an ex post measure of the quality of the primal solutions. In the next two sections, we outline the relaxation and decomposition schemes and their corresponding algorithms. Subsequently, we present an experimental design for testing the algorithms and discuss computational results for 270 randomly generated test problems. We conclude the paper with a discussion of the results obtained.
3. The Lagrangian relaxation scheme In the Lagrangian relaxation approach, we view constraints (4) as the complicating constraints. We relax (4) and place it in the objective function with multipliers Xitk. We obtain two subproblems SPi and SP,. SPi is a transportation problem with the following objective function:
s.t. SP,:
(2), (31, (51, (6).
12(n) = MinCxJ,yi, i
f
(7) (8)
H.H.
s.t.
Millar,
M. YanglInt.
_ht = fSir - C 71ifkdik)Yit3
J. Production
(9)
(10) Yit
E(o3l)
SP, is a simple integer program which can be solved by inspection of the coefficients. Iffit d 0, then y, is set to 1. If the lower bound on the number of setups for item i is not satisfied, then we add setups for item i in ascending order of f;:I until the minimum number of setups is satisfied. Constraints (10) establishes a minimum number of setups (y;) for item i. We have
Yi
=
~4klmW{ct} [
f
1
(11)
.
We observe that each solution to feasible. We not only generate a solution at every iteration, but also to evaluate quality of the primal lower bound r(n), is given by
SPi is primal feasible primal a lower bound solution. The
(12) The corresponding
Lagrangian
problem
is given
by LR:
Y* = Max r(rr) S.t. iTirk3
34 (1994J
I-15
the constraints to create the subproblems we want to exploit (see (15)-(22)). By relaxing the coupling constraint between the copies and placing them in the objective function as a penalty term, the original problem decomposes into the two desired subproblems, each one defined over one of the group of constraints from the original constraint set. Unlike Lagrangian relaxation which removes the complicating constraints from the original problem, Lagrangian decomposition retains all of the original constraints. This technique was pioneered by Glover and Mulvey [lo], and has since been applied by other researchers such as Glover and Klingman [l 11, Guignard and Kim and Opaswongkarn [ 131, c121, Guignard Guignard and Rosewein [14], and recently by Millar and Yang [6]. We illustrate the decomposition process as follows. First, Pi is reformulated as shown below. In this formulation, we make copies of the variable Xitk, which we represent by Kirk, and use the copies to split the original constraint into two sets of constraints represented by (15)-(17) and (19)-(22). Note that we have duplicated constraint (9) which we represent by (19). Though it appears to be redundant at this time, it becomes useful when P2 is decomposed over the two set of constraints. Constraint (18) links the two groups of constraints and is regarded as the complicating constraint.
(13) P2: 0. up
4. The decomposition
Economics
scheme
The Lagrangian decomposition approach attempts to exploit two embedded well-defined and well-solved optimization problems. The two embedded problems are: (1) a transportation problem, and (2) N independent single-item uncapacitated lot-sizing problems with backorders. To extract and exploit these problems, we introduce copies of the production variables (Zifk) along with a constraint set which equates the copies (Xitk = Zitk). We then use the copies to reorganize
=
Min
TxF, i f
Xifk
hitk
+
cc i
k
s.t. TF
t E T,
ai Xirk d c,,
CXitk
3
Xirk
2
0,
Xitk
=
Zitk,
dik,
i
Sir Yit
E
(14)
f
N, k
E
(15)
T,
i E N, t E T, k E T, i E N, t E T, k E T,
(16) (17) (18)
H.H. Millar, M. Yang/In!. J. Producrion Economics 34 (1994)
Zitk
G
dik
Zitk
2
0,
icN,teT,keT,
(20)
i E N, t E T, k E T,
(21)
i E N, t E T.
(22)
Yit,
yi, {0, 1)
ul(A) = Min ~.~~ i
t
(hifk
-
Min~~~AtkXitk
s.t.
(24)
~~SitYir
k
i
f
(19)-(22).
The Lagrangian as follows: LD:
+ f
“‘i,
k, t E T,
W-4
t 2
where
1 k=f,j-
dik
iEN,t
C dikr
(27)
k=rj
1
T,
k=r,t
iEN,t
SPz:
i
d
(25)
k
s.t. (15)-(17).
uz(A) =
pitk
t d k, t E T,
(23)
Airk)Xirk
5
Lagrangian costs coefficients) while keeping the same setups (yt = 1) may improve the primal solution. We strengthen the contribution of SP4 to the lower bound u(A) by adding two simple potentially useful inequalities in the form of surrogate capacity constraints. The constraints are as follows:
FZit, ,< W/t, We obtain separability by relaxing (18) and placing it in the objective function with multipliers 1itk to yield the subproblems SP, and SP,. Subproblem SP, is a transportation problem as in the case of the Lagrangian relaxation approach. It is solved quite efficiently by several available network codes, such as that of Ahrens and Finke [lS]. Subproblem SP, can be decomposed into N independent unlot-sizing problems single-item capacitated each of which can be solved by Zangwill’s [16] algorithm.
I-IS
decomposition
problem
is stated
v* = Maxu(i) s.t. iitk unconstrained,
where u(n) = ~~(2) + u2(A), and v*, the optimal Lagrangian solution, is a lower bound to the primal problem. Each solution of SP, is primal feasible since it satisfies both demand and capacity constraints. Additionally, adjustment of the production quantities (using the true arc costs and not the
T,
where wi: represents an integer number of period demands that can be inventoried starting from period t, which will just exceed the actual capacity limit of period t. Similarly, wi: represents an integer number of period demands that can be backordered starting from period 1, which will just exceed the actual capacity limit of period t. During implementation, the surrogate capacities may be adjusted to reflect shelf-life and backorder limits if they exist. The equalities (25) and (26) are potentially quite useful, since in the optimal solution, the total production in period t will never exceed c,. Further, they both allow the optimality property for the single-item problem to be valid (i.e. they do not change the structure of the problem). While they will allow an integer number of periods of demand to be satisfied, they will prohibit the production of unnecessarily large lot sizes for each item particularly lot sizes that are much
6
H.H. Miller,
M. Ymgllnt.
J. Production
larger than the capacity restriction in the original problem. This has the potential to improve computational efficiency as we explore a smaller solution space. Further, since this is an improvement over the capacitated version of the problem, there is possibility of producing tighter bounds. These constraints may be particularly useful when capacities are tight, the planning horizon is long and the setup cost/holding cost ratio is high.
Economic.v 34 11994) I-15
The above partitioning scheme approach was not employed by Thizy and van Wassenhove [S]. They restricted the primal partition to activated setups to form the solution of the N WagnerWhitin problems subproblems (see also Wagner Whitin [17]). This was a reason for the failure of their algorithm. The primal transportation problem has the following formulation and cost structure.
5. Primal Partitioning Primal partitioning refers to the use of a given set of activated setups (4’ir = 1) to attempt to find a feasible production schedule. In our methodology, we use the setups activated in the solutions to the subproblems to define a corresponding primal transportation problem. In the case of SPi and SP,, only the setups activated are needed to produce the primal solution. This is because, their solutions are already primal feasible. Also, we note that their corresponding schedules are based on Lagrangian coefficients. In the primal partitioning procedure, we are attempting to improve the schedule by using the true cost coefficients. Hence, any adjustment in the production quantities imply an improvement of the primal solution. In the case of the SP2 and SP, the setups activated, more often than not, are insufficient to generate a primal feasible solution. It is not difficult, however, to produce a primal feasible solution by using the activated setups plus a minimum number of additional setups necessary to produce a feasible solution. To do that we simply devise a cost structure that allows the model to choose the additional setups necessary for feasibility. The cost structure gives priority to previously activated setups, and considers the use of non-activated setups at a premium, only if they are needed to produce a primal feasible primal solution. Our solution scheme, therefore, guarantees an additional feasible primal solution at every iteration of the Lagrangian implementation. Consequently, this produces a strong possibility of finding the optimal or near-optimal solution to the problem.
where
(29)
The role of m is to lower the priority of the arc corresponding to .& since yi, = 0, and to give priority to those arcs associated with positive setups (i.e. Yit = 1). Increasing the cost of each of these arcs by a fixed amount m does not affect their attractiveness relative to each other. In this way a feasible solution is always produced by the primal partition, and its true cost is easily determined by using the correct cost coefficients. Note that the term sit/hitk in gitk attemptS t0 reflect the setup cost associated with producing in periods where no production took place. This has the effect of influencing the selection of setups in a manner which is sensitive to the true setup cost.
