Copyrig:ht © IF .\( : I .<lrgl' S(;ilt, \\'arsaw , Poland I ~Ht S\' 1t' 1l1 ' 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 •