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OPTIMAL MULTI-PERIOD PLANNING FOR
MANUFACTURING PLANT - GENERAL PROBLEMS
T. Trzaskalik
Aostract. The paper aints at intro ducing a formalized description of
work in an indus tria 1 en terprise. In tha t description basic input and
OLl tpu t varia bles, rela tions and connexioIls wi th envirom~
t, a ll
i lilpo r tan t technological dependences and decision functions are taken
into consideration. That formalized description is then used for
l ay ing out optimal mul ti-period econoIJic plans.
Industrial enter prises differ, for instance in size, assor t ment of
production etc. Ne wan t to obta in a solution useful for varied ty pes of
enterprises a nd that is why we introduce the concept of a wanufa cturing
plant. IV e define a manufacturing plant as a set of features describing
really existing enterprises. tie consider the work of 0 manufa cturing
pln nt in t he fixed space of time in the f\,;.ture, divided into a detert;Jined nwnber of T basic periods.
Tne article considers a deterministic and stocIlastic app roach to these
pro ble ms . In the deterministic case a forr;la lized model of lllul ti-perio d
planning is shown. The me thod of solving the problem of r:lul ti-period
planning in the situation when the set of conditions is contradictory
is also considered. In the stochastic case the author considers models
with information received ex ante and models with information received
ex post. In this case the problem of constructing a deterliiinistic
initial plan is also formulated.
Keywords. Optimal systems, discrete time systems, dynamic programming,
manufacturing processes, manufacturing plant.
INTRODUCTION
The main aim of an industrial enterprise is
manufacturing profitable products, which
meet receiver's needs. In this paper we
describe the problem of choosing the optimal
schedule for an industrial enterprise in
the fixed space of time in the future
divided into a finite number of basic
periods and we name it the problem of
optimal multi-period planning.
Industrial enterprises differ, for instance
in size, a ssortment of production etc. We
want to obtain a solution of problema
formulated below, useful for varied types
of enterprises and that is why we introduce
the ooncept of a manufacturing plant. We
define a manufacturing plant as a set of
features describing really existing
enterprises.
There are many publications describing
mathematical aspects of planning for industrial enterprises. It is due to a similar
way of formulating the problems in question
481
that we mention works written by Zavelski
(1970), Pervozvanski(1975), Pawlowski
(1977), Drelichowska and Drelichowski
(1981). The solution of problems taken
into consideration in this work is based
on Bellman's (1957) principle of optimality and equations of optimality.
The aim of this paper is to formulate the
problem of multi-period planning for
manufacturing plant and presentation in
a synthetic form methods of its solution.
It seems that the described methods can
be applied for instance in engineering
and mining industries. The deterministic
model was practically utilized in the
method of seeking the optimal coal~ine
production plan which was worked out by
Trzaskalik(1977). A fu11 description
of the proposed methods can be found in
Trzaskalik's (1980) doctoral dissertation.
T. Trzaskalik
482
ASSUlW'TIONS
DETERMINISTIC APPROACH
1. The Vlork of the manufacturing plant is
considered in the fixed space of time in
the future divided into a determined number
of T basic periods. During the considered
space of time the manufacturing plant leads
activity aimed at turning
out products
of specified sorts.
Multi-period planning in the manufacturing plant is a deterministic problem,
if activity condition vectors are known
for all the basic periods before the
considered space of time. The gain in the
fixed space of time is taken as the
objective function. The optimal multiperiod plan is a sequence of permissible
decisions u~"
•• '~T
which maximizes
the value of the objective function.
2. Froduction technologies are used for
laying out products. Production technology
is a logical and usefully well-ordered set
of operations, as a result of which we
obtain a particular product. It is possible
to fix the set of different kinds of
i)roduc ts which can be manufac tured and the
set of production technologies which can be
used in the production process in the
fixed space of time.
Let Yt be the set of all possible state
variables in the period t and Ut (y t)the set of permissible decision vectors
for the period t, if at the beginning
of the period t the manufacturing plant
found itself in the possible state Y •
t
and et is the activity condition vector
3. rtelutions between the manufacturing
plan t and the environlllen t take pla ce. The
lllanufHcturing plant receives fron, the
environment new production technologies,
materials used in the production process
and manpower, and delivers manufactured
produc ts to the environment.
