Journal of the Franklin Institute 348 (2011) 15231536

A linear optimization approach to the combinedproduction planning model$

Maria Cristina N. Gramania,, Paulo M. Franc-ab,1,

a planning horizon, the well-known lot sizing problem. The other process consists of cutting raw

www.elsevier.com/locate/jfranklin

3

arenales@icmc.usp.br (M.N. Arenales).1Tel.: 55 18 3229 5385; fax: 55 18 3229 5353.2Tel.: 55 16 3373 9655; fax: 55 16 3373 9751.0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jfranklin.2010.05.010

The use of the new name Insper Institute of Education and Research in the Schools academic documents,

replacing the name Ibmec S~ao Paulo, is pending approval by the Brazilian Ministry of Education.materials in stock in order to produce smaller parts used in the assembly of nal products, the well-

studied cutting stock problem. In this paper the decision variables of these two problems are

dependent of each other in order to obtain a global optimum solution. Setups that are typically

present in lot sizing problems are relaxed together with integer frequencies of cutting patterns in the

cutting problem. Therefore, a large scale linear optimizations problem arises, which is exactly solved

by a column generated technique. It is worth noting that this new combined problem still takes the

trade-off between storage costs (for nal products and the parts) and trim losses (in the cutting

process). We present some sets of computational tests, analyzed over three different scenarios. These

$This paper was presented in a preliminary form at the proceedings of the Third International Conference on

Modeling, Simulation and Applied OptimizationICMSAO 2009.Corresponding author. Tel.: 55 11 4504 2436; fax: 55 11 4504 2388.E-mail addresses: mariacng@insper.edu.br (M.C. Gramani), paulo.morelato@fct.unesp.br (P.M. Franc-a),Marcos N. Arenalesc,2

aInsper Institute of Education and Research,3 Rua Quat a, 300 - Vila Olmpia, 04546-042 S ~ao Paulo - SP, BrazilbFCT, Universidade Estadual PaulistaUNESP, Av. Roberto Simonsen, 305, 19060-900 Presidente Prudente - SP,

BrazilcICMC, Universidade de S ~ao PauloUSP, 13560-970 Caixa Postal, 668 S ~ao Carlos - SP, Brazil

Received 26 May 2009; received in revised form 26 February 2010; accepted 17 May 2010

Available online 4 June 2010

Abstract

Two fundamental processes usually arise in the production planning of many industries. The rst

one consists of deciding how many nal products of each type have to be produced in each period of

results show that, by combining the problems and using an exact method, it is possible to obtain

signicant 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

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523153615241. 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 optimization-based planning, industries have beenlooking for more advanced global methods. Instead of determining optimal solutions ofisolated 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 difcult 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 others

industrial settings are straightforward. Given a demand of nal 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)) inorder to minimize the trim loss during the cutting process. Therefore, two problems of highcomplexity arise in this production planning. The rst one, the lot sizing problem, whichconsists of planning the quantity of each type of nal product to be produced in each period.Setup costs may be associated with production decision for each nal 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 nalproducts, the well-known rectangular guillotineable two-dimensional cutting stock problem(see [22]). In this way, combining these two problems, the issue consists of the trade-offanalysis 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 oflots of parts or nal products, increasing the storage costs, but reducing losses in cuttingprocess as well as decreasing the quantity of setups. Thus, three economical factors haveinuence on the combined problem: the trim loss, storage and setup costs.Fig. 1. (a) Rectangular plates to be cut, (b) rectangular parts and (c) nal products.

