MODELLING A PRODUCTION SYSTEM AS A COMPETITIVE …lsc.fie.umich.mx/~juan/PS/ModProdSist.pdf ·...

AMSE (MS’05Morelia, México)

MODELLING A PRODUCTION SYSTEM AS A COMPETITIVE

STRATEGY

González Santoyo F.*, Flores Romero B.*, Gil la Fuente A.M.**,,

Flores Juan J.***

Universidad Michoacana de San Nicolás de Hidalgo. México.

*Facultad de Contaduría y Ciencias Administrativas

*** Facultad de Ingeniería Eléctrica

*Universidad de Barcelona. España.

{fsantoyo, juanf}@zeus.umich.mx, [email protected],

amgil@ub,edu

ABSTRACT

In this paper, an efficient heuristic algorithm to solve the

problem of Hierarchical Production Planning is proposed.

Mixed Integer-Linear Programming (MILP) was used for the

solution, using Benders ´ decomposition techniques. We

present an application example involving a sawmill and the

corresponding results.

KEYWORDS: Hierarchical production, MILP, Langrangean, sub-

gradient.

INTRODUCTION

Mexican companies are immerse in world wide range

competition. González S. F. (1995, 2004,2005) establishes that

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in face of a global market, a company will struggle to remain in

the market and distribute its products if it does not keep up

with modern techniques. This implies presenting a competitive

infrastructure in both human and material resources. It will also

need a control system that makes it highly efficient, according

to our current reality.

Under these circumstances, an optimal production planning and

scheduling is crucial to take a competitive advantage over other

companies in the market, considering its products and/or

services are certified and compete in price and quality.

The production planning problem addressed in this paper is a

NP-Complete problem. The goal is to plan and schedule the

production in time. According to González S.F. et al (1996)

production planning has to be performed keeping in mind

several fundamental objectives. Planning determines the

requirements, the point in time when these occur, and the

order in which demands have to be satisfied in the planning

horizon. Scheduling determines how the available production

sources act in the immediate period of time, locating the

individual products provided to consumers involving a minimum

production cost.

The problem aims to reach a production level for the period of

analysis, implying high efficiency levels, high utility, and

lowering the production and operation costs. It also seeks to

guarantee an efficient way for the company to stay in the

global markets.

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This paper states the problem of sawmills, we develop a

mathematical model, and present a solution algorithm. We

present the results of a run of the algorithm, providing a

production plan. Finally, we draw conclusions and cite the

bibliographic references.

PROBLEM STATEMENT

One of the economically important activities in the state of

Michoacan, Mexico is the industrialization of the wood

resources. From the mechanical point of view, the first stage

in this process, is the sawmills. The stages that compose this

process González Santoyo F. (1987) are: reception and

classification of raw materials, conveyor cart, main saw, edger,

multiple pitcher, sorting table, and storage. The representative

variables in this problem are: raw materials, sawing time,

available processing time, labor, spendables, electricity,

maintenance, and sawyer products market.

The main problem in this kind of processes is the raw

material, logs come in different diameters, lengths, and

uniformities, so they need to be sorted according to these

variables. This allows us to optimize the process, selecting

optimal cut designs in the main saw and in subsequent stages.

The general problem for the state of Michoacan, Mexico, is

associated to the length of the logs, which come in 8, 10, 12,

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14, 16, 18, 20, and 22 feet, with diameters from 12 to 30

inches. Commercial cuts are ½, ¾, 1 1/2, 3, and 3 1/2

inches thick, and 8, 10, and 12 inches wide. This scheme

allows us to use Mixed Integer-Linear Programming (MILP).

The result will produce 8 kinds with 1, 2, and 4 families; this

yields 160 different products of sawed wood for the company.

The solution of the problem is produced using MILP.

Production of different items is aggregated in families, and the

families in kinds of products. This aggregation structure

González S.F. (1995) lies in the line of research known as

hierarchical production. The original production is divided in a

hierarchy of sub-problems; in making the production plans we

consider that individual parts and final products can be

aggregated (grouped together). The aggregation criterion

follows Bitran (1981) using the description provided by

González S.F. (1995): Items, are final products requested by

the market in a given unit of time, Kinds of Products, are

groups of final products with similar production costs, which

exhibit the same demand model and production range (the

kinds will be characterized according to log lengths), Families

of Products, are represented by a set of items in a kind,

sharing common characteristics (for this case, the cuts width).

The general MILP problem is characterized as a large scale

problem, which will be solved heuristically, based in Bender ´s

decomposition theory González S.F. et al (1996).

