A Stochastic Multi-item Lot-sizing Problem
with Bounded Number of Setups
Etienne de Saint Germain
1,2
, Vincent Leclère
2
and Frédéric Meunier
2
1
Argon Consulting, 122 Rue Édouard Vaillant, 92300 Levallois-Perret, France
2
CERMICS, Cité Descartes, 6-8 Avenue Blaise Pascal, 77455 Champs-sur-Marne, France
Keywords:
Lot-sizing, Stochastic Optimization, Sample Average Approximation, Simulation.
Abstract:
Within a partnership with a consulting company, we address a production problem modeled as a stochastic
multi-item lot-sizing problem with bounded numbers of setups per period and without setup cost. While this
formulation seems to be rather non-standard in the lot-sizing landscape, it is motivated by concrete missions
of the company. Since the deterministic version of the problem is NP-hard and its full stochastic version
clearly intractable, we turn to approximate methods and propose a repeated two-stage stochastic programming
approach to solve it. Using simulations on real-world instances, we show that our method gives better results
than current heuristics used in industry. Moreover, our method provides lower bounds proving the quality of
the approach. Since the computational times are small and the method easy to use, our contribution constitutes
a promising response to the original industrial problem.
1 INTRODUCTION
Fixing the production level for the forthcoming week
is a basic decision to be taken when managing a pro-
duction line. Usually, a demand has to be satisfied
at due dates but the limited capacity of the line pre-
vents last minute production. On the other hand, too
early productions may lead to unnecessary high in-
ventory costs. The challenge of this kind of pro-
blems, known as lot-sizing problems in the operatio-
nal research community, consists in finding a trade-
off between demand satisfaction and holding costs.
This is a well-studied topic, with many variations (de-
terministic/stochastic, single/multi item, etc.). Re-
cent surveys have been proposed: see (Gicquel et al.,
2008, Quadt and Kuhn, 2008) for the determinis-
tic version and (Mula et al., 2006, Aloulou et al.,
2014, Díaz-Madroñero et al., 2014) for the stochastic
version. When several references can be produced on
a same line the so-called multi-item lot-sizing pro-
blem –, the capacity is often all the more reduced as
the number of distinct references produced over the
current week is high. Indeed, changing a reference in
production stops the line for a moment. This additi-
onal capacity reduction is usually modeled by setup
costs contributing to the total cost.
The present work introduces a stochastic multi-
item lot-sizing problem met by the authors within
a partnership with a consulting company. A non-
standard feature of the problem is that the capacity
reduction due to reference changes is not modeled by
setup costs but instead by an explicit upper bound on
the total number of references that can be produced
over a week. According to the consulting company,
many clients aim at minimizing mainly their inven-
tory costs while keeping the number of distinct refe-
rences produced over each week below some thres-
hold. This is essentially because, contrary to inven-
tory costs, setup costs are hard to quantify and a max-
imal number of possible changes per week is easy
to estimate. To the best of authors’ knowledge, the
problem addressed in the present work is original and
such a bound on the number of distinct references pro-
duced over a week has not been considered by acade-
mics yet, with the notable exception of (Rubaszewski
et al., 2011) but, contrary to our problem, their bound
is an overall bound for the whole horizon and they
still consider setup costs.
We consider two versions of the problem. In a first
one, backorder costs are present and the objective is
to minimize the overall inventory costs (sum of hol-
ding and backorder costs). In a second version, there
are only holding costs and there is a service level con-
straint to be satisfied over the whole horizon. We pro-
pose for these two versions a method that can be ea-
sily used and maintained in practice. The efficiency
de Saint Germain E., Leclère V. and Meunier F.
A
Stochastic
Multi-item
Lot-sizing
Problem
with
Bounded
Number
of
Setups.
DOI:
10.5220/0006622501060114
In
Proceedings
of
the
7th
International
Conference
on
Operations
Research
and
Enterprise
Systems
(ICORES
2018),
pages
106-114
ISBN:
978-989-758-285-1
Copyright
c
2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
of the method is proved via extensive numerical ex-
periments on real industrial data. In particular, we
compare our results with current heuristics used in in-
dustry and with lower bounds.
2 PROBLEM FORMULATION
AND MODEL
The assembly line produces a set R of references over
T weeks. The number of distinct references produced
over a week cannot exceed N. There is also an upper
bound on the total week production (summed over all
references). We normalize all quantities so that this
upper bound is equal to 1.
