Journal of the Franklin Institute 348 (2011) 1523–1536
A linear optimization approach to the combinedproduction planning model
$
Maria Cristina N. Gramani
a,
, Paulo M. Franc

a
b,1
,Marcos N. Arenales
c,2
a
Insper Institute of Education and Research,
3
Rua Quat
a, 300  Vila Ol ´ ımpia, 04546042 S
~
ao Paulo  SP, Brazil
b
FCT, Universidade Estadual Paulista
—
UNESP, Av. Roberto Simonsen, 305, 19060900 Presidente Prudente  SP,Brazil
c
ICMC, Universidade de S
~
ao Paulo
—
USP, 13560970 Caixa Postal, 668 S
~
ao Carlos  SP, Brazil
Received 26 May 2009; received in revised form 26 February 2010; accepted 17 May 2010Available online 4 June 2010
Abstract
Two fundamental processes usually arise in the production planning of many industries. The ﬁrstone consists of deciding how many ﬁnal products of each type have to be produced in each period of a planning horizon, the wellknown lot sizing problem. The other process consists of cutting rawmaterials in stock in order to produce smaller parts used in the assembly of ﬁnal products, the wellstudied cutting stock problem. In this paper the decision variables of these two problems aredependent of each other in order to obtain a global optimum solution. Setups that are typicallypresent in lot sizing problems are relaxed together with integer frequencies of cutting patterns in thecutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solvedby a column generated technique. It is worth noting that this new combined problem still takes thetradeoff between storage costs (for ﬁnal products and the parts) and trim losses (in the cuttingprocess). We present some sets of computational tests, analyzed over three different scenarios. These
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&
2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.jfranklin.2010.05.010
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This paper was presented in a preliminary form at the proceedings of the Third International Conference onModeling, Simulation and Applied Optimization
—
ICMSAO 2009.
3
The use of the new name Insper Institute of Education and Research in the School’s academic documents,replacing the name Ibmec S
~
ao Paulo, is pending approval by the Brazilian Ministry of Education.
Corresponding author. Tel.:
þ
55 11 4504 2436; fax:
þ
55 11 4504 2388.
Email addresses:
mariacng@insper.edu.br (M.C. Gramani), paulo.morelato@fct.unesp.br (P.M. Franc

a),arenales@icmc.usp.br (M.N. Arenales).
1
Tel.:
þ
55 18 3229 5385; fax:
þ
55 18 3229 5353.
2
Tel.:
þ
55 16 3373 9655; fax:
þ
55 16 3373 9751.
results show that, by combining the problems and using an exact method, it is possible to obtainsigniﬁcant gains when compared to the usual industrial practice, which solve them in sequence.
&
2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Keywords:
Lot sizing; Cutting stock; Column generation technique; Linear optimization approach
1. Introduction
Due to economical aspects and computational advances, the complexity of optimizationmodels in industrial processes has been increasing considerably. It is still common thatmost research is focused on solving isolated industrial problems. But with the growth andgeneral dissemination of computer and optimizationbased planning, industries have beenlooking for more advanced global methods. Instead of determining optimal solutions of isolated problems, people of industries are looking for integrated solutions that representthe industrial activities more accurately. For instance, managers that have tools to solveseparately logistics and production planning, now are interested in treating both problemsin conjunction, to obtain a global cost minimization. Obviously, by joining variousproblems of high complexity, one is faced with a problem much more difﬁcult to solve,since there are additional coupling constraints, and a global optimum solution to thecombined problem is not in general a straight composition of each optimal solution of theisolated problems.In this paper we focus on the case of furniture industries, but extensions to many othersindustrial settings are straightforward. Given a demand of ﬁnal products (Fig. 1(c)) over aplanning horizon, the issue addressed consists of deciding what to produce in each periodover the planning horizon and how to arrange the parts (Fig. 1(b)) in plates (Fig. 1(a)) in
order to minimize the trim loss during the cutting process. Therefore, two problems of highcomplexity arise in this production planning. The ﬁrst one,
the lot sizing problem
, whichconsists of planning the quantity of each type of ﬁnal product to be produced in each period.Setup costs may be associated with production decision for each ﬁnal product in any givenperiod. The second problem that arises in the case of furniture industries consists of cuttingrectangular plates in order to produce smaller rectangular parts used in the assembly of ﬁnalproducts, the wellknown
rectangular guillotineable twodimensional cutting stock problem
(see [22]). In this way, combining these two problems, the issue consists of the tradeoff
analysis existent when we solve the cutting stock problem taking into account the productionplanning for various periods. Probably it would be worth of anticipating the production of lots of parts or ﬁnal products, increasing the storage costs, but reducing losses in cuttingprocess as well as decreasing the quantity of setups. Thus, three economical factors haveinﬂuence on the combined problem: the trim loss, storage and setup costs.
