~ Pergamon Computers ind. Engng Vol. 32. No. 2. pp. 399 417, 1997 © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved PII: S0360-8352(96)00294-X 0360-8352/97 $17.00 + 0.00

A G E N E R A L I Z E D C O N T R O L M O D E L F O R

M A N - M A C H I N E P R O D U C T I O N S Y S T E M S W I T H

D I S T U R B A N C E S

DIMITRI GOLENKO-GINZBURG and VLADIMIR KATS Department of Industrial Engineering and Management, Ben-Gurion University of the Negev,

Beer-Sheva 84105, Israel

(Received 28 August 1996)

Abstract--A control model for a two-level man-machine production system is considered. The system comprises a section and several production units. Within the planning horizon the section is faced with manufacturing several different products with planned target amounts. Each unit can manufacture all kinds of products. In the course of manufacturing, each unit utilizes different types of non-consumable resources which may be reallocated among the units. Each production unit can manufacture a product at several possible speeds which correspond to one and the same resource capacities. Those speeds depend only on the degree of intensity of manufacturing and are subject to random disturbances. To carry out the process of manufacturing, the products have to be rescheduled among the units. This means that for each unit and for each product assigned to that unit the corresponding planned amount and the planning horizon have to be determined. Controlling the system is carried out at two levels: the section level and the unit level. At the unit level all production units are controlled separately. For each unit and for each product manufactured by that unit decision-making centers on determining: (i) control points to observe the product's output; (ii) the speeds to manufacture the product. If at a routine control point it is anticipated that a unit is unable to meet its deadline on time, emergency is called. The section level is then faced with the problem of both resource and target amount reallocation among the units. New resource capacities and target amounts for each product and each production unit are decision variables to be determined. The objective is to maximize the probability of the slowest unit to accomplish the planned amounts of its products by the due date. The problem is too difficult to obtain a precise or even an approximate solution. Heuristic algorithms are outlined on both levels. The model's performance is verified via extensive simulation. © 1997 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION

A number of recent papers have demonstrated the advantage of using planning and control models for man-machine production systems with disturbances (see Bitran et al. [1, 2], Dillenberger et al.

[3], Gabbay [4], Golenko-Ginzburg [5], Golenko-Ginzburg and Sims [6], Golenko-Ginzburg and Sinuany-Stern [7], Lefkowitz [9], Mesarovic [11], Pervozvanskiy and Sheynis [12], Sawik [13], etc.). Those models can be used in various man-machine systems, e.g., building systems, mining, forestry, agriculture, metallurgy, etc., where control actions are introduced by decision-makers since the plants are not fully automatic.

This paper is a further development of our previous publications [5-7], where two-level production systems have been considered. Those systems comprise several production units on the lower level and a section on the upper one. In the process of manufacturing, each unit utilizes non-consumable resources which may be reallocated among the units. A unit can manufacture different types of products. For each product the unit has several possible speeds of manufacturing that product. Those speeds depend only on the degree of intensity of manufacturing and are subject to random disturbances. For each unit and for each product assigned to that unit the corresponding target amount and the planning horizon have to be determined.

On the unit level at each control point, decision-making centers on determining both the next control point and the speed to proceed with up to that point. The section level is faced with problems of either reallocating resources among the section's units or reassigning the remaining target amounts among the units.

It can be well-recognized that control models considered in publications [5-7] are not generalized: each of them fits only certain manufacturing scenarios and covers only specific control actions. For

399

400 Dimitri Golenko-Ginzburg and Vladimir Kats

example, in [5] resources are not reallocated and only one product with target amount reassignment among the units is considered. Paper [6] considers several types of products with target amount reassignment, but resources are not taken into account. In [7] both resource reallocation and target amount reassignment have been carried out by separate optimization problems, etc. Thus, the control model needs generalization.

The main goal of this paper is to develop a generalized two-level control model for man-machine production systems comprising a section and several production units. All the recently developed production control models will be particular cases of that model. The model will comprise multiple resources together with different products being manufactured by several production units. At the lower level all production units work independently and are controlled separately. Both resource capacities for each unit and target amounts for each product to be manufactured by that unit are fixed and remain unchanged until either:

--the unit will finish manufacturing, or --at a routine control point it is anticipated that the unit is unable to meet the due date on time

without additional help from other units.