The production must satisfy a random demand.
The demand of reference r over week t is a random
parameter d
r
t
, whose realization is known at the end
of week t. When production of a reference r is not
used to satisfy the demand, it can be stored but incurs
a unit holding cost h
r
> 0 per week. When a demand
is not satisfied by the production of the current week
or by inventory, it can be satisfied later but incurs a
unit backorder cost γ per week for some coefficient
γ > 0. Note that there is no setup cost, as discussed
in the introduction. For each reference r, there is an
initial inventory s
r
0
R
+
.
At the beginning of each week, before the demand
of each reference is revealed, the production of the
week has to be fixed. The objective is to minimize
the total expected inventory cost (holding cost plus
backorder cost) over the whole horizon of T weeks.
Regarding randomness, we assume that for any r
and t, realizations of (d
r
t
,d
r
t+1
,...,d
r
T
) have finite ex-
pectation and can be efficiently sampled, knowing a
realization of d
r
t1
.
Before turning to the modeling of the problem, we
make two remarks. The first one is about its position
in the literaure. Our problem (in its deterministic ver-
sion) is a variation of the Capacitated Lot-Sizing Pro-
blem (see (Karimi et al., 2003) for a review of mo-
dels and algorithms). The single differences with the
usual basic version of that problem is that there is no
setup cost and that there is this upper bound on the
total number of distinct references that can be produ-
ced over a week. The second remark is about the so-
mehow non-conventional assumption that the demand
is revealed at the end of the week. There are indeed
industrial cases where this occurs, like in computer
manufacturing. However, the methods developed in
this paper can be easily adapted to the case where the
demand is revealed at the beginning of the week.
The production problem at the beginning of week
t can be modeled as the following stochastic program
(S):
min E
"
T
t
0
=t
rR
(h
r
˜
s
r
t
0
+ γb
r
t
0
)
#
s.t.
s
r
t
0
=
˜
s
r
t
0
b
r
t
0
t
0
Jt,TK, r R
s
r
t
0
= s
r
t
0
1
+ q
r
t
0
d
r
t
0
t
0
Jt,TK, r R
rR
q
r
t
0
1 t
0
Jt,TK
q
r
t
0
x
r
t
0
t
0
Jt,TK, r R
rR
x
r
t
0
N t
0
Jt,TK
x
r
t
0
{
0,1
}
t
0
Jt,TK, r R
q
r
t
0
,
˜
s
r
t
0
, b
r
t
0
0 t
0
Jt,TK, r R
σ(q
r
t
0
) σ
(d
r
0
t
,...,d
r
0
t
0
1
)
r
0
R
t
0
Jt,TK, r R ,
where the variable
˜
s
r
t
0
(resp. b
r
t
0
) models the inventory
(resp. the backorder) of reference r at the end of week
t
0
, the variable q
r
t
0
models the quantity of reference r
produced over week t
0
, and the variable x
r
t
0
takes the
value 1 if the reference r is produced over week t
0
and 0 otherwise. All these variables are random. The
last constraint of the program, written as a measurabi-
lity constraint, means that the values of the variables
q
r
t
0
can only depend on the values taken by the de-
mand before time t
0
(the planner does not know the
future). The reader would have probably expected si-
milar constraints for the other variables, but it is easy
to see that adding them changes neither the optimal
value, nor the production decisions. Note that in this
program the inventory s
r
t1
is not a variable and has a
known deterministic value at the beginning of week t.
A feasible solution (q
r
t
0
)
t
0
t,rR
of (S) provides a
deterministic production (q
r
t
)
rR
for the current week
t.
3 FOCUS ON THE
DETERMINISTIC VERSION
The deterministic version of (S) is obtained by remo-
ving the measurability constraint and by considering
all d
r
t
s deterministic, denoted then simply by d
r
t
. The
variables are then also denoted with non-bold charac-
ters. The main message of this section is that this ver-
sion is already difficult.
The deterministic version is NP-hard in the strong
sense for any fixed N 3, since there is a straight-
forward reduction from 3-PARTITION. Given a set
of 3n positive integers {a
1
,...,a
3n
}, this latter pro-
blem consists in deciding whether this set admits a
partition into n triples of same sum. It is a notorious
strongly NP-complete problem (Garey and Johnson,
1979). The problem 3-PARTITION can be reduced to
the case N = 3 where each index in J1,3nK is a dis-
tinct reference r whose demand d
r
t
is equal to 0 for all
period t, except for the last period T = n where it is
equal to
n
3n
r
0
=1
a
r
0
a
r
. (The cases N 4 are dealt with
via a similar reduction.) The complexity status of the
deterministic version when N = 1 or N = 2 seems to
be a challenging open question.