6. The subgradient procedure We solve both the Lagrangian decomposition (LD) and relaxation (LR) problems by a subgradient algorithm due to Held et al.
Cl0
H.H. MiNar, M. YanglInt. J. Production Economies 34 (1994)
6.1.
The relaxation
The Lagrangian lows: r& ’ = Max[O,
I-15
algorithm multipliers
{&
+ nj(,&
where uj is the step length
are updated
- dikyit)}],
as fol-
(30)
Initialize the Lagrangean multipliers and iteration counux
given by
where yj is a scalar set to 2 initially and reduced by half if the lower bound fails to improve after a fixed number of iterations, X$ is the optimal solution to SP, , y{* is the optimal solution to SP,, r* is the value of the current best primal r(rrj) is the current lower feasible solution, bound, and & are the multipliers for the current iteration.
6.2.
The decomposition
The Lagrangian as follows:
algorithm
multipliers
for LD are updated Fig. 2. A schematic
&: 1 = ;l;,k + uj(zjtk - x!J,
(32)
where uj is the step length ,j = yj(U* - r(~j))/~,~~ i
given by
(dk f
-
dtk)2,
(33)
k
where yj is a scalar set to 2 initially and reduced by half if the lower bound fails to improve after a fixed number of iterations, XiZk is the optimal solution to SP,, zitk is the optimal solution to SP,, u* is the value of the current best primal feasible o(Aj) is the current lower bound, solution, and & are the multipliers for the current iteration.
6.3. Summary
qf the subgradient
algorithm
The implementation of the subgradient algorithm (see Fig. 2) for both Lagrangian problems are identical, except for the use of different para-
of the subgradient
algorithm
meters. We summarize the procedure for LD. The appropriate modifications are made to implement the algorithm for LR. The subgradient algorithm is summarized below. Step 1. Initialize j, I, y, u*. Step 2. Solve SP, and SP,. Determine the lower bound ~(2’). Step 3. Let yi: and yi: denote the setup variables for SPJ and SP,, respectively, and let Y’ and Y2 be the sets of setups corresponding to the optimal solutions of SP3 and SP,, respectively. Determine the following coefficients:
8
H.H. Millar, M. Yang/Inf.
J. Production
Step 4. Solve PP using the two sets of coefficients. Let F, and F, be the primal values of the corresponding optimal solutions. Let ~‘j= min{F,, F2, u*). Set t’* = uj. Step 5. Determine the dual gap as follows. 0 = 2.O(v* - a(I,j)/(u* + u(ij)). Stop if a prespecified iteration limit is reached, or 0 < E, where E is the prespecified tolerance. Otherwise, update the Lagrangian multipliers using (32) and (33) and go to Step 2.
7. Computational
results
We found no well-known set of test problems with backorder costs which could be used for assessing the algorithm. We adapt an experimental design described in Maes and Van Wassenhove [4] because it is an accepted design for testing lot-sizing problems.
7.1. The experimental
design
A total of 270 test problems were generated. A summary of the parameters of the experimental design is found in Table 1. In the deTable I A summary
of the problem
parameters
Parameter
Value
Number of periods Number of items Holding cost Backorder cost
Constant Constant Constant Low Medium High Low Medium High Constant Low High Low Medium High Norma1
Capacity
tightness
Absorption rate Time between orders Lumpiness
Demand
distribution
(TBO)
(6)
(5) (1) WI
(4h) (6h) (111%) (135%) (200%) (1) (l-3) (2 -6) (0%) (10%) (20%) (100,25)
Economics
34 (1994)
I-15
sign, we vary four parameters: capacity tightness, setup cost through time between orders (TBO), demand lumpiness, and backorder costs. Capacity tightness is represented by the ratio of total aggregate capacity to total aggregate demand. Setup cost is related to the square of the TBO, and is obtained through the economic order quantity (EOQ) formula. The lumpiness parameter is defined as the probability of a demand occurring in a given period. Once the demand is permitted, it is generated according to a normal distribution with a mean demand of 100, and a standard deviation 25. For the purpose of the experiment, we define backorder cost as a linear function of the carrying cost (b = fh), wheref 2 1. At this point in time, the actual value of the backorder cost is not the primary issue, but rather, the quality of the performance of the algorithms when backordering is allowed. The four factors, capacity tightness, TBO, lumpiness, and backorder costs result in (3 x 2 x 3 x 3) = 54 combinations. For each combination, 5 random problems were generated each of size 5 items x 6 periods. We generate solutions for these test problems using the relaxation and decomposition algorithms which we will refer to as RELAX and DECOMP, respectively. Further, we solve the no-backorder restriction of the problems using a decomposition algorithm described by Millar and Yang [S]. We refer to this algorithm as the pure inventory model (PIM). The purpose of generating these solutions is to provide an additional scheme for evaluating the algorithms, and to allow the assessment of the potential of a backordering policy. The multipliers for all the algorithms are initialized with a value of 0. We measure the quality of the solutions generated by the three algorithms by computing a dual gap as follows: Gap = 2oo x (upper
bound
- lower
bound)
(upper
bound
+ lower
bound) ’
The lower bound used is the better of the lower bounds produced by RELAX and DECOMP.
H.H. Millar, M. YanglInt. J. Production Economics 34 (1994)
In summarizing the results, we compute the means and standard deviations of the dual gaps for the 5 problems in each of the 54 categories. The standard deviations provide a measure of the stability of the algorithms, a characteristic that is extremely important in the design of heuristics. These results are summarized in Tables 5-10 in the appendix.
40
l-15
*RELAX + DECOMP _) PIM
1 30 -
20 -
10 -
7.2. The effect of capacity tightness and setupjholding cost ratios
0
I 0
Figure 3 shows a graphical representation of the performance of the three algorithms vs. problem class when the lumpiness is O%, and the backordering unit cost is four times that of the holding cost. A problem class refers to specific capacity/setup cost combination. The most stable algorithm appears to be RELAX followed by DECOMP. In Figure 4, we observe a general saw-tooth pattern of performance related to the alternating low and high setup costs. Both DECOMP and PIM appear to perform better on low setup cost problems than on high setup cost problems. On the other hand, RELAX appears to do better on the problems with high setup costs. When capacity is loose, the pure inventory model (PIM) performs extremely well compared to RELAX and DECOMP. This is because there is little opportunity for backordering. Because of sufficient capacity, inventory will tend to be preferable to backorders for eliminating setups. Because PIM searchs a smaller solution space than RELAX and DECOMP, it often produces a better solution using the same amount of computational effort. We believe that RELAX and DECOMP would need significantly greater effort to come up with a comparable solution. As we move from problem class 1 to 6, the tightness of the capacity increases. PIM shows a dramatic change in performance with capacity tightness but for those cases with high setup costs. The performance of RELAX and DECOMP improves notably. This can be explained by the fact that with increase in capacity tightness, there is greater potential for backordering, pareticularly
.
1
I 2
.
I 3
.
I 4
.
I 5
.
1 6
.
I I
PROBLEM CLASS Fig. 3. Size of the dual gaps vs. problem ness = 0%).
60 50
4
PIMib=Zh
n
PIM/bdh
A
PIM/b=bh
class (b = 4h, lumpi-
40 a 3 a %
30
20
10
0 1
2
3
4
5
6
PROBLEM CLASS
Fig. 4. The ness = 0%).
effect
of
backorder
costs
on
PIM
(lumpi-
particularly when backorder costs are inexpensive. This potential, however, is tempered by the size of the setup costs. If capacity is tight, setup cost is high and backorder cost is low to moderate, there is a significant potential for using backorders to eliminate setups. By searching only the pure inventory solution space, PIM misses out on cheap solutions
10
H.H. Millar, M. Yang/hi.
J. Production
that incorporate backorders. However, when setup cost is low, the potential use of backorders as a means of eliminating setups is greatly reduced. In fact, due to the tightness of capacity, the lot-for-lot solution may even be a viable alternative. If production shifts are necessary for reasons related to capacity restrictions, then inventory would be preferable alternative to backordering. Though the above phenomenon is well illustrated when b = 4h, (see Fig. 3) it is even more dramatic when b = 2h (see Table 5 in the appendix). For high capacity tightness, low setup costs and b = 2h, PIM obtains a mean gap of 53.73% compared with 8.61% for the case with low setup costs. RELAX and DECOMP, on the other hand, perform reasonably well with tight capacity, high setup costs, and low backorder costs. We note that the same general saw-tooth pattern of performance repeats itself as we vary backorder costs from 2h to 6h with increasing improvement in the performance of all the algorithms. This is because as backorder cost increases, even with tight capacity and high setup costs, the potential for backordering diminishes. Given there are few good backorder solutions, all the algorithms including PIM tend to perform well. In most cases, the optimal solution will tend to be a pure inventory solution.