In the latter case emergency is called and the section level is faced with the generalized problem of both resource and target amount reallocation among the units. Thus, new resource capacities and target amounts for each product and each production unit are decision variables to be determined. The objective is to maximize the probability of the slowest unit to meet its deadline on time, i.e., to maximize the probability of that unit to accomplish the planned amounts of its products by the due date. The problem is a minimax-type stochastic optimization problem which is too difficult to obtain a precise or even an approximate solution. A heuristic solution may be obtained by substituting the stochastic objective outlined above for a similar deterministic one. Such modified reallocation problem (of minimax-type too) will be developed. Its solution can be obtained by using methods of non-linear programming.

The developed control model can be used for partially automated plants where the output can be measured only at preset points. The model complements an important class of JIT systems with uncertainty for which the number of publications remains very scanty (see [8]).

The structure of the paper is as follows. In Section 2 we will present the system's description, while Section 3 considers a notation being used on both hierarchical levels. In Section 4 we will consider the on-line control at the unit level. In Section 5 we will present a generalized two-level stochastic optimization problem, while in Section 6 we suggest a heuristic algorithm for solving the problem. In Section 7 the NLP method to solve the reallocation problem will be outlined, while Section 8 presents a numerical example. In Section 9 extensive experimentation based on computer usage will be undertaken. Section 10 presents conclusions and future research. The Appendix presents some numerical material to describe the two-level production system.

2. DESCRIPTION OF THE SYSTEM

The structure of the two-level production system is as follows. A section comprises n production units and has to produce, by a given due date, b different types of products with planned target amounts. Each unit can manufacture all types of products. In the course of manufacturing, the unit utilizes f types of different resources. All resources are restricted and are at the section's disposal before the system begins to work. Each unit can manufacture a certain product at m possible alternative speeds which are subject to disturbances and differ for different units and different products. It is assumed that when manufacturing a certain product by a certain unit with a certain speed the latter depends linearly on the resource capacities which have been allocated to that unit.

To control the process of manufacturing we have to determine the order of manufacturing different products by production units.

Man-machine production systems 401

Two different assumptions may be introduced:

I. the order of manufacturing different products is one and the same for all production units entering the section;

II. each production unit has its individual order of producing different products within the time interval [0, T], where T is the section's due date.

Assume for the sake of simplicity that at t = 0 case I takes place. Thus, we can enumerate the sequentially manufactured products by 1, 2 . . . . . b. Later on we will consider the possibility of changing that order. The new order will also be common for all production units (see Section 6).

To control a production unit which produces several different products we have to determine for each product c which is manufactured by unit k two planned values as follows: -- target amount Vk, to be manufactured; --planning horizon T,c for that target.

Note that for some values k and c relations Vk~ = 0 hold, i.e., product c is not manufactured by unit k. For such a "flow-shop" production system obvious relations Tk~ = T, 1 ~< k ~< n, hold. At t = T all production units terminate manufacturing. Each unit k stops manufacturing product c at moment Tk, even in the case of not accomplishing target Vkc on time.

Emergency is called in two cases:

(a) if it is anticipated at any routine control point that target Vk, will not be reached at Tk, on the average;

(b) if at moment Tk~ the actual output of product c is less than Vk~.

In the case of emergency call from any production unit k the section level will be faced with stochastic optimization problems outlined in Section 5.

3. NOTATION

Let us introduce the following terms:

3.1. Product ion unit level

Vk,--production plan of the cth product for unit k, 1 ~< c ~< b, 1 ~< k ~< n; it is assumed that each production unit can manufacture all kinds of products;

Tkc--planning horizon for production plan V~c; Rka--the dth type of resource capacity allocated to unit k, 1 ~< d ~<f, 1 ~< k ~< n;

n--number of production units; b--number of different products; f - -number of different resources.

