Discrete Industrial Process

SCHEDULING A CONflNUOUS/DISClffiTE INDUSTRIAL PROCESS

A. Ashimov, K.S.

Sa~ngaliev

V.I. Lenin Kazakh Polytechnical Institute, Alma-Ata, USSR

ABSTRACT

continuous/discrete industrial process marketin(j.

Self-consistent scheduling of an industrial system is discussed. The system cons~ ists of a continuous, a discrete, and a marketing subsystems. The second one comprises aggregates operating in parallel. Models are proposed for the continuous/discrete process and the multi-product marketiDg; estimates for the subsystem schedules have been worked out.

1.'"

2. PROBLEM m'A'l'ED

The system under discussion may be described as follows. The discrete subsystem consists of p similar aggregates operatiDg in parallel and is characterised by the .JK K parameter CA (ci is the extent of conversion of the semi-finished product k to the commodity k) and the technological delay Some of the results have been tested on a Tt. The latter defines the unit of schenon-ferrous metallurgy plant. The methods duling time. Each of the discrete subsysproposed have been found effective. tem aggregates produces batches of commodities of only one type at the end of each 1. INTRODUCTION of the schedulin(j intervals, the batch is a unit in which amounts of commodities, A widespread industrial chemical metallursemi-finished products, and raw materials gy system is a sequence of the following are measured, in other words, these amounts subsystems: raw materials resource accuare measured in units of efficience of a mulation, a continuous/discrete process discrete subsystem aggregate per unit time. (including a continuous subsystem, a stoAny batch of the finished product is prorage, and a discrete subsystem), and maI'duced once for a sch eduling interval; losK ketine;. ses due to readjustiU[, the aggregates, depend on the finished product type, k, the Mathematical models and algorithms have readjustment time is negligible compared been proposed for the raw materials accuwith the scheduling interval; no standsmulation~,2], continuouJ3,4J or discretJ5,6] till is allowed. processes, and one-product marketinJ7,~ .

t

The continuous Subsystem continuously transforms material fluxes. Its parameters are as follows: the conversion factors o(K and erI (they express the extent of conver-

This paper deals with multi-aggre gate processes, multi-product marketing, and an algorithm to estimate scheduling of a

J(

60

sion of the overall raw material amount to the overall amount of the semi-finished product k, and the same for the component f of the overall raw material and the component f of the semi-finished product k), the technological delay time 'l:H that is a multiple of the scheduling interval, and the efficience 6 K • 'l'he continuous subsystem meets operating requirements of the discrete one; it feeds the storage so that the semi-finished product level therein be above one batch equal to the efficience of a discrete subsystem aggregate.

(2) t: = 1,.t, _0,7' j

o~ d!ft'<

$

61(

K= "f,2, - _/?Z-j

en

t=~~.,~

(4)

I(

JC

Xjtt'I)~X.it~O K=f,,,,)m.;j=~~ . ,I'> t=t;"; o,'7' (5)

where cfr:-.... is the vector plan of the continuous s ubsystem for the first t scheduling inter.... f $ A< AJ:) vals, ,1d'f: = (Alt.l A'yt') . _.) A~t) _.. ) A~t! is the vector plant of the continuous subsystem for the t-th scheduling interva l; some components of the vector are zero if the respective semi-finished species is not produced during the t-th interval. °

The continuous and discrete subsystems are connected through the non-interchangeable storages of the capacity (J~ •

0

0

o·

°

Quality ratings for semi-finished products Quality rating of the commodities is achieved by satisfying quality ratings of semi-finished products characteristics of the subsystems, and the quality ratings, are constants within the scheduling period T. The latter is assumed finite and equal to the period for which the plant operates in accordance with the account parameters given beforehand. No delay is allowed in shipping finished products to consumers. The plant makes its finished products in the following way. The starting raw material (its amount is t;c ) enters the stock according to an integral schedule. The K raw material (its amount is now X/t, viz., the amount of the raw material j consumed by producing the semi-finished product k for the first t scheduling intervals) en~ ters the continuous subsystem. At non-zero amounts of the raw material in the stock, the raw material dynamics may be written down in the following form. ""

I(

O(&jt f~·O :~X;·t ~

OJ j= 1,2,

/l-; t:=

~.e,

0 '1'.1

are K I( ~ K /C g;i~ ~lX.t hJBli7°t ~ elJ'-~ J<

(6) JC

-I(

where 1;1 and 2.1 are lower and upper admissible values of the factor f, ~II- are contents of the substance f. The semi-finished product k leaves the continuous SUbsystem for an intermediate storage described by the following stock dynamics, O~)f:I-'yolC - ?r./~8K K=-f,,!, .._,we,· t-=t;.2.. .. :1' (7)