It is also worth noting that the optimal value of
the continuous relaxation is independent of N. This
can be seen by considering the mixed integer program
obtained by removing the constraints
rR
x
r
t
0
N:
any feasible solution of that program provides a fe-
asible solution of the original program of same va-
lue, simply by redefining x
r
t
0
= q
r
t
0
for all t
0
and all r if
necessary. In an attempt of improving the quality of
the continuous relaxation, one may consider the ex-
tended formulation with the binary variables y
p
t
for
each p
R
N
and each t J1,T K in place of the x
r
t
s
(indicating the references produced on period t) but it
can be shown with a nontrivial proof that it does not
improve the quality of the continuous relaxation. Si-
milarly, the extended formulation with the binary va-
riables z
r
τ
for each τ J1,TK (indicating the periods
of production of the reference r) does not improve
the quality of the relaxation (this time, with an easy
proof).
4 METHOD
As explained in Section 3, the deterministic version of
(S) is difficult. Therefore, we cannot expect a quick
algorithm solving exactly the problem, and this holds
especially for the full stochastic version. Moreover,
one of the requests of the partner was to have an easy
to understand method, which can be used and main-
tained in practice, with short computation times. We
propose a two-stage approximation consisting in re-
placing the measurability constraint by
σ(q
r
t
) σ() r R
σ(q
r
t
0
) σ

d
r
0
t
,...,d
r
0
T
r
0
R
t
0
t + 1, r R ,
which provides a relaxation of the initial program: the
production decisions for the current week t can still
not depend on the future, but now the subsequent pro-
duction decisions depend on the future demand. We
denote this relaxation by (2SA).
This approximation is a two-stage approximation
as we distinguish between two levels of information
over the uncertainty: production decisions for the first
week are the first stage variables, while all other de-
cisions are second stage variables. Three-stages or
more generally multistage approximation would give
better approximations of (S) but increases exponenti-
ally the number of variables. We chose for practicabi-
lity reasons to stick to the two-stage approximation.
The (2SA) relaxation is then solved by a classical
sample average approximation, see (Kleywegt et al.,
2002) for a presentation of the method. We build a
set of m scenarios sampled uniformly at random.
Each of these scenarios is a possible realization of
(d
r
t
,d
r
t+1
,...,d
r
T
) for each r. The parameter m is fixed
prior to the resolution.
We get the following mixed integer program
(2SA-m), solved by any standard MIP solver.
min
1
m
ω
T
t
0
=t
rR
h
r
˜s
r
t
0
,ω
+ γb
r
t
0
,ω
s.t.
s
r
t
0
,ω
= ˜s
r
t
0
,ω
b
r
t
0
,ω
t
0
Jt,TK, r R , ω
s
r
t
0
,ω
= s
r
t
0
1,ω
+ q
r
t
0
,ω
d
r
t
0
,ω
t
0
Jt,TK, r R , ω
rR
q
r
t
0
,ω
1 t
0
Jt,TK, ω
q
r
t
0
,ω
x
r
t
0
,ω
t
0
Jt,TK, r R , ω
rR
x
r
t
0
,ω
N t
0
Jt,TK, ω
x
r
t,ω
= x
r
t
r R , ω
q
r
t,ω
= q
r
t
r R , ω
x
r
t
, x
r
t
0
,ω
{
0,1
}
t
0
Jt,TK, r R , ω
q
r
t
, q
r
t
0
,ω
, ˜s
r
t
0
,ω
, b
r
t
0
,ω
0 t
0
Jt,TK, r R , ω
At week t, the production is then set to be the solution
(q
r
t
)
rR
found by the solver.
The validity of this method for solving (2SA) is
supported by the following proposition.
Proposition 1. The following three properties hold
when m goes to infinity:
(i) The value of (2SA-m) converges almost surely to
the optimal value of (2SA).
(ii) For every m, we consider the values
( ˆq
r
t,m
, ˆx
r
t,m
)
rR
of the decision variables for
week t of an optimal solution of (2SA-m). Any
limit point of these values is an optimal solution
of (2SA).
