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The discrete lot-sizing and scheduling problem with sequence-dependent setup costs
European Journal North-Holland
of Operational
Research
395
75 (1994) 395-404
The discrete lot-sizing and scheduling problem with sequence-dependent setup costs
Bernhard
Fleischmann
Lehrstuhl fiir Produktion und Logistik, Universitiit Augsburg, Memminger Strasse 14, 86159 Augsburg, Germany Received
March
1992; revised
September
1992
Abstract: We consider the problem of scheduling several products on a single machine so as to meet the
known dynamic demand and to minimize the sum of inventory costs and sequence-dependent setup costs. The planning interval is subdivided into many short periods, e.g. shifts or days, and any lot must last one or several full periods. We formulate this problem as a travelling salesman problem with time windows and present a new procedure for determining lower bounds using Lagrangean relaxation as well as a heuristic. Computational results for problems with up to 10 products and 150 periods are reported. Keywords: Manufacturing;
Lot-sizing; Scheduling; Lagrange multipliers
1. Introduction
Planning lot-sizes for many products with dynamic demand under tight capacity restrictions is an important and difficult task in manufacturing industries, which is neglected in the standard MRP based production and scheduling systems, but has recently become a subject of extensive research. A recent survey of various lot-sizing models is given by Salomon in [13]. Besides the usual capacitated lot-sizing problem (CLSP), models with ‘small time buckets’ are useful for developing short-term production schedules for single facilities. In these models, the planning interval is subdivided into many short periods, shifts or days, and only one product can be processed per period on a certain facility. A lot then consists in the production of the same product in one or several consecutive periods, and a setup occurs at the beginning of the first period of every lot. Models of this kind with continuously varying lot-sizes are considered in [4,9] for a Correspondence to: Prof. B. Fleischmann, Lehrstuhl fiir Produktion und Logistik, Universitiit Augsburg, Memminger Strasse 14, 86159 Augsburg, Germany. 0377.2217/94/$07.00 0 1994 - Elsevier SSDI 0377.2217(93)E0202-9
Science
single product and in [lo] for the multi-product case. Discrete lot-sizing and scheduling problems (DLSP) are based on the ‘all or nothing’ assumption, i.e. in any period the facility either processes one product at full capacity or it is completely idle. Variants of the DLSP are considered by Gascon and Leachman [71 for a special case, by Magnanti and Vachani [ll] with distinct setup costs and change-over costs, by Salomon [13] with setup costs and setup times, by van Hoesel [S] for a single product with setup and start-up costs and by the author [5]. A major advantage of the small-time-bucket models against the CLSP is the exact control of the sequence of lots and, hence, the possibility to include sequence dependent setup costs. Nevertheless, all solution methods presented so far for the DLSP are restricted to sequence independent setup costs, because a preferred procedure consists in decomposing the problem into singleproduct problems, either by Lagrangean relaxation [5] or by column generation [131. This product decomposition, however, is obstructed by the additional interdependence of the products caused by sequence dependent setup costs. Also the integer programming model [II] and its facet
B.V. All rights reserved
B. Fleischmann / Sequence-dependent setup costs
396
inequalities do not admit sequence dependent costs. As an exception, we m a d e an attempt [6] to generalize the product decomposition by splitting the setup costs into a sequence dependent and an independent part, but with rather poor results. In this paper, we consider the multi-product single-machine DLSP with sequence-dependent setup costs and present a new solution procedure, based on the equivalence of the DLSP and a travelling salesman problem with time windows the (TSPTW). This equivalence is explained in Section 3. In Section 4, we develop a Lagrangean relaxation of the T S P T W into a shortest path problem with time windows (SPPTW) and determine lower bounds for the DLSP. Section 5 describes a heuristic for the TSPTW, and Section 6 reports on computational results. First, we summarize the basic notations in the next section.
A typical structure of dynamic demand in practice is not based on the (small) DLSP periods, but on larger macroperiods (e.g. weeks or months),
Ts={ts_l+l ..... ts}, s = l , . . . , N ' , where t o = 0, which may contain a different number of periods, and the demand is concentrated in the last period of every macroperiod, i.e. djt = 0 for t 4= t s. The holding costs hj also refer to the macroperiods and we set h 0 = 0. Let Yjt denote the decision variable indicating whether product j is produced in period t (yit = 1) or not (Yjt = 0) and Ijs the inventory level at the end of macroperiod s. We transform the total holding cost as follows: ts
Ehjlj = E
j,s
j,s
We consider a planning interval with periods t = 1 , . . . , N of equal length and products j = 1. . . . , M processed on a single machine. In addition, we introduce the ' p r o d u c t ' j = 0, which is ' p r o d u c e d ' when the machine is idle. Let d~t denote the net demand and Pi the output per period of product j > 0. H e n c e the required number of periods for production of product j up to period t is
1
djr/Pj ,
j>0,
where [.] is the integer rounding-up operator, and for j = 0 we set /)0t = 0,
t=l ..... N-l, M
150N = N -
E if)iN" j=l
As there is only one machine, the production requirements up to period t can be strengthened into Djt for j = 0 . . . . , M as follows: DiN=DiN,
Djt=max{ff)jt, Dj,t+l-1),
hj E ( P j Y j t - djt) t=l
= E hj ~., ( N ' - s + l)(pjYjt-djt)
2. Problem formulation
Djt =
j,s
,=N-
1,...,1.
t~ Ts
= EcjtYjt-Ho, j,t
with the constants
%t=hjpy(N'-s+l)
for t E Ts
(1)
and
Ho= ~.,hj E ( N ' - s + l)djt" j,s
t~Ts
Note that the macroperiod structure in no way restricts the validity of our model, but is an arbitrary additional time structure, which is useful to model real data situations (cf. Section 6). As a special case, the macroperiods can be equated with the DLSP periods. The setup cost S u arises at the beginning of a lot of product j when product i is produced in the immediately preceding period. For convenience we s e t Sii = O. This definition includes S0j and Sio, the latter term can be considered as shut down cost. This setup cost model is restrictive in so far as it does not apply to the case that the setup cost after some idle periods depends on the last product before the interruption, i.e. that there is no cost difference between the sequences i ~ 0 ~ j and i ~ j . This case would require a more complicated model and methods, since it involves a process with longer memory.
B. Fleischmann / Sequence-dependent setup costs
Note that S~j need not satisfy the triangle inequality, i.e. there may be products i, j, p such that
397
following latest and earliest period for element (j, k): tj~a× = min{t • Dj, = k},
(5)
Sip -'F Spj < Sij.
tj~in=min{t:r - ~_~Dir>k for r = t . . . . . N } , A particular case is a 'rinsing product' p in the chemical industry, with Sly and Sm being zero or very small compared to Sit. In this case it is advantageous to split the production of product p into many small lots each of which removes a major setup. Thus, the setup cost structure may be an incentive for small lots of some products, as opposed to sequence independent setup costs. Summarizing the above notations we obtain the following model:
i4-j
(6) which are also used in [5]. Note that definition (5) is valid since for every j, D~1 4 1 and 0 ~
t = 2 . . . . . n.
Equations (5) and (6) constitute time windows, TWjk= [ti~ i", tj~~x]
for ( j , k) ~ J K ,
with the following properties: (DLSP) Min
Lemma 1. min , and (i) tj~i" < t~,k+
E Sij max( Yi,t-I ~- Y j t - 1, O)
tjn~ax < t j,max k + 1 for every j and k < DiN. (ii) DLSP has a feasible solution if and only if TWik 4=0 for every (j, k) e JK. (iii) A one-to-one assignment of the elements' (j, k ) ~ JK to the periods t constitutes a feasible solution to DLSP if t <~tT~a× for every (j, k) assigned to r
i ,j,t
+ ~_~c jr y j, - H o j,t l E Yjr ~ Ojt
s.t
for all j, t,
(2)
r=l
~yj,=
1
for all t,
(3)
J
yj, ~ {0, 1},
(4)
where Yj0 = 1 if j is the initial product i0, and for j 4: i o.