J:

where is the starting stock of the semi-product k, is the semi-product k amount consumed by the discrete subsystem for the first t scheduling intervals

zr:

The semi-product is then fed to the discrete subsystem that produced the finished IC product k (its amount is i!d ) according to

(1)

The continuous subsystem transforms the raw material and produces the following amount of the semi-finished product k,

61

K

(11)

It

~

K

tervals, with at =L.. fl.f't" ; (iii) const
( 13)

~/o=O

j:

where is the overal~ consumer demand for the commodity k, Zd is the cor!ID1 odity k amount produced by the i-th discrete subsystem ue;gregate for the first t scheduling intervals.

where Gt is time slack of the station,j3K labour consumption factor for delivery, of the k-th product; (iv) constraints on the product stock O~

The schedulin;;; problem (Problem I) may be stated as follows: find a L1aximally profitable schedule of the continuous/discrete process (x;t] and [i! i;], ",p

KK

'"

J(

Itt-!

le

J(

P==P,ffi:;~j2e~iT-L~xirJ-1. ~Sdi!ir)t 11:=1 Ft J:t le, J £'1

P I.. 2
K.

Llpt ~ 01<

K= (,.t.,

0 00 ,

In-

j t:~.t, 00 7)

(11::l)

ptf~K

where 0;< is the stock size for the k-th product. A finished product is assumed realised i f the sum payable by the consumer for the product delivered to him has been received by the supplier's bank, -K _ lI;t ) If Zp ~ 'r-t ; ",LL i p 0 )i/ (;.f'.>r-Cj

I<

(14)

satisfyinr; (1)-(13), where eKand Sjare the unit prices of the commodity j( andSj the raw material j respectively, a;(-2£~) is the number of readjustments of the i-th discrete subsystem aggregate with respect to the product k.

(Ll

K== 1,2,,= j'pE;;

91(; t= 0.e,

000)

T;

(19)

where Zf is the time of document cycle with the.F -th consumer, or the time lag between the shipment and the realisation, -I( ..1 Up [ is the k-th product amount_sold to the .P -th consumer by the time t and shipped at the time t.

Each k-th product is supplied to all the consuraers .FE. C)'" according to their orders. IX; doine; so the supplier should meet (1.) schedules of supplyine; to the"p -th consumer the k-th product over the period

'r , le

-K

I(

lpT~{jJ"r~9f'r I(

K=-f,"ro,M j

pE:.

Q

I()

The schedulinG problem for multicommodity marketing (Problem 11) is formulated as follows: find a schedule of marketing a product, {utt], giving the maximal price of the realised product,

(15)

-K

where !Jpr and 9-'pr are lower and upper bounds of the total shipment to the .p -th consumer of the k-th product over the /C period 'f, Up., is amount of the k-th product shipments to the p -th consumer over the period 'f ; (ii) consumer requirements to the shipments

Zp >r-t

j

ZJ'~r-t; .:

where ~ it is the'p -th consumer demand for the k-th product over the first t in-

This is a linear problem with a special structure of t ile matrix /1 Cp~ II and the

62

knowledge of its specifics leads to a relatively simple algorithm.

until t=1. Because shipments ahead of time are admissible (16), the compulsory preplanned shipments 11 tUpt//:u-e included in final shipments. At the fourth step the schedule for a product shipped in terms of "goods delivered" is determined. At this stage the final consumptions start with consumers with the maximal Zy • The estimation of the schedule for Problem II is r educed to solution of independent one-product Problems 1 the above method. An ordered series of products in decreasing order of the ratione(j3'" is formed and the solution starts with the product which is the first in this series; then the algorithm takes up the second one, etc.

It is clear tha t the scheduling/marketing problem is an optimation problem involving a vector criterion with the components (24) and (20) and the limitations (1 )-(13) and (15)-(19) (Problem Ill). Specific of the overall algorithm suggests that the industrial and the marketins algorithms should be considered separately.

3.

'l.'HE f,iAR.K]i;TING PHOBLEM SOLVED

Let us first consider an algorithm for solution of a one-product version of Problem II[8](Problem 1) whose specific is in determining the solution matrix /lUptllelement by element. This is a triangular matrix because of the delivery conditions (16), and the nonzero elements are below the main diagonal.