Note that values V,,., T~,. and Rka are decision variables which are repeatedly recalculated and reassigned to all production units from the section level. This is carried out by solving an optimal control problem at t = 0 (before the system starts manufacturing) and at each emergency call. Thus, V,,., T~,. and Rkd are control actions introduced at the section level.

Vk , ( t ) - - t he actual output of product c manufactured by unit k, observed at moment t;

t , , - - the ith inspection moment (control point) of unit k when manufacturing product c, 1 ~ i <~ N,,., 0 <~ t , , ~< T,,.; note that for every new product value i starts from 1;

N,,.--number of control points of unit k to manufacture product c (a random variable determined by the control model);

Ak,.--minimal value of the closeness of the inspection moment t,,.~ to the planning horizon T,,., 1 <~ c <~ b, 1 <~ k <<. n;

dk,--minimal given time span between two consecutive control points t,,., and t,,.~ +, (in order to force convergence);

vjk,.(R,~ . . . . . R,/)---the j th speed of unit k to manufacture product c, 1 <<.j<~m, 1<<. k <<. n, 1 ~< c ~< b. As mentioned above, each production unit k can manufacture a

402 Dimitri Golenko-Ginzburg and Vladimir Kats

product at several possible speeds which correspond to one and the same resource capacities R,, . . . . . Rkr and depend only on the degree of intensity of manufacturing. Speeds vi*,. are subject to disturbances and are random values. It is assumed that they depend on resource capacities Rk,~ linearly, i.e., vj,,.(Rk, . . . . . Rkf) = Y/a~ l(ajk,.d'Rka) hold. Coefficients a/k,.d, 1 <~ j <~ m , 1 <~ k <~ n, 1 ~< c ~ b, 1 ~< d ~< f , are pregiven random values.

m--number of possible speeds common to all units in the section; ~, , (R,~ . . . . . R,u)--the average of speed Vi,,(Rk~ . . . . . Rkd). Values ~k,(R,~ . . . . . R,,~) are sorted in

ascending order, i.e., for j, <j2 relation f j ,k,(Rk, . . . . . Rk~) < fj,*,.(Rkz . . . . . Rk , ) holds. Note that f j , , (Rk, . . . . . RkU) = Et,)=,(ajk,a'Rkd). Each value 8jk,, is an average of ajk,.~.

jk,,--index of the speed introduced by unit k at the control point tk,., to manufacture product c, up to the next control point t,,,.,, + ,. As mentioned above, on the unit level decision-making centers on determining both the next control point tk,.,+~ and the speed vj,,. to proceed with up to that point. The integer number j,,.,, 1 ~< j,,~ ~< m, is determined by solving the control problem on the unit level at moment t = t,,.,. To simplify the notations we will henceforth use index j;

v/,,.(Rk~ . . . . . Rkt)--the lower bound of random speed Vjk,.(Rk, . . . . . Rkr). This is the production rate which is realized under most unfavourable circumstances;

Vjk,.(t)---the average amount of product c manufactured by unit k during interval of length t given that only speed vjk,. has been used throughout; note that usually v j , , ( t ) = ~i,,.t.

Xk,.(t)--the future amount of product c to be manufactured by unit k during interval of length t (a random variable determined by the control model);

raita Rkd , R~,)"X--lower and upper bounds of value Rka (in the case of R~, < Rkm~ ° unit k is unable to work, in the case of Rka > R~a "x there are redundant resources);

3.2. The sec t ion level W,.--production plan of the section for product c, 1 ~< c ~< b (pregiven);

T-- the due date (planning horizon) of the section (pregiven); Re--total available resources of type d (usually Rd = ~",= ~Rkd) at the section's disposal

(pregiven).

4. ON-LINE CONTROL AT THE UNIT LEVEL

At the unit level all the units first work independently and are controlled separately. The decisions are done on-line, in real time. For each unit k, 1 ~< k ~< n, and for each product c manufactured by that unit, the problem is to determine at each routine control point tk,.~ both the index of the new speed j and the next control point kk,..~+ ,. The problem is

- - to minimize the number of control points

Min N,, (1)

- - and to maximize the probability of accomplishing the planned amount Vk,. not later than at the planning horizon Tk,.