Proof. (i): For s (6),
=
min tj.k+ I we have, by definition
r-~Dir>k+l>k
for r = s . . . . . N
i4=j
Yjo = 0
and
3. TSP model with time windows Any feasible solution to DLSP assigns a product to every period. However it is useful to distinguish the production activity in a single period t not only by the product j, but also by a counter k = Etr_ I Y~r specifying the number of occurrences of product j up to the considered period. The set of all pairs (j, k) occurring in a feasible solution is JK={(j, which Then ment 1. . . . .
k) : j = O , . . . , M ;
k = I , . . . , D j N },
has exactly N elements, as E~DjN=N. a solution becomes a one-to-one assignof the elements of JK to the periods t = N. The constraints (2) obviously imply the
s-l-
~_~Di,s_l>/s- ~ _ , D i s - l > ~ k + l - 1 i~j
i~j
=k, hence tjn~in <~ S --
1.
The second inequality is obvious. (ii): DLSP has a feasible solution if and only if
~,,D#<~t for t = l . . . . . N. J This condition implies, for any (j, k ) ~ JK and max r >~ tjk ,
r- EDi~=ri4~j
EOi~+Osr>Dj~>k, i
hence t~ i" ~
398
B. Fleischmann / Sequence-dependent setup costs
If DLSP is not feasible, there are some t and j such that EiDit > t and Djt > O. For k Oil we have t~ ax ~< t and
and
t-
The arcs bear the setup costs, i.e. Si0j for an arc /2 --->(j, 1), Sij for (i, k) ~ (j, l) and 0 for (j, k) --->O. Since an arc represents the transition from the production in one period to the production in the following one, the travel time through any arc is 1. Thus, any TSP tour on G can be viewed as an assignment of JK to the periods, (j, k) being assigned to t if (j, k) is 'reached at time t' in the tour. The holding costs c jr arise in any node (j, k) depending on the arrival time t. By the above considerations and Lemma 1 we have the following result:
( j , k) ~ $~ if k = DiN and tj~ ax = g .
=
~Dit
hence t~ in > t and TWjk = ¢. (iii): The statement is trivial if only those assignments are considered where the elements (j, k) occur in the right order, i.e. (j, k) is assigned to an earlier period than (j, k + 1). However, this is not necessary. Consider any (j, k) assigned to period t being in fact the kl-th occurrence of product j. If k I >/k, we have max . t ~< tj~ax ~< tyk~
If k 1 < k, there is some k 2 ~ t, hence max max t < t 2 <~tjk 2 <~ tjk ~ .
In both cases, the production of product j in period t is in time. [] Note that the lower limits of the time windows tj~in are redundant with respect to the feasibility of an assignment. However, they are useful in strengthening relaxed formulations, as presented in [5] and in the next section. The higher the utilization of the machine, the stronger is the strengthening effect. E.g. in the problems T V 1 1 TV14 (see Section 6), the density of the time windows, i.e. the average ratio of the time window length to the total planning horizon, varies only between 56% and 60% for the time windows [1, .t~x], but is reduced, for the time windows [t~ 'n, t~ax], to 19% in case of the highest utilization (97%, T V l l ) and to 51% in case of the lowest utilization (64%, TV14). Unfortunately, the assignment structure only describes the feasibility of a DLSP solution, but cannot express the sequence-dependent setup costs. We therefore have to formulate a TSPTW. We define a directed graph G = (Jr', sO) with the node set ,¢" = JK U {g2}, where /2 is the origin of the TSP tour, and the following arcs: /2 --* ( j , 1)
if t~in = 1,
(j,k)--->(j,k+l) (i, k) ~ ( j , l)
Lemma 2. DLSP is equivalent to the asymmetric T S P T W on G with time windows TWy~ for the nodes (j, k) ~ JK and time-dependent costs c jr if
(j, k) is reached at time t. The T S P T W can be formulated as follows using the assignment variables U j k t ( ( j , k) ~ JK, t TWo.k) and the TSP variables xij t indicating whether the tour goes from product i in period t - 1 to product j in period t:
E CjtUjkt + E Sijxijt
Min
j,k,t
(7)
i,j,t
s.t.
EUjkt
=
1
for ( j , k) ~ JK,
(8)
=
1
for all t,
(9)
t
EUjkt j,k
E Uik,t- 1 At- E Ujkt -- Xijt ~ 1 k k
t >~ 2,
EUjkl --Xiojl 4 0
for all i, j a n d
(lOa) for all j,
(10b)
k Ujk,, Xij t ~ {0, 1}.
(11)
4. A Lagrangean relaxation
if k
such that i ~ j, ti~in ~< ti~ax - 1 and ti~ax >1 tjn~in - 1;
The DLSP with sequence dependent setup costs is a very hard problem which seems unlikely to be solvable by exact methods in practical cases.
B. Fleischmann / Sequence-dependentsetup costs We therefore prefer to solve it by heuristics and to try to compute tight lower bounds for the minimal costs. This section presents a new procedure for determining lower bounds based on a Lagrangean relaxation. T h e r e is little work in the literature on lower bounds for the TSPTW. An exception is the p a p e r of Desrosiers, Sauv6 and Soumis [3]. We follow their idea to relax the T S P T W into a SPPTW. This can be achieved by relaxing the constraints (8) of the T S P T W (7)-(11), i.e. the conditions that the node (j, k) is visited exactly once for every (j, k ) ~ J K and assigning it the Lagrange multiplyer Ark. The resulting relaxed problem SPP(A) has the objective function (7) with crt replaced by c# + Ark and the constraints (9)-(11). It can be viewed as an SPPTW in the following way: SPP(A). Find a path on G from ~ to $2: - through exactly N (not necessarily distinct) nodes (j, k) ~ JK; - considering the time windows TWrk; - m i n i m i z i n g the arc costs and the time dependent node costs cr, + Ark, if node (j, k) is reached at time t. For any solution to SPP(A), we have a subgradient g with the components gik = ( n u m b e r of occurrences of ( j , k ) in the path) - 1 and £(~,k)~JKgrk = 0. If we update Ajk by subgradient iterations, starting from a - 0, we therefore have always E(j,k)~jKark = 0, Hence the cost of an optimal solution to SPP(A) is a lower bound to the minimal cost of the DLSP. For solving an SPPTW Desrochers and Soumis propose an algorithm in [2]. Our problem SPP(A) however is easier than the general S P P T W since the travel times are equal to 1 in every arc. In a DP algorithm for the general SPPTW the arrival time at the current node is needed as an additional state component, besides the node itself, whereas in our case the stage number t, which counts the n u m b e r of nodes reached so far, is just the arrival time. As there is no delay in the nodes, the arrival time is equal to the start time
399
in any node. Thus, our DP algorithm uses the state space
JKt={(j,k)~JK:t~TWjk},
t=l,...,n,
and JK 0 = {g2}, and the value functionf,(j, k ) = minimal cost of a path from (j, k) E JKt to g2 with start time t. It can be computed recursively as follows:
fN ( J' k) = cjx + Ajk for k = DiN and j such that ti~ a× = N,
f~(i, k) =cit +Aik + min{f,+
(12)
,(j, l)
+ Sij" ( j, l) ~ J ( i, k) C3JK,+,},
(13)
for (i, k) ~ JKt, t = N - 1. . . . . 1, where Y ( i , k) is the set of immediate successors of (i, k) in G, and
T0(S?) =
m i n { f l ( j , 1) + Si.j: t j~in = 1}.