Both problems considered separately have been well studied in literature. Previousresearch on the lot sizing problem has focused on single and multi-levels, as well ascapacitated and uncapacitated resources. There are a number of studies in the literature(see for example, [1,1113,20,21]). 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 in

many industries, only a few papers have been published to nd good mathematical models and,hopefully, efcient 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 ofoff-roads trucks. Reinders [19] studied the process of cutting tree trunks to assortments andboards for various markets. Hendry et al. [9] proposed a two-stage solution to solve thecombined problem for a copper industry; rst 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 nal products.Only few papers in the literature approach directly the combined lot sizing and cutting

stock problem. Gramani and Franc-a [7] analyzed the trade-off 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 lot-sizing 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 nal products.Gramani et al. [8] proposed a heuristic method based on Lagrangian relaxation to the

coupled lot sizing and cutting stock problem. The difculty faced by the Lagrangiansolution approach is that the resulting Lagrangian subproblems are NP-hard capacitatedlot sizing problems.In this paper, we address the model as Gramani et al. [8] by relaxing setups but

including 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 incor-poration 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 trade-off between storage and trim losses stillremains.This paper is organized as follows: Next section presents the mathematical model

proposed 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, rst 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 nally some concluding remarks are given in

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 15231536 1525Section 6.

2.

y, where goodssuc tes are cut intopa furniture. Theov utting problemari

problem (for

fro at the part demandsno In order to solve thecu nd, that is, the waythe n be used to denefea uillotineable cutting

36152patterns. A (orthogonal) guillotine cut applied on a plate produces two newintermediate rectangles. This is the rst 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 two-stage guillotineable. For sake oftime consuming, it is usual that the cutting patterns accepted in furniture industriesare two-stage. It is also possible to be still more restricted (see [16]). A plate cut by apattern j produces parts, denoted by aij 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, LW.Let us consider the following notation.Indexes

i=1,y,M: number of different ordered nal productsp=1,y,P: number of different types of required parts to be cutt=1,y,T: number of planning periodsj=1,y,N: number of cutting patterns

Parameters

cit: unit production cost of a nal product type i in period thit: unit inventory cost of a nal product type i in the nal of period tcp: unit plate costhppt: unit inventory cost of part type p in the nal of period tdit: demand of nal product i in period trpi: number of parts type p necessary to a unit of the nal product ivj: time spent to cut a single plate by using pattern jbt: saw capacity (in hours) for period tm ordinary cutting stock problem found in the literature is thw are not known, since they depend on early lot sizing decisions.tting stock problem, a number of cutting patterns should be plates are cut to produce the parts. Of course, many rules casible cutting patterns. In this paper, we consider two-stage gdetails, see [6,15]). Suppose a number of parts (rectangles with length li and width wi)are to be cut from plates (rectangles with length L and width W). The main differenceThe combined lot sizing and cutting problem mathematical modeling

In this paper we focus on a production planning from furniture industrh as wardrobes, beds, shelves, etc., should be produced. Large wood plarts, which are assembled (after drilling, painting, etc.), to the pieces oferall problem can be seen as a multi-level lot sizing problem, where a cses in the rst level.Before writing down the model, a few words on the cutting stock

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523156apj: number of parts type p in pattern j

jt

(2) , i.e., number N is inpr of patterns consistsof them implicitly andge ration technique byGi

Next, we give a rst approach to solve the problem, a simple strategy usually employed

by people in industry to handling the overall problem. This approach solves the problem in

a sthe enormous quantity of cutting patterns that could be generatedactice very large. A possible strategy to deal with this huge numberxing a reduce number of them. In this paper, we consider all ofnerate whenever is needed, which consists of the column genelmore and Gomory.Variables

xit: number of nal product i produced in period tIit: number of nal product i in inventory in the nal of period tIPpt: number of parts type p in inventory in the nal of period tyjt: 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):

