There is great concern to improve the production and management process

11/12/2006 – In preparation – Restricted Information

A lot-scheduling model applied to a soft drink company Deisemara Ferreira

Reinaldo Morabito

Departamento de Engenharia de Produção – UFSCar 13565-905 - São Carlos, SP– Brasil

deise@dep.ufscar.br, morabito@power.ufscar.br Socorro Rangel

DCCE/IBILCE/UNESP

Rua Cristóvão Colombo, 2265; 15054-000 – S. J. Rio Preto, SP– Brasil socorro@ibilce.unesp.br

December, 2006

Abstract

In this paper we present a mixed integer model that integrates lot sizing and lot scheduling decisions for the production planning of a Brazilian soft drink company. Preliminary computational results for a fictitious small example are presented. The results indicate the difficulties of the proposed model and that special purpose techniques are needed to solve practical instances.

Key words: Mixed integer programming, Production Planning, Lot sizing and scheduling Models.

1. Introduction

Brazil is one of the largest producers of soft drinks in the world with more than 800 plants, and a consumer market of more than 12 millions of liters a year (ABIR, 2006). There is a high potential for market growth considering the country population and a propitious climate conditions for soft drink consumption. A typical Brazilian consumer drinks only 65 liters a year while Portuguese and Spanish consumers drink 84 and 109 liters respectively.

The market growth potential together with the increasing numbers of new products and the competition for market share has posed several challengers to the producers. There is great concern to improve the production and management process. The development and solution of mathematical programming models applied to real world problems has proved to be useful as decision support system. A mixed integer programming model for the production planning of a Brazilian soft drink company is discussed in this paper.

The company produces soft drink bottles in different flavors and sizes. A production line is made up of a conveyor belt and machines that wash the bottles, fill them with a combination of syrup and carbonic gas, and then seal, label and pack them. The production of a different flavor and/or size involves machine set up time and costs. The proposed model integrates the decision of lot sizing and scheduling of the soft drink production taking into account the capacity of the production lines, and the capacity and availability of syrup mixing tanks.

The paper is organized as follows. In section 2 we briefly describe the production system of the company studied. In section 3 we define the problem and develop the model. Preliminary results and analyses of a small computational study are given in section 4.

2. The soft drink production process

The soft drink Company studied is a medium-large size industry situated at the state of São Paulo-Brazil. The production process of this company is very similar to others companies in the soft drink sector, although the number of production lines, and syrup tanks can be very different.

The company produces drinks in several flavors and different bottle types (disposable and recycled). The production process is carried out in two stages: liquid (syrup) preparation (Stage I) and bottling (Stage II).

In Stage I, syrup preparation, the flavor ingredients are mixed together in a pre-specified quantity according to the number of lots of final product. The composed ingredient is then stored in a tank that will receive sugar (or any other type of sweetener) and water to become a syrup of a given flavor. The tanks should work with minimum liquid quantity to guarantee the syrup homogeneity. To properly mix the syrup ingredients the tank propeller must be completely covered.

The tanks are connected to the production lines through special links. The syrup receives carbonic gas and water to become the liquid that supplies the bottling machine.

The bottling machines, Stage II, are initially adjusted to produce soft drinks of a given flavor in a given bottle size. To produce soft drinks of another flavor and/or bottle size it is necessary to stop the production and make all necessary adjustments for the production of another bottle size and/or syrup flavor. A production line is made up of a conveyor belt and diverse machines arranged in series. The machines are used to sterilize the bottles, fill them with liquid, close, label, codify and pack the drinks. The production is carried out in the order described above.

If for any reason it is necessary to remove a bottle from the conveyor belt, it will be done at the end of the production process, before packaging. There is only one entry for the bottles in the production line. For this reason we will treat each production line as a single machine. One tank can supply syrup for several production lines simultaneously, but the production line can receive syrup from only one tank at a time. A schematic representation of the production process is illustrated in Figure 1 below.

Figure 1 – Production process of soft drinks

Stage I – Syrup preparatioon

Stage II - Bottling Production Line 1 Production Line2 ^... Production Linem

An important factor in the production planning is the synchrony between the two stages. In practice if the syrup is not ready (Stage I) the machine (Stage II) must wait for the syrup preparation. In the same way, the tank must wait the machine setup before releasing the liquid.

Figures 2 and 3 below show the production planning for five types of soft drinks (items 1-5) produced from four types of syrups (a-d). The rectangles represent the lot size and the spaces between them in Figure 2 the setup time in machine m and the setup time in tankm. In Figure 3, the black rectangles represent the waiting time.

