STRATEGIES TO MODEL SCHEDULING DECISIONS TO PLAN

THE SOFT DRINK PRODUCTION PROCESS

Socorro Rangel and Michelli Maldonado

UNESP, S˜ao Paulo State University, Rua Crist´ov˜ao Colombo, 2265, 15054-000, S˜ao Jos´e do Rio Preto, SP, Brazil

Keywords:

Production planning, Integrated lot sizing and scheduling models, Asymmetric traveling salesman problem.

Abstract:

In this paper we present a mixed integer model that integrates lot sizing and lot scheduling decisions for the

production planning of a soft drink company. The main contribution of the paper is to present a model that

differ from others in the literature for the constraints related to the scheduling decisions. The proposed strategy

is compared to other strategies presented in the literature.

1 INTRODUCTION

The lot sizing and scheduling problems has received

a lot of attention given its relevance to the indus-

trial process. A recent trend has been on mathemati-

cal models that capture the relationship between both

problems (Clark et al (2011)). Integrated models have

been proposed for several industrial contexts. For ex-

ample, Almada Lobo et al (2007) study the problem

for the glass container industry, Toso et al (2009) for

the animal feed supplements industry, and Ferreira et

al (2009, 2010) for the softdrink industry. Two main

strategies have been used to model the scheduling de-

cisions. The ﬁrst one is a small bucket strategy in

which each period of the planning horizon is divided

into subperiods. For each subperiod only one item

can be produced. This strategy is based on the GLSP

model ((General Lotsizing and Scheduling Problem)

(Fleischmann and Meyer (1999)). The second strat-

egy is a bigbucket one and allows the production of

several items in a given period. To obtain the pro-

duction sequence constraints based on the asymmet-

ric traveling salesman problema (ATSP) are added to

the lot sizing formulation.

The paper is organized as follows. In Section 2

a brief description of the production process of soft

drinks according to visits to small and medium scale

soft drinks plants in Brazil and of the one stage one

machine model P1S1MTS given in the literature is

presented. In Section 3 a alternative to strengthen the

P1S1MTS model is proposed. Section 4 presents ﬁnal

remarks.

2 BRIEF DESCRIPTION OF

PREVIOUS WORK FOR

PLANNING THE SOFT DRINK

PRODUCTION PROCESS

In this section we review the mathematical model

P1S1MTS proposed by Defalque et al. (2010) to rep-

resent the production process of small scale soft drink

plants. The production process of soft drinks in dif-

ferent sizes and ﬂavours is carried out in two stages:

liquid ﬂavor preparation (Stage I) and bottling (Stage

II). The model P1S1MTS considers that there are J

soft drinks (items) to be produced from L liquid ﬂa-

vors (syrup) on one production line (machine). To

model the decisions associated with Stage I, it is sup-

posed that there are several tanks to store the syrup

and that it is ready when needed. Therefore, it is not

necessary to consider the scheduling of syrups in the

tanks, nor the changeover times since it is possible

to prepare a new lot of syrup in a given tank, while

the machine is bottling the syrup from another tank.

However, the syrup lot size needs to satisfy upper and

lower bound constraints in order to not overload the

tank and to guarantee syrup homogeneity. In Stage

II, the machine is initially adjusted to produce a given

item. To produce another item, it is necessary to stop

the machine and make all the necessary adjustments

(another bottle size and/or syrup ﬂavor). Therefore, in

this stage, changeover times from one product to an-

other may affect the machine capacity and thus have

to be taken into account. The P1S1MTS model ad-

dresses the problem of deﬁning the lot size and lot

schedule taking into account the demand for items

181

Rangel S. and Maldonado M..

STRATEGIES TO MODEL SCHEDULING DECISIONS TO PLAN THE SOFT DRINK PRODUCTION PROCESS.

DOI: 10.5220/0003761801810185

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 181-185

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

and the capacity of the machine and syrup tanks, min-

imizing the overall production costs. It assumes that

there is an unlimited quantity of other supplies (e.g.

bottles, labels, water).

2.1 The P1S1MTS Model

In the P1S1MTS model the decisions associated with

lot sizing are based on the Capacitated Lot Sizing

Problem (CLSP) (e.g. Karimi et al (2003)). The

scheduling decisions use the ATSP approach with the

MTZ constraints to eliminate subtours. Some simpli-

ﬁcations of the production process have been made.

Only one production line (machine) is considered and

it is also supposed that there are several tanks dedi-

cated to it. Therefore, it is not necessary to consider

the scheduling of liquid ﬂavor in the tanks, nor the

changeovertimes. It is possible to prepare a new lot of

liquid ﬂavor in a given tank, while the machine is bot-

tling the liquid ﬂavor from another tank. However, the

stage I constraints cannot be completely discarded.