Economics
30
34 (1994)
I-IS
+
1
a-
RELAX/h=2h
+
RELAX/lHh RELAX/b=6h
20 3 % a a
10 -
0; 0
.
(
.
1
,
.
2
,
,
3
I
4
-
5
I
I
.
6
7
PROBLEM CLASS Fig. 5. The effect ness = 0%).
of backorder
costs
on
30 _
RELAX
(lumpi-
4
DECOMP/b=2h
+
DECOMP/b=4h DECOMP/b=6h
4
10 -
7.3. The impact qf backorder costs 0 0
Backorder costs impact on the performance of each algorithm differently. When backorder costs are small, the performance of the algorithms depend heavily on the setup costs and capacity tightness (in essence the opportunity for backorders). If capacity is loose, despite high or low setup costs, the use of inventory is preferable to backorders. Consequently, PIM outperforms RELAX and DECOMP in this case. When capacity is tight, the setup cost becomes very important. If setup cost is high then the use of backorders to eliminate setups is desirable. If setup cost is low, then the opportunity for backorders is reduced, and capacity “crunches” are better handled by using inventory wherever possible.
1 I
*
I 2
.
I 3
.
1 4
*
1 s
*
’ 6
’
’ 1
PROBLEM CLASS Fig. 6. The effect of backorder ness = 0%).
costs
on DECOMP
(lumpi-
When backorder costs are high, there is little opportunity for the use of backorders to eliminate setups. Consequently, all the algorithms perform comparatively well. Figures 4-6 show the performance behavior of each of the algorithms as a function backorder costs and problem class. We observe that all of the algorithms display their most erratic performance under low backorder costs. This behavior stabilizes as backorder costs increase.
H.H. Millar. M. YanglInt. J. Producrion Economies 34 (1994)
7.4. The stability
of the algorithms
Table 3 One-way
Tables 8810 in the appendix display the standard deviations of the dual gaps for each of the 54 problem categories. The variability which exist does not seem to show any peculiar pattern. It would appear that all of the algorithms are fairly stable. There is one class of problems in Table 10 which show a standard deviation of 20.98%. This can be attributed to a single problem for which none of the algorithms was able to produce a suitable lower bound. Table 2 shows a summary of statistics for each of the algorithms. We note that performance of RELAX and DECOMP appears to be comparable, though RELAX has a lower mean performance. PIM has the worst mean and standard deviation of the dual gaps. This is an indication of the value of an algorithm specifically designed to handle the lot-sizing problem with backorders. Tables 3 and 4 are ANOVA tables highlighting which factors affect the performance of the algorithms. From the tables, it would appear that setup cost is an important factor, particularly so for PIM and DECOMP. Demand lumpiness did not have a significant impact on all of the algorithms, and capacity had only a moderate impact on RELAX and DECOMP, but a significant impact on PIM. Backorder costs played a major role in the success of RELAX and DECOMP. From Table 3, the results suggest that both setup cost and backorder cost interacting together can significantly explain the performance of RELAX and PIM. However, the interaction of the two factors does not seem to play a significant role in explaining the performance of DECOMP.
Table 2 Summary
statistics
Sample size Minimum Maximum Mean value Standard deviation
for the algorithms RELAX
DECOMP
PIM
54 2.4 1 22.50 8.20 4.08
54 2.44 22.95 10.43 4.72
54 0.07 53.13 12.36 13.11
ANOVA
Factors
Lumpiness Capacity
Table 4 Two-way
cost
ANOVA
Factors
Backorder
F-value P-value F-value P-value F-value P-value F-value P-value
cost
Setup cost* Backorder cost
Algorithms RELAX
DECOMP
PIM
5.98 0.018 1.32 0.271 0.62 0.541 13.04 0.000
2.23 0.139 0.56 0.576 0.39 0.678 22.58 0.000
24.16 0.000 0.06 0.94 5.56 0.007 5.77 0.008
results for setup vs. backorder Statistics
Setup cost
11
results Statistics
Setup cost
Backorder
I-15
F-value P-value F-value P-value F-value P-value
costs
Algorithms RELAX
DECOMP
PIM
11.52 0.001 18.89 0.000 7.116 0.002
4.27 0.044 24.16 0.000 1.15 0.325
40.93 0.000 10.92 0.000 8.32 0.001
8. Conclusions Few approaches to the CMLSP-B have been developed to date. We have presented two effective algorithms which guarantee finding a feasible schedule if one exists, and which also provide an ex post measure of the quality of the solutions generated. The formulation provided allows the user to specify several constraints on the problem without increasing its complexity. These constraints include: production and inventory constraints at an aggregate level, production and inventory constraints at an item level, shelf-life limits for inventoried items, limits on the number of backorder periods allowed, and preventive maintenance. Traditional formulations make it difficult to incorporate these constraints.
12
H.H. Millar, M. YanglInt.
J. Production
We execute the Lagrangian procedure for 150 iterations. We can increase the number of runs to any number. Computational requirements per execution are very reasonable, with a mean of 12.33 s and standard deviation of 0.47 s for the given problem size of 5 items and 6 periods. The problems were solved on a VAXJVMS 785 computer, which is a relatively slow computer. Computational times could be greatly improved through the use of a faster computer, and by improving the efficiency of the programming code. With regards to computational complexity, it is well-known that problem size and the S/H ratios are indicators of computational complexity for the CMLSP. It would appear that we need to generalize the S/H ratio to include backorder costs. If we accept the argument that a backorder is a mirrored
Economics
34 (1994)
I-15
view of an inventoried item, then S/H ratios and S/B ratios are equivalent. We may consider a measure given by S’/(HB). However, this would have to be tested. Recognizing the fact that existing myopic heuristics for the CMLSP are not easily modified, we believe our heuristics are a useful contribution to reducing the scarcity of effective algorithms for the multi-item lot-sizing problem with backorders.
Acknowledgement This research is partially funded by the Natural Science and Engineering research Council of Canada under grant 0GPIN020.