Figure 2 – Non-synchronized schedule

Figure 3 – Synchronized schedule

Note in Figure 2 that there is no synchrony between the two stages. In the beginning of the schedule a set up time for syrup a is necessary in the tank (Stage I) while the production line is ready to produce item 1. In the first changeover (from syrup a to b, and item 1 to 2), although the set up time is the same in both stages, the machine (Stage II) did not wait for syrup b, necessary for the production of item 2, to be ready. The third changeover time in the machine is smaller than the changeover in the tank, but the machine schedule did not consider that before beginning the production of item 3 (which needs syrup c). To produce the lot of item 3 it is necessary a second lot of syrupc, and again, the schedule of the machine did not take that in account. The production of items 4 and 5 uses syrupd. However, the tank schedule did not wait the machine set up for item 5 before releasing the liquid. If the necessary waiting time is taken into account, a synchronized production plan is showed in Figure 3. Note that in this case the spare time capacity in both stages is smaller than the spared time capacity associated with the non-synchronized plan showed in Figure 2. In some cases, the non-synchronized schedule may become infeasible in practice.

Therefore, the synchrony between the two stages must be taken into account while the lot sizing and the production schedule is being planned (Ferreira et al, 2005).

3. Model development

The problem of the soft drink company described in the previous section can be stated as follows. Define the lot size and lot schedule of the products taking into account the synchrony between the two stages, the items demands and the capacity of the production line and syrup tanks, minimizing the overall production costs. The mixed integer mathematical programming model presented considers that there is one syrup tank dedicated to each machine (production line) and that there is an unlimited quantity of bottles, labels and water.

Let the following parameters define the problem size:

J = number of soft-drinks (items);

M = number of machines (and tanks);

L = number of syrup flavors;

T = number of macro-periods ;

N = number of micro-periods (i.e. number of setups in each macro-period);

and let (i,j,m,k,l,t,s) be the index set defined as:

} ..

1 {

} ..

1 {

} ..

1 { ,

} ..

1 {

} ..

1 { ,

M m

N s

L l k

T t

J j i

∈

∈

∈

∈

∈

.

We will also consider that the following sets and data are known:

Sets:

St = set of micro-periods in each macro-period t;

Pt = first micro-period of period t;

∆l = set of items that needs syrup l;

λj = set of machines that can produced item j;

αm = set of items that can be produced on machine m;

βm = set of syrup flavors that can be produced on tank m;

γml = set of items that can be produced on machine m and need syrup l.

The data and variables described below with superscript I relate to Stage I (syrup preparation) and with superscript II relate to Stage II (bottling).

Data

djt = demand for item j in period t;

hj = non-negative inventory cost for item j;

gj = non-negative backorder cost for item j;

s^Ikl = changeover cost from syrup k to l;

s^IIij = changeover cost from item i to j;

b^Ikl = changeover time from syrup k to l;

b^IIij = changeover time from item i to j;

a^IIj = production time for one lot of item j;

K^Im = total capacity of tank m, in liters of syrup;

II

Kmt = total time capacity in machine m in period t;

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

I⁺j0 = Initial inventory for item j;

Variables:

I⁺jt = inventory for item j at the end of period t;

I^-jt = backorder for item j at the end of period t;

x^IImjs = production quantity in machine m of item j in micro period s;

v^IIms = waiting time in machine m in micro period s;







=

otherwise 0

period micro in syrup for up set is tank the if

1 m l s

y_mls^I







=

otherwise 0

period micro in item for up set is machine the

if

1 m j s

y_mjs^II







=

otherwise 0

period micro in to syrup from in tank

chanveover is

there if

1 m k l s

z_miks^I







=

otherwise 0

period micro in to item from machine

in chanveover is

there if

1 m i j s

z_mijs^II

The mathematical formulation for the Two-Stage Multi-Machine Lot-Scheduling Problem (TSMM- LS) is then:

(1)

∑∑ ∑∑ ∑ ∑ ∑∑ ∑ ∑

= = ∈ ∈

= = ∈ ∈

= =

−

+ + ++ +

= ^M

m N

s i j

II mijs II ij M

m N

s k l

I mkls I kl j

T

jt j jt j

m m

m m

z s z

s I

g I h Min

1 1

1 1

J 1 t 1

) (

Z

α α β β

Subject to:

Stage I (syrup preparation) (2) _m^I _mls^I

j

II mjs

ijx K y

r

ml

∑

≤

∈γ

m=1...M ,l∈β_m,s=1,...,N; (3)