Max Pr{ V , , ( t = T , , ) >>. V,., }, (2)

s. t . Vk,(t,,.,) = 0, (3)

" Tk,. i f V,,.(t,,.,) < Vk,.Vi:tk,., < Tk,. t,.,.+,., = [ tk , . , (<Tk, . ) i f Vk,.(tk,.,)>~ V,, . ,C>~ 1, (4)

tk,.~+ , -- tk,.~ >t dk, V i : l ~< i < Ark,., (5)

Man-mach ine production systems 403

Tk, . - tk,..i+! >1 A,,Yi:I ~< i < N, , , (6)

f j -Lk, "(Tk, -- t,c,) < V,,. -- Vk,(tk~,) <~ fj ,~'(T,, -- t,,.,), 1 ~ k ~ n, 1 ~ c ~ b. (7)

Restriction (3) means that for each production unit k the starting time of manufacturing product c is the first control point t,,~. Decision-making at that control point centers on determining the next control point tk,.z and the index of the speed, j , to be introduced from the beginning up to the second control point t,,2.

Restriction (4) means that for the next product (c + 1) its first control point tk.~+ H coincides with the planning horizon T,, for the cth product, c/> 1, only in the case if that product has been inspected within the whole planning horizon. Otherwise, if at a control point t, , < T,c it has been observed that the cth product has met its target, then the next (c + 1)th product starts manufacturing from that point on.

Note that if in the latter case the system is not under emergency (see Section 6), the gained time interval [t,,.,, T,~] has to be distributed among the remaining products (c + 1) . . . . . b. Thus, new planning horizons Tk.~ +, . . . . . T,b have to be recalculated on the basis of remaining target amounts Vk.,+~ . . . . . Vkb and resources {Rk~}. We suggest to calculate those planning horizons by rescheduling the remaining time T - tk,.~ in proportion to values

V k , c + I Vkb

Omk,c+ , (Rkl , R, 2 . . . . . Rkf) . . . . . ~mkb(Rk,, Rk2 . . . . . R,r)"

Those values are the average time the unit needs to manufacture products c + 1, c + 2 . . . . . b, given that only m a x i m a l speeds v,,c will be actually used throughout. Thus, the new planning horizons T,.c+ ~ . . . . . T,b will be as follows:

q=c+ I f,,kq(Rkl, • • • , R , t ) Tk~ = t,~ + (T -- tka) , c + 1 ~< s ~< b. (8)

b Vkq X" o~+ ~ f, .kq(R,,, • . . , R,r)

Restriction (5) ensures the time span between two consecutive control points to be restricted from below (in order to force convergence). Restriction (6) enables the closeness of the inspection moment to the due date. Restriction (7) means that at all control points t,ci index j denotes the min ima l speed that on average guarantees completion Vkc by the due date Tk,. Thus, the on-line control model at the unit level prohib i t s using unnecessari ly high speeds.

Stochastic optimization problems (1-7) cannot be solved in the general case; it allows only a heuristic solution. The algorithm outlined below determines at each control point tkc, for unit k which manufactures product c:

- - the speed vjk,. to be introduced until the next control point tk,..i + I. The index of the speed j is the minimal value satisfying (7);

- - the next control point tk,..i + ~ which is determined by using risk adverse decision-making. Assume that due to most unfavourable circumstances the unit will produce product c according to the minimal rate vjk,. until the next control point. This point is determined so that, by applying the average m a x i m a l speed f,,k,, from that point on, there will be enough time to meet target Vk,. on time. The steps of the on-line algorithm to control the unit k are as follows:

Step 0. Given at t = tk,~ (for c = 1, t,,.~ = 0): - -plan targets Vk,, 1 ~ c <<. b; --planning horizons Tk, 1 ~< c ~< b; --average values fjk,.(R,~ . . . . . Rkl) and minimal values v j k , . (R ,~ , . . . , Rkt) for all speeds

1 ~<j~< m with fixed capacity resources R,~ . . . . . R,r, 1 <~ c <~ b. For the sake of simplicity we will use in this section terms gjk,. and v~,..