4.

This problem (Problem lA) may be stated as follows: find the finished production scheduling (~i~ ]that minimises the readjustment losses,

The al~orithm consists of four main steps. At the first step a series of consumers in increasing order of Z,p is established. At the second step the delivery of compulsory shipments {ipt]is established starE;ing with a consUmer with a minimal Z1' • 'rhe delivery time t,p is given by the condition t = T- 'l:.,p pE qK

K

K

(21 )

"et i= 1

with (19)-(11) being satisfied. We will use the "branch and boundary" methoJ.10J. This require s a technique to divide the scheduling set into subsets and make estimates over the subsets.

This will result in 3. matrix 1/1.u"t//whose nonzero elements are arbitrarily spaced. At the third step the r emainder of the shipment schedule is made starting with a consumer of the . minimal '2:1' , in the range of free balance. If for all the consumers under consideration the condition

pEQI<

81(

.... p

-P,,:LL i(t!:i.r)~

f

t,p~T-c:p

THE DISCIlli'1'.£i; PROBLEM SOLVED

The algorithm sta rts with finding the local-optimum schedule with respect to each k-th finished product for the discrete subsystem aggregates (Problem 2). Estimating the schedule with respect to the k-th product ne glects production of other finished species. Problem 2 may be formulated as follows: find the local-optimum schedule for the product k, i~ that minimises the local readjustment losses

,

is valid, then the final shipment is scheduled for a consumer with a larger Z.f' otherwise a leftward shift along the time axis is made for amounts to be shipped to all cons umers, for whom final shipments were planned earlier. '1'hese are performed

{&

A

....

P

"K

AI(

-~ =6 ~ 6~ (~£'r)

63

J,

(22)

-I(

bound is found for t

'td

starting from

= T (t = 'r , T-1, . • • , 2, 1) A'~=mVt-[~i~) pt-I cjr,c - £ Cf:'J l..'~t

<'I i

K'.:;L

1:'/oK

(29)

Then the lower admissible bound is found, starting from t = 'l'(t = 'r, 'r_1, . . . , 2,

(23)

1)1(

-K

K

J

jdit =o~a.x[CYit ) ...J!i(tlt)-1.

(0) _ le

-I{

where Cfit is the i-th aggregate charge with respect to the k-th product, the charge including consumer demands and operating conditions of the aggregates.

f i-/t 1 and p

Let us calculate readjustments number

A::

0 i

Sometimes we may encounter ~ it
x

If

estim'lte the

((Cl,.)} .

"" J

The fifth step. To lower the intermediate information anount due to the branching, we look for whether an aggregate might be specialised with respect to one of the products. The procedure starts from the first product of the ordered set K by fixing the aGgregates at the product. To implement this, we compare the new upper -It: admissible bounds [oft it} calculated through (29) at p = p - Pi (Pi is the overall amount of the specialised aggregates at the precedinC; iteration associated vlith the local-optimum plans {;,~) found at the fourth step). In other words, the following equation should hold for all times and products.

(24)

The second step. Starting from the finished product k (k = 1, 2, . . . , m) and from the time t (t = 1, 2, . , T), -le" the estim'lte 9-'d is made for t h e i-th aggregate (i = 1, 2, p).

-t "] )'j,t

- K

(25)

Then we calculate le

Ll t "

{9-: -t , '-I 0

,

. ,,/

(26)

Now, for all 'l:~t the earlier demand values f are corrected, Z=- t/~ 1+2 J

•. ' J

T;

(31)

At the fourth steE5~ \'fe find the readjustments number due to the local-optimum schedule {~ir . All the numbers obtained for parts of tile same product are summarised.

X4

~t"t ::min. L

- fl Lt

to be included in the remainder of the product through (28).

At the first step, construct an ordered set of indices of the products for which the demand exceeds the aggregate efficience for the period to be scheduled. /;

-A:

I(

.At -= /lit

"le"

K:{I
I<

(27)

"K -K i!£·t . ~.ftif

and the remainder of the product schedule is

t::. 1,Z.... ,T,

c' o: -f,2,···d~-PL

(32)

Its physical meaning is that the readjustments number estiGJates should become invariable after having specialised SQue of the a ggresutes.