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MATHEMATICAL MODEL OF THE PROBLEM

A Mathematical model for the problem can be stated as:

PS Z ps Min .∑

tC t Ot∑

ihit I it ∑

j∑

tS jt X jt

∑j∈T i

FP−P=0,∀ i,t

∑ι

K i P it−Ot≤r t ,∀ t,

∑j∈T i

FI jt−I it=0,∀ i,t

FP jtFI j,t−1−FI jt=d jt ,∀ j,tFP jt−m jt X jt≤0,∀ j,tOt ,P tι ,I it ,FP jt ,FI jt≥0,∀ i,j,t

X jt∈{0,1 } ,∀ j,t

Where t iterates over time periods, Ct is the hourly cost of

overtime in period t, hit is the stock cost for kind i, in period t,

Sjt is the preparation cost for family j in period t, dit (djt) is the

demand for kind I, family j during period t, Ki is the unit

production time needed for kind i, T(i) is the set of families

that belong to kind i, mjt is the production upper bound for

family j in period t, rt is the regular production time in period t.

The model decision variables are: C= overtime in hours for

production during period t, Iit (FIjt) = stock of kind i, family j,

during period t, Pit (FPjt) = production of kind i, family j during

period t, Xjt = Boolean variable that indicate update of family j

in period t.

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The solution process is based in Benders ´ decomposition

techniques. Some variables in PS are classified as complex;

these are: (Xjt , FIjt , FPjt). These variables allow us to

express a problem at the level of kind, or at the level of

family. For this partition of the variables, Benders ´ su-

problem is:

PSUBT

Z PSUB t =Min .C t Ot∑i

hit I it

P= ∑j∈T i

FPjt ,∀ i

∑i

K i P it−Ot≤r t

I it= ∑j∈T i

FI it ,∀ i

Ot , I it , Pit≥0,∀ i

An upper bound for the optimal value is provided by the sub-

problem, where the constant ∑j∑

tS jt X jt , is added to

∑t

Z PSUB t , so this is a PS constraint. The sub-problem

has a feasible primal solution, found by inspection by:

Pit= ∑j∈T i

FP jt ,∀ i , t

I= ∑j∈T i

FI jt ,∀ i , t

Ot=Max . {0,∑i

K i ∑j∈T i

FP jt −r t},∀ t

Let Uit, Vt, Wit, be the dual variables corresponding to the

constraints of the primal, formulated in PSUBt. PSUBt´ s dual

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problem can be expressed as:

DPSUBt

Z DPSUB t =Max .∑i

U it ∑j∈T i

FP jt V t r t∑i

W it ∑j∈T i

FI it

U itK i V t≤0,∀ i−V t≤C t

W it≤hit ,∀ iV t≤0U it ,W it , irrestrictas ,∀ i

The solution to the dual problem, provided by inspection, is:

W it=hit ,∀ i , t

−C t si Ot0,∀ t0 si Ot=0,∀ t

V t=¿ {¿ }¿{}U it=−K i V t ,∀ i , t

DPSUB, has multiple solutions, given that PSUB has a

degenerate solution. An alternative is to make Uit y Wit zero, if∑

j∈T i FP jt y ∑

j∈T i FI jt are zero, respectively.

For the case where Ot =0, w can get Benders ´ cutZ t≥∑

ihit ∑

j∈T i FI jt , where Zt, is an auxiliary variable in

Benders ´ master problem.

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Cuts of this kind are favored and included in the master

problem. Let U*it y V*

t , be the dual solutions corresponding to

the case where Ot > 0. We can assume that PSUB has been

solved for a sequence of given variables and that the case Ot

> 0 may occur at least once for periods T ¿⊆{1, . . . ,T } .

The master problem is described as:

PM 1Z PM=Min .∑

j∑

tS jt X jt∑

tzt

zt≥∑i

U it¿ ∑

j∈T i FP jt V t

¿r t∑i

hit ∑j∈T i

FI jt , t∈T ¿

z≥∑i

hit ∑j∈T i

FI jt ,∀ t

FP jtFI j , t−1−FI jt=d jt ,∀ j , tFP jt−m jt X jt≤0,∀ j , tOt , Pit , FP jt , FI jt , I it≥0,∀ i , j , t

X jt∈{0,1 }

By variable substitution, we obtain the following reformulated

master problem.