Fig. 1. (a) Rectangular plates to be cut, (b) rectangular parts and (c) ﬁnal products.
M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523–1536
1524
Both problems considered separately have been well studied in literature. Previousresearch on the lot sizing problem has focused on single and multilevels, as well ascapacitated and uncapacitated resources. There are a number of studies in the literature(see for example, [1,11–13,20,21]). Also, the cutting stock problem has been greatlystudied since the seminal paper of Gilmore and Gomory [5] (see for example[3,5,6,10,14,15,22]).Even though the cutting stock problem coupled in the production planning problem arises inmany industries, only a few papers have been published to ﬁnd good mathematical models and,hopefully, efﬁcient solution methods. Drexl and Kimms [4] remarked that this combinedproblem (they called coordination problem) as a ‘‘most crucial goal for future research’’. Nonasand Thorstenson [17] studied the combined cutting stock and lot sizing problem to a company of offroads trucks. Reinders [19] studied the process of cutting tree trunks to assortments andboards for various markets. Hendry et al. [9] proposed a twostage solution to solve thecombined problem for a copper industry; ﬁrst they found the best cutting patterns minimizingthe waste, and then these patterns are given as input to the second stage, which provides thedaily production planning. This decomposition approach was also used by Poltroniere et al. [18]to solve a combined problem in the paper industry. However, none of them studied the assemblyof ﬁnal products.Only few papers in the literature approach directly the combined lot sizing and cuttingstock problem. Gramani and Franc

a [7] analyzed the tradeoff that arises when solving thecutting stock problem by taking into account the production planning for various periods.The goal was to minimize the trim loss costs in the cutting process, the inventory costs (forthe parts) and the setup costs. The authors formulated a mathematical model of thecombined cutting stock and lotsizing problem and proposed a solution method based onan analogy with the network shortest path problem, comparing its results with the onessimulated in the industrial practice. However, this paper does not consider the productionand inventory costs of ﬁnal products.Gramani et al. [8] proposed a heuristic method based on Lagrangian relaxation to thecoupled lot sizing and cutting stock problem. The difﬁculty faced by the Lagrangiansolution approach is that the resulting Lagrangian subproblems are NPhard capacitatedlot sizing problems.In this paper, we address the model as Gramani et al. [8] by relaxing setups butincluding the storage of parts. The aims with this new approach are, on the one hand,a simpler model solvable by available solvers, and, on the other hand, the incorporation of part inventories, a practical and relevant issue, which can be seen asusable leftovers (see [2] for other approaches). We solve this combined model tooptimality by using the CPLEX package with the column generation technique. It is worthof noting that even relaxing setups the tradeoff between storage and trim losses stillremains.This paper is organized as follows: Next section presents the mathematical modelproposed in Gramani et al. [8] by relaxing setups but including the storage of parts. Then adecomposition approach is presented to solve the problem in two steps, ﬁrst solving the lotsizing problem and then the cutting stock problem. In Section 4 we present a columngeneration method to solve the combined model. In Section 5 we take two sets of instancesanalyzed over three different aspects to be used in computational tests, which support theadvantage of the combined model, and ﬁnally some concluding remarks are given inSection 6.