404 Dimitri Golenko-Ginzburg and Vladimir Kats

Step 1.

Step 2.

Step 3.

Step 4.

- -actual amounts inspected V~,.(tk,.,); note that Vk,.(t~,~) = 0, 1 ~< k ~< n, 1 ~< c ~< b; --values of closeness Ak,. and dk,. for all c. Start with c = 1, i = 1.

Stopping rules: (a) If Vk,.(tk,,) + V~k, ' (Tk , . - tk,.~) >>. V~, set j = 1 and tk,..,+ j = Tk,~; go to 9. (b) If Vk,.(tk,~) + f , , k , ' ( T k , - tk,.~) < Vk,, go to 10. Stopping rule (a) is carried out under most favourable circumstances when even applying the minimal rate of the lowest speed enables meeting the deadline on time. Thus j = 1, and the product 's output is inspected at t = Tk,.. If within the planning horizon a breakdown occurs, step 1 has to be applied anew. In the case of stopping rule (b) the product cannot be accomplished on time by even introducing the highest speed. Emergency has to be declared and decision-making is carried out at the section level.

Determining the production speed:

j = min[qVq:l <~ q <~ m,Vk,(tk, . ,) + fqkc ' (Tk , . - tk,,) /> Vk,.]. (9)

Determining the next control point tk,.,i+l: Find tk,..i+ ~ which satisfies

Vk,(tk,.~) + Vf "(tk~j+, -- tk,,) + 6m'(Tk, - tk, j+ ,) = Vk~. (lO)

The step carries out the risk averse decision-making which has been outlined above.

Closeness to the planning horizon and the next control point: (a) if Tkc -- tk,.i+ ~ <<. Ak~, then reset tkc~j + ~ = T,c and go to 9. (b) if tk,.j + ~ > Tkc, then reset tk,.j + ~ = Tk, and go to 9. (c) if tk~,,+ ~ - tkc~ < dk,, then reset tk,.j+~ = tk,~ + dkc and go to the next step.

Step 5. Observe Vkc(tk,.j+ ~). If Vk~(tk, j+ ~) >/ Vk,. apply the next step. Otherwise set i = i + 1 and go to 1.

Step 6. Recalculate planning horizons Tk., c + 1 ~< s ~< b, according to (8).

Step 7. Set t,.,+j.~ according to (4).

Step 8. S e t i = 1, c = c + l ; i f c ~ < b g o t o 1. Otherwise go to 11.

Step 9. Observe Vk,.(Tk,). If Vk,.(Tk,.) >1 Vk,. go to 7. Otherwise apply the next step.

Step 10. Emergency is called since either: (a) production unit k has not accomplished its target amount for product c on time; or (b) it is anticipated that product c cannot be accomplished on time by even introducing

the highest speed.

Step 11. Stop.

5. GENERAL TWO-LEVEL CONTROL MODEL

A two-level production control model is suggested where each level faces a stochastic optimization problem. At the section level at moment t, 0 ~< t < T, the problem is to determine optimal target amounts Vk,. >1 0 and resource capacities R,,I >>. O, 1 <~ k <<. n, 1 <~ c <~ b, 1 <~ d <~ f , to maximize the probability of the slowest unit to meet its deadline on time

subject to

Man-machine production systems 405

J = Max Min F ( ] Pr{V, , ( t )+ X , , . ( T - t)>>. Vk,.}l (11) [ V~, ,R~, ~ k [._--c = I A

Vk,,= Wc- ~ V,,(t), 1 <~ c <~ b, (12) k = l k = l

• R,d <. Ra, 1 <~ d <~ f, (13) k = l

R~J"<~R,d<~R,~d ax, l <~d<~f, l <~k<~n, (14)

values X , , ( T - t) being determined by solving problems (1-7) at the unit level with new target amounts V,c and new resource capacities R,,..