The liquid ﬂavor lot size needs to satisfy upper and

lower bound constraints in order to not overload the

tank and to guarantee liquid homogeneity. The prob-

lem considered in this paper can thus be stated as: de-

ﬁne the lot size and lot schedule of the products taking

into account the items demands and the capacity of

the production line and syrup tanks, minimizing the

overall production costs. It is also supposed that there

is an unlimited quantity of bottles, labels and water.

To present the model, let the following parame-

ters deﬁne the problem size: J is the number of soft-

drinks (items); L is the number of syrup ﬂavors and

T is the number of periods; in the planning horizon.

Let (i, j, k,l,t) be the index set deﬁned as: i, j,k ∈

{1,.. .,J};l ∈ {1,.. .,L};t ∈ {1, ...,T}. The data and

variables are described in Table 1. The superscript I

relates to Stage I (syrup preparation) and with super-

script II relates to Stage II (bottling).

The optimization criterion (1) is to minimize the

overall costs taking into account inventory, backorder

and machine changeover costs.

Min Z =

∑

j=1

∑

t=1

+ g

−

) +

∑

t=1

∑

i=1

∑

j=1, j6=i

ijt

(1)

The lotsizing decisions in Stage I, as deﬁned by

constraints (2)-(5) control the syrup production. Con-

straints (2) guarantee that if the tank is ready for pro-

duction of syrup l, then there will be production of

item j and the quantity produced uses all the syrup

prepared in that period. The variables n

allow par-

tial use of the tank and is controlled to respect the

minimum amount needed to ensure syrup homogene-

ity, as speciﬁed by constraints (3). Constraints (4) en-

sure that there is production of the syrup l only if the

tank is prepared. According to constraints (5), the to-

tal number of tanks produced in period t is limited by

the maximum number of tank setups.

Stage I: Syrup Preparation

∑

j∈γ

l j

= K

− n

), ∀l, ∀t. (2)

≤ 1 −





, ∀l,∀t. (3)

≤ w

≤ S

, ∀l,∀t. (4)

∑

l∈L

≤ S

; ∀t. (5)

The lotsizing decisions in Stage II are deﬁned by

constraints (6)-(9). Constraints (6) represent the ﬂow

conservation of each item in each time period. Con-

straints (7) represent the machine capacity in each

time period. Constraints (8) guarantee that there is

production of item j only if the machine is prepared.

Note that the setup variable is considered implicitly in

terms of the changeover variables and that production

may not occur although the machine might be pre-

pared. Constraints (9) control the maximum number

of setups in each period.

Stage II (bottling) - Lot Sizing:

j(t−1)

+ I

−

+ x

− I

−

j(t−1)

= d

, ∀ j,∀t (6)

∑

j=1

∑

i=1

∑

j=1, j6=i

ijt

≤ K

, ∀t (7)

≤ K

∑

i=1,i6= j

ijt

, ∀ j,∀t (8)

∑

i=1

∑

j=1, j6=i

ijt

≤ S

, ∀t (9)

Constraints (10)-(14) model the order in which the

items will be produced in a given period t. They are

based on the ATSP model. Constraints (10) consider

that in each period the machine is initially setup for a

ghost item i

. The changeover costs associated with

the ghost item are zero and do not interfere in total so-

lution cost. Constraints (11) guarantee that each item

j is produced at maximum once in each period t. Con-

straints (12) conserve ﬂow and ensure that if there is

a changeover from an item i to any item k then there

is a changeover from that item k to an item j.

Constraints (10) and (12) alone might generate

subtours, that is disconnected subsequences, and thus

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

182

Table 1: Model data and variables.

Data Name Meaning

J number of soft-drinks (items)

L number of syrup ﬂavors

T number of periods

machine production time for one lot of item j;

machine changeover time from item i to j;

demand for item j in period t;

non-negative backorder cost for item j;

non-negative inventory cost for item j;

initial inventory for item j;

−

initial backorder for item j;

total time capacity of the machine in period t;

machine changeover cost from item i to j;

maximum number of tank setups in perod t;

total capacity of the tank, in liters of syrup;

minimum quantity of syrup l to guarantee homogeneity;

l j

quantity of syrup l necessary for the production of one lot of item j;

set of items that need syrup l;

Variable Name Meaning

inventory for item j at the end of period t;

−

backorders for item j at the end of period t;

production quantity of item j in period t;

ijt

changeover on machine (stage II) from item i to item j in period t.

auxiliary variable - might be used to indique the production order of item j in period t;

number of tanks to be prepared with syrup l in period t;

fraction of tank capacity used to produce syrup l in period t;

is equal to 1 if the tank is setup for syrup l in period t;

do not guarantee a proper sequence of the items. The

MTZ type subtour elimination constraints (13) avoid

this situation. With the inclusion of constraints (14)

the variable u

gives the order position in which item

j is produced. Finally constraints (15) deﬁne the vari-

ables’ domain.