4. Froduction possibilities of the manufacturing plant are determined in each
period t (t = 1, ••• , ':1' '\ by ac ti vi ty condi tiOll
vector et' Components of that vector
concern possibilities of :
- using production technologies,
storage materials and products,
putting new production technologies
in to service,
bUJI ing mB terials,
utilization of labour resources,
- selling manufactured products,
- values of technological coefficients,
- prices of lull teriuls und produc ts,
- costs per unit of production.
5. State of the manufacturing plant at the
beginning of each period t is fixed by
state variable Y • Components of that
t
vector are
- stored
materials of pa rticular sorts,
stored
products of particular sorts,
- un accuoul a tod amount of products
man;;.fac tured by the use of separa te
technoloBies before the beginning of
the period t.
6. Decision for the period t is a vector
Ut' wh o se c omp o nents
are:
- lots of materials of particular sorts
which are to be bought,
- lots of products of particular sorts
which are to be sold,
- lots of products manufactured by the
use of particular production technologies.
for this period. The state Y
depends
t +1
E Ut(Y t} )
on Y t and taken decision Ut
that is
wher~t
is the transformation o£ the
state in t th period. The analytiC form
of this transformation omitted in this
paper includes technological dependences
of the manufacturing plant.
Let ,?:Ybe the objective function,!3t( y ,uJ
t
- gain for the period t, if at the
beginning of the period t the manufacturing plant found itself in the state Yt
and decision utE Ut (y t;
I'/as taken,
qt - bank rate for the period t (~
•
The form of the objective function is
, ~=
L
T
t=1
( ~
0).
t
) /)tYt'ut ! (2)
1 :q.
~=1
~-1
According to Bellman's equations of
optimality one obtains
,?t
x
utfUt(Y t}
+
for t
1t
r
m a
Yt
•••
,T t
..
1+ q t
~
. T+1
=0
a
:) t ( y t' Ut ": +
'1
t+,(y t+1
\l
(3)
.
?t Yt is the maximum value of gain for
the periods from t to T (with the natural
numbering of periods ) . Let u;(Y ) be the
t
function which assigns optimal decisions
for particular states Y • Assuming that
t
initial state Y, is given,
that
it follows
483
Optima l Multi-period Planning for Manufacturing Plant
Let U:(Y t ) be the function which assigns
on the basis of (6) and (7) optimal
decisions for particular states Y • For
Y';..1 ..
.QT(Y~')
Let us suppose now that before the beginning
of the period .1 ( 1 <.. !. <. T) it is known tha t
conditions of activity for the manufC~lrig
plant in the periods t, ••• ,t (t~
t<T) will
change as canpare:i wi tn the comi tions which
were taken into consideration in the prooedure of finding the optimal solution before
the beginning of the fixed space of time.
The size of these changes is specified by
the sequence of vectors ~e!,
••• , lIe t •
At the beginning of the period t the manufacturing plant finds itself in the state Y~
optimal in the original c~nditos.
Such
a sequence of decisions ut, ••• ,u T is looked
for to ma~e
the objective function
in the periods from ! to T.
t
be the set of all the possible
states in changed conditions of activity
( t ~ !) , Uf(y t) - the set of permissible
decisions in changed conditions (t ~),
6~
- gain of the manufacturing plant in the
"-
period t in changed conditions,{2t transformation in the period t in changed
condi tions ( ! { t ~ i).
If
(5 )
utilizstion of computations made in the
original conditions is possible. On the
ground of Bellman's equations of optimality
t i(Yi) = ui£U
m a",x
[~(Yt,U)
t(Yi)
+
1+ q i
+
~
............
l\
j!
The sequence of decisions u , ••• ,u repreT
1
sents the optimal solution, while the sequence of states Y1'~
••• 'Y;+1 represents
the optimal trajectory with the initial
sta te Yl '
yl>
Cl
l>
Ut • u;(y;)
( 4)
Let
t
••• ,t
~,
t •
t+1 (Y t +1)]
0
Ut •
u~(Y
Y!.+1
.