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PM 2Z PM=Min .∑

j∑

tS jt X jt∑

i∑

thit ∑

j∈T i FI jt qt

qt≥∑i

U it¿ ∑

j∈T i FP jt V t

¿r t , t∈T ¿

qt≥0,∀ tFP jtFI j , t−1−FI jt=d jt ,∀ j , tFP jt−m jt X jt≤0,∀ j , tOt , Pit , FP jt , FI jt , I it≥0,∀ i , j , t

X jt∈{0,1 } ,∀ j , t

The master problem is a relaxed version of PS, so ZPM is a

lower bound for ZPS. González S.F. (1995) suggests using

lagrangean relaxation as a good strategy to solve PM2

problems. So, relaxing Benders ´ cuts with λ t multipliers, we

can structure the objective function for the case as:

Min .∑j∑

tS jt X jt∑

i∑

thit ∑

j∈T i FI jt ∑

tqt∑

t∈T ¿

I t gt FP , FI −qt

Dónde: gFP , FI =∑i

U it¿ ∑

j∈T i FP jt V t

¿r t

Each qt is a continuous non-negative variable that does not

appear in the rest of the constraints. Taking λ t ≤ 1 and

t∈T*, qt´ s coefficient becomes non-negative. Then qt´ s

optimal value may be zero as long as the objective function is

minimized.

PM2 ´s lagrangean relaxation is stated as:

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LRPMZ LRPM λ =Min .∑

j∑

tS jt X jt∑

i∑

thit ∑

j∈T i FI jt ∑

j∈T i λ t gt FP , FI

FP jtFI j , t−1−FI jt=d jt ,∀ j , tFP jt−m jt X jt≤0,∀ j , tOt , Pit , FP jt , FI jt , I it≥0,∀ i , j , t

X jt∈{0,1 } ,∀ j , t

We relaxed PM because LRPM, the resulting problem is

separated in families in a set of problems of convenient size

without a capacity constraint. This kind of problem, Wagner,

Whitin (1958), can be solved efficiently using dynamic

programming. Using the lagrangean duality for integer

programming, we find that ZLRMP is a lower bound in ZPM. The

largest available lower bound is obtained from the solution of:

DZ D=Max . Z LRPM λ

0≤λ ≤1, t∈T ¿

which is the lagrangean dual with respect to Benders’ relaxed

cuts. This name is used because ={ λ λ tt} plays a similar roll

on LRPM with lagrangean multipliers used in the continuous

problem. The dual objective function ZLRPM( ) is continuous,λ

concave, piecewise linear, and sub-differentiable.

A standard procedure to solve the dual problem is the sub-

gradient optimization algorithm, which generates dual solutions

according to the following rule:

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λ tl1=λ t

lθγ tl , t∈T ¿ , l=0,1, . . .

Where, for a particular value of ,λ

γ ={

γ t} is a sub-gradient of

ZLRPM ( ) and λ Θ l is the step.

DESCIPTION OF THE ALGORITHM

1. Assume that T* Benders’ cuts are generated and that

(PM2) is solved using lagrangean relaxation and sub-

gradient optimization.

2. The vector λ t is updated a predetermined number of

times according to equation.

λl1=λ t

lθ 1γ tl , t∈T ¿ , l=0,1, . . .

For a particular value of ,λ

γ ={

γ t} is a sub-

gradient of Zlrpm ( ) and λ Θ 1 is the step. (LRPM) is

solved to provide a lower bound for Zps.

3. The values of Xjt, FIjt, FPjt, obtained for the solution of

LRPM, are used to feed PSUB, while the set of

updated variables is improved for the heuristic

exchange.

4. The optimal dual variables towards PSUB are based in

new Benders’ cuts generated using the above

procedure. Nevertheless, it may happen that a cut

already included in PM2, be generated again as a

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candidate. For this case, we know that a cut that has

already been generated cannot be included again in

PM2.

5. An upper bound on Zps is given by the value obtained

through the heuristic exchange for the solution of PSUB

and depends on the lower bound.

In the following iteration, the sub-gradient procedure for

PM2 continues starting from the last solution to LRPM. The

procedure is initialized when the initial values for {Xjt,} are

generated. The algorithm works until a determined number

of Benders’ sub-problems has been explored, or the

variation between the upper and lower bounds is small

enough.

The dual variables Uit, Wit are necessary as input elements

for the objective function of LRPM. These variables can

be seen as production and storage costs for the stock

corresponding to the families that belong to a given kind of

product.

STUDY CASE

Problem including 1,2, and 4 families for the 8 kinds in the

problem were processed. We included all 160 different

products (commercial cuts) that can be obtained in the

sawmill with the supply mentioned above. Modelling a

family includes 5 commercial cuts, so to model 4 families,

we need 20 different cuts for each kind we included.

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On each production plan, the commercial cut is specified,

and its place with respect to the family and kind where it

is located. The cut also contributes to the objective

function, minimizing cost. To satisfying demand, we specify

its production level, as well as the stock level in the

planning horizon (12 months). This process is a dynamic

analysis, allowing the manager to define clearly the way the

production plant will operate. We will present as an

example, the results for the production plan for kind 1,

characterized by 8 ´ long logs and for 4 families (different

commercial cuts made with the 8 ´ long logs and different

diameters, according to the classification of commercial

cuts, cited in the following tables. For convenience, the

plan information is presented in P.U. (per unit).