MinXM

i1

XT

t1citxit hitIit

XN

j1

XT

t1cpyjt

XP

p1

XT

t1hpptIPpt 1

subject to

xit Ii;t1Iit dit; i 1; . . .; M; t 1; . . .; T 2

XN

j1apjyjt IPp;t1IPpt

XM

i1rpixit; p 1; . . .; P; t 1; . . .; T 3

XN

j1vjyjtrbt; t 1; . . .; T 4

yjtZ0 integer j 1; . . .; N; t 1; . . .; T 5

xit; IitZ0 and Ii0 given i 1; . . .; M; t 1; . . .; T 6

IPptZ0 and IPp0 given p 1; :::; P; t 1; :::; T 7

The objective function (1) represents the costs involved in production and storage ofnal products, trim loss and storage of the parts. Eq. (2) denote the inventory balances forthe nal products, which together with (6) ensure that the demands of nal 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 sizing

and cutting decisions.There are two great difculties in order to solve this model: (1) the integrality of y and

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 15231536 1527eparated way.

i.e.T c4. An exact approach to the combined model

Despite the fact that the setups were disregarded, the uncapacitated combined model(1)(7) still capture the trade-off between the storage and plate costs that arise byandcoXN

j1vjyjtrbt; t 1; . . .; T 13

yjtZ0 integer j 1; . . .; N; t 1; . . .; T 14

IPptZ0 and IPp0 given p 1; . . .; N; t 1; . . .; T 15

Observe that the lot sizing problem (M1) isolated will always generate a lot for lot solution,, the lot sizes in each period is equal the corresponding demand. And then, it remains to solvelassical cutting stock problems (M2), one per period, taking into account the demand of partsthe saw capacity, which can be solved with the column generation technique.3. A decomposition approach

In daily practice in industry, the lot sizing problem and the cutting stock problem aresolved separately and in sequence as the following way:

1. First of all, solve the uncapacitaded lot sizing problem, named (M1). The aim here is todetermine the optimal solution xit from Model (M1).Model (M1)

MinXM

i1

XT

t1citxit hitIit 8

subject to:

xit Ii;t1Iit dit; i 1; . . .; M; t 1; . . .; T 9xit; IitZ0 and Ii0 given i 1; . . .; M; t 1; . . .; T 10

2. Then, solve the cutting stock problem, named M2. Notice that to solve the cutting stockproblem, xit is substituted by the parameter xit at constraint (3). Then, we determine yjt,that is, the quantity of plates in pattern j at period t used.Model (M2)

MinXN

j1

XT

t1cpyjt

XP

p1

XT

t1hpptIPpt 11

subject to:

XN

j1apjyjt IPp;t1IPpt

XM

i1rpixit; p 1; . . .; P; t 1; . . .; T 12

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 152315361528mbining the lot sizing and cutting stock problems. The analysis if it is advantageous to

anticipate the production of certain lots of parts or nal products, increasing the storagecosts but obtaining a worthwhile prot in the cutting process, still remains.The combined model has two difculties: Firstly, the integrality of the yjt variables,

which are relaxed due to the high demands. The second difculty involves the enormousquantity of cutting patterns that could be generated. To overcome this problem, use thecolumn generation technique. The complexity of the column generation process consists ofsolving a great number of knapsack problems. Therefore, this technique enables theresolution of problems with millions of variables, since very few columns are stored(corresponding the basic variables) and the remaining ones are generated only if necessary.For the sake of notation, we re-write the Master Problem (1)(7) as follows:

MinXT

t1ctxt htIt

XT

t1cpyt

XT

t1hptIPt 16

subject to:

dt xt It1It; t 1; . . .; T 17

0 Rxt Ayt IPt1IPt; t 1; . . .; T 18

CtZVyt; t 1; . . .; T 19

ytZ0; t 1; . . .; T 20

xt; ItZ0 and I0 given t 1; . . .; T 21

IPtZ0 and IP0 given t 1; . . .; T 22where ct=(cit), xt=(xit) and It=(Iit) are M-vectors; yt=(yjt) and V=(vj) are N-vectors;IPt=(IPpt), dt=(dpt) are P-vectors; and R=(rpi) is a PM matrix. Table 1 depicts thesparsity structure coefcient matrix.The column generation technique works, at each iteration, with a Restricted Master

Problem, where only a few columns are explicitly considered. Initially, only homogeneouscutting patterns are taken, that is, in matrix A only columns type: (0yaiiy0)

T areconsidered, where aii=bL/licbW/wic. The pricing subproblem consists of solving arectangular guillotineable two-stage two-dimensional cutting problem, which gives anew entering column to the next Restricted Master Problem.A owchart of the column generation technique is given in Fig. 2.