(14)
As there are N stages and the size of the state space is IJK t [ ~
Step 1. For every j = 0 . . . . . M: g* = m i n { f , + , ( j , / ) : l = 1 . . . . . DiN; ( j , l) ~JK,+,}, the minimum being attained for l Step 2. For every i = 0 . . . . , M:
=
1a*.
f/* = min{gj* + Sij : j = 0 . . . . . M; j 4=i}, the minimum being attained for j = Ji*. Step 3. For every (i, k) ~ JK,: i f ( i , k + 1) ~ JKt+ 1 and ft+l(/, k + 1) ~ * : f t ( i , k) = f t + , ( i , k + 1) + Aik + cit,
B. Fleischmann / Sequence-dependent setup costs
400
the optimal successor of (i, k) is (i, k + 1); otherwise:
L ( i , k) =f~* + aik + Cit , the optimal successor of (i, k) is (Ji*, lj?.). The number of operations required is O ( N ) in Step 1, since EsDju = N , O ( M z) in Step 2 and O ( N ) in Step 3.
5. Heuristics
Many heuristics are known for the TSPTW, mainly improving heuristics which are popular in vehicle scheduling problems for postoptimizing the sequence of customers in a single tour [1,16]. According to our experience the final result achieved by these heuristics hardly depends on the quality of the initial sequence. We therefore use a rather simple initial heuristic followed by an improving heuristic based on the OR-opt principle [12]. Both parts will be described in this section. We now disregard the lower l i m i t s tjn~in of the time windows, which were only introduced for the relaxation. The initial heuristic is of the 'best-successor' type, only we construct the solution backwards from the last to the first period (or from the last to the first node in the 'tour'), because then the feasibility of the solution is easier to control. As opposed to the last section, the following algorithm yields a tour that contains every node (j, k) JK exactly once and, for every j, in the order of increasing k: Initial heuristic. Start: k~ = DsN (j = 0 . . . . . M).
For t = N , . . . , 1: Select any j such that kj > 0 and
t; ax
(15)
Assign (j, k~.) to period t; k s = k s - 1. Note that there is always a j satisfying (15) if the DLSP has a feasible solution. This would not be true, if the periods were processed forwards, because the Dst (and hence the t~ ax) do not remain valid for the partial problems over the
later periods when fixing the products in the earlier periods. The criterion for selecting product j in period t in the above heuristic should reflect the costs affected by this decision. If t < N and Jt+ 1 is the product assigned to period t + 1, we select j that minimizes Sj,jt+l 71- 6#,
(16)
cjt is an appropriate term of the holding costs, but not equal to c jr; because this would favor j = 0, as Cot = 0, and proceeding backwards would lead to idle periods in the latest positions and hence cause high holding costs. We have tested several criteria, among which the following gave the best results: where
Cjt=(ejt--Ct)--(Cjtl--Cti) ~
min
with tl=tyk, ,
where 1
M
?t =- -M j~=l Cjr" The motivation of this criterion is that, due to (1), the ratio %t/Pt = aj does not depend on t and hence
(j,= (~',- :,,)(aj-- 1). As t >~tt, the first factor is negative, so that products with high holding costs are prefered. Cjt represents the holding cost difference between assigning (j, k s) to period t and assigning it to the earliest possible period t~, supposed that all other products have average holding costs. For t = N , i.e. the first step of the initial heuristic, we try every j satisfying (15) and run both the initial heuristic and the improvement heuristic for every j again. The improvement heuristic is basically on Oropt procedure with time windows, as described in [1,14,16]. An Or-opt operation consists in cutting off a piece of the current tour and inserting it at another (earlier or later) position. The maximal length of the considered pieces affects the quality of the solution as well as the computation time. We found that a limit of 6 periods is reasonable for the test problems investigated (cf. Section 6), as it covers nearly all possible improvements. Our search strategy consists, at every step, in checking all possible Or-opt operations and then performing the best one. This strategy proved to yield
B. Fleischmann / Sequence-dependent setup costs
40l
periods and a capacity utilization of 97%, 95%, 76% and 64%, respectively. Various setup cost matrices (Sij) have been considered for these problems: SO denotes the original (sequence independent) setup costs of [17]. S1 is the matrix shown in Table 1. The entries are generated at random from the interval [0,600] and thus have the same order of magnitude as in SO. The entries of $2 are taken at random from the values 0, 100, 200 . . . . . 600, the entries of $3 from [0,300]. $4, $5 and $6 represent typical situations in practice with only two kinds of setups, a major setup (cost 500) and a minor setup (cost 100). $4 has the entries
considerably better results than performing every first improving operation found, but it also takes much more computation time. Additionally, we investigate the exchange of single periods (which is not an Or-opt operation for non-neighbored periods). We summarize the procedure as follows:
Step 1. Search for the minimal cost exchange of two single periods; if it reduces the cost, perform it and repeat Step 1; otherwise go to Step 2. Step 2. Search for the minimal cost Or-opt operation with pieces of up to 6 periods; if it reduces the cost, perform it and go to Step 1; otherwise stop.
100
Sgj = ~ 500
O u r implementation of the Or-opt search is different from that proposed in literature: It proceeds by increasing first the length of the piece to be displaced, while fixing the first node of this piece, as opposed to Savelsbergh [14,15], who varies the start node while fixing the length of the piece. Details of the implementation will be published in a separate paper.
for i < j , for i > j
where i, j >/1, i.e. there is a natural order of the products (e.g. from light to dark colors) with only minor setups, but any deviation from this order, in particular the start of a new cycle, requires a major setup. $5 corresponds to four product families (1, 2}, {3, 4}, {5, 6} and {7, 8} with minor setups within and major setups between, $6 to the two product families {1, 2, 3, 4} and (5, 6, 7, 8}. The additional entries of $4, $5 and $6 are S0i = 500, Sio = 0 (i >~ 1). The initial product is icl = 0 in all problems. The problems P R 1 - P R 4 are practice cases in the food industry, also considered in [6]. The n u m b e r of products and periods is shown in Table 3. P R I and PR2 have a planning interval of 8 weeks (macroperiods) and differ only in the available capacity: PR1 is based on a 3-shift system (120 hours per week), PR2 on a 2-shift system (80 hours per week), the periods are shifts. PR3 and
6. Computational results The procedure of Sections 4 and 5 has been tested by means of three classes of problems: The problems T V l l - T V 1 4 correspond to the data sets 11-14 of [17] for the CLSP with 8 products and macroperiods, which have been subdivided into DLSP periods of equal capacity (50 units per period), as described in Section 2. The 4 problems T V l l - T V 1 4 differ only in the capacity of the macroperiods and have 63, 64, 80 and 96 Table | The setup cost matrix $1 from product
to product 0 1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8
0 0 0 0 0 0 0 0 0
69 375 0 121 471 593 545 69 172
25 80 123 0 403 25 515 117 490
116 386 385 116 0 548 236 211 193
97 397 268 301 367 0 97 441 278
32 467 32 502 581 97 0 446 95
121 121 541 401 441 255 357 0 259
0 0 422 19 140 72 505 488 0
136 0 297 161 545 136 554 409 530
B. Fleischmann / Sequence-dependent setup costs
402
PR4 concern two separate lines in a factory and have a planning interval of one year, the macroperiods lasting 2 weeks and the periods being days. The third class of problems has been generated using a random data generator provided by M. Salomon (private communication). It generates problems with the number of products and periods and the capacity utilization specified. The demands djt are 0 or 1, the output rate pj = 1 for all products, i.e. the output per period is used as the quantity unit. The holding costs per period are 1, the average setup cost Si is derived, for every product j, from the optimal time between orders (TBO), which itself is a random number, equally distributed in a specified interval. Sit are normally distributed random numbers with mean Sj and a specified standard deviation. For these problems no particular macroperiods are defined, we equate the macroperiods with the periods. All problems considered have 10 products and 150 periods. Starting from a basic problem with a
utilization of 90% and TBO interval [3, 10 periods], we varied the utilization and the TBO interval as shown in Table 4. The standard deviation of Sit was fixed to 10. Tables 2 - 4 present the results for the three problem classes. For every problem, the gap (UB - L B ) / U B between the lower bound LB and the upper bound UB (cost of the best solution found by the heuristic of Section 5) and the percentage of the setup costs within UB is shown. For the random problems the results in every line of Table 4 are averages of three problems with identical parameters. For the problems TV, the data of which is published completely, Table 2 indicates LB, UB and the setup costs explicitly. This is to enable other researchers to compare these results with those from alternative methods. Nevertheless, the data and detailed results of all problems are available from the author. The computation times are split into the parts required for LB and for the heuristic. They refer to an implementation in MS Fortran 5.1 on a PC
Table 2 Results for problems T V l l - T V 1 4 Problem
TV11 TV12 TV13 TV14 TVll TV12 TV13 TV14 TVll TV12 TV13 TV14 TV11 TV12 TV13 TV14 TV11
a a a a b b b b
setup cost matrix
LB
$1 S1 S1 S1 S1 S1 S1 $1 S1 81 81 Sl SO SO SO SO $2 $3 $4 $5 $6
5152 4195 3362 3272 3821 2241 1174 898 6464 5992 4703 4559 8339 7539 7321 7334 6750 3066 7140 8420 5207
a holding cost = 0. b holding cost doubled.