Appendix Table 5 The means
of the dual gaps in each category
Methods
h = 2h
Capacity L Setup
H Setup
M Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
19.29 18.80 1.94*
12.99 12.57 9.29*
22.50 22.95 8.43*
11.18* 21.97 40.23
12.46 13.02 8.61*
7.22’ 10.44 53.73
10%
RELAX DECOMP PIM
16.51 16.42 2.10*
7.88* 7.95 14.33
17.35 18.30 4.26*
8.13* 13.87 30.45
11.73* 12.53 16.36
7.05* 16.08 50.16
20%
RELAX DECOMP PIM
8.68 8.54 1.24*
7.36* 18.88 15.18
12.19 12.00 1.80*
6.36* 14.11 37.04
12.73 10.82 10.66*
5.61* 17.85 47.15
H.H. Millar. M. Yanglint. J. Production Economics 34 (1994)
Table 6 The means of the dual gaps in each category Methods
I-15
13
b = 4h
Capacity L Setup
M Setup
H Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
7.74 8.32 0.07*
7.14* 13.09 9.81
7.40 8.13 2.68;
6.31* 10.11 14.20
8.34 8.64 8.00*
5.74* 9.77 30.83
10%
RELAX DECOMP PIM
5.59 5.18 1.31:
2.80; 8.84 10.68
7.65 6.79 2.64+
4.96* 8.17 20.80
10.87* 10.87* 12.09
6.11* 9.67 27.05
20%
RELAX DECOMP PIM
3.20 4.26 2.18*
4.00* 10.99 5.06
8.07 7.14 3.71*
6.85* 13.10 19.09
6.40 7.35 4.62*
10.74* 13.53 18.36
Table 7 The means of the dual gaps in each class (b = 6h) Methods
used
Capacity L Setup
M Setup
H Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
2.74 2.44 1.37:
6.67* 11.50 6.76
8.10 5.70 4.35*
8.05* 12.82 17.51
8.18 7.16 5.99*
6.81* 7.66 8.19
10%
RELAX DECOMP PIM
4.04 4.04 0.60*
3.83* 5.47 4.47
7.28 7.26 3.00*
5.40* 6.13 6.11
4.47* 5.3 1 4.47*
12.35* 15.60 18.37
20%
RELAX DECOMP PIM
9.40* 9.40* 9.40*
2.4 1 6.93 2.07*
5.10 4.48* 4.50
5.86 5.21* 9.23
8.35* 9.96 8.72
6.52 4.95* 6.37
H.H. Millar. M. YanglInt. J. Production Economics 34 (1994)
14
Table 8 The standard Methods
deviations
used
of the dual gaps in each category
l-15
b = 2h
Capacity L Setup
M Setup
L
H
L
H Setup H
L
H
0%
RELAX DECOMP PIM
5.91 6.01 2.67
5.58 3.41 3.37
4.97 4.96 1.80
2.52 1.54 8.35
2.82 2.66 4.51
3.87 3.54 12.49
10%
RELAX DECOMP PIM
4.04 4.71 1.81
4.30 4.87 6.07
1.61 2.42 1.31
4.18 7.43 13.91
5.34 3.34 8.66
3.44 6.84 11.05
20%
RELAX DECOMP PIM
2.82 2.85 2.06
3.27 4.21 7.99
2.53 2.96 2.99
3.37 4.86 14.87
4.4 1 5.21 3.90
2.17 3.14 19.38
Table 9 The standard
deviations
Algorithms
of the dual gaps in each category
b = 4h
Capacity L Setup
H Setup
M Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
3.52 3.82 0.25
2.55 9.94 13.10
1.95 2.70 3.08
3.48 3.99 10.28
3.26 3.61 3.99
4.75 7.04 17.91
10%
RELAX DECOMP PIM
4.35 5.15 2.21
3.36 7.01 13.15
3.81 2.25 3.39
3.61 4.95 8.42
5.60 5.60 4.19
3.00 3.23 12.78
20%
RELAX DECOMP PIM
1.40 1.48 1.49
3.63 3.96 6.24
3.49 4.95 2.30
3.24 8.48 1.17
2.51 3.62 2.55
6.86 4.66 9.05
H.H. MiNar. M. YanglInt. Table 10 The standard Methods
deviations
used
of the dual gaps in each category
J. Production
Economic.! 34 (1994J
I-15
15
b = 6h
Capacity L Setup L
M Setup
H Setup
H
L
H
L
H
0%
RELAX DECOMP PIM
2.84 2.32 1.77
2.76 3.41 4.56
3.38 4.11 4.44
6.05 6.01 9.35
4.97 3.00 6.35
5.45 9.34 4.44
10%
RELAX DECOMP PIM
5.37 5.37 1.10
2.80 5.44 5.99
5.73 5.75 2.87
8.24 8.36 7.84
5.56 5.32 5.56
8.33 10.49 11.66
20%
RELAX DECOMP PIM
20.98 20.98 20.98
2.17 4.94 1.62
5.18 5.64 5.62
7.73 4.40 9.42
13.51 12.83 13.22
8.77 9.06 8.87
References Cl1 Dixon, P. and Silver, E., 1981. A heuristic solution procedure for the multi-item, single-level, limited capacity, lot sizing problem, J. Oper. Mgmt.,2: 23-39. PI Bahl, H., Ritzman, L. and Gupta, J., 1987. Determining lot sizes and resources requirements: A review. Oper. Res., 35: 329-345. c31 Maes, J. and van Wassenhove, L., 1988. Multi-item singlelevel dynamic lot sizing heuristics: A general review. J. Oper. Res. Sot., 39: 991-1004. M. and Vanderveken, H., 1979. Heuristic c41 Lambrecht, procedures for the single operation multi-item loading problem. A.1.I.E Trans., 11: 319-326. relax[51 Thizy, J. and van Wassenhove, L., 1985. Lagrangian ation for multi-item capacitated lot sizing problem. A.I.I.E. Trans., 17: 3088313. of Lagrangian 161 Millar, H. and M. Yang, An application decomposition to the capacitated multi-item lot sizing problem. Comput. Oper. Res., 20(4) (1993) 409420. c71 Pochet, Y. and L. Wolsey, 1988. Lot size models with backlogging: Strong reformulations and cutting planes. Math. Progr., 40: 317-335. PI Eppen, G. and Martin, R., 1987. Multi-item capacitated lot sizing using variable redefinition. Oper. Res., 35: 832-848. c91 Barany, I., Van Roy, T.J. and Wolsey, L.A., 1984. Strong formulations for multi-item lot sizing. Mgmt. Sci., 30: 1255-1261.
[lo]
[l l]
[12]
[13]
[14]
[15]
[ 161
[17] [18]
Clover, F. and Mulvey, J., 1980. The equivalence of the O-1 integer programming problem in discrete generalized and pure network models. Oper. Res., 28: 829-933. Clover, F. and Klingman, D., 1988. Layering strategies for creating exploitable structure in linear and integer problems. Math. Progr., 40: 165-182. Guignard M. and Kim, S., 1987. Lagrangian decomposition: A model yielding stronger Lagrangian bounds. Math. Progr., 39: 2155228. Guignard, M. and Opaswongkarn, K., 1987. Lagrangian dual ascent algorithms for computing bounds in capacitated plant location problems. Report #57, The Whaton School, University of Pennsylvania. Guignard, M. and Rosewein, M., 1989. An applicationoriented guide for designing Lagrangian dual ascent algorithms. Eur. J. Oper. Res., 43: 1977205. Ahrens, J. and Finke, G., 1980. Primal transportation and transshipment algorithms. Zeitschrit fur Oper. Res., 24: l-32. Zangwill, W., 1966. A Deterministic multi-period production scheduling model with backlogging. Mgmt. Sci., 13: 1055119. Wagner, H. and Whitin, T., 1985. Dynamic version of the economic lot size model. Mgmt. Sci., 5: 89-96. Held, M., Wolfe, P. and Crowder, H., 1974. Validation of subgradient optimization, Math. Progr., 6: 62-88.
ELSEVIER
Economics
34 (1994) 1-15
Lagrangian heuristics for the capacitated multi-item lot-sizing problem with backordering Harvey
H. MillaP*,
Minzhu
Yangb
“Finance and Management Science Department, Saint Mary’s University. Halifax, Nova Scotia, Canada, B3H 3C3 bDepartment of Management Engineering, Xian Jiao-tong University. Xian, China (Received
17 August
1992; accepted
12 August
1993)
Abstract This paper presents two algorithms for solving a network-based formulation of the capacitated multi-item lot-sizing problem with backordering. We employ Lagrangian decomposition and Lagrangian relaxation techniques which exploit the underlying network structure of the problem. In both approaches, we exploit a transportation subproblem which guarantees a primal feasible solution at every iteration of the procedure. Further, we use a primal partitioning scheme to produce additional primal feasible solutions. Valid lower bounds are obtained at each iteration of the algorithms, thereby providing a readily available ex post measure of the quality of the primal solutions. Computational analysis shows that both algorithms are quite effective, particularly when item setup and unit backorder costs are high. We also provide a means of evaluating the potential impact of permitting backorders under various problem characteristics.