Stage II (bottling) - Scheduling:

∑

j=1, j6=i

≥

∑

i=1,i6=k

ikt

,∀k,∀t (10)

∑

j=1, j6=i

ijt

≤ 1, ∀i,∀t (11)

∑

i=1,i6=k

ikt

∑

j=1, j6=k

kjt

,∀k,∀t (12)

≥ u

+ 1 − (J− 1)(1− z

ijt

);

∀i,∀j;i 6= j;∀t (13)

1 ≤ u

≤ J − 1∀ j, ∀t (14)

≥ 0, z

ijt

, y

= 0/1,

∈ Z

, n

≥ 0,

∀i, j;∀t;∀l. (15)

The complete description of the P1S1MTS model is

given by expressions (1)-(15). More details on the

P1S1MTS model can be obtained from Defalque et

al. (2011). Other formulations of the soft drink pro-

duction process can be found in Toledo et al (2007),

Ferreira et al (2009 and 2010).

3 THE MULTICOMMODITY

FLOW BASED MODEL

In the model P1S1MTS the constraints associated

with the scheduling decisions are formulated based on

the constraints proposed by Miller, Tucker and Zem-

lim (MTZ) to eliminate subtours, constraints (13).

These constraints are of polynomial order, thus al-

lowing their inclusion a priori. However, the MTZ

constraints produce a weak linear relaxation of the

associated formulation. Motivated by this fact, sev-

eral authors have proposed different approaches to

strengthen the ATSP mathematical formulation. On-

can et al (2009) reviews and compares several math-

ematical models for the ATSP. The review focuses on

STRATEGIES TO MODEL SCHEDULING DECISIONS TO PLAN THE SOFT DRINK PRODUCTION PROCESS

183

how the formulations compare to one another as re-

gard to the strengthen of the associated linear relax-

ation.

The main difference among the various formula-

tions for the ATSP relate to the constraints used to

eliminate subtours. The multi-commodity-ﬂow for-

mulation proposed by Claus (1984) has been used by

Clark et al (2011) to model the scheduling decisions

in the presence of non-triangular setups times. The

main idea of the proposed formulation is to ensure

that, in any period t, there is always a path from the

initial product s to any other product r in the periodt’s

sequence. In this work the multi-commodity-ﬂowfor-

mulation is also used to eliminate subtours. However,

the objectiveis to obtain a formulation that is stronger

than others from the literature, and therefore might

have a better computational behavior when solved by

a general purpose software.

To obtain the new formulation, it is necessary to

deﬁne a new index r = {1, ...,J}, and a new set of

variables. The continuous variables, m

rijt

, are used

to formulate subtour elimination constraints based on

the multi-commodity-ﬂow formulation for the ATSP.

The idea behind this formulation is that there are J

commodities available at node i

and a demand of one

unit of commodity j at node j. If m

rijt

= 1 then the

ﬂow of commodity r ﬂows from node i

no node r

through arc (i, j). In terms of the items sequence in

period t, it means that if product r is included in the

production sequence, then product j follows product

i in such sequence. The constraints (16)-(19) elimi-

nates disconnected subsequence of items.

Since only the items which are produced (i.e.

> 0) should be sequenced, constraints (16) and

(17) take place only when the machine is prepared

for item r. These constraints guarantee that if product

r is included in the sequence at least one other item

should be also included.

∑

j=1

−

∑

j=1

rji

∑

j=1, j6=r

jrt

, ∀r,∀t (16)

∑

j=1

rjrt

−

∑

j=1

rr jt

= −

∑

j=, j6=r

jrt

, ∀r,∀t (17)

Constraints (18) are the ﬂow conservation constraints,

for all but product r in node r. And constraints (19)

states that item j should follow item i in the sequence

that includes item r only if there is a changeover from

product i to product j.

∑

i=1

rijt

∑

i=1

rjit

, ∀r,∀ j; j 6= r,∀t (18)

rijt

≤ z

∀i, j, r;∀t (19)

The multi-commodity-ﬂow model for the sin-

gle stage single machine lot scheduling problem

(MM1S1M) is deﬁned by the objective function (1),

the Stage I constraints (2)-(5), the Stage II con-

straints (6)-(12), the subtour elimination constraints

(16)-(19), and the domain constraints (20).