Here

b

I-I Pr{V,c(t) + X , ~ ( T - t) >1 V,,.} (15) c = l

is the probability of unit k to accomplish all the products in time subject to new target amounts Vk,. Thus objective (11) maximizes the probability of the slowest unit k to meet its due date T on time. Restriction (12) means that for each product c, 1 ~< c ~< b, the sum of target amounts V~,, obtained by solving problems (11-14), must be equal to the non-accomplished part of the section's target for that product at moment t. Restrictions (13-14) mean that in the course of manufacturing all the available non-consumable resources at the section's disposal are reallocated among the units honoring boundary values R~" and R~d dx.

Reallocation problem (11-14) is a complicated stochastic optimization problem. A heuristic solution is suggested, based on replacing all stochastic terms in objective (11) by deterministic ones. The idea is as follows:

Relation (15) can be regarded as the ability of unit k to meet the deadline on time. It can be well-recognized that a unit with the lowest ability is at the same time the slowest unit to accomplish all its remaining target amounts. Thus we suggest to determine the unit's ability by calculating the average time the unit needs to meet its deadline on time, given that only maximal speeds v,,k,, will be actually used throughout. Such an average time for each unit k can be determined by

b Wk,. Z = Z em,,(R,,, Rkr)

c = | " ' " ~

We suggest to replace objective (11) by a new one: Determine optimal values Vk,, Rkd, 1 <~ k <~ n, 1 <~ c <~ b, 1 <. d <~ f , to minimize the average

completion time of the slowest unit, i.e.

J = Min Max T, = Min Max - , , , '~,~--- " l V*,.,Rkal k { V,,..Rkd] k c = I " '

Using relation f.,,,.(R,I, . . . . R,j) = Z~=~(&,,,.~.R,a) (see Notation) we obtain the modified

406 Dimitri Golenko-Ginzburg and Vladimir Kats

objective as follows:

I Vk,..Rk,t] k ,' = t f

d = l

Problem (11", 12-14), unlike problem (11-14), does not comprise probabilistic terms and is an NLP problem. An outline of the method used to solve that problem will be given below.

Note, in conclusion, that planning horizons Tk,., 1 ~< k ~< n, 1 ~< c ~< b, can be calculated on the basis of values Rk~ and Fkc obtained by solving problem (I 1", 12-14). Using (8) we obtain

s = 1

T k , = t + ( T - t ) , 1 ~<c~<b. (16) b

s = I

6. TWO LEVEL ON-LINE CONTROL ALGORITHM

The goal of the section is to unify all optimization problems on both levels to consider an on-line control algorithm for the two-level man-machine system. This algorithm unifies the on-line control model (1-7) at the unit level and reallocation problem (11", 12-14) which is solved at the section level. The structure and the performance of the two-level algorithm are presented in Fig. 1. Two additional heuristic options will be imbedded in the decision-making at the section level:

(a) It can be well-recognized that after enumerating the products at moment t = 0 products with lower numbers are in a better condition that those with higher ones. If a product with a low index being manufactured by a certain unit fails to meet its deadline on time, then an emergency is called and the remaining time is rescheduled among the products which have not yet been accomplished. Thus, such failing products which are processed first have an opportunity to consume additional time reserves the system may possess while for products of higher indices these reserves may be already exhausted. In order to prevent such cases we recommend to change the enumeration order after resolving problems (11", 12-14) in the course of the system's work by sending the overdue product to the end of the products' list (for all production units). Thus the new order of manufacturing is the same for all units. Although such a control action may cause the overdue product to delay further it enables the same conditions for all products to be manufactured.

(b) If solving reallocation problem (11", 12-14) after a recent emergency call at moment t results in obtaining J > T - t, the system is unable to accomplish all products in time even when introducing the highest production speeds with the utmost intensity. In such a case an overall emergency is called and we suggest decision-making as follows:

1. Calculate for each non-accomplished product c, 1 ~< c ~< b, the value

L = T - t ~ Vk,. k = l

W , . - Vk,.(t) , . , r ~=, y~ (~.,~.~" R~)

d = l

Value L denotes the ratio of two values: --the average output of product c which can be manufactured by all units within their

corresponding planning horizons, on condition that the units will produce that product with maximal speeds;

--the remaining amount of product c at moment t to be manufactured within the period It, T].

M a n - m a c h i n e product ion systems

Thus L can be regarded as the product's delivery performance.