1. Introduction
The mathematical model for the multi-item lot-sizing problem with (CMLSP-B) can be stated as follows:
capacitated backorders
Xirk
G
dik Yit,
Xitk
2
0,
yit{O,l)
ieN,tET,keT,
iEN,tET,kET ieN,tET,
(4) (5) (6)
P,:
up= Min CCC hitkxitk+ CC sitYit i
‘.”
f
FT
k
at&k
i
< Ct,
Exit, > diky i f
* Corresponding
t
t E
E
T,
N, k E T,
(3)
author
0925-5273/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0925-5273(93)E0076-8
where Xitk is the quantity of product i in units produced in period t for use in period k, hitk is the carrying/backorder cost of one unit of item i produced in period t for use in period k, where t > k implies backordering, and t < k implies a carrying cost, sit is the setup cost for item i in period t, N is the set of items as well as its cardinality, ai is the capacity absorption rate in time per unit for item i, c, is the capacity available in period t in time units, T is the set of periods as well as its
2
H.H. Miller, M. Yung,iht.
cardinality, di, is the demand in period k and
in units
J. Production
Economics
34 11994)
l-15
for item i (
C(l)
1 if xxi,, Yir =
i
d(l.1)
> 0,
W,l)
f
1 0 otherwise. The objective function requires the minimization of carrying costs, backorder costs, and setup costs. We assume that the unit production cost is timeinvariant. As a result, we ignore the production costs. Production costs, however, can easily be incorporated in hitk. Constraints (2) imply that the capacity in each period cannot be exceeded, (3) imply that the demand must be met, and (4) are logical constraints which force a setup to be incurred if production of item i takes place in period t. The model assumes that d, reflects net demand for item i in period k. That is, initial inventory for item i is used to reduce corresponding demands until all the initial inventory is consumed. Figure 1 shows a well-defined underlying network structure for the lot-sizing problem. The first column of nodes represents the capacities available in each period. The capacity in a given period is used to supply demand for products in that period, or to satisfy demand in future periods (incurring inventory), or satisfy demands in earlier periods (incurring backorders). The quantity C(t) represents the available capacity in period t. The quantity u(t) represents the amount of the capacity C(t) that is allocated to backorders and inventory. An upper bound on u(t) is easily incorporated in the network. The amount v(i, t) is the total amount of u(t) allocated for backorder or inventory for item i in period t. As in the case of u(t), an upper bound on the amount v(i, t) is easily incorporated in the network. Finally, the quantity x(i, t, k) is the amount of u(i, t) that is used to satisfy demand in period k. If k > t, then x(i, t, k) is inventory, if k < t, then x(i, t, k) is a backorder quantity. Once again, the network structure easily facilitates upper bounds on x(i, t, k). The quantity d(4,3), referred to as PM, is used to illustrate how a required shutdown (for preventive maintenance for example) can be modelled. We treat the shutdown as a demand for product N + 1,
d(3,1)
Aggregate Inventory/ Backorders
N12)
C(2)
NW Item
Inventory/
KW
Backordeis
d(L3)
C(3)
d(2.3) d(3.3) d(4.3) (PM)
Fig. 1. The underlying
network
B&order structure
Arcs
of the lotsizing problem.
which in our example is item 4. This item would have zero carrying cost and a large setup cost. The large setup cost is used to force a single setup (shutdown) for PM. By using this approach, we can strategically place PM so as to minimize the total cost of the production schedule.
2. The literature The CMLSP-B, unlike the no-backorder case (which will be referred to as CMLSP), has received very little attention. There are at least two possible reasons: (1) the complexity of the problem makes it difficult to produce effective simple heuristics, and (2) assigning unit backorder costs. Though the use of backorders is not uncommon, assigning a unit backorder cost is situation-specific. It is influenced by the impact on customer goodwill if the product is unavailable, the risk of obsolescence, the impact on future operations if the item is needed in an
H.H. Millur. M. Yangjlnt. J. Production Economics 34 (1994) I-15
assembly process, the cost of the part (backordering cheap parts may not be sensible), the physical characteristics of the part (large voluminous parts may have to always be backordered due to physical restrictions), and the desired customer service policy. Inspite of these issues, careful analysis can produce estimates of the cost impact of backordering. However, we note that it is generally felt that the unit backordering cost for an item is more expensive than the corresponding unit carrying cost (i.e. it is better to have the part in stock than to be without it). Given the fact that backorder situations do arise (e.g. in MRP systems), there is a need for effective approaches to solve production lot-sizing problems in which backorders are permitted. Dixon and Silver [l] has shown that the CMLSP is NP-hard. It follows that CMLSP-B is also NP-hard, since CMLSP is a restriction of CMLSP-B. Further, every solution to CMLSP is also a solution to CMLSP-B. Several of the myopic heuristics and math programming approaches developed for the CMLSP (see surveys by Bahl et al. [2] and Maes and van Wassenhove [3]) may or may not easily be modified for the CMLSP-B. Maes and van Wassenhove [3] illustrated the inability of the Dixon-Silver [l] and the Lambrecht-Vanderveken [4] algorithms to handle adequately shelf-life restrictions for example. In many cases, the approaches do not guarantee of finding a feasible solution even if one exists. Similarly, the algorithm proposed by Thizy and van Wassenhove [S] would have to be modified. We note that as it currently stands, it does not even guarantee a feasible solution to the CMLSP. There is a need for approaches that specifically take into account backordering, shelf-life restrictions, shutdowns, etc. These features are easily handled by our formulation which has been shown to be quite effective [6]. Pochet and Wolsey [7] report the only significant piece of work we found on the CMLSP-B. In fact, Pochet and Wolsey [7] made no reference to other approaches to the CMLSP-B. The authors addressed the CMLSP-B using two formulations. One approach used a shortest-path reformulation similar to the variable reformulation approach of
3
Eppen and Martin [S] and Barany et al. [9] for fixed charge network-flow problems. This formulation is solved by standard IP codes. The second approach used a plant location reformulation for which a cutting plane algorithm is developed. Both approaches produce near-optimal solutions to large problems. However, the computational effort is quite significant. It is in this light that we believe that the algorithm presented in this paper is a significant contribution to “faster” approaches to the CMLSP-B. In this paper we present two Lagrangian-based algorithms to solve the CMLSP-B. One algorithm is based on Lagrangian decomposition, while the other is based on Lagrangian relaxation. The solution schemes produce feasible primal solutions at every iteration of the corresponding Lagrangian procedure, and guarantee finding a feasible solution if one exists. Further, valid lower bounds are available at each iteration thereby providing an ex post measure of the quality of the primal solutions. In the next two sections, we outline the relaxation and decomposition schemes and their corresponding algorithms. Subsequently, we present an experimental design for testing the algorithms and discuss computational results for 270 randomly generated test problems. We conclude the paper with a discussion of the results obtained.
3. The Lagrangian relaxation scheme In the Lagrangian relaxation approach, we view constraints (4) as the complicating constraints. We relax (4) and place it in the objective function with multipliers Xitk. We obtain two subproblems SPi and SP,. SPi is a transportation problem with the following objective function:
s.t. SP,:
(2), (31, (51, (6).
12(n) = MinCxJ,yi, i
f
(7) (8)
H.H.
s.t.
Millar,
M. YanglInt.
_ht = fSir - C 71ifkdik)Yit3
J. Production
(9)
(10) Yit
E(o3l)
SP, is a simple integer program which can be solved by inspection of the coefficients. Iffit d 0, then y, is set to 1. If the lower bound on the number of setups for item i is not satisfied, then we add setups for item i in ascending order of f;:I until the minimum number of setups is satisfied. Constraints (10) establishes a minimum number of setups (y;) for item i. We have
Yi
=
~4klmW{ct} [
f
1
(11)
.
We observe that each solution to feasible. We not only generate a solution at every iteration, but also to evaluate quality of the primal lower bound r(n), is given by
SPi is primal feasible primal a lower bound solution. The
(12) The corresponding
Lagrangian
problem
is given
by LR:
Y* = Max r(rr) S.t. iTirk3
34 (1994J
I-15
the constraints to create the subproblems we want to exploit (see (15)-(22)). By relaxing the coupling constraint between the copies and placing them in the objective function as a penalty term, the original problem decomposes into the two desired subproblems, each one defined over one of the group of constraints from the original constraint set. Unlike Lagrangian relaxation which removes the complicating constraints from the original problem, Lagrangian decomposition retains all of the original constraints. This technique was pioneered by Glover and Mulvey [lo], and has since been applied by other researchers such as Glover and Klingman [l 11, Guignard and Kim and Opaswongkarn [ 131, c121, Guignard Guignard and Rosewein [14], and recently by Millar and Yang [6]. We illustrate the decomposition process as follows. First, Pi is reformulated as shown below. In this formulation, we make copies of the variable Xitk, which we represent by Kirk, and use the copies to split the original constraint into two sets of constraints represented by (15)-(17) and (19)-(22). Note that we have duplicated constraint (9) which we represent by (19). Though it appears to be redundant at this time, it becomes useful when P2 is decomposed over the two set of constraints. Constraint (18) links the two groups of constraints and is regarded as the complicating constraint.