(iii) Let ε > δ > 0. Assume that the random demand
(d
r
t
0
)
t
0
t,rR
is such that
C,K, u R, E[e
ukdk
] Ce
u
2
K
. (1)
Denote by Q
δ
m
(resp. Q
ε
) the set of all possible va-
lues of ( ˆq
r
t,m
)
rR
in a δ-optimal solution of (2SA-
m) (resp. in an ε-optimal solution of (2SA)). Then
for every α (0,1), we have P(Q
δ
m
Q
ε
) > 1 α
for m large enough.
If the random demand d is bounded or Gaussian
then it satisfies (1). (Some companies work with pos-
sibly negative demands. Assuming a normal distribu-
tion is hence not irrelevant.)
The proof of part (iii) relies on the following
technical lemma.
Lemma 2. Consider g(d) = inf
yY
G(y,d), whereY is
non-empty and where the function G is non-negative
and κ-Lipschitz with respect to d. If the random va-
riable d satisfies (1), then g(d) also satisfies (1) with
C
0
= max{1,e
G(y
0
,0)
C} and K
0
= Kκ + G(y
0
,0), for
y
0
Y .
Proof. For u 0 we take y
0
Y yielding g(d)
G(y
0
,0) + κkdk. We then have E[e
ug(d)
]
E[e
uG(y
0
,0)
e
κukdk
] C
0
e
K
0
u
2
. For u 0, by non-
negativity of G we have g(d) 0, hence E[e
ug(d)
]
1.
Proof of Proposition 1. Let Q
t
be the (bounded) set
of feasible values for the first-stage variables q
t
=
(q
r
t
)
rR
in (2SA). Denote by F(q
t
,d) the minimal
cost of (2SA) that can be reached when the first stage-
variables are fixed to q
t
Q
t
and the realization of
the demand is d = (d
r
t
0
)
t
0
t,rR
. We introduce the
map f : q
t
7→ E[F(q
t
,d)], which associates to a gi-
ven choice of q
t
Q
t
the expected minimal cost, and
similarly the map
ˆ
f
m
, which associates to a q
t
Q
t
the minimal cost of (2SA-m) when the first-stage va-
riables are set to this q
t
.
The map f is continuous and Q
t
is compact. Thus
f is bounded, and, by (Shapiro et al., 2009, Theorem
7.48), we have that (
ˆ
f
m
(q
t
))
mZ
+
converges to f (q
t
)
uniformly on Q
t
. Then (i) and (ii) are direct conse-
quences of (Shapiro et al., 2009, Theorem 5.3).
By Lemma 2, there exist K and C such that for
any q
t
Q
t
, F(q
t
,d) satisfy (1). Consequently there
exists σ > 0 such that for all q
t
,q
0
t
Q
t
, the random
variable [F(q
t
,d) f (q
t
)] [F(q
0
t
,d) f (q
0
t
)] is σ-
subgaussian, (see e.g., (Vershynin, 2010)). Further-
more, for any demand d, the map F(·, d) is Lipschitz-
continuous on Q
t
. Then, according to (Shapiro et al.,
2009, Theorem 5.18), for every α (0,1), there exists
M Z
+
such that for m M, we have P(Q
δ
m
Q
ε
) >
1 α.
5 NUMERICAL EXPERIMENTS
5.1 Instances
C++11 has been chosen for the implementations and
Gurobi 6.5.1 was used to solve the model on a PC
with Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz
and 8Go RAM.
The instances used are realistic and have been pro-
vided by a client of the partner. The client gave actu-
ally the figures of seven assembly lines but we give
the results for only two of them: L2 and L6. The line
L2 experiences overcapacity: the ratio expected de-
mand over total capacity is smaller than 1 (actually
equal to 0.7). The line L6 experiences undercapacity:
the ratio expected demand over total capacity is lar-
ger than 1 (actually equal to 1.1). The horizon T is
the typical one used in practice by this client, namely
T = 13 weeks (a quarter). The demand is obtained
via d
r
t
= (
¯
d
r
t
+ e
r
t
)
+
, where the
¯
d
r
t
s are historical data
and where e
r
t
is a generalized autoregressive process
so that e
r
t
= 0.25e
r
t1
+ 0.75ε
t
, where ε
t
is a Gaussian
white noise process with zero mean and standard de-
viation equal to v
¯
d
r
t
where v is the “volatility” and is
chosen in
{
0.2,0.5
}
. The initial inventory is set to
s
r
0
=
1
3
(
¯
d
r
1
+
¯
d
r
2
+
¯
d
r
3
). The other parameters are pro-
vided in Table 1. In particular, for each value of v,
we have considered three possible values of the unit
backorder cost γ, which have been determined follo-
wing a procedure described in Section 6. At the mo-
ment, we do not discuss further these values and take
them as part of the input, as required by the problem
formulation. The parameter C is the capacity of the
line before normalization. (Recall that problem and
the model have been formulated in Section 2 after nor-
malization.) In the column
˜
h
r
, we indicate the range
of the holding costs before normalization. We obtain
the h
r
s by dividing these costs by C.