≥ 0, m

rijt

≥ 0, z

ijt

, y

= 0/1, w

∈ Z

≥ 0, ∀i, j,r;∀t;∀l. (20)

4 CONCLUSIONS

In this paper a new formulation for the single stage,

single machine lotscheduling problem has been pro-

posed. This model might be useful for building

decision support systems for the production plan-

ning that arises in the soft drink production of small

and medium sized plants. The main feature of

the the model MM1S1M proposed in Section 3 is

that it includes multi-commodity-ﬂow constraints to

model the sequence at which the items should be

produced. In spite of the fact that the MM1S1M

model has a higher number of constraints to eliminate

subtours than the model P1S1MTS, the number of

these constraints is still polynomial. Moreover, these

constraints provide a stronger formulation since the

ATSP formulation using these type of constraints is

stronger than the MTZ formulation that is used in the

P1S1MTS model. A computational experiment using

data from the literature is being prepared to compare

the MM1S1M model with other models from the liter-

ature and to evaluate its computational behavior when

solved by general purpose software (e.g. Cplex (IBM,

2011), Gurobi (Gurobi, 2011)).

ACKNOWLEDGEMENTS

This research was partly supported by the Brazilian

research agencies Capes, CNPq and Fapesp. It also

receivedpartial support from the FP7-PEOPLE-2009-

IRES Project (no. 246881).

REFERENCES

Almada-Lobo, B., Klabjan, D., Oliveira,J. F., Caravilla, J.

F.. Single machine multiproduct capacitated lot sizing

with sequence-dependent setups. International Jour-

nal of Production Research. v.45, n.20, p.4873-4894,

2007.

Clark, A. R., Almada-Lobo, B. and Almeder, C.. Edito-

rial: Lot sizing and scheduling - industrial extensions

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

184

and research opportunities, special issue on lot siz-

ing and scheduling, International Journal of Produc-

tion Research 49(9): 2457-2461, 2011.

Clark, A. R., Mahdieh, M., Rangel, S.. Production Lot Siz-

ing and Scheduling with Non-Triangular Sequence-

Dependent Setup Times, Technical report, UWE -

University of West England, 201.

Claus, A. A new formulation for the travelling salesman

problem, SIAM Journal on Algebraic and Discrete

Methods 5: 21-5, 1984.

Defalque, C. M.; Ferreira, D.; Rangel, S. Usando o ATSP na

modelagem do problema integrado de produo de be-

bidas, TEMA - Tendˆencias em Matem´atica Aplicada,

to appear, 2011.

Ferreira, D. Morabito, R,, Rangel, S.. Solution approaches

for the soft drink integrated production lot sizing

and scheduling problem, European Journal of Oper-

ational Research, 196, 2, 2009, 697-706.

Ferreira, D. Morabito, R,, Rangel, S.. Relax and ﬁx heuris-

tics to solve one-stage one-machine lot-scheduling

models for small-scale soft drink plants, Computers

and Operations Research, 37, 4, (2010), 684-691.

Karimi, B.; Fatemi Ghomi, S. M. T.; Wilson, J. T.. The ca-

pacitated lot sizing problem: a review of models and

algorithms. Omega International Journal of Manage-

ment Science. v.31, p.365-378, 2003.

Fleischmann, B.; Meyr H. The General Lotsizing and

Scheduling Problem. OR Spektrum. v.19, p.11-21,

1997.

Gurobi Optimization Inc.. Gurobi optimizer version 4.5.0.

http://www.gurobi.com, 2011.

IBM, IBM-ILOG-CPLEX (2011), Mathematical Program-

ming Optimizers, http://www-01.ibm.com/software/

integration/ optimization/ cplex-optimizer/, 2011.

Oncan, T.; Altinel, K.; Laporte, G. A comparative analysis

of several asymetric traveling salesman problem for-

mulations. Computers and Operations Research, v.36,

p. 637-654, 2009.

Toledo, C. F. M. , Frana, P. M. , Morabito, R. , Kimms,

A.. Um Modelo de Otimizao para o Problema Inte-

grado de Dimensionamento de Lotes e Programao da

Produo em Fbrica de Refrigerantes, Pesquisa Opera-

cional, 27, 155-186, 2007.

Toso, E. A. V.,Morabito, R. and Clark, A. R.. Lot-sizing

and sequencing optimisation at an animal-feed plant,

Computers and Industrial Engineering 57: 813-821,

2009.

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