UB
5194 4381 3509 3379 3904 2470 1527 1285 6484 6054 4899 4769 8940 8340 8040 7890 7090 3078 7290 11190 7640
setup cost
3904 2691 2119 1989 3904 2470 1527 1285 3904 2774 2119 1989 7450 6450 6200 6250 5700 1738 5800 9800 6300
gap (%)
0.8 4.2 4.2 3.2 2.1 9.3 23.1 30.1 0.3 1.0 4.0 4.5 6.7 9.6 8.9 7.0 4.8 0.4 2.1 24.8 31.8
comp. time (seconds) LB
UB
15 18 20 32 16 18 24 23 13 19 23 32 17 19 24 21 16 13 10 18 19
23 63 136 326 45 118 249 364 22 45 85 122 41 77 244 455 76 120 83 52 43
B. Fleischmann / Sequence-dependent setup costs
403
Table 3 Results for practice problems Problem
PR PR PR PR
M
1 2 3 4
N'
9 9 3 4
8 8 26 26
N
120 80 230 225
utilization
setup
(%)
cost (%)
66 99 91 95
with 486/33 M H z processor. The subgradient iterations are organized as in [5]: the step size is UB - LB Ov= l'Z
Y'jkg2k
w h e r e / z has the starting value 2 and is divided by a factor f > 1 every Q iterations without increase of LB; the standard values are f = 1.2 and Q = 4. The iterations terminate when o- falls below a certain limit e. The total n u m b e r of iterations was between 100 and 200 in most cases. As the results of the heuristic were unsatisfactory in many cases, we tried to repeat the heuristic in every subgradient iteration, in a similar way as in [5], adding the current Aiks to the criterion (16). Unfortunately, the observation made for the product decomposition algorithm [5], that multipliers A which give a high LB also yield good heuristic solutions, proved not to be true in this algorithm. Only the increased n u m b e r of trials of the heuristic improved the UB sometimes. We therefore included the repetition of the heuristic in the first 20 subgradient iterations only. The computation time for UB in Tables 2 - 4 comprises the time for all trials.
Table 4 Results for r a n d o m problems with 10 products and 150 periods; every line shows the average results for 3 problems utilization (%)
TBO interval
setup cost (%)
gap (%)
comp. time (seconds) LB
UB
80 85 90 95 99 90 90
[3,10] [3,10] [3,10] [3,10] [3,10] [5,20] [5,30]
71 68 64 61 58 79 85
5.2 6.7 5.0 5.8 6.9 10.7 16.7
55 56 48 45 40 61 89
122 107 95 104 105 128 135
17 9 16 20
gap (%)
1.0 1.1 0.7 3.5
comp. time (seconds) LB
UB
56 26 132 181
310 45 96 306
Overall, the results have improved remarkably c o m p a r e d to those obtained by means of an extended product decomposition algorithm for the problems T V and PR [6]. However, the gaps between LB and UB of up to 30% are still unsatisfactory. Even the majority of cases with gaps between 4% and 7% still requires improvement of the procedure. Whether a large gap is due to a weak LB or a bad heuristic solution or both, is an open question. In the only cases with known optimal solution, the problems TV with SO [5], however, the LB is very tight (less than 0,3% below the minimal cost in all 4 cases). This suggests the conjecture that the SPPTW relaxation yields tight lower bounds for the DLSP and that primarily the heuristics need to be improved. The quality of the results seems to depend little on the capacity utilization of the problem, as opposed to most capacitated lot-sizing algorithms which work worse in case of higher utilization [5]. T h e r e is even a slight reduction in the LB computing time in case of higher utilization, due to the reduced solution space. A crucial factor, however, is the percentage of the setup costs: the largest gaps and the highest computing times of the heuristic arise for the problems with zero holding cost, i.e. pure scheduling problems, and the random problems with high TBO; the easiest problems are those with low relative setup costs, e.g. the problems PR and T V with doubled holding costs. As to the different setup cost matrices there is little difference in the results for S1, $2 and $3; the cyclic setup costs $4 seem to be easier and the family setup costs $5 and $6 extremely difficult. For sequence independent setup costs (SO) the T S P T W heuristic is obviously not adequate, but the SPPTW relaxation gives an excellent LB, as explained above. Important tasks of further research on the DLSP are the determination of exact solutions
404
B. Fleischmann / Sequence-dependent setup costs
for sample problems, e.g. by means of branching, in order to give a more profound evaluation of the lower bounds, and the development of faster and better heuristics.
References [1] Desrochers, M., Lenstra, J.K., Savelsbergh, M.W.P., and Soumis, F., "Vehicle routing with time windows: Optimization and approximation", in: B.L. Golden and A.A. Assad (eds.), Vehicle Routing: Methods and Studies, North-Holland, Amsterdam, 1988, 65-84. [2] Desrochers, M., and Soumis, F., "A generalized permanent labeling algorithm for the shortest path problem with time windows", INFOR 26 (1988) 191-212. [3] Desrosiers, J., Sauv6, M., and Soumis, F., "Lagrangean relaxation methods for solving the minimum fleet size multiple travelling salesman problem with time windows", Management Science 34 (1988) 1005-1022. [4] Eppen, G.D., and Martin, R.K., "Solving multi-item capacitated lot-sizing problems using variable redefinition", Operations Research 35 (1987) 832-848. [5] Fleischmann, B., "The discrete lot-sizing and scheduling problem", European Journal of Operational Research 44 (1990) 337-348. [6] Fleischmann, B., and Popp, Th., "Das dynamische Losgr6Benproblem mit reihenfolgeabh~ingigen Riistkosten", in: Operations Research Proceedings 1988, SpringerVerlag, Heidelberg, 1989, 510-515. [7] Gascon, A., and Leachman, R.C., "A dynamic programming solution to the dynamic multi-item, single-machine scheduling problem", Operations Research 36 (1988) 5056.