5. Computational results

The algorithm proposed was programmed in C. Two sets of tests were analyzed overthree different aspects: (1) comparison between the results obtained using the combinedmodel with tight and normal saw capacity; (2) comparison between the results obtainedwith the decomposition approach from Section 3 and the exact method from Section 4;and nally (3) we present an analysis of the solutions quality when decreasing the platescost. The CPLEX tolerance was 103.The parameters concerned to the dimensions of the plates, number of parts that

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 15231536 1529constitute the nal products, and the demand of nal products, were randomly generated

Table 1

Coefcients matrix for the model (16)(22).

dt x1 x2 y xT I1 I2 y It1 It y IT y1 y2 y yT IP1 IP2 .. IPt1 IPt y IPT

c1 c2 y cT h1 h2 y ht1 ht y hT cp cp y cp hp1 hp2 .. hpt1 hpt y hpT

t=1 d1I0 = I 0 y 0 I 0 y 0 0 y 0 0 y 0 0 y 0t=2 d2 = 0 I y 0 I I y 0 0 y 0 0 y 0 0 y 0y y y y y y y y y 0 y 0 0 y 0t=t-1 dt1 = 0 0 y 0 0 0 y I 0 y 0 0 y 0 0 y 0t=t dt = 0 0 y 0 0 0 y I I y 0 0 y 0 0 y 0y y y y y y y y y y y y y y yt=T dT = 0 0 y I 0 0 y 0 0 y I 0 y 0 0 y 0t=1 0 = R 0 y 0 0 y 0 A 0 y 0 I 0 .. 0 0 y 0t=2 0 = 0 R y 0 0 y 0 0 A y 0 I I .. 0 0 y 0y y y y y y 0 y 0 y y y y y .. .. y y y yt=t1 0 = 0 0 y 0 0 y 0 0 0 y 0 0 0 .. I 0 y 0t=t 0 = 0 0 y 0 0 y 0 0 0 y 0 0 0 .. I I y 0t=T 0 = 0 0 y R 0 y 0 0 0 y A 0 0 .. 0 0 y It=1 C1 Z 0 0 y 0 0 y 0 V 0 y 0 0 0 .. 0 0 0 0t=2 C2 Z 0 0 y. 0 0 y 0 0 V y 0 0 0 .. 0 0 0 0y y y y y y y y y y y y y y .. y y y yt=T CT Z 0 0 y 0 0 y 0 0 0 y V 0 0 .. 0 0 0 0

M.C

.N.

Gra

ma

ni

eta

l./

Jo

urn

al

of

the

Fra

nk

linIn

stitute

34

8(

20

11

)1

52

3

15

36

1530

wi

wi

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 15231536 1531thin the following bounds:

width (w) and length (l) of parts: [25,75];width (W) and length (L) of the plates: 120 and 100, respectively;number of parts (r) that constitute the nal products: [0,5];demand (d) of nal products: [0,500];

The storage and plates cost were considered in the following way:Let li, wi, i=1,y,P be the width and length of parts types i, respectively, and L, W be thedth and length of the plates, respectively.

Parts cost: [(liwi)/(L

W)];Final products cost: The sum of the parts cost that assembly the nal products;Storage cost for the parts: (parts cost)103;Storage cost for the nal products: (nal product cost)103;Plates cost: (LW)103;Capacity: Calculated as the average of the capacities consumed in all periods of theplanning horizon using the decomposition method. Two levels were considered: tightcapacity denominates the average of the capacity consumed in all periods of theplanning horizon, using the decomposition method, plus 20%, and normal capacitycorresponds to the same but using an one hundred percent increase.