The number m of scenarios used to solve (2SA-m)
is fixed to 20, determined by preliminary experiments
showing that it is a good trade-off between accuracy
and tractability. The time limit of the solver has been
set to 90 seconds.
5.2 Other Heuristics
Our method is compared with three other heuristics.
The first heuristic is the deterministic version
of (S), where the random demand is replaced by its
expectation.
The second one, the lot-size heuristic, consists in
determining before the first week once and for all a
value `
r
for each reference r R . At time t, if the in-
ventory of reference r is below a precomputed safety
level, the quantity q
r
t
is chosen so that the inventory of
reference r exceeds the safety level of exactly `
r
. In
case of capacity issues, the production is postponed
and thus backorder costs appear. In addition, if some
capacity issues are easily anticipated, the production
of a reference r can be activated even if the inventory
Table 1: Instance characteristics.
Instances Instance characteristics
|
R
|
max(
¯
d
r
t
) C N
˜
h
r
v γ
L2_v20_13 21 4992 10562 7 35–61 0.2 13
L2_v20_81 81
L2_v20_203 203
L2_v50_48 0.5 48
L2_v50_154 154
L2_v50_341 341
L6_v20_3 22 8640 13299 8 16–23 0.2 3
L6_v20_19 19
L6_v20_55 55
L6_v50_11 0.5 11
L6_v50_42 42
L6_v50_98 98
is not below the safety level.
The third one, the cover-size heuristic is almost
the same, but instead of precomputing a fixed quantity
for each reference, a duration T
r
is fixed before the
first week. When the inventory of reference r is below
the safety level, the quantity q
r
t
is computed so that the
inventory of reference r exceeds the safety level of the
expected demand for the next T
r
weeks.
The values `
r
and T
r
are determined as follows.
(T
r
)
rR
is actually chosen to be the optimal solution
of the following convex program, which somehow
considers the problem at a “macroscopic” level. (Si-
milar convex programs in the same context have been
considered in the literature; see (Ziegler, 1982) for ex-
ample.)
min
rR
h
r
¯
d
r
T
r
s.t.
rR
1
T
r
N
T
r
> 0 r R ,
where
¯
d
r
= E
T
t=1
d
r
t
.
The parameter `
r
of the lot-size heuristic is then
set to
¯
d
r
T
r
.
The safety levels have been provided by the part-
ner and are those used in practice.
The cover-size heuristic adapts the production to
the realization of the demand, contrary to the lot-
size heuristic. According to our partner, it makes the
cover-size heuristic more suitable for situations with
low short term volatility of demand or for overcapa-
citated lines, while the lot-size heuristic is expected
to behave better with high short term volatility of de-
mand or for undercapacitated lines.
Notice that the backorder costs are not taken into
account at all for determining the values of the pa-
rameters `
r
and T
r
. But playing with safety levels
allows to prevent too large backorder costs. However,
in real life it is usually the other way round: the com-
pany does not associate costs to backorder and aims
at keeping the total amount of unsatisfied demand be-
low some predetermined level. We come back to this
point later in Section 6.
5.3 Results
The results are provided in Table 2. All quantities
are in Me and are given with a confidence interval at
95%.
The column LB provides the lower bound obtai-
ned by the optimal value at time t = 1 of program
(2SA-m) (with m = 1000 and a time limit of 24 hours
for the solver).
The column 2SA-m is the cost of the method pro-
posed in Section 4. (We remind the reader that we
propose m = 20 in this case.) The three next columns
provide the results for the three heuristics described
in Section 5.2. For these four columns, between 27
and 30 runs have been used for each instance.
Complementary numerical results of these expe-
riments are provided in Tables 3 and 4, discussed in
Section 6 where other criteria are considered. Table 3
can already by of interest since it deals with holding
costs obtained for the L2 instances.