[8] S. van Hoesel, "Models and algorithms for single-item lotsizing problems", Doctoral Dissertation, Erasmus University Rotterdam, 1991. [9] Karmarkar, U.S., Kekre, S., and Kekre, S., "The dynamic lot-sizing problem with startup and reservation costs", Operations Research 35 (1987) 389-398. [10] Karmarkar, U.S., and Schrage, L., "The deterministic dynamic product cycling problem", Operations Research 33 (1985) 326-345. [11] Magnanti, T.L., and Vachani, R., "A strong cutting plane algorithm for production scheduling with changeover costs", Operations Research 35 (1987) 832-848. [12] Or, I., "Traveling salesman-type combinatorial problems and their relation to the logistics of blood banking", Ph.D. Thesis, Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL, 1976. [13] Salomon, M., Deterministic Lotsizing Models for Production Planning, Springer-Verlag, Heidelberg, 1991. [14] Savelsbergh, M.W.P., "Local search for routing problems with time windows", Annals of Operations Research 4 (1986) 285-305. [15] Savetsbergh, M.W.P,, "An efficient implementation of local search algorithms for constrained routing problems", European Journal of Operational Research 47 (1990) 7585. [16] Solomon, M.M., Baker, E.K., and Schaffer, J.R. "Vehicle routing and scheduling problems with time window constraints: Efficient implementations of solution improvement procedures", in: B.L. Golden and A.A. Assad (eds.), Vehicle Routing." Methods and Studies, North-Holland, Amsterdam, 1988, 85-105. [17] Thizy, J.M., and van Wassenhove, L.N., "Lagrangean relaxation for the multi-item capacitated lot-sizing problem: A heuristic implementation", liE Transactions 17 (1985) 308-313.
of Operational
Research
395
75 (1994) 395-404
The discrete lot-sizing and scheduling problem with sequence-dependent setup costs
Bernhard
Fleischmann
Lehrstuhl fiir Produktion und Logistik, Universitiit Augsburg, Memminger Strasse 14, 86159 Augsburg, Germany Received
March
1992; revised
September
1992
Abstract: We consider the problem of scheduling several products on a single machine so as to meet the
known dynamic demand and to minimize the sum of inventory costs and sequence-dependent setup costs. The planning interval is subdivided into many short periods, e.g. shifts or days, and any lot must last one or several full periods. We formulate this problem as a travelling salesman problem with time windows and present a new procedure for determining lower bounds using Lagrangean relaxation as well as a heuristic. Computational results for problems with up to 10 products and 150 periods are reported. Keywords: Manufacturing;
Lot-sizing; Scheduling; Lagrange multipliers
1. Introduction
Planning lot-sizes for many products with dynamic demand under tight capacity restrictions is an important and difficult task in manufacturing industries, which is neglected in the standard MRP based production and scheduling systems, but has recently become a subject of extensive research. A recent survey of various lot-sizing models is given by Salomon in [13]. Besides the usual capacitated lot-sizing problem (CLSP), models with ‘small time buckets’ are useful for developing short-term production schedules for single facilities. In these models, the planning interval is subdivided into many short periods, shifts or days, and only one product can be processed per period on a certain facility. A lot then consists in the production of the same product in one or several consecutive periods, and a setup occurs at the beginning of the first period of every lot. Models of this kind with continuously varying lot-sizes are considered in [4,9] for a Correspondence to: Prof. B. Fleischmann, Lehrstuhl fiir Produktion und Logistik, Universitiit Augsburg, Memminger Strasse 14, 86159 Augsburg, Germany. 0377.2217/94/$07.00 0 1994 - Elsevier SSDI 0377.2217(93)E0202-9
Science
single product and in [lo] for the multi-product case. Discrete lot-sizing and scheduling problems (DLSP) are based on the ‘all or nothing’ assumption, i.e. in any period the facility either processes one product at full capacity or it is completely idle. Variants of the DLSP are considered by Gascon and Leachman [71 for a special case, by Magnanti and Vachani [ll] with distinct setup costs and change-over costs, by Salomon [13] with setup costs and setup times, by van Hoesel [S] for a single product with setup and start-up costs and by the author [5]. A major advantage of the small-time-bucket models against the CLSP is the exact control of the sequence of lots and, hence, the possibility to include sequence dependent setup costs. Nevertheless, all solution methods presented so far for the DLSP are restricted to sequence independent setup costs, because a preferred procedure consists in decomposing the problem into singleproduct problems, either by Lagrangean relaxation [5] or by column generation [131. This product decomposition, however, is obstructed by the additional interdependence of the products caused by sequence dependent setup costs. Also the integer programming model [II] and its facet
B.V. All rights reserved
B. Fleischmann / Sequence-dependent setup costs
396
inequalities do not admit sequence dependent costs. As an exception, we m a d e an attempt [6] to generalize the product decomposition by splitting the setup costs into a sequence dependent and an independent part, but with rather poor results. In this paper, we consider the multi-product single-machine DLSP with sequence-dependent setup costs and present a new solution procedure, based on the equivalence of the DLSP and a travelling salesman problem with time windows the (TSPTW). This equivalence is explained in Section 3. In Section 4, we develop a Lagrangean relaxation of the T S P T W into a shortest path problem with time windows (SPPTW) and determine lower bounds for the DLSP. Section 5 describes a heuristic for the TSPTW, and Section 6 reports on computational results. First, we summarize the basic notations in the next section.
A typical structure of dynamic demand in practice is not based on the (small) DLSP periods, but on larger macroperiods (e.g. weeks or months),
Ts={ts_l+l ..... ts}, s = l , . . . , N ' , where t o = 0, which may contain a different number of periods, and the demand is concentrated in the last period of every macroperiod, i.e. djt = 0 for t 4= t s. The holding costs hj also refer to the macroperiods and we set h 0 = 0. Let Yjt denote the decision variable indicating whether product j is produced in period t (yit = 1) or not (Yjt = 0) and Ijs the inventory level at the end of macroperiod s. We transform the total holding cost as follows: ts
Ehjlj = E
j,s
j,s
We consider a planning interval with periods t = 1 , . . . , N of equal length and products j = 1. . . . , M processed on a single machine. In addition, we introduce the ' p r o d u c t ' j = 0, which is ' p r o d u c e d ' when the machine is idle. Let d~t denote the net demand and Pi the output per period of product j > 0. H e n c e the required number of periods for production of product j up to period t is
1
djr/Pj ,
j>0,
where [.] is the integer rounding-up operator, and for j = 0 we set /)0t = 0,
t=l ..... N-l, M
150N = N -
E if)iN" j=l
As there is only one machine, the production requirements up to period t can be strengthened into Djt for j = 0 . . . . , M as follows: DiN=DiN,
Djt=max{ff)jt, Dj,t+l-1),
hj E ( P j Y j t - djt) t=l
= E hj ~., ( N ' - s + l)(pjYjt-djt)
2. Problem formulation
Djt =
j,s
,=N-
1,...,1.
t~ Ts
= EcjtYjt-Ho, j,t
with the constants
%t=hjpy(N'-s+l)
for t E Ts
(1)
and
Ho= ~.,hj E ( N ' - s + l)djt" j,s
t~Ts
Note that the macroperiod structure in no way restricts the validity of our model, but is an arbitrary additional time structure, which is useful to model real data situations (cf. Section 6). As a special case, the macroperiods can be equated with the DLSP periods. The setup cost S u arises at the beginning of a lot of product j when product i is produced in the immediately preceding period. For convenience we s e t Sii = O. This definition includes S0j and Sio, the latter term can be considered as shut down cost. This setup cost model is restrictive in so far as it does not apply to the case that the setup cost after some idle periods depends on the last product before the interruption, i.e. that there is no cost difference between the sequences i ~ 0 ~ j and i ~ j . This case would require a more complicated model and methods, since it involves a process with longer memory.