407

2. Single out all products c~, c~ . . . . . Cq with L.,/> 1, 1 ~< r ~< q, and all other products c*, c2" . . . . c,* w i t h l : * < l , l ~ <g~ <v .

3. Introduce for all v "tense" products c*, c~* . . . . c,* at each production unit only maximal speeds

Algorithm I

Min Nkc { tkci, j }

Max Pr{Vkc,(Tkc) > Vkc} { tkci, j }

s.t. (3-7).

> o

: 7 v ,~ ,,I ]>

>

Observe output Vkc (t) of

/ Uni t 1 • • •

Vt

>

product c at the routine

control point t = tkc i. In case

tkc i < Tkc,

Vkc(Tkc i) + ~mkc'(T-tkci) ~ Vkc,

introduce proper speed j and proceed manufacturing.

In case Vkc(tkci) = Vkc,

tkc i < Tkc, recalculate

Vkq, Tkq, c + 1 S q g h .

Min Max, b Vk¢

{Vkc, Rkd } k c~=l ~ (ammkc d .Rk c d=l

Uni t k

8 o

Algorithm II ("tense" products) j = m ,

tkc I = T k c ;

maximal speed is introduced to inspect at

tkc l = Tkc.

e.~

Introduce the maximal

speed V * and inspect mkcg

the product's output at the planning horizon Tkc ~.

;> m

z

[ -

• • • Uni t n

<

+

Algorithm III ("non-tense" products)

Min Nkc

{ tkci , j = m }

M a x P r { V k c , ( T k c ) -> V k c }

{ tkci ' i=m } s.t. (3-7)

I I , I '

I "" II

I

i . ' ~ ~ I

I I

Introduce only maximal

speeds v m and inspect the

output eke(t) at each routine

control point t = tkcri. In case

tkcri < T k c r, V k c r ( t k c r i ) : V k c r,

apply the section level for

solving optimal reallocation

problem.

O~

~r

m m

o a~

x

[ -

r~ 0 a~ eL

Fig. 1. A two-level generalized control model (emergency call at moment t).

408 Dimitri Golenko-Ginzburg and Vladimir Kats

v,, , ,q(R,i . . . . . R,t), 1 ~< g ~< v, without intermediate control points. Inspect the manufactured amounts Vk,~(Tkq), 1 <<. k <~ n, 1 <~ g <<. v, at the ends of the planning horizons.

4. For other q non-tense products ct, c2 . . . . Cq at each production unit introduce maximal speeds v,,,,,r(R,~ . . . . . Rkr), 1 <~ r <~ q, 1 <<. k <~ n. Determine control points according to step 3 of the unit-level algorithm (see Section 4) but for j = m only. Thus, step 2 is temporarily cancelled. If at a recent control point t,,~ < Tk,, it is observed that product c~ is accomplished, problems (11", 12-14) have to be resolved in order to reschedule the gained time interval [tk,~,, Tk,,] among the "tense" products.

During the overall emergency the order of the products must not be changed. Henceforth the unit-level algorithm outlined in Section 4 will be denoted Algorithm I.

Denote its two modifications: - -only maximal speeds are introduced and the only control point is the planning horizon Tk,.; - -only maximal speeds are introduced but control points are determined and the product's output

is inspected,

by Algorithms II and III, correspondingly. The enlarged steps of the two-level algorithm are as follows (see Fig. 1):

Step 0. At moment t = 0 the input data (see Section 3) is externally given: --values R~" and R,"2 x, 1 ~< k ~< n, 1 ~< d ~< f ; --values Akc and dkc; --values ajkcd, l <~ j <<. m , l <<. k <~ n, l <<. c <<. b, l <~ d <~ f ; --values vf.,~(R,,, . . . , R ,r) , 1 <~ j <. m ; --values W,., 1 <<. c <~ b; --values T, f , b, n and m; --values Rd, 1 <<. d <<.f; --products ' enumeration from 1 to b.

Step 1.

Step 2.

Step 3.

Step 4.

Step 5.

At t = 0 solve reallocation problems (11", 12-14) to determine values I1",, R,d, Tkc.