5.4 Comments
Our method clearly outperforms lot-size and cover-
size heuristics and is better than the deterministic ap-
proximation for all but one instance. By running our
method instead of a usual heuristic at the beginning
of each week, the inventory costs can be reduced of-
ten by more than 50%. For the instance L2_v20_13,
the inventory costs have been divided by more than 6
(which corresponds to several Me).
Table 2: Results - Inventory costs (in Me).
Instances LB 2SA-m Det. Cover-size Lot-size
L2_v20_13 0.53 0.89 ± 0.03 1.17 ± 0.10 6.95 ± 0.17 7.79 ± 0.14
L2_v20_81 0.94 2.29 ± 0.06 2.36 ± 0.07 8.12 ± 0.19 9.65 ± 0.14
L2_v20_203 1.00 3.05 ± 0.07 3.25 ± 0.08 9.35 ± 0.29 10.99 ± 0.19
L2_v50_48 0.97 2.73 ± 0.11 3.06 ± 0.21 8.03 ± 0.26 8.37 ± 0.21
L2_v50_154 1.36 4.54 ± 0.20 5.06 ± 0.33 10.83 ± 0.53 11.20 ± 0.38
L2_v50_341 1.51 5.91 ± 0.25 7.90 ± 0.66 15.17 ± 1.21 14.65 ± 0.77
L6_v20_3 0.54 0.61 ± 0.01 0.70 ± 0.02 1.71 ± 0.08 1.74 ± 0.08
L6_v20_19 1.41 1.81 ± 0.06 1.86 ± 0.06 3.51 ± 0.12 3.20 ± 0.08
L6_v20_55 2.67 3.57 ± 0.24 3.71 ± 0.30 7.49 ± 0.39 6.24 ± 0.34
L6_v50_11 1.33 2.00 ± 0.11 2.14 ± 0.12 3.42 ± 0.15 3.03 ± 0.13
L6_v50_42 2.99 4.45 ± 0.53 4.48 ± 0.51 7.99 ± 0.62 6.57 ± 0.60
L6_v50_98 6.13 8.29 ± 1.23 7.94 ± 1.04 16.34 ± 1.61 12.96 ± 1.45
We may note that the cover-size heuristic behaves
better than the lot-size heuristic on the L2 instances,
while the converse holds on the L6 instances. It is
in line with our partner’s opinion regarding their be-
haviors with respect to the capacity of the line (see
Section 5.2). Regarding their behaviors with respect
to the volatility, it does not seem to be possible to
draw any concrete conclusion.
The 2SA-m algorithm requires 90 seconds to out-
put a solution, while lot-size and cover-size heuris-
tics take less than a second and the deterministic ap-
proximation less than 10 seconds. The 2SA-m algo-
rithm is thus slower, but note that 90 seconds to be
run only once at the beginning of each week remains
very short. Moreover, even with improved computers,
the lot-size and cover-size heuristics and the determi-
nistic approximation will not change their output (in
all our experiments, Gurobi always found the optimal
solution of the deterministic approximation). This is
not the case for 2SA-m, which means that it would
benefit from improved computational capacities.
Otherwise, complementary figures not provided
here show that the deterministic approximation and
2SA-m use almost every available setups (between
97% and 100%) whereas lot-size and cover-size heu-
ristics cannot take full advantage of this flexibility
(only 84% to 98% of setups are used).
Our method dramatically reduces the inventory
costs, while remaining a quite simple approach. It
provides without any doubt a positive answer to the
industrial request. Moreover, as explained in the next
section, our method can be used to address the pro-
blem of minimizing the holding costs, while keeping
the backorder at a reasonable level.
6 CONTROLLING THE FILL
RATE SERVICE LEVEL
6.1 The Problem
In practice, except when they are enshrined through
contracts with the client, backorder costs can be hard
to estimate. Thus, we consider here an alternative ver-
sion of the problem, which often meets the objecti-
ves in industry. Everything remains the same, except
that there is no backorder cost anymore, which means
that we remove the corresponding part in the objective
function of (S) and that the following constraint is ad-
ded:
E
"
rR
T
t=1
min(d
r
t
,q
r
t
+
˜
s
r
t1
)
rR
d
r
#
β, (2)
where β [0,1] is the desired fill rate service level de-
fined by the company, and where d
r
=
T
t=1
d
r
t
. This
constraint implies that, in expectation, the proportion
of the production delivered on-time is at least β. In-
deed, at week t, we deliver on-time the minimum bet-
ween the demand and the sum of the real inventory at
the end of week t 1 and of the production of week t.