"'( I<
.at
4)
(8)
A)
~(l>
<l
)
Yl'u~
Yt + 1 - J2i Yi·Ui
and for t • t+1 ••••• T
U~+1
=
u~+1(Yt)
Y;+2 • Q t+1 (yi+1
.u~+1)
(9)
sequence of decisions ut,
'" •••• u 'T" represents
the optimal solution in-changed conditions
of activity. Comparison of vslues ~t(Y)
i
and
t (';1) indica tes whe ther the chan8es
exert positive or negative influence on
the work of the manufacturing plant.
When the set of the original conditions
is contradictory the problem of seeking
the optimal multi-period plan can be
solved, if we assume that changes of
constraiDB are possible. Let K(u •••• ,u )
T
1
be the sum of violating original constraints
in particular periods with regard to
given coefficients in the considered
multi-period plan. including decisions
u ' •••• u T• Let U(~)
be the set of decision
1
vector sequences (u1 ••••• ~)
fulfilling
the condition
It
U(K
).
~ (u 1 •••• ,UT) IK(U 1 , ••• ,UT) ~
min]
(10 )
The problem is solved in two stages. At
the first stage the set of sequences
U(K*) is calculated. At the second stage
an optimal plan u~ ••••• u; in accordance
with the condition
t3(u~
, ••• ,u;)
-
ID
a
x P.>(u\! •••• u T )
(u 1 ••••• ~)EU
~AI')
(11)
+
t .. ~,
••• , t-1
is chosen. The numerioal procedure is also
based on dynamic programming.
T. Trzaskalik
484
STOCEASTIC APPROACH
j,.ul ti-period planning in the manufac turing
plant is a stochastic problem if activity
condition vectors are not known for all
the basic periods before the fixed space
of time. The gain of the manufacturing plant
is a random function now and its values
depend on realizations of activity condition vectors in separate basic periods and
on made decisions. In stochastic models
the mean value of the gain is considered
as the objective function. The optimal
multi-period plan is now a strategy.
In the stochastic approach ex ante we
aSSWlle tha t before the beginning of the
considered space of time it is possible
to formulate for all the basic periods
distributions of activity condition
vectors. Besides, it is assumed that at
the beginning of the period t the activity
condi tion vec tor for the period t is known.
Let Lt be the set of all a ctivity condition
vectors and et - activity condition vector
for the period t
~t
~t:
t
= 1 , ••• ,T
•
Suppose that all sets L' t [ire finite. Let
p(e ) be the distribution in ;"t' Ut (Y t,9 )
t
t
represents the set of permissible decisions
for the period t if at the beginning of
this period the manufacturing plant found
itself in the state Yt and et is the known
activity condition vector for the period t.
The fonn of the transformation for the
period t is
We will discuss the problem of constructing
a deterministic initial plan now. Because
it is a de terminis.tic plan. it includes
a sequence of T decisions. The gain of
the manufacturing plant in the mean activity conditions is taken as the objective
function in the procedure of seeking
the optimal initial plan. On the basis of
this plan the construction of the optimal
schedule connected with the development
of production technology is possible.
which is particulary important if the
re81ization periods for individual
undertakings are longer then the basic
period. BiB differences between initial
decisions and obligatory decisions taken
after receiving exact predictions of
activity condition vectors are inadvisable.
Thus. the mean value of gain reduced by
the costs of changing initial decisions
is taken as the objective function in the
process of seeking optimal obliga tory
decisions. r"oreover. there is a possibility
of correcting the initial plan for periods
t+1 ., ••• T on the basis of realizing the
activity condition vector in the period t.
All the numerical procedures solving the
problems fonnulated above are founded on
Bellman's equations of optimality.
In the stochastic approach ex post we
3SSUlue tha t before the beginning of the
fixed space of time it is possible to
fonnulate for all the basic periods
distributions of activity condition vectors
P(e t ) ;t = 1, •••• T). Besides. it is assumed
that at the beginning of the period t the
activity condition vector for the period t
is unknown.
If a t the beginning of the period t the
manufacturing plant found itself in the
state Yt' a set of really penuissible
The form 01' the objective function is
~i
~'n
T
t
.
i=1
Hq.
)' E ~t(Y,u.e)
•
~-1
decisions which can be taken in the
existing conditions should be determined.
Some propositions of constructing this
set are given below.
The 1·st method. Let us assume that gt is
given (1 ~ t<' T.