(13) P2: 0. up
4. The decomposition
Economics
scheme
The Lagrangian decomposition approach attempts to exploit two embedded well-defined and well-solved optimization problems. The two embedded problems are: (1) a transportation problem, and (2) N independent single-item uncapacitated lot-sizing problems with backorders. To extract and exploit these problems, we introduce copies of the production variables (Zifk) along with a constraint set which equates the copies (Xitk = Zitk). We then use the copies to reorganize
=
Min
TxF, i f
Xifk
hitk
+
cc i
k
s.t. TF
t E T,
ai Xirk d c,,
CXitk
3
Xirk
2
0,
Xitk
=
Zitk,
dik,
i
Sir Yit
E
(14)
f
N, k
E
(15)
T,
i E N, t E T, k E T, i E N, t E T, k E T,
(16) (17) (18)
H.H. Millar, M. Yang/In!. J. Producrion Economics 34 (1994)
Zitk
G
dik
Zitk
2
0,
icN,teT,keT,
(20)
i E N, t E T, k E T,
(21)
i E N, t E T.
(22)
Yit,
yi, {0, 1)
ul(A) = Min ~.~~ i
t
(hifk
-
Min~~~AtkXitk
s.t.
(24)
~~SitYir
k
i
f
(19)-(22).
The Lagrangian as follows: LD:
+ f
“‘i,
k, t E T,
W-4
t 2
where
1 k=f,j-
dik
iEN,t
C dikr
(27)
k=rj
1
T,
k=r,t
iEN,t
SPz:
i
d
(25)
k
s.t. (15)-(17).
uz(A) =
pitk
t d k, t E T,
(23)
Airk)Xirk
5
Lagrangian costs coefficients) while keeping the same setups (yt = 1) may improve the primal solution. We strengthen the contribution of SP4 to the lower bound u(A) by adding two simple potentially useful inequalities in the form of surrogate capacity constraints. The constraints are as follows:
FZit, ,< W/t, We obtain separability by relaxing (18) and placing it in the objective function with multipliers 1itk to yield the subproblems SP, and SP,. Subproblem SP, is a transportation problem as in the case of the Lagrangian relaxation approach. It is solved quite efficiently by several available network codes, such as that of Ahrens and Finke [lS]. Subproblem SP, can be decomposed into N independent unlot-sizing problems single-item capacitated each of which can be solved by Zangwill’s [16] algorithm.
I-IS
decomposition
problem
is stated
v* = Maxu(i) s.t. iitk unconstrained,
where u(n) = ~~(2) + u2(A), and v*, the optimal Lagrangian solution, is a lower bound to the primal problem. Each solution of SP, is primal feasible since it satisfies both demand and capacity constraints. Additionally, adjustment of the production quantities (using the true arc costs and not the
T,
where wi: represents an integer number of period demands that can be inventoried starting from period t, which will just exceed the actual capacity limit of period t. Similarly, wi: represents an integer number of period demands that can be backordered starting from period 1, which will just exceed the actual capacity limit of period t. During implementation, the surrogate capacities may be adjusted to reflect shelf-life and backorder limits if they exist. The equalities (25) and (26) are potentially quite useful, since in the optimal solution, the total production in period t will never exceed c,. Further, they both allow the optimality property for the single-item problem to be valid (i.e. they do not change the structure of the problem). While they will allow an integer number of periods of demand to be satisfied, they will prohibit the production of unnecessarily large lot sizes for each item particularly lot sizes that are much
6
H.H. Miller,
M. Ymgllnt.
J. Production
larger than the capacity restriction in the original problem. This has the potential to improve computational efficiency as we explore a smaller solution space. Further, since this is an improvement over the capacitated version of the problem, there is possibility of producing tighter bounds. These constraints may be particularly useful when capacities are tight, the planning horizon is long and the setup cost/holding cost ratio is high.
Economic.v 34 11994) I-15
The above partitioning scheme approach was not employed by Thizy and van Wassenhove [S]. They restricted the primal partition to activated setups to form the solution of the N WagnerWhitin problems subproblems (see also Wagner Whitin [17]). This was a reason for the failure of their algorithm. The primal transportation problem has the following formulation and cost structure.
5. Primal Partitioning Primal partitioning refers to the use of a given set of activated setups (4’ir = 1) to attempt to find a feasible production schedule. In our methodology, we use the setups activated in the solutions to the subproblems to define a corresponding primal transportation problem. In the case of SPi and SP,, only the setups activated are needed to produce the primal solution. This is because, their solutions are already primal feasible. Also, we note that their corresponding schedules are based on Lagrangian coefficients. In the primal partitioning procedure, we are attempting to improve the schedule by using the true cost coefficients. Hence, any adjustment in the production quantities imply an improvement of the primal solution. In the case of the SP2 and SP, the setups activated, more often than not, are insufficient to generate a primal feasible solution. It is not difficult, however, to produce a primal feasible solution by using the activated setups plus a minimum number of additional setups necessary to produce a feasible solution. To do that we simply devise a cost structure that allows the model to choose the additional setups necessary for feasibility. The cost structure gives priority to previously activated setups, and considers the use of non-activated setups at a premium, only if they are needed to produce a primal feasible primal solution. Our solution scheme, therefore, guarantees an additional feasible primal solution at every iteration of the Lagrangian implementation. Consequently, this produces a strong possibility of finding the optimal or near-optimal solution to the problem.
where
(29)
The role of m is to lower the priority of the arc corresponding to .& since yi, = 0, and to give priority to those arcs associated with positive setups (i.e. Yit = 1). Increasing the cost of each of these arcs by a fixed amount m does not affect their attractiveness relative to each other. In this way a feasible solution is always produced by the primal partition, and its true cost is easily determined by using the correct cost coefficients. Note that the term sit/hitk in gitk attemptS t0 reflect the setup cost associated with producing in periods where no production took place. This has the effect of influencing the selection of setups in a manner which is sensitive to the true setup cost.
6. The subgradient procedure We solve both the Lagrangian decomposition (LD) and relaxation (LR) problems by a subgradient algorithm due to Held et al.
Cl0
H.H. MiNar, M. YanglInt. J. Production Economies 34 (1994)
6.1.
The relaxation
The Lagrangian lows: r& ’ = Max[O,
I-15
algorithm multipliers
{&
+ nj(,&
where uj is the step length
are updated
- dikyit)}],
as fol-
(30)
Initialize the Lagrangean multipliers and iteration counux
given by
where yj is a scalar set to 2 initially and reduced by half if the lower bound fails to improve after a fixed number of iterations, X$ is the optimal solution to SP, , y{* is the optimal solution to SP,, r* is the value of the current best primal r(rrj) is the current lower feasible solution, bound, and & are the multipliers for the current iteration.
6.2.
The decomposition
The Lagrangian as follows:
algorithm
multipliers
for LD are updated Fig. 2. A schematic
&: 1 = ;l;,k + uj(zjtk - x!J,
(32)
where uj is the step length ,j = yj(U* - r(~j))/~,~~ i
given by
(dk f
-
dtk)2,
(33)
k
where yj is a scalar set to 2 initially and reduced by half if the lower bound fails to improve after a fixed number of iterations, XiZk is the optimal solution to SP,, zitk is the optimal solution to SP,, u* is the value of the current best primal feasible o(Aj) is the current lower bound, solution, and & are the multipliers for the current iteration.
6.3. Summary
qf the subgradient
algorithm
The implementation of the subgradient algorithm (see Fig. 2) for both Lagrangian problems are identical, except for the use of different para-
of the subgradient
algorithm
meters. We summarize the procedure for LD. The appropriate modifications are made to implement the algorithm for LR. The subgradient algorithm is summarized below. Step 1. Initialize j, I, y, u*. Step 2. Solve SP, and SP,. Determine the lower bound ~(2’). Step 3. Let yi: and yi: denote the setup variables for SPJ and SP,, respectively, and let Y’ and Y2 be the sets of setups corresponding to the optimal solutions of SP3 and SP,, respectively. Determine the following coefficients:
8
H.H. Millar, M. Yang/Inf.
J. Production
Step 4. Solve PP using the two sets of coefficients. Let F, and F, be the primal values of the corresponding optimal solutions. Let ~‘j= min{F,, F2, u*). Set t’* = uj. Step 5. Determine the dual gap as follows. 0 = 2.O(v* - a(I,j)/(u* + u(ij)). Stop if a prespecified iteration limit is reached, or 0 < E, where E is the prespecified tolerance. Otherwise, update the Lagrangian multipliers using (32) and (33) and go to Step 2.
7. Computational
results
We found no well-known set of test problems with backorder costs which could be used for assessing the algorithm. We adapt an experimental design described in Maes and Van Wassenhove [4] because it is an accepted design for testing lot-sizing problems.