B. Fleischmann / Sequence-dependent setup costs
Note that S~j need not satisfy the triangle inequality, i.e. there may be products i, j, p such that
397
following latest and earliest period for element (j, k): tj~a× = min{t • Dj, = k},
(5)
Sip -'F Spj < Sij.
tj~in=min{t:r - ~_~Dir>k for r = t . . . . . N } , A particular case is a 'rinsing product' p in the chemical industry, with Sly and Sm being zero or very small compared to Sit. In this case it is advantageous to split the production of product p into many small lots each of which removes a major setup. Thus, the setup cost structure may be an incentive for small lots of some products, as opposed to sequence independent setup costs. Summarizing the above notations we obtain the following model:
i4-j
(6) which are also used in [5]. Note that definition (5) is valid since for every j, D~1 4 1 and 0 ~
t = 2 . . . . . n.
Equations (5) and (6) constitute time windows, TWjk= [ti~ i", tj~~x]
for ( j , k) ~ J K ,
with the following properties: (DLSP) Min
Lemma 1. min , and (i) tj~i" < t~,k+
E Sij max( Yi,t-I ~- Y j t - 1, O)
tjn~ax < t j,max k + 1 for every j and k < DiN. (ii) DLSP has a feasible solution if and only if TWik 4=0 for every (j, k) e JK. (iii) A one-to-one assignment of the elements' (j, k ) ~ JK to the periods t constitutes a feasible solution to DLSP if t <~tT~a× for every (j, k) assigned to r
i ,j,t
+ ~_~c jr y j, - H o j,t l E Yjr ~ Ojt
s.t
for all j, t,
(2)
r=l
~yj,=
1
for all t,
(3)
J
yj, ~ {0, 1},
(4)
where Yj0 = 1 if j is the initial product i0, and for j 4: i o.
Proof. (i): For s (6),
=
min tj.k+ I we have, by definition
r-~Dir>k+l>k
for r = s . . . . . N
i4=j
Yjo = 0
and
3. TSP model with time windows Any feasible solution to DLSP assigns a product to every period. However it is useful to distinguish the production activity in a single period t not only by the product j, but also by a counter k = Etr_ I Y~r specifying the number of occurrences of product j up to the considered period. The set of all pairs (j, k) occurring in a feasible solution is JK={(j, which Then ment 1. . . . .
k) : j = O , . . . , M ;
k = I , . . . , D j N },
has exactly N elements, as E~DjN=N. a solution becomes a one-to-one assignof the elements of JK to the periods t = N. The constraints (2) obviously imply the
s-l-
~_~Di,s_l>/s- ~ _ , D i s - l > ~ k + l - 1 i~j
i~j
=k, hence tjn~in <~ S --
1.
The second inequality is obvious. (ii): DLSP has a feasible solution if and only if
~,,D#<~t for t = l . . . . . N. J This condition implies, for any (j, k ) ~ JK and max r >~ tjk ,
r- EDi~=ri4~j
EOi~+Osr>Dj~>k, i
hence t~ i" ~
398
B. Fleischmann / Sequence-dependent setup costs
If DLSP is not feasible, there are some t and j such that EiDit > t and Djt > O. For k Oil we have t~ ax ~< t and
and
t-
The arcs bear the setup costs, i.e. Si0j for an arc /2 --->(j, 1), Sij for (i, k) ~ (j, l) and 0 for (j, k) --->O. Since an arc represents the transition from the production in one period to the production in the following one, the travel time through any arc is 1. Thus, any TSP tour on G can be viewed as an assignment of JK to the periods, (j, k) being assigned to t if (j, k) is 'reached at time t' in the tour. The holding costs c jr arise in any node (j, k) depending on the arrival time t. By the above considerations and Lemma 1 we have the following result:
( j , k) ~ $~ if k = DiN and tj~ ax = g .
=
~Dit
hence t~ in > t and TWjk = ¢. (iii): The statement is trivial if only those assignments are considered where the elements (j, k) occur in the right order, i.e. (j, k) is assigned to an earlier period than (j, k + 1). However, this is not necessary. Consider any (j, k) assigned to period t being in fact the kl-th occurrence of product j. If k I >/k, we have max . t ~< tj~ax ~< tyk~
If k 1 < k, there is some k 2 ~
In both cases, the production of product j in period t is in time. [] Note that the lower limits of the time windows tj~in are redundant with respect to the feasibility of an assignment. However, they are useful in strengthening relaxed formulations, as presented in [5] and in the next section. The higher the utilization of the machine, the stronger is the strengthening effect. E.g. in the problems T V 1 1 TV14 (see Section 6), the density of the time windows, i.e. the average ratio of the time window length to the total planning horizon, varies only between 56% and 60% for the time windows [1, .t~x], but is reduced, for the time windows [t~ 'n, t~ax], to 19% in case of the highest utilization (97%, T V l l ) and to 51% in case of the lowest utilization (64%, TV14). Unfortunately, the assignment structure only describes the feasibility of a DLSP solution, but cannot express the sequence-dependent setup costs. We therefore have to formulate a TSPTW. We define a directed graph G = (Jr', sO) with the node set ,¢" = JK U {g2}, where /2 is the origin of the TSP tour, and the following arcs: /2 --* ( j , 1)
if t~in = 1,
(j,k)--->(j,k+l) (i, k) ~ ( j , l)
Lemma 2. DLSP is equivalent to the asymmetric T S P T W on G with time windows TWy~ for the nodes (j, k) ~ JK and time-dependent costs c jr if
(j, k) is reached at time t. The T S P T W can be formulated as follows using the assignment variables U j k t ( ( j , k) ~ JK, t TWo.k) and the TSP variables xij t indicating whether the tour goes from product i in period t - 1 to product j in period t:
E CjtUjkt + E Sijxijt
Min
j,k,t
(7)
i,j,t
s.t.
EUjkt
=
1
for ( j , k) ~ JK,
(8)
=
1
for all t,
(9)
t
EUjkt j,k
E Uik,t- 1 At- E Ujkt -- Xijt ~ 1 k k
t >~ 2,
EUjkl --Xiojl 4 0
for all i, j a n d
(lOa) for all j,
(10b)
k Ujk,, Xij t ~ {0, 1}.
(11)
4. A Lagrangean relaxation
if k
such that i ~ j, ti~in ~< ti~ax - 1 and ti~ax >1 tjn~in - 1;
The DLSP with sequence dependent setup costs is a very hard problem which seems unlikely to be solvable by exact methods in practical cases.