(28) At the third step, the upper admissible

64

Let the specialised machines number be P1. Then, at the sixth step, the lower and upper admissible bounds are finally calculated as follows, ~ I

ijt~)?U"-[.Ji(~H»)(b-P1)t-2... v r k~t I<

J.!-tcl(

where 1l OtIC

/i: ]

-I(

~I(

(33) -K

where Cf-c is the demand schedule remainders which failed to be assigned to aggregates. The solution subsets arc obtained at (t-1) within the bounds (33) through

-::2t

- .tl"it-

a

structure at

~-t

•

Now, for each of the subsets at the time t calculate the current estimate.

" f J:1t t. K "K cP"t k =ln4x L_"K O(o,{t-tJj Ll (f.-t) +- Ct,r];tFu.,t)J

~

__ ( Ro{;"L)

-~:-l)

,

if

Pn

- 1007.

where Pn is an apl>roximation of the purpose function, p~ is the starting e s timate based on the exact algorithm.

P(~-l);> 0:-,1) j

(t-l)

)

otherwise;

The current estimate of the purpose function of Problem 1a is unambiguously made through the formula (21) including the readjustments number estimates (35) for all subsets of the set of solutions (of (34».

.... E.~ PH-PI(

(35)

where

jK

q;.t) >p~6( j

'llhus, processing the problem up to the time t = 0 results in an optimal solution of Problem 1a. The algorithm described above yields an exact solution of the discrete problem. If the problem dimensionality is high, the algorithm may be improved by specialising a great number of the aggregates in order to reduce the intermediate information amount, This results in an approximation in which the accuracy estimate is

(34 1 ) where ~t is the vector plan of production for the first t scheduling intervals, ~~t is the same for the t-th interval, 4.2:: is the k-th component of the vector Ll

)

if

'rhe best estimate, min (-fi), gives the new branching point.

(34)

by altering the Ll r

)

The q uantity cf(l»(t-l)J is the estimate for the readjustments number at the interval [0, (t-1)J ' it has been found at the fourth step.

'tIC

./::.'.,. =/ ' 7' = Cf-'1' j

2(t_-J.)

(

Pr.f~OIJ -p~1( "'K

~ ffUIK L'j/') .A!(~.iJ -(p -P')] )

1/1(

=

otherwise;

-(+-.1.) =a K - ) - J! (t-.t) K R , U I The quantity p!;-:t) is the number of aggK

To solve Problem I, let us use the branch and boundary method[10].

regates producing the k-th species during the (t-1)-th scheduling interval; the readjustments number at the interval [t,T] may be found from the branching tree through the recursion below.

Let us start with estimating the purpose function (14). To begin with, the schedule for the period T is estimated with the criterion (14) by solvine the partially integer proble~10,11J

65

(Problem 3). i\laximise PI.

P

Ft =1. [Z el<.~/~r

Io-.tr

-2-. ;;j x) 1'1 j=i

Then the specifYLne is processed with the new ['::/(1'1 and (Xd-~}values.

(36) at (1)-(13) valid for the time t = 'r, and maximise FToblem 2 via the procedure described in Section 4. <.'{

(=L

6. THE r.IAIN PHO Blli 1;1 SOLV'.t:D

The cost nature of the continuous/discrete productionEcmarketing criteria prompts the following structure for the general criterion [9]

Now, specify the period T schedule for the intervals t (t = '1'-1, T-2, . . . , 2, 1). Divide the solutions set into subsets, and fix the finished product allount produced by the discrete subsystem for the first (t-1) intervals. 'rhis is made via (34) at (9)-(13) and

r:{t-).)P+:A:D where

Vfuen dividing the solutions set into subsets, the in-sequence structure of the system discussed is taken irrt 0 account.

At the known [2~:_1) ,[xJt} , and l//:~) solve Problem 4, a priority problem described by (1)-(7) valid at [0, (t-1)] , [b,t] , and

K i

K. €.

f

(38)

The current estimate for (14) is made through A [.111. P .. PcH) :::P,( r"" "!'" 1 f,fttJ[cJtilt-Jl)fIJto, (li"-'?«r-!,)] (9) p

K

where

p:,(r) = Pt

.A. ~ 1

Problem III is solved with the method [10] •

(37)

j =f,2,""". It,;

0 ~

(41)

is the value of t he first

«i'oiL-t "I(

addend in (14), has been found f - 1 ») by solving Problem 2 for [0, (t-1)] , and

Si I(U(~ -~ ~t-.f»)

is found at the tree of the solutions. Further, the branching point is found through the estimate max (t-1).