First, note that this constraint does not make a dif-
ference between a one-week delay and a two-week
delay. Second, this constraint is a global constraint to
satisfy over the whole horizon: it does not depend on
the current week. This is a matter of modeling point
of view since we were not able to get a precise formu-
lation from our partner and from its clients. A draw-
back of this formulation with respect to the one given
in Section 2 is that this additional constraint cannot
always be satisfied. In the next section, we propose a
way to address this new constraint in an approximate
way, while being able to provide solutions in any case.
6.2 Surrogate Backorder Costs
We address the alternative formulation by defining a
surrogate backorder coefficient γ before the first week,
with the idea to use the algorithm of Section 4 and to
heuristically entice it to choose solutions satisfying
constraint (2).
γ := max
rR
γ
r
γ
r
:=
P[d
r
q
r
(β)]
P[d
r
> q
r
(β)]
q
r
(β) := min
q R
+
E
min(d
r
,q)
d
r
β
(3a)
(3b)
(3c)
where d
r
=
T
t=1
d
r
t
is the demand of reference r ag-
gregated over time. Since d
r
is integrable, q
r
(β) is
well-defined (we set
0
0
= β so that references with no
demand would not impact the constraint). Computing
an approximate value of q
r
(β) at an arbitrary preci-
sion can easily be performed by binary search.
To justify this choice, consider the original pro-
blem of Section 2 with only one reference and for a
horizon of one week. Assuming no initial inventory,
it takes then the form of the famous newsvendor pro-
blem (see e.g., (Shapiro et al., 2009, Chapter 1))
min
q0
E
h
r
(q d
r
)
+
+ γ
r
(d
r
q)
+
, (4)
where γ
r
is a unit backorder cost specific to reference
r. The next proposition means that with the right
choice for γ
r
, the alternative version with a desired
fill rate service level β (which takes the form of (3b)
since h
r
> 0) is equivalent to the original one (4). Of
course, it holds only for the case with one reference
and a horizon of one week. Many references lead to
several possible values γ
r
for the surrogate backorder
cost. The choice in (3a) of taking the maximum of
them for our actual γ aims at facing the additional un-
certainty induced by the limited number of setups per
week and the length of the horizon.
Proposition 3. Define γ
r
as in (3b). Then q
r
(β) is the
smallest optimal solution to (4).
Proof. The aggregated production problem (4)
is known to have the optimal solution q
r
=
F
1
d
r
(γ
r
/(γ
r
+h
r
)), where F
1
d
r
is the left-inverse of the
cumulative distribution function of d
r
, i.e., F
1
d
r
(κ) =
inf{q | P(d
r
q) κ}. Since we have set γ
r
=
P[d
r
q
r
(β)]
P[d
r
>q
r
(β)]
, we have q
r
= inf{q | P(d
r
q) P(d
r
q
r
(β)}, which implies that q
r
q
r
(β). Now if
this inequality were strict, then it would mean that
P(d
r
(q
r
,q
r
(β))) = 0, which contradicts the mi-
nimality assumption in the definition of q
r
(β) (Equa-
tion (3c)).
Note that this formulation does not take into ac-
count the capacity constraint. It is therefore probably
better suited to overcapacited production lines.
Remark. If instead of controlling the fill rate service
level, we want to control the cycle service level, defi-
ned as the probability of satisfying the whole demand,
then we can choose
γ =
β
1 β
max
rR
h
r
. (5)
Indeed, in this case, the optimal solution q
r
of (4)
satisfies P(q
r
d
r
) = β. Interestingly, Equation (5)
does not depend on the distribution of the demand,
which contrasts with the fill rate service level.
6.3 Numerical Results
We discuss briefly the results of the experiments des-
cribed in Section 5 in the new context of this section.
Tables 3 and 4 provide the numerical results: the first
table shows the holding costs obtained when using the
method proposed in this work; the second table shows
the average (over the 30 runs) of the fill rate service
level measured ex post, which we define by the reali-
zation of
rR
T
t=1
min(d
r
t
,q
r
t
+
˜
s
r
t
)
rR
d
r
.
We plot on Figure 1 the values of the holding costs
obtained for each of the 30 runs for the instance L2
with β = 95% and v = 0.5. On Figure 2, we plot the
average of these values for three desired fill rate ser-
vice levels (85%, 95%, and 98%).