7.1. The experimental
design
A total of 270 test problems were generated. A summary of the parameters of the experimental design is found in Table 1. In the deTable I A summary
of the problem
parameters
Parameter
Value
Number of periods Number of items Holding cost Backorder cost
Constant Constant Constant Low Medium High Low Medium High Constant Low High Low Medium High Norma1
Capacity
tightness
Absorption rate Time between orders Lumpiness
Demand
distribution
(TBO)
(6)
(5) (1) WI
(4h) (6h) (111%) (135%) (200%) (1) (l-3) (2 -6) (0%) (10%) (20%) (100,25)
Economics
34 (1994)
I-15
sign, we vary four parameters: capacity tightness, setup cost through time between orders (TBO), demand lumpiness, and backorder costs. Capacity tightness is represented by the ratio of total aggregate capacity to total aggregate demand. Setup cost is related to the square of the TBO, and is obtained through the economic order quantity (EOQ) formula. The lumpiness parameter is defined as the probability of a demand occurring in a given period. Once the demand is permitted, it is generated according to a normal distribution with a mean demand of 100, and a standard deviation 25. For the purpose of the experiment, we define backorder cost as a linear function of the carrying cost (b = fh), wheref 2 1. At this point in time, the actual value of the backorder cost is not the primary issue, but rather, the quality of the performance of the algorithms when backordering is allowed. The four factors, capacity tightness, TBO, lumpiness, and backorder costs result in (3 x 2 x 3 x 3) = 54 combinations. For each combination, 5 random problems were generated each of size 5 items x 6 periods. We generate solutions for these test problems using the relaxation and decomposition algorithms which we will refer to as RELAX and DECOMP, respectively. Further, we solve the no-backorder restriction of the problems using a decomposition algorithm described by Millar and Yang [S]. We refer to this algorithm as the pure inventory model (PIM). The purpose of generating these solutions is to provide an additional scheme for evaluating the algorithms, and to allow the assessment of the potential of a backordering policy. The multipliers for all the algorithms are initialized with a value of 0. We measure the quality of the solutions generated by the three algorithms by computing a dual gap as follows: Gap = 2oo x (upper
bound
- lower
bound)
(upper
bound
+ lower
bound) ’
The lower bound used is the better of the lower bounds produced by RELAX and DECOMP.
H.H. Millar, M. YanglInt. J. Production Economics 34 (1994)
In summarizing the results, we compute the means and standard deviations of the dual gaps for the 5 problems in each of the 54 categories. The standard deviations provide a measure of the stability of the algorithms, a characteristic that is extremely important in the design of heuristics. These results are summarized in Tables 5-10 in the appendix.
40
l-15
*RELAX + DECOMP _) PIM
1 30 -
20 -
10 -
7.2. The effect of capacity tightness and setupjholding cost ratios
0
I 0
Figure 3 shows a graphical representation of the performance of the three algorithms vs. problem class when the lumpiness is O%, and the backordering unit cost is four times that of the holding cost. A problem class refers to specific capacity/setup cost combination. The most stable algorithm appears to be RELAX followed by DECOMP. In Figure 4, we observe a general saw-tooth pattern of performance related to the alternating low and high setup costs. Both DECOMP and PIM appear to perform better on low setup cost problems than on high setup cost problems. On the other hand, RELAX appears to do better on the problems with high setup costs. When capacity is loose, the pure inventory model (PIM) performs extremely well compared to RELAX and DECOMP. This is because there is little opportunity for backordering. Because of sufficient capacity, inventory will tend to be preferable to backorders for eliminating setups. Because PIM searchs a smaller solution space than RELAX and DECOMP, it often produces a better solution using the same amount of computational effort. We believe that RELAX and DECOMP would need significantly greater effort to come up with a comparable solution. As we move from problem class 1 to 6, the tightness of the capacity increases. PIM shows a dramatic change in performance with capacity tightness but for those cases with high setup costs. The performance of RELAX and DECOMP improves notably. This can be explained by the fact that with increase in capacity tightness, there is greater potential for backordering, pareticularly
.
1
I 2
.
I 3
.
I 4
.
I 5
.
1 6
.
I I
PROBLEM CLASS Fig. 3. Size of the dual gaps vs. problem ness = 0%).
60 50
4
PIMib=Zh
n
PIM/bdh
A
PIM/b=bh
class (b = 4h, lumpi-
40 a 3 a %
30
20
10
0 1
2
3
4
5
6
PROBLEM CLASS
Fig. 4. The ness = 0%).
effect
of
backorder
costs
on
PIM
(lumpi-
particularly when backorder costs are inexpensive. This potential, however, is tempered by the size of the setup costs. If capacity is tight, setup cost is high and backorder cost is low to moderate, there is a significant potential for using backorders to eliminate setups. By searching only the pure inventory solution space, PIM misses out on cheap solutions
10
H.H. Millar, M. Yang/hi.
J. Production
that incorporate backorders. However, when setup cost is low, the potential use of backorders as a means of eliminating setups is greatly reduced. In fact, due to the tightness of capacity, the lot-for-lot solution may even be a viable alternative. If production shifts are necessary for reasons related to capacity restrictions, then inventory would be preferable alternative to backordering. Though the above phenomenon is well illustrated when b = 4h, (see Fig. 3) it is even more dramatic when b = 2h (see Table 5 in the appendix). For high capacity tightness, low setup costs and b = 2h, PIM obtains a mean gap of 53.73% compared with 8.61% for the case with low setup costs. RELAX and DECOMP, on the other hand, perform reasonably well with tight capacity, high setup costs, and low backorder costs. We note that the same general saw-tooth pattern of performance repeats itself as we vary backorder costs from 2h to 6h with increasing improvement in the performance of all the algorithms. This is because as backorder cost increases, even with tight capacity and high setup costs, the potential for backordering diminishes. Given there are few good backorder solutions, all the algorithms including PIM tend to perform well. In most cases, the optimal solution will tend to be a pure inventory solution.
Economics
30
34 (1994)
I-IS
+
1
a-
RELAX/h=2h
+
RELAX/lHh RELAX/b=6h
20 3 % a a
10 -
0; 0
.
(
.
1
,
.
2
,
,
3
I
4
-
5
I
I
.
6
7
PROBLEM CLASS Fig. 5. The effect ness = 0%).
of backorder
costs
on
30 _
RELAX
(lumpi-
4
DECOMP/b=2h
+
DECOMP/b=4h DECOMP/b=6h
4
10 -
7.3. The impact qf backorder costs 0 0
Backorder costs impact on the performance of each algorithm differently. When backorder costs are small, the performance of the algorithms depend heavily on the setup costs and capacity tightness (in essence the opportunity for backorders). If capacity is loose, despite high or low setup costs, the use of inventory is preferable to backorders. Consequently, PIM outperforms RELAX and DECOMP in this case. When capacity is tight, the setup cost becomes very important. If setup cost is high then the use of backorders to eliminate setups is desirable. If setup cost is low, then the opportunity for backorders is reduced, and capacity “crunches” are better handled by using inventory wherever possible.
1 I
*
I 2
.
I 3
.
1 4
*
1 s
*
’ 6
’
’ 1
PROBLEM CLASS Fig. 6. The effect of backorder ness = 0%).
costs
on DECOMP
(lumpi-
When backorder costs are high, there is little opportunity for the use of backorders to eliminate setups. Consequently, all the algorithms perform comparatively well. Figures 4-6 show the performance behavior of each of the algorithms as a function backorder costs and problem class. We observe that all of the algorithms display their most erratic performance under low backorder costs. This behavior stabilizes as backorder costs increase.
H.H. Millar. M. YanglInt. J. Producrion Economies 34 (1994)
7.4. The stability
of the algorithms
Table 3 One-way
Tables 8810 in the appendix display the standard deviations of the dual gaps for each of the 54 problem categories. The variability which exist does not seem to show any peculiar pattern. It would appear that all of the algorithms are fairly stable. There is one class of problems in Table 10 which show a standard deviation of 20.98%. This can be attributed to a single problem for which none of the algorithms was able to produce a suitable lower bound. Table 2 shows a summary of statistics for each of the algorithms. We note that performance of RELAX and DECOMP appears to be comparable, though RELAX has a lower mean performance. PIM has the worst mean and standard deviation of the dual gaps. This is an indication of the value of an algorithm specifically designed to handle the lot-sizing problem with backorders. Tables 3 and 4 are ANOVA tables highlighting which factors affect the performance of the algorithms. From the tables, it would appear that setup cost is an important factor, particularly so for PIM and DECOMP. Demand lumpiness did not have a significant impact on all of the algorithms, and capacity had only a moderate impact on RELAX and DECOMP, but a significant impact on PIM. Backorder costs played a major role in the success of RELAX and DECOMP. From Table 3, the results suggest that both setup cost and backorder cost interacting together can significantly explain the performance of RELAX and PIM. However, the interaction of the two factors does not seem to play a significant role in explaining the performance of DECOMP.