M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523–1536
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2. The combined lot sizing and cutting problem mathematical modeling
In this paper we focus on a production planning from furniture industry, where goodssuch as wardrobes, beds, shelves, etc., should be produced. Large wood plates are cut intoparts, which are assembled (after drilling, painting, etc.), to the pieces of furniture. Theoverall problem can be seen as a multilevel lot sizing problem, where a cutting problemarises in the ﬁrst level.Before writing down the model, a few words on the cutting stock problem (fordetails, see [6,15]). Suppose a number of parts (rectangles with length
l
i
and width
w
i
)are to be cut from plates (rectangles with length
L
and width
W
). The main differencefrom ordinary cutting stock problem found in the literature is that the part demandsnow are not known, since they depend on early lot sizing decisions. In order to solve thecutting stock problem, a number of cutting patterns should be ﬁnd, that is, the waythe plates are cut to produce the parts. Of course, many rules can be used to deﬁnefeasible cutting patterns. In this paper, we consider twostage guillotineable cuttingpatterns. A (orthogonal) guillotine cut applied on a plate produces two newintermediate rectangles. This is the ﬁrst stage. Each intermediate rectangle is cutsuccessively by guillotine cuts until parts are obtained. If the guillotine cut appliedon an intermediate plate is orthogonal to the previous one (the one made to obtainthe intermediate plate), one adds the number of stages. If the number of stages is limitedby two, we say the obtained cutting pattern is twostage guillotineable. For sake of time consuming, it is usual that the cutting patterns accepted in furniture industriesare twostage. It is also possible to be still more restricted (see [16]). A plate cut by apattern
j
produces parts, denoted by
a
ij
that means the number of parts type
i
obtainedwhen a plate is cut according to the pattern
j
. The number of possible cutting patterns is inpractice very large. If there are available diverse sizes of plates, different cutting patternsare built for each one, and what follows is straightforward extensible. Therefore, for sakeof simplicity, we assume there is only one type of plate in stock,
L
W
.Let us consider the following notation.Indexes
i=
1,
y
,
M
: number of different ordered ﬁnal products
p
=1,
y
,
P
: number of different types of required parts to be cut
t=
1,
y
,
T
: number of planning periods
j
=1,
y
,
N
: number of cutting patternsParameters
c
it
: unit production cost of a ﬁnal product type
i
in period
th
it
: unit inventory cost of a ﬁnal product type
i
in the ﬁnal of period
tcp
: unit plate cost
hp
pt
: unit inventory cost of part type
p
in the ﬁnal of period
td
it
: demand of ﬁnal product
i
in period
tr
pi
: number of parts type
p
necessary to a unit of the ﬁnal product
i v
j
: time spent to cut a single plate by using pattern
j b
t
: saw capacity (in hours) for period
ta
pj
:
number of parts type
p
in pattern
j
M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523–1536
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Variables
x
it
: number of ﬁnal product
i
produced in period
tI
it
: number of ﬁnal product
i
in inventory in the ﬁnal of period
tIP
pt
: number of parts type
p
in inventory in the ﬁnal of period
t y
jt
: number of plates cut according to pattern
j
in period
t
.Now we are ready to write down the combined
lot sizing and cutting problem
(MasterProblem):Min
X
M i
¼
1
X
T t
¼
1
ð
c
it
x
it
þ
h
it
I
it
Þ þ
X
N j
¼
1
X
T t
¼
1
cpy
jt
þ
X
P p
¼
1
X
T t
¼
1
hp
pt
IP
pt
ð
1
Þ
subject to
x
it
þ
I
i
;
t
1
I
it
¼
d
it
;
i
¼
1
;
. . .
;
M
;
t
¼
1
;
. . .
;
T
ð
2
Þ
X
N j
¼
1
a
pj
y
jt
þ
IP
p
;
t
1
IP
pt
¼
X
M i
¼
1
r
pi
x
it
;
p
¼
1
;
. . .
;
P
;
t
¼
1
;
. . .
;
T
ð
3
Þ
X
N j
¼
1
v
j
y
jt
r
b
t
;
t
¼
1
;
. . .
;
T
ð
4
Þ
y
jt
Z
0 integer
j
¼
1
;
. . .
;
N
;
t
¼
1
;
. . .
;
T
ð
5
Þ
x
it
;
I
it
Z
0 and
I
i
0
given
i
¼
1
;
. . .
;
M
;
t
¼
1
;
. . .
;
T
ð
6
Þ
IP
pt
Z
0 and
IP
p
0
given
p
¼
1
; :::;
P
;
t
¼
1
; :::;
T
ð
7
Þ
The objective function (1) represents the costs involved in production and storage of ﬁnal products, trim loss and storage of the parts. Eq. (2) denote the inventory balances forthe ﬁnal products, which together with (6) ensure that the demands of ﬁnal products aremet. The initial inventories are considered zero without loss of generality. Eq. (3) denotethe inventory balances for the parts, where the RHS gives the internal demand of them,which together with (7) ensure they are met. Constraints (4) are due to saw capacity.Observe also that the constraints (3) are the only coupling constraints, linking lot sizingand cutting decisions.There are two great difﬁculties in order to solve this model: (1) the integrality of
y
jt
and(2) the enormous quantity of cutting patterns that could be generated, i.e., number
N
is inpractice very large. A possible strategy to deal with this huge number of patterns consistsof ﬁxing a reduce number of them. In this paper, we consider all of them implicitly andgenerate whenever is needed, which consists of the column generation technique byGilmore and Gomory.Next, we give a ﬁrst approach to solve the problem, a simple strategy usually employedby people in industry to handling the overall problem. This approach solves the problem ina separated way.
M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523–1536
1527