Again, the inventory costs (which are in this ver-
sion also the holding costs) are dramatically better
than the ones obtained by the heuristics used by the
clients of our partner. These two heuristic are howe-
ver able to provide almost always very good fill rate
service level but it comes at the price of very high
holding costs
When β {95%, 98%}, the deterministic approx-
imation reaches better holding costs than 2SA-m but
worse fill rate service levels. This is especially true
when the volatility is high (v = 0.5). Further experi-
ments would probably be necessary to understand bet-
ter how the deterministic approximation and 2SA-m
compare but since they both use the surrogate backor-
der cost, their behavior regarding the fill rate service
level shows that the way we compute it is relevant.
For high desired fill rate service levels ( 95%),
the method 2SA-m behaves well: the fill rate service
level measured in practice is close to the desired level.
Since constraint (2) is an approximation of what is
sought in practice, these results can be considered as
Table 3: Results - Holding costs (in Me).
Instances Results
name β γ 2SA-m Det. Cover-size Lot-size
L2_v20 85% 13 0.44 ± 0.04 0.82 ± 0.12 6.85 ± 0.18 7.70 ± 0.15
95% 81 1.58 ± 0.07 1.56 ± 0.11 7.49 ± 0.20 9.14 ± 0.17
98% 203 2.34 ± 0.10 1.97 ± 0.10 7.79 ± 0.21 9.73 ± 0.19
L2_v50 85% 48 1.55 ± 0.12 1.78 ± 0.18 6.96 ± 0.30 7.58 ± 0.24
95% 154 2.94 ± 0.19 2.55 ± 0.19 7.47 ± 0.32 8.80 ± 0.28
98% 341 3.87 ± 0.17 2.68 ± 0.19 7.75 ± 0.34 9.30 ± 0.34
Table 4: Results - Fill rate service level (in %).
Instances Results
name β γ 2SA-m Det. Cover-size Lot-size
L2_v20 85% 13 76.3 ± 0.7 80.3 ± 0.8 96.2 ± 0.4 94.7 ± 0.6
95% 81 92.5 ± 0.5 92.0 ± 0.7 96.2 ± 0.4 95.1 ± 0.6
98% 203 96.8 ± 0.4 94.9 ± 0.5 96.2 ± 0.4 95.1 ± 0.5
L2_v50 85% 48 80.5 ± 1.1 81.2 ± 1.6 92.0 ± 1.1 90.7 ± 1.0
95% 154 90.9 ± 0.9 87.5 ± 1.2 92.1 ± 1.1 91.3 ± 0.9
98% 341 94.6 ± 0.6 88.2 ± 1.2 92.1 ± 1.1 91.3 ± 0.9
80 85 90 95 100
0
2
4
6
8
Fill rate service level (in %)
Holding costs (in Me)
2SA-m
Det.
Cover-size Lot-size
Figure 1: Results for the 30 runs of L2 with v = 0.5 and
β = 95%.
positive and the method of Section 4 combined with
our rule of thumb for determining γ forms an effective
way to compute the production levels in real-world
settings.
7 CONCLUSION
In this paper, we focus on a multi-item lot-sizing pro-
blem with a constraint not yet addressed by academic
75 80 85 90 95 100
0
2
4
6
8
85%
95%
98%
85%
95%
98%
85%
95%
98%
85%
95%
98%
Fill rate service level (in %)
Holding costs (in Me)
2SA-m
Det.
Cover-size Lot-size
Figure 2: Average results for L2 for various values of desi-
red service level β and v = 0.5.
works, while being often met in practice: an explicit
upper bound on the weekly number of setups. We mo-
del this problem as a stochastic program that includes
this constraint and we propose a repeated two-stage
approximation to solve it. Our method proves its ef-
ficiency on real-word instances and outperforms the
heuristics currently used.
The costs in the previous model originate from
holding and backorder. According to the consulting
partner we work with, companies approach their pro-
duction decisions by considering only holding costs,
while trying to keep the backorder below some thres-
hold. Building on the method proposed for the previ-
ous model, we propose a rule of thumb for determi-
ning surrogate backorder costs that bias the solutions
towards the desired service levels. This rule relies on
the famous newsvendor problem, which is the special
case of our problem when there is one reference and
one week. There exist generalizations of the newsven-
dor problem with more references or more weeks; it
would be worth checking whether they could improve
the way this surrogate cost is computed.
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