Fig. 2. Schematic representation of the exact method for the combined problem.

2 18,225 16,860 47 5.463 6307 5740 47 6.31

4 Infeasible 13,729 48

5 Infeasible 16,729 45

6 14,543 12,960 47 8.12Table 2

10 parts, 4 nal products and 4 periods.

Instances Tight capacity Normal capacity % global solution

#plates used #plates used %capacity utilization

1 15,321 14,125 52 5.55

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 152315361532Each Tables 25 is titled as the number of different parts, nal products and periodsconsidered. For example, the set of tests used in Table 2 considers 10 different parts, 4different nal products and a 4-period planning horizon.

5.1. Saw capacity: normal tight

Tables 2 and 3 compare the quality of the solutions, using the exact approach, regardingthe normal and tight capacity. The columns #plates used represent the quantity of platesused to meet the demand of nal products. The column global solution denotes the sum ofthe storage and plates cost for each problem. The columns %capacity utilization denotethe percentage of the capacity utilization of the machines. Finally, the column %global

7 Infeasible 5701 48

8 11,888 11,510 43 2.18

9 13,241 12,258 46 5.35

10 Infeasible 25,079 48

Average 47 5.50

Table 3

50 parts, 10 nal products and 6 periods.

Instances Tight capacity Normal capacity % global solution

#plates used #plates used %capacity utilization

1 Infeasible 28,394 48

2 Infeasible 32,139 42

3 Infeasible 29,752 48

4 40,581 34,148 47 11.98

5 143,900 123,798 47 10.32

6 Infeasible 132,225 48

7 175,440 146,229 48 12.73

8 Infeasible 142,875 48

9 Infeasible 157,540 48

10 Infeasible 148,864 49

Average 47 11.67

Table 4

10 parts, 4 nal products and 4 periods.

Instances Separated method Exact method % global solution

#plates used #plates used $ storage

1 15,321 14,125 4.18 8.46

2 17,776 16,860 14.63 5.42

3 6031 5740 5.59 5.06

4 14,281 13,729 14.26 4.02

5 18,352 16,729 10.26 9.70

6 13,636 12,960 13.10 5.21

7 5973 5701 8.51 4.55

8 11,888 11,510 4.23 3.28

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 15231536 1533solution shows the gain (in terms of storage and plates cost) which would be obtained if thecapacity had been considered normal.Tables 2 and 3 clearly show that when we deal with normal capacity, the number of

plates used is lower compared to the tight capacity; this occurs because when consideringnormal capacity, the problem is more exible regarding the storage, leading to a betterglobal solution. And most instances provide infeasible solutions when considering tightcapacity. These tables show that the normal capacity provides a gain up to 12.7% over thetight capacity. This gain could be interpreted as fewer shift turns in the planning horizon.

5.2. Comparison between the exact method and the decomposition approach

Tables 4 and 5 illustrate the performance of the exact method. The test sets were thesame of those of Tables 2 and 3 considering normal capacity. The column separated

9 13,194 12,258 6.02 7.64

10 26,156 25,079 6.17 4.30

Average 5.76

Table 5

50 parts, 10 nal products and 6 periods.

Instances Separated method Exact method % global solution

#plates used #plates used $ storage

1 29,213 28,394 31.72 2.87

2 33,747 32,139 36.40 5,00

3 31,093 29,752 28.66 4.50

4 36,445 34,148 26.51 6.72

5 130,208 123,798 160.57 5.17

6 137,451 132,225 107.23 3.95

7 151,273 146,229 81.28 3.45

8 147,950 142,875 159.05 3.54

9 162,665 157,540 157.42 3.25

10 153,378 148,864 56.67 3.03

Average 4.15

method denotes the number of plates used (#plates used) when we apply the decompositionmethod (usually used in practice), and the column exact method shows the number ofplates used (#plates used) and the storage cost ($storage) when we apply the exactproposed algorithm. The column %global solution shows the gain, in terms of storage andplates cost, between the combined algorithm and the separated method.It is evident from Tables 4 and 5 that the combined problem anticipates the production

of some lots of parts, increasing the storage costs, but getting an advantageous decrease inthe number of plates used, leading to an overall signicant gain, as compared to theseparated method.