∑

∈ml

j

II mjs ijx r

γ

I mls I

m y

K

≥ 8 m=1...M ,l∈β_m,s=1,...,N;

(4)

∑ ∑

∈

∈ − ≥

m

m l

I mls l

I s

ml y

y

β β ^{( 1)}

, m=1,...,M;t =1,...,T;s=S_t −

{ }

P_t . (5) z_mkls^I ≥ y_mk^I _(s−1) +y_mls^I −1, m=1,...,M;k,l∈β_m;s=2,...,N. (6) ≥

∑

_{( 1)} + −1

∈ − I

mls j

II s mj I

mkls y y

z

γmk

, m=1,...,M;k,l∈β_m;t =2,...,T;s=P_t

(7) _ml^I

k I

mkl y

z

m

1 1≥

∑

∈β

,

(8)

∑ ∑

₁ ≤1

∈ _m ∈ _m

k l

I

zmkl β β

,

Stage II (bottling) (9) I⁺_j(t−1)+I⁻_jt+

∑∑

∈ _j∈ _t

m s S

II

xmjs λ

=I⁺_jt+I⁻_j(t−1)+d_jt, j=1,...,J, ;t=1,...,T (10) I⁺_jt ≥d_j(t+1), j=1,...,J, ;t =1,...,T

(11) _mt^II

S s

II ms

i j s S

II mijs II ij

j s S

II mjs II

j x b z v K

a

t

m m t

m t

≤ +

+

∑ ∑ ∑ ∑

∑ ∑

∈α ∈ ∈α ∈α ∈ ∈

, m =1...M, t =1...T;

(12)

∑ ∑ ∑ ∑

∈ ∈

∈ ∈

−

≥

m m

m m i j

II mijs II ij

k l

I mkls I kl II

ms b z b z

v

α α

β β

, m=1...M, s=1...N; (13) _{II mjs}^II

j II II mt

mjs y

a

x ≤ K , m=1,...,M ; j∈α_m;s=1,...,N;

(14)

∑

⁼¹

∈ _m j

mjsII

y

α

, m=1...M, s =1...N;

(15) z_mijs^II ≥ y_mi^II_(s−1)+y_mjs^II -1, m=1,...,M ;i,j∈α_m;s = 2,...,N;

(16) _mj^II

k II

mij y

z

m

1 1≥

∑

∈α

, m=1,...,M;j∈α_m; (17)

∑ ∑

∈ m ∈ m

i j

II

zmijs α α

≤1, m=1,...,M , s=1,...,N ;

(18) I⁺_jt, I⁻_jt, x_mjs^II , v_ms^II , z_mijs^II , z_mkls^I ≥0; y_mjs^II , y_mls^I =0/1;

; ,...,

1 J

j= l=1,...,L; t=1,...,T ; m=1,...,M ; i ; and j∈α_m;kandl∈β_m s∈S_t.

; , ,...,

1 Ml _m

m= ∈β

N s

M

m=1,..., , =1,...,

The objective function (1) contains items storage and backorder costs, and machine and tank changeover costs. In Stage I, the demand for syrup l is computed in terms of the production variables. That is the demand for syrup l in each tank m in each micro period s is given by

∑

∈ _ml j

II mjs ljx r

γ

. The constraints (2), that are similar to constrains (13) in Stage II, together with constraints (3) guarantees that if the tank m is setup for syrup l in micro period s (y_mls^I =1) there will be production of syrup l (between the minimum quantity necessary for syrup homogeneity and the tank capacity). In the company studied, there is a syrup flavor, p, with high demand that is continuously produced. If that is taken into account, constraints (2) and (3) are not generated for

p

l= . The constraints (4) forces the idle micro periods to happen at the end of the associated macro period. Constraints (5), similar to constraints (14) in Stage II, controls the syrup changeover. Note that the tank setup does not hold from one macro period to another if the setup variable in the last micro-period is zero. Therefore, constraints (6) are needed to count the changeover between macro periods, given that the setup variables in stage II indicate which syrup was prepared in the last non- idle micro period of each period. Constraints (7) count the first changeover of each tank.

Constraints (8), similar to constraints (16) in Stage II, guarantee that there is at most one changeover in each tank m in each micro period s.