B. Fleischmann / Sequence-dependentsetup costs We therefore prefer to solve it by heuristics and to try to compute tight lower bounds for the minimal costs. This section presents a new procedure for determining lower bounds based on a Lagrangean relaxation. T h e r e is little work in the literature on lower bounds for the TSPTW. An exception is the p a p e r of Desrosiers, Sauv6 and Soumis [3]. We follow their idea to relax the T S P T W into a SPPTW. This can be achieved by relaxing the constraints (8) of the T S P T W (7)-(11), i.e. the conditions that the node (j, k) is visited exactly once for every (j, k ) ~ J K and assigning it the Lagrange multiplyer Ark. The resulting relaxed problem SPP(A) has the objective function (7) with crt replaced by c# + Ark and the constraints (9)-(11). It can be viewed as an SPPTW in the following way: SPP(A). Find a path on G from ~ to $2: - through exactly N (not necessarily distinct) nodes (j, k) ~ JK; - considering the time windows TWrk; - m i n i m i z i n g the arc costs and the time dependent node costs cr, + Ark, if node (j, k) is reached at time t. For any solution to SPP(A), we have a subgradient g with the components gik = ( n u m b e r of occurrences of ( j , k ) in the path) - 1 and £(~,k)~JKgrk = 0. If we update Ajk by subgradient iterations, starting from a - 0, we therefore have always E(j,k)~jKark = 0, Hence the cost of an optimal solution to SPP(A) is a lower bound to the minimal cost of the DLSP. For solving an SPPTW Desrochers and Soumis propose an algorithm in [2]. Our problem SPP(A) however is easier than the general S P P T W since the travel times are equal to 1 in every arc. In a DP algorithm for the general SPPTW the arrival time at the current node is needed as an additional state component, besides the node itself, whereas in our case the stage number t, which counts the n u m b e r of nodes reached so far, is just the arrival time. As there is no delay in the nodes, the arrival time is equal to the start time
399
in any node. Thus, our DP algorithm uses the state space
JKt={(j,k)~JK:t~TWjk},
t=l,...,n,
and JK 0 = {g2}, and the value functionf,(j, k ) = minimal cost of a path from (j, k) E JKt to g2 with start time t. It can be computed recursively as follows:
fN ( J' k) = cjx + Ajk for k = DiN and j such that ti~ a× = N,
f~(i, k) =cit +Aik + min{f,+
(12)
,(j, l)
+ Sij" ( j, l) ~ J ( i, k) C3JK,+,},
(13)
for (i, k) ~ JKt, t = N - 1. . . . . 1, where Y ( i , k) is the set of immediate successors of (i, k) in G, and
T0(S?) =
m i n { f l ( j , 1) + Si.j: t j~in = 1}.
(14)
As there are N stages and the size of the state space is IJK t [ ~
Step 1. For every j = 0 . . . . . M: g* = m i n { f , + , ( j , / ) : l = 1 . . . . . DiN; ( j , l) ~JK,+,}, the minimum being attained for l Step 2. For every i = 0 . . . . , M:
=
1a*.
f/* = min{gj* + Sij : j = 0 . . . . . M; j 4=i}, the minimum being attained for j = Ji*. Step 3. For every (i, k) ~ JK,: i f ( i , k + 1) ~ JKt+ 1 and ft+l(/, k + 1) ~ * : f t ( i , k) = f t + , ( i , k + 1) + Aik + cit,
B. Fleischmann / Sequence-dependent setup costs
400
the optimal successor of (i, k) is (i, k + 1); otherwise:
L ( i , k) =f~* + aik + Cit , the optimal successor of (i, k) is (Ji*, lj?.). The number of operations required is O ( N ) in Step 1, since EsDju = N , O ( M z) in Step 2 and O ( N ) in Step 3.
5. Heuristics
Many heuristics are known for the TSPTW, mainly improving heuristics which are popular in vehicle scheduling problems for postoptimizing the sequence of customers in a single tour [1,16]. According to our experience the final result achieved by these heuristics hardly depends on the quality of the initial sequence. We therefore use a rather simple initial heuristic followed by an improving heuristic based on the OR-opt principle [12]. Both parts will be described in this section. We now disregard the lower l i m i t s tjn~in of the time windows, which were only introduced for the relaxation. The initial heuristic is of the 'best-successor' type, only we construct the solution backwards from the last to the first period (or from the last to the first node in the 'tour'), because then the feasibility of the solution is easier to control. As opposed to the last section, the following algorithm yields a tour that contains every node (j, k) JK exactly once and, for every j, in the order of increasing k: Initial heuristic. Start: k~ = DsN (j = 0 . . . . . M).
For t = N , . . . , 1: Select any j such that kj > 0 and
t; ax
(15)
Assign (j, k~.) to period t; k s = k s - 1. Note that there is always a j satisfying (15) if the DLSP has a feasible solution. This would not be true, if the periods were processed forwards, because the Dst (and hence the t~ ax) do not remain valid for the partial problems over the
later periods when fixing the products in the earlier periods. The criterion for selecting product j in period t in the above heuristic should reflect the costs affected by this decision. If t < N and Jt+ 1 is the product assigned to period t + 1, we select j that minimizes Sj,jt+l 71- 6#,
(16)
cjt is an appropriate term of the holding costs, but not equal to c jr; because this would favor j = 0, as Cot = 0, and proceeding backwards would lead to idle periods in the latest positions and hence cause high holding costs. We have tested several criteria, among which the following gave the best results: where
Cjt=(ejt--Ct)--(Cjtl--Cti) ~
min
with tl=tyk, ,
where 1
M
?t =- -M j~=l Cjr" The motivation of this criterion is that, due to (1), the ratio %t/Pt = aj does not depend on t and hence
(j,= (~',- :,,)(aj-- 1). As t >~tt, the first factor is negative, so that products with high holding costs are prefered. Cjt represents the holding cost difference between assigning (j, k s) to period t and assigning it to the earliest possible period t~, supposed that all other products have average holding costs. For t = N , i.e. the first step of the initial heuristic, we try every j satisfying (15) and run both the initial heuristic and the improvement heuristic for every j again. The improvement heuristic is basically on Oropt procedure with time windows, as described in [1,14,16]. An Or-opt operation consists in cutting off a piece of the current tour and inserting it at another (earlier or later) position. The maximal length of the considered pieces affects the quality of the solution as well as the computation time. We found that a limit of 6 periods is reasonable for the test problems investigated (cf. Section 6), as it covers nearly all possible improvements. Our search strategy consists, at every step, in checking all possible Or-opt operations and then performing the best one. This strategy proved to yield
B. Fleischmann / Sequence-dependent setup costs
40l
periods and a capacity utilization of 97%, 95%, 76% and 64%, respectively. Various setup cost matrices (Sij) have been considered for these problems: SO denotes the original (sequence independent) setup costs of [17]. S1 is the matrix shown in Table 1. The entries are generated at random from the interval [0,600] and thus have the same order of magnitude as in SO. The entries of $2 are taken at random from the values 0, 100, 200 . . . . . 600, the entries of $3 from [0,300]. $4, $5 and $6 represent typical situations in practice with only two kinds of setups, a major setup (cost 500) and a minor setup (cost 100). $4 has the entries
considerably better results than performing every first improving operation found, but it also takes much more computation time. Additionally, we investigate the exchange of single periods (which is not an Or-opt operation for non-neighbored periods). We summarize the procedure as follows:
Step 1. Search for the minimal cost exchange of two single periods; if it reduces the cost, perform it and repeat Step 1; otherwise go to Step 2. Step 2. Search for the minimal cost Or-opt operation with pieces of up to 6 periods; if it reduces the cost, perform it and go to Step 1; otherwise stop.