5.3. Comparison of the number of plates used when decreasing the plates cost

Figs. 3 and 4 show the performance of the number of plates used when the plates costdecreases. The test sets used were the same of those of Tables 2 and 3. The bar (area of theplate)103 represents the number of plates used when we apply the exact method for theplate costs equal to (LW103). The bar $storage of the piece denotes the number ofplates used when the plate costs is taken as the average of the storage costs of parts.Finally, the bar (0.000022) represents the number of plates used for the plate costs equal to

10 parts, 4 final products and 4 periods

M.C.N. Gramani et al. / Journal of the Franklin Institute 348 (2011) 1523153615341 2 3 4 5 6 7 8 9 10Instances

(area of the plate)*10-3 $ storage of the part 0.0000221 2 3 4 5 6 7 8 9 10Instances

(area of the plate)*10-3 $ storage of the part 0.000022

Fig. 3. Ten parts, 4 nal products and 4 periods

50 parts, 10 final products and 6 periodsFig. 4. Fifty parts, 10 nal products and 6 periods

0.000022 (this cost is much lower than the average of the storage parts). In all cases thecapacity considered is normal.Figs. 3 and 4 show that the number of plates used increases when the plates cost

decreases. This is obvious, because when the plates cost increases considerably theprogram should try to rearrange the parts on the plates in order to obtain the minimumnumber of plates used.

6. Concluding remarks

This paper provides a new methodology to solve in conjunction two complex problemsthat arise in furniture companies. The rst one is the problem of cutting rectangular platesin order to produce smaller rectangular parts, the well-known cutting stock problem. Theother is the problem of planning the quantity of nal products to be produced in eachperiod of the planning horizon, while satisfying all demand requirements and optimizingthe production and storage costs, the lot sizing problem.Our approach is inspired in a combined model proposed in [8] which showed to be successful

in capturing the trade-off between the setup and plate costs. In the new model the nal productsetup cost is disregarded but the part inventory cost is now considered, which maintains a trade-off between the storage and plates cost. As stated previously, the aim with this new approach istwo-fold: the simpler model permits the use of a commercial solver to achieve optimal solutionswhile it allows for the incorporation of parts inventories, a practical and relevant issue in mostindustries. Toward this, we proposed an exact approach to solve the combined model using thesimplex method with the column generation technique. We show that, when solving theseproblems in conjunction, even disregarding the setup constraints, it could be benecial toanticipate the production of some lots of parts or nal products, increasing the storage costs, butobtaining an advantageous cost decrease in the trim loss through a better composition of thecutting patterns.Computational results obtained when the new method is executed over a set of

generated instances clearly show its applicability. We present two sets of computationaltests, analyzed over three different aspects. In one case, the results of the exact methodcompared with the solutions obtained by the method used in usual practice (separatedmethod) show that we were able to obtain gains up to 12.7%. Another aspect concernedthe saw capacity shows that when the capacity is considered tight, the solutions use moreplates in comparison when we use a normal capacity, leading to costly global solutions.The third aspect considered is that, if the plates costs increases, the quantity of plates useddecreases considerably, using better cutting patterns.

Acknowledgments

This research was partially supported by grants from CNPq and FAPESP, Brazil.

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A linear optimization approach to the combined production planning modelIntroductionThe combined lot sizing and cutting problem mathematical modelingA decomposition approachAn exact approach to the combined modelComputational resultsSaw capacity: normaltimestightComparison between the exact method and the decomposition approachComparison of the number of plates used when decreasing the plates cost

Concluding remarksAcknowledgmentsReferences