Cuts Cuts Cuts Cuts

½ ´ ´ x 6

´ ´ x 8

½ ´ ´ x 8

´ ´ x 8

½ ´ ´ x 10

´ ´ x 8

½ ´ ´ x 12

´ ´ x 8

¾ ´ ´ x 6

´ ´ x 8

¾ ´ ´ x 8

´ ´ x 8

¾ ´ ´ x 10

´ ´ x 8

¾ ´ ´ x 12

´ ´ x 8

11/2 ´ ´ x 6

´ ´ x 8

11/2 ´ ´ x 8

´ ´ x 8

11/2 ´ ´ x 10

´ ´ x 8

11/2 ´ ´ x 12

´ ´ x 8

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2 ´ ´ x 6 ´ ´ x 8 2 ´ ´ x 8 ´ ´ x 8 2 ´ ´ x 10

´ ´ x 8

2 ´ ´ x 12

´ ´ x 8

31/2 ´ ´ x 6

´ ´ x 8

31/2 ´ ´ x 8

´ ´ x 8

31/2 ´ ´ x 10

´ ´ x 8

31/2 ´ ´ x 12

´ ´ x 8

RESULTS

Production Plan for Kind 1

KIND I

Family 1 Family 2 Family 3 Family 4

T D P I D P I D P I D P I

1 40 125 0 40 90 0 65 65 0 70 120 0

2 60 0 85 50 0 5012

0120 0 50 0 50

3 25 0 25 50 90 012

0145 0

10

0190 0

4 65 65 0 40 0 40 25 0 25 90 0 90

512

0120 0 70 120 0 60 120 0 80 120 0

612

0145 0 50 0 50 60 0 60 40 0 40

7 25 0 2510

0190 0 85 125 0 60 85 0

8 60 120 0 90 0 90 40 0 40 25 0 25

9 60 0 60 80 120 0 40 90 0 65 65 0

10 85 165 0 40 0 40 50 0 5012

0120 0

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11 40 0 80 60 85 0 50 90 012

0145 0

12 40 0 40 25 0 25 40 0 40 25 0 25

T= Time (months), D = Demand, P = Production Volume

(pieces), I = Stock Level.

The characterization of the kind of product, the family it

belongs to, and the classification for Plan 1, characterized by 8

´ long logs in different widths and thicknesses is presented in

the following table.

FAMILIES PRODUCT t(min) COST ($)

1 (3/4” x 6” x 8”)

2 (1/2” x 8” x 8”)

3 (2” x 10” x 8”)

4(3/4” x 12” x

8”)

.826 3,801.00

These results allow the manager to make efficient and timely

decisions associated to the plan’s operation cost and kinds of

commercial cuts requested by the market. Among the

decisions made by the manager, we find setting an adequate

stock level, satisfying the market’s requirements under

contingency conditions, with the least amount of resources

invested in it.

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CONCLUSIONS

From the production plan for kind 1, we conclude that for

large scale problems, the proposed algorithm is computationally

efficient. In 4 iterations we obtained the solution.

The algorithm presents a practical flexibility for decision making

in the company.

The algorithm was easily programmed in a PC and presents a

greater flexibility than the commercial software available in the

market for solving this kind of problems.

BIBLIOGRAPHY

Bitran G.R., Haas E.A., Hax A.C. (1981). Hierarchical

Production Planning: a single stage system. Operations

Res.

González S.F. (1987). Modelado Matemático de un

Aserradero Estándar en el Estado de Michoacán. Bol. 10,

CIC- UMSNH. México.

González S.F. (1995). Planeación de la Producción

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Jerárquica empleando Técnicas de Descomposición. PhD

Tesis. Universidad Nacional Autónoma de México.

González S.F., Pérez Morelos G. (1996). Planeación de la

Producción en la Pequeña Empresa Mexicana. Economía y

Sociedad. Facultad de Economía UMSNH-México.

González S.F., Brunet I.I, Flores B, Chagolla M. (2004).

Diseño de Empresas de Orden Mundial. URV España)-

UMSNH (México), FeGoSa-Ingeniería Administrativa (México).

González S.F., Terceño G.A., Flores B, Diaz R. (2005).

Decisiones Empresariales en la Incertidumbre. URV

España)-UMSNH (México), FeGoSa-Ingeniería Administrativa

(México).

Wagner M, Harvey and Whithin M. (1958). Dynamic

Version of the Economic Lot Size Model. Management

Science.

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