100
Sgj = ~ 500
O u r implementation of the Or-opt search is different from that proposed in literature: It proceeds by increasing first the length of the piece to be displaced, while fixing the first node of this piece, as opposed to Savelsbergh [14,15], who varies the start node while fixing the length of the piece. Details of the implementation will be published in a separate paper.
for i < j , for i > j
where i, j >/1, i.e. there is a natural order of the products (e.g. from light to dark colors) with only minor setups, but any deviation from this order, in particular the start of a new cycle, requires a major setup. $5 corresponds to four product families (1, 2}, {3, 4}, {5, 6} and {7, 8} with minor setups within and major setups between, $6 to the two product families {1, 2, 3, 4} and (5, 6, 7, 8}. The additional entries of $4, $5 and $6 are S0i = 500, Sio = 0 (i >~ 1). The initial product is icl = 0 in all problems. The problems P R 1 - P R 4 are practice cases in the food industry, also considered in [6]. The n u m b e r of products and periods is shown in Table 3. P R I and PR2 have a planning interval of 8 weeks (macroperiods) and differ only in the available capacity: PR1 is based on a 3-shift system (120 hours per week), PR2 on a 2-shift system (80 hours per week), the periods are shifts. PR3 and
6. Computational results The procedure of Sections 4 and 5 has been tested by means of three classes of problems: The problems T V l l - T V 1 4 correspond to the data sets 11-14 of [17] for the CLSP with 8 products and macroperiods, which have been subdivided into DLSP periods of equal capacity (50 units per period), as described in Section 2. The 4 problems T V l l - T V 1 4 differ only in the capacity of the macroperiods and have 63, 64, 80 and 96 Table | The setup cost matrix $1 from product
to product 0 1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8
0 0 0 0 0 0 0 0 0
69 375 0 121 471 593 545 69 172
25 80 123 0 403 25 515 117 490
116 386 385 116 0 548 236 211 193
97 397 268 301 367 0 97 441 278
32 467 32 502 581 97 0 446 95
121 121 541 401 441 255 357 0 259
0 0 422 19 140 72 505 488 0
136 0 297 161 545 136 554 409 530
B. Fleischmann / Sequence-dependent setup costs
402
PR4 concern two separate lines in a factory and have a planning interval of one year, the macroperiods lasting 2 weeks and the periods being days. The third class of problems has been generated using a random data generator provided by M. Salomon (private communication). It generates problems with the number of products and periods and the capacity utilization specified. The demands djt are 0 or 1, the output rate pj = 1 for all products, i.e. the output per period is used as the quantity unit. The holding costs per period are 1, the average setup cost Si is derived, for every product j, from the optimal time between orders (TBO), which itself is a random number, equally distributed in a specified interval. Sit are normally distributed random numbers with mean Sj and a specified standard deviation. For these problems no particular macroperiods are defined, we equate the macroperiods with the periods. All problems considered have 10 products and 150 periods. Starting from a basic problem with a
utilization of 90% and TBO interval [3, 10 periods], we varied the utilization and the TBO interval as shown in Table 4. The standard deviation of Sit was fixed to 10. Tables 2 - 4 present the results for the three problem classes. For every problem, the gap (UB - L B ) / U B between the lower bound LB and the upper bound UB (cost of the best solution found by the heuristic of Section 5) and the percentage of the setup costs within UB is shown. For the random problems the results in every line of Table 4 are averages of three problems with identical parameters. For the problems TV, the data of which is published completely, Table 2 indicates LB, UB and the setup costs explicitly. This is to enable other researchers to compare these results with those from alternative methods. Nevertheless, the data and detailed results of all problems are available from the author. The computation times are split into the parts required for LB and for the heuristic. They refer to an implementation in MS Fortran 5.1 on a PC
Table 2 Results for problems T V l l - T V 1 4 Problem
TV11 TV12 TV13 TV14 TVll TV12 TV13 TV14 TVll TV12 TV13 TV14 TV11 TV12 TV13 TV14 TV11
a a a a b b b b
setup cost matrix
LB
$1 S1 S1 S1 S1 S1 S1 $1 S1 81 81 Sl SO SO SO SO $2 $3 $4 $5 $6
5152 4195 3362 3272 3821 2241 1174 898 6464 5992 4703 4559 8339 7539 7321 7334 6750 3066 7140 8420 5207
a holding cost = 0. b holding cost doubled.
UB
5194 4381 3509 3379 3904 2470 1527 1285 6484 6054 4899 4769 8940 8340 8040 7890 7090 3078 7290 11190 7640
setup cost
3904 2691 2119 1989 3904 2470 1527 1285 3904 2774 2119 1989 7450 6450 6200 6250 5700 1738 5800 9800 6300
gap (%)
0.8 4.2 4.2 3.2 2.1 9.3 23.1 30.1 0.3 1.0 4.0 4.5 6.7 9.6 8.9 7.0 4.8 0.4 2.1 24.8 31.8
comp. time (seconds) LB
UB
15 18 20 32 16 18 24 23 13 19 23 32 17 19 24 21 16 13 10 18 19
23 63 136 326 45 118 249 364 22 45 85 122 41 77 244 455 76 120 83 52 43
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Table 3 Results for practice problems Problem
PR PR PR PR
M
1 2 3 4
N'
9 9 3 4
8 8 26 26
N
120 80 230 225
utilization
setup
(%)
cost (%)
66 99 91 95
with 486/33 M H z processor. The subgradient iterations are organized as in [5]: the step size is UB - LB Ov= l'Z
Y'jkg2k
w h e r e / z has the starting value 2 and is divided by a factor f > 1 every Q iterations without increase of LB; the standard values are f = 1.2 and Q = 4. The iterations terminate when o- falls below a certain limit e. The total n u m b e r of iterations was between 100 and 200 in most cases. As the results of the heuristic were unsatisfactory in many cases, we tried to repeat the heuristic in every subgradient iteration, in a similar way as in [5], adding the current Aiks to the criterion (16). Unfortunately, the observation made for the product decomposition algorithm [5], that multipliers A which give a high LB also yield good heuristic solutions, proved not to be true in this algorithm. Only the increased n u m b e r of trials of the heuristic improved the UB sometimes. We therefore included the repetition of the heuristic in the first 20 subgradient iterations only. The computation time for UB in Tables 2 - 4 comprises the time for all trials.
Table 4 Results for r a n d o m problems with 10 products and 150 periods; every line shows the average results for 3 problems utilization (%)
TBO interval
setup cost (%)
gap (%)
comp. time (seconds) LB
UB
80 85 90 95 99 90 90
[3,10] [3,10] [3,10] [3,10] [3,10] [5,20] [5,30]
71 68 64 61 58 79 85
5.2 6.7 5.0 5.8 6.9 10.7 16.7
55 56 48 45 40 61 89
122 107 95 104 105 128 135
17 9 16 20
gap (%)
1.0 1.1 0.7 3.5
comp. time (seconds) LB
UB
56 26 132 181
310 45 96 306
Overall, the results have improved remarkably c o m p a r e d to those obtained by means of an extended product decomposition algorithm for the problems T V and PR [6]. However, the gaps between LB and UB of up to 30% are still unsatisfactory. Even the majority of cases with gaps between 4% and 7% still requires improvement of the procedure. Whether a large gap is due to a weak LB or a bad heuristic solution or both, is an open question. In the only cases with known optimal solution, the problems TV with SO [5], however, the LB is very tight (less than 0,3% below the minimal cost in all 4 cases). This suggests the conjecture that the SPPTW relaxation yields tight lower bounds for the DLSP and that primarily the heuristics need to be improved. The quality of the results seems to depend little on the capacity utilization of the problem, as opposed to most capacitated lot-sizing algorithms which work worse in case of higher utilization [5]. T h e r e is even a slight reduction in the LB computing time in case of higher utilization, due to the reduced solution space. A crucial factor, however, is the percentage of the setup costs: the largest gaps and the highest computing times of the heuristic arise for the problems with zero holding cost, i.e. pure scheduling problems, and the random problems with high TBO; the easiest problems are those with low relative setup costs, e.g. the problems PR and T V with doubled holding costs. As to the different setup cost matrices there is little difference in the results for S1, $2 and $3; the cyclic setup costs $4 seem to be easier and the family setup costs $5 and $6 extremely difficult. For sequence independent setup costs (SO) the T S P T W heuristic is obviously not adequate, but the SPPTW relaxation gives an excellent LB, as explained above. Important tasks of further research on the DLSP are the determination of exact solutions
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B. Fleischmann / Sequence-dependent setup costs
for sample problems, e.g. by means of branching, in order to give a more profound evaluation of the lower bounds, and the development of faster and better heuristics.
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