Table 2 Summary
statistics
Sample size Minimum Maximum Mean value Standard deviation
for the algorithms RELAX
DECOMP
PIM
54 2.4 1 22.50 8.20 4.08
54 2.44 22.95 10.43 4.72
54 0.07 53.13 12.36 13.11
ANOVA
Factors
Lumpiness Capacity
Table 4 Two-way
cost
ANOVA
Factors
Backorder
F-value P-value F-value P-value F-value P-value F-value P-value
cost
Setup cost* Backorder cost
Algorithms RELAX
DECOMP
PIM
5.98 0.018 1.32 0.271 0.62 0.541 13.04 0.000
2.23 0.139 0.56 0.576 0.39 0.678 22.58 0.000
24.16 0.000 0.06 0.94 5.56 0.007 5.77 0.008
results for setup vs. backorder Statistics
Setup cost
11
results Statistics
Setup cost
Backorder
I-15
F-value P-value F-value P-value F-value P-value
costs
Algorithms RELAX
DECOMP
PIM
11.52 0.001 18.89 0.000 7.116 0.002
4.27 0.044 24.16 0.000 1.15 0.325
40.93 0.000 10.92 0.000 8.32 0.001
8. Conclusions Few approaches to the CMLSP-B have been developed to date. We have presented two effective algorithms which guarantee finding a feasible schedule if one exists, and which also provide an ex post measure of the quality of the solutions generated. The formulation provided allows the user to specify several constraints on the problem without increasing its complexity. These constraints include: production and inventory constraints at an aggregate level, production and inventory constraints at an item level, shelf-life limits for inventoried items, limits on the number of backorder periods allowed, and preventive maintenance. Traditional formulations make it difficult to incorporate these constraints.
12
H.H. Millar, M. YanglInt.
J. Production
We execute the Lagrangian procedure for 150 iterations. We can increase the number of runs to any number. Computational requirements per execution are very reasonable, with a mean of 12.33 s and standard deviation of 0.47 s for the given problem size of 5 items and 6 periods. The problems were solved on a VAXJVMS 785 computer, which is a relatively slow computer. Computational times could be greatly improved through the use of a faster computer, and by improving the efficiency of the programming code. With regards to computational complexity, it is well-known that problem size and the S/H ratios are indicators of computational complexity for the CMLSP. It would appear that we need to generalize the S/H ratio to include backorder costs. If we accept the argument that a backorder is a mirrored
Economics
34 (1994)
I-15
view of an inventoried item, then S/H ratios and S/B ratios are equivalent. We may consider a measure given by S’/(HB). However, this would have to be tested. Recognizing the fact that existing myopic heuristics for the CMLSP are not easily modified, we believe our heuristics are a useful contribution to reducing the scarcity of effective algorithms for the multi-item lot-sizing problem with backorders.
Acknowledgement This research is partially funded by the Natural Science and Engineering research Council of Canada under grant 0GPIN020.
Appendix Table 5 The means
of the dual gaps in each category
Methods
h = 2h
Capacity L Setup
H Setup
M Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
19.29 18.80 1.94*
12.99 12.57 9.29*
22.50 22.95 8.43*
11.18* 21.97 40.23
12.46 13.02 8.61*
7.22’ 10.44 53.73
10%
RELAX DECOMP PIM
16.51 16.42 2.10*
7.88* 7.95 14.33
17.35 18.30 4.26*
8.13* 13.87 30.45
11.73* 12.53 16.36
7.05* 16.08 50.16
20%
RELAX DECOMP PIM
8.68 8.54 1.24*
7.36* 18.88 15.18
12.19 12.00 1.80*
6.36* 14.11 37.04
12.73 10.82 10.66*
5.61* 17.85 47.15
H.H. Millar. M. Yanglint. J. Production Economics 34 (1994)
Table 6 The means of the dual gaps in each category Methods
I-15
13
b = 4h
Capacity L Setup
M Setup
H Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
7.74 8.32 0.07*
7.14* 13.09 9.81
7.40 8.13 2.68;
6.31* 10.11 14.20
8.34 8.64 8.00*
5.74* 9.77 30.83
10%
RELAX DECOMP PIM
5.59 5.18 1.31:
2.80; 8.84 10.68
7.65 6.79 2.64+
4.96* 8.17 20.80
10.87* 10.87* 12.09
6.11* 9.67 27.05
20%
RELAX DECOMP PIM
3.20 4.26 2.18*
4.00* 10.99 5.06
8.07 7.14 3.71*
6.85* 13.10 19.09
6.40 7.35 4.62*
10.74* 13.53 18.36
Table 7 The means of the dual gaps in each class (b = 6h) Methods
used
Capacity L Setup
M Setup
H Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
2.74 2.44 1.37:
6.67* 11.50 6.76
8.10 5.70 4.35*
8.05* 12.82 17.51
8.18 7.16 5.99*
6.81* 7.66 8.19
10%
RELAX DECOMP PIM
4.04 4.04 0.60*
3.83* 5.47 4.47
7.28 7.26 3.00*
5.40* 6.13 6.11
4.47* 5.3 1 4.47*
12.35* 15.60 18.37
20%
RELAX DECOMP PIM
9.40* 9.40* 9.40*
2.4 1 6.93 2.07*
5.10 4.48* 4.50
5.86 5.21* 9.23
8.35* 9.96 8.72
6.52 4.95* 6.37
H.H. Millar. M. YanglInt. J. Production Economics 34 (1994)
14
Table 8 The standard Methods
deviations
used
of the dual gaps in each category
l-15
b = 2h
Capacity L Setup
M Setup
L
H
L
H Setup H
L
H
0%
RELAX DECOMP PIM
5.91 6.01 2.67
5.58 3.41 3.37
4.97 4.96 1.80
2.52 1.54 8.35
2.82 2.66 4.51
3.87 3.54 12.49
10%
RELAX DECOMP PIM
4.04 4.71 1.81
4.30 4.87 6.07
1.61 2.42 1.31
4.18 7.43 13.91
5.34 3.34 8.66
3.44 6.84 11.05
20%
RELAX DECOMP PIM
2.82 2.85 2.06
3.27 4.21 7.99
2.53 2.96 2.99
3.37 4.86 14.87
4.4 1 5.21 3.90
2.17 3.14 19.38
Table 9 The standard
deviations
Algorithms
of the dual gaps in each category
b = 4h
Capacity L Setup
H Setup
M Setup
L
H
L
H
L
H
0%
RELAX DECOMP PIM
3.52 3.82 0.25
2.55 9.94 13.10
1.95 2.70 3.08
3.48 3.99 10.28
3.26 3.61 3.99
4.75 7.04 17.91
10%
RELAX DECOMP PIM
4.35 5.15 2.21
3.36 7.01 13.15
3.81 2.25 3.39
3.61 4.95 8.42
5.60 5.60 4.19
3.00 3.23 12.78
20%
RELAX DECOMP PIM
1.40 1.48 1.49
3.63 3.96 6.24
3.49 4.95 2.30
3.24 8.48 1.17
2.51 3.62 2.55
6.86 4.66 9.05
H.H. MiNar. M. YanglInt. Table 10 The standard Methods
deviations
used
of the dual gaps in each category
J. Production
Economic.! 34 (1994J
I-15
15
b = 6h
Capacity L Setup L
M Setup
H Setup
H
L
H
L
H
0%
RELAX DECOMP PIM
2.84 2.32 1.77
2.76 3.41 4.56
3.38 4.11 4.44
6.05 6.01 9.35
4.97 3.00 6.35
5.45 9.34 4.44
10%
RELAX DECOMP PIM
5.37 5.37 1.10
2.80 5.44 5.99
5.73 5.75 2.87
8.24 8.36 7.84
5.56 5.32 5.56
8.33 10.49 11.66
20%
RELAX DECOMP PIM
20.98 20.98 20.98
2.17 4.94 1.62
5.18 5.64 5.62
7.73 4.40 9.42
13.51 12.83 13.22
8.77 9.06 8.87
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