In Stage II, constraints (9) represent the flow conservation constraints for each item in each time period. Since the production variable is defined for each micro period, to obtain the total production of item j in a given period t it is necessary to add its production over all machines where it can be produced (m∈λ_j) and micro periods (s∈S_t) of period t. In the company studied, the inventory level in a given period must be enough to cover the demand in the next period. To have a fair comparison, constraints (10) were included in the model. Constraints (11) represent the machine capacity in each time period. Note here the inclusion of the waiting variable, v_ms^II , to ensure that the lot schedule will be feasible. The waiting time in machine m in each micro period s is the difference between the changeover time in tank m (

∑ ∑

∈ _m∈ _m

k l

I mkls I klz b

β β

) and the changeover time in

machine m (

∑ ∑

∈ _m ∈ _m

i j

II mijs II ij z b

α α

), constraints (12). Constraints (13) guarantees that there is production of item j only if the associated setup variable is set to one, and constraints (14) and (15) counts the changeover in each machine m in each micro period s. Constraints (16) refers to a single mode production in each micro period s. Note that production may not occur although the machine is always ready to produce an item. Constraints (17) guarantee that there is at most one changeover for each machine m in each micro period s.

Finally constraints (18) define the nonnegativity and integrality restrictions. Note that the changeover variables z_mkls^I and z^II_mijsare continuous. Constraints (4), (5), (14), (16), and the optimization sense (minimization) guarantee that these variables will take only 0 or 1 values.

Looking at constraints (2) to (17) it is possible to identify elements of several mathematical models given in the literature: ‘Multi-item capacitated lot-sizing with set up costs and times’

(Constantino (1996), Barany, van Roy e Wolsey (1984)); ‘Production Planning with set-up and change-over costs’ (Wolsey (1997), Magnanti e Vachani (1990)), ‘The General Lot Sizing and Scheduling Problem’ (Fleishmann e Meyr (1997), Drexl e Kimms (1997), Meyr( 2002)). Therefore, solution methods for these models might be also useful for the solution of model TSMM-LS.

Other mixed integer programming models for the production planning of soft drinks have been proposed in the literature (e.g. Rangel e Ferreira, 2003; Clark, 2001; Gutiérrez, e Pizzolato, 2004, Toledo et al., 2006). The model given above has certain similarities with the mathematical model proposed in Toledo et al. (2006). The TSMM-LS model is also a two level, multi-machine model, however the tanks are dedicated to the machines, the micro period size is flexible (depends

on the lot size) and the total number of micro periods in both stages is the same. The setup costs are included in the changeover costs.

4 – Preliminary Computational Study

The TSMM-LS model presented in section 3 was defined considering the production of J items, in M machine, using L flavors, in T periods divided in

∑

=

= ^T

t

St

N

1

micro periods of variable size. These parameters might generate large-scale instances that proved to be very hard to solve by commercial optimization software (Ferreira et al., 2006).

To illustrate the difficulties of the model, a small, fictitious example was generated (see details in Table 1). Three different combinations between the machine and tank capacities were studied. In the first instance of the example, Ex1, the machine and tank capacities are higher than the necessary capacity to meet the demand. In the second instance, Ex2, the machine capacity is the same as in Ex1 but the tank capacity is reduced by half, so that there is a bottleneck in Stage I. The third instance, Ex3, has a bottleneck in Stage II, the tank capacity is the same as in Ex1 and the machine capacity is reduced by half.

The model was coded using the Xpress-Mosel (v. 1.6.1) modeling language and solved by Xpress-MP(v. 16.10.02) (Dash optimization, 2004) and Cplex 10.0 (ILOG, 2006). Both solvers were applied with its default settings. The runs were executed using machine Intel Pentium 4, 3.2 GHz, 2.0 GB RAM.

Table 1 – Example Dimension

J M L T N

Total Number Variables

Total Number of

Binary Variables

Total Number of Constraints

4 2 2 2 12 760 144 776

Table 2 shows the results of the Xpress-MP and Cplex 10.0 solution process in terms of CPU time in seconds, total number of nodes to prove optimality and total number of cutting planes applied. It is interesting to note that, although the three instances are very small compared to practical instances, it took more than one minute for both systems to solve EX1 and Ex3. The system Xpress-MP had a better performance than Cplex 10.0 both in terms of CPU time and total number of nodes generated. The number of cutting planes applied by Cplex 10.0 is higher than Xpress-MP. The instance Ex2 was easier to solve by Xpress-MP than the other two. Instance Ex3 was the harder one for both systems.

Table 2 – Computational Results – Optimal Solution Found

CPU(seconds) Total Nodes Total Cutting Planes Name

Optimal

value Xpress_MP Cplex 10.0 Xpress_MP Cplex 10.0 Xpress_MP Cplex 10.0 Ex1 21 67 97.59 9641 33153 25 100

Ex2 12244.3 27 434.25 2619 117819 84 72

Ex3 125318.95 257 735.81 72622 261402 33 72

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