According to Bellman's equations of
optimali ty one obtains (6 T.. ,,, r;) :
as follows:
El t+1 ''
?t'yt.e t : is the maximum value of gain for
the period t and mean value of gain for the
periods t+1, •••• T. Let u~ (y t'~
be the
function which assigns. according to (14 )
optimal decisions to particular pairs (yt.ei •
• is the optimal
The sequence of functions Ut
5
tra tegy
,
,
t,. 1 ••••• T .•
O{gt~
1 ) . Sets
--:
gt\" U" (,yt.g t \, . Uu (yt.g t ) are defined
JUt (
485
Optimal Multi-period Planning for Manufacturing Plant
Se-t Un or Uu is taken as the
really permissible decisions.
SJI t
of
The 2·nd method. The vector
(18 ,
e·t -
is defined •. Let us assume tha t in each
period vectors s •• defining maximum
deviations from mean value et are given.
The following sets are defined l
ii)et.s t ) -
~eiMtl
let -
e)~
sd
'19'
of realization of really permissible decisions
depends on conditions of activity for
the manufacturing plant that come into
being in the period t. I f informations
about the realization of activity condition
vector for the period t. inflowing
during this period make it clear that
a deoision taken at the beginning of
the period t from the set Ur(Y t ) is not
permissible. then it must be modified.
In the period t two sorts of decisions
are taken into consideration I really
permissible decision Ut' taken at the
beginning of the period t and realized
decision Ut' which is the result of
the accepted method of adaptation for
the situation arising in the period t.
Let us assume that in the accepted
method of adaptation the set of really
permissible decisions and the set of
realized decisions are the same for the
period t. and distributions p(U t ut.eJ.
representing probability of realization
Ut 1t at the beginning of the period t
decision Ut was taken and vector et
realized, are known for t - 1 ••••• T.
We obtain
' 21
Set
u(\
(25
or U u is taken as the set of
really permissible decision.
The 3·rd method. The functions
We can receive the solution of the
stochastic problem ex post on the grounds
of Bellman's equations of optimality :
wllere
are defined. Let us assume that gt is
given (1 ~ t ~T.
0 ~gt
~ 1). Set u&lY t )
is taken as the set of really permissible
decisions and is defined in the following
tel'Ull!! :
~
T+1 : O•
.h (Y ) is the maximum mean value of gain
'+'t t
in the periods from t to T. Let u~ ( Yt )
be the function which assigns according
to (26) really permissible decisions to
particular states y t€Y t' Sequence of
functions U;(Y ) for t - 1 ••••• T is
t
the optimal solution of the problem
ex post.
Let Ur(y ) be the set of really permissible
t
deoisiOJls obtained by the use of one of the
methods desoribed above. !he possibility
T. Trzaskalik
486
REFBRENCES
Bellman, R.E. (1957). Dynamic Programm1ug.
Princenton University Press, Princenton
New Yersey.
DrelichOlVska ,J. and Drelichowski ,1. (1981).
Concept of Deterministic and Dynamic
ProGramming 1: odels in a Union of
Enterprises. In A. Straszak and
A. ZioLkowsld Ed , System Analysis
and Its Applications to Technology
and Economics. Proceedings of the
1 st Finnish-Polish Symposium Zaborow,
November 25-28,1980 Part one.
fawlowski, Z. {1977 ' . Discriminating
Frediction und Economic Process Control.
Przeglgd Statysty czny, R.XXIV-2 ,
161 - 177. (In Polish'.
Fervozvanski, A.~1975)
1wthematical j,"odels
in Production Control. Nauka, t.i oscow.
(In Russian I.
'T rzaskalik, T. ',1977 ) . A L.ethod of Seeking
for the Optimal Coal-I.;ine Production
Plan Subj ec t to the Coal-ldne Development Assumptions. Przeglad sta~ycznY'
R. X:c IV-3, 385 - 397. (In Polish.
Trzaskalik, T.1980 ' . l~athem];ic
l.:odels
of IIJul ti-Stage Planning for Industrial
Enterprise. Doctoral Dissertation,
Academy of Economics, Katowice.
(In Polish \ .
Zavelski, 1.1. \1970 ' • .Optimal Flanning
for Indurtrial EnterPrise. Nauka,
h;oscow. In Russian •