P

'1'he purpose fune t ion is estimated with " "' ... f =(: -::.) P + A!! (42) where P and P are the purpose function estimates obtained for Problems I and 11 "respectively. 'rhe estimate P is calculated through (9) t see Section 5. Vfuen estimati~ the Problem 11 purpose function we assume that the whole production of the period 'r has been realised. Hence, the starting estimate is " Ill. p (43) CJ(2;,

P=LL t le·

(=1

where i~tr} has been found by solving Problem I. Now, specify the 'I' schedule for the scheduling intervals by fixing the finished product amount produced by the discrete system; use (4) and (37). Find shipping schedules by the method described in Sec-

When Problem 4 has no solution tIle current estinates of (14) are assumed to equal a high negative number P =-i'v1, and the partially integer problem[10,11) (Problem 5)

tion 3 for [t,T]; for [0,(t-1)], the product is assumed to have been realised completely. Then, the current estimate for Problem 11 may be vITitten in the following

is solved. Iv:aximise

.... P " I t I()'" '" " 17 P,=X[I. e (Z,"t +LJl'INJ -L~j(Xjt +-.1Xi{r-O/J I(.J (.t ,roJ

(40)

form. (t-i)

3

at (1)-(13) valid for [O,t] and [t,T], where

66

PI-

P

'T'-;:

=K.' L. [.f.C~i:f-t)+L -r'PeI<4 Upi] {ptri/< t

Id

(44)

The new branching point is delected on I' the basis of the estimate max

(4) Pervozvanskii A.A., Dinamicheskaya Model' ITpravleniya Sistemoi i Ee Priblizhennaya Optimizatsiya (Optimising an Industrial Dynamical Control System), Avtomatika i Telemekhanika, No .8, 1971.

t

Thus, using our algorithm to solve Problem III up to the time t = 0 gives us an optimal solution of the main problem.

(5) Mitsumori s., OptiIIIUIJl Production Scheduling of Multicommodity in Flow Line, IEEE Transactions on Systems, Man and Cybernetics, Vol.smc-2, No.4, 1972.

7. CONCLUSION 1. A matherratical model has been proposed for a system cons isting of a sequence of the following subsystems: continuous production, discrete production with parallel agGregates, and multi-product marketing.

(6) Eilon S. I Multi-product Scheduling in a Chemical Plant, Management Science, Vol.15, No.6, 1969.

2. Algorithms have been worked out for the component scheduling problems such as: multi-product narketing with a maximal realisation criterion, discrete production on parallel aggregates with a minimal readjustment loss criterion, discrete/continuous production with a maximal profit criterion, and discrete/continuous production&marketing with a vector criterion.

(7) Kuzin V.P., Upravlenie Sbytom v ASUP (Marketing Control in an Industrial Optimal Control System), the Energiya Publishers, 1973. (8) Sagyngaliev K.S., Kulkabaev N., Ashi-

mov A.A., Algorith~ Resheniya Zadach Kalendarnogo Planirovaniya Sbyta Gotovoi Produktsii Odnogo Tipa (Algorithms for Schedulinc One-Product Merketting), in Avtomatika i Kibernetika, the V.I. Lenin Kazakh NrI Press, Alma-Ata, issue 1, 1973.

3. Some of t he results have been tested on a non-ferrous metallurgy plant. The methods proposed have been found effective. (9)

REFElllil"C.l!:S (1) IIadley G., ~'lhitin T.lii., Analysis of Inventory Sys tems, Prentice-Hall, Inc., Englewood Cliffs, N.Y., 1963 (the Russian translation: the Nauka Publishers, 1969) .

Zak Yu.A., Modeli i Metody Postroeniya Komromisnykh Planov v Zadachakh Matematicheskogo Progran~irovaniya s Neskol' kimi 'r selevymi Funktsiyami (Construing Compromise Plans in Programming Problems with Several Purpose Functions), Kibernetika, Uo.4, 1972.

(10) Korbut A.A., Finkel'shtein Yu.Yu., Diskretnoe Programmirovanie (Discrete Pro gramming), the Nauka Publishers, 1969.

(2) Hyzhykov Yu.I., Upravlenie Zapasami (Resource Control), the Nauka Publishers, 1969. (3) Dudnikov ~ .E., Detalizatsiya Optimal'nO Go 'rekushchego Plana Haboty Promyshlennogo Kompleksa (SpecifyiD(j on Optimal Schedule in Industry), Avtomatika i 'r elemekhanika, No. 5, 1973

67