If J > T - t an overall emergency is called; go to the next step. Otherwise, i.e., in case J ~< T - t, use Algorithm I to control each production unit k, 1 ~< k ~ n, with values V,a, Rka and Tk, obtained on step 1 (at t = 0) or step 4 (at t > 0). All the units work independently and are controlled separately. Decision-making is carried out on-line, in real time.

Subdivide all the products which have not yet been finished manufacturing into two groups: the "tense" {c* } and the "non-tense" group {cr} (see control actions 2-4 outlined above). For each product of the first group to be m a n u f a c t u r e d b y al l p r o d u c t i o n uni ts apply Algorithm II. For each product of the non-tense group apply Algorithm III.

This step is applied at moment t > 0 in two cases: ----emergency is called at step 10 of Algorithm I; --if , in the course of realizing Algorithm III on step 3 for non-tense products {c,}, it is

observed at a routine control point tk,~i < Tk, r that unit k has accomplished its target amount Vk,r, the gained time interval [tk,.ri, Tk, r] has to be rescheduled among the tense products.

In both cases solve reallocation problem (11", 12-14) and examine value J. If J > T - t go to 3. Otherwise apply the next step.

Change the products' order by sending the overdue product to the end of the enumeration list. Go to 2 with values V, , , R,d and Tp,. obtained at step 4 by solving the reallocation problem at the section level. Apply Algorithm I for all units and proceed manufacturing.

Man-machine production systems 409

Step 6. At time T after all production units are inspected the algorithm terminates. The algorithm is performed in real time; namely, each iteration of the algorithm can be

performed only after values Vk,.(tk,.i) are actually inspected. The control points cannot be predetermined. However, if we want to evaluate the efficiency of the two-level control model we can simulate the system's work by random sampling of the actual output between two adjacent control points tk~ and tk,.,+ ~. By simulating the system's work many times, the probability of meeting the due date on time, the average number of control points N = Z~ = ~ Y.~ = ~ Nk, and other parameters may be evaluated.

7. APPROXIMATE METHOD FOR SOLVING THE REALLOCATION PROBLEM

Let G be the set of variables (Vkc, Rka) satisfying linear constraints (12-14). Denote: X = (xt . . . . . x,), r = nb + nf, the vector of variables (V.,,c, Rk,~) to be optimized, and

I s Vkc t f(x, . . . . , X r ) = Max Y. J'

d=l

Since

b Vkc 2 c= I .f

2 ( a m k , . a ' R k a )

d = I

1 ~< k ~< n, are continuous functions of variables (Vkc, R,a), functionf(x~ . . . . . xr) is continuous too. Thus, reallocation problem (11", 12-14) boils down to minimize a non-linear continuous function f ( x , , . . . , Xr) with linear constraints (12-14), i.e.,

Min f (x , . . . . . Xr). (17) (xl . . . . . x r ) eG

Note that f ( x , . . . . . xr) is not a convex function and may possess several local minimums• Moreover, the function is not differentiable everywhere on G. This makes the problems more general and complicated.

It can be clearly recognized that for reallocation problem (11", 12-14) only (n - 1)(b + f ) < r variables are independent• Other (b + f ) variables, namely, V,c, 1 <~ c <<. b, and R,~, 1 <. d <<. f , are dependent ones and satisfy

f n-l R.a = Rd-- ~ Rke, k = l

n - I

v . c=w, - yvk , . k = l

(18)

To solve (17) we have chosen one of the coordinate descent methods which are very attractive because of their easy implementation ([10], pp. 158-161). Given a point X(x~ . . . . . Xq) e G, with independent variables, ci, 1 ~ i ~ q, q = (n - 1)(n - f ) , descent with respect to the coordinate x, (i fixed) means that problem

M!n f (x , . . . . . xq) (19) A" i, i ~ ~ I ,qj

is solved. Thus changes in the single component xj are allowed in seeking a new and better vector X. By sequentially minimizing with respect to different components, a relative minimum of f might ultimately be determined.

The approximate solution of the reallocation problem is obtained via the cyclic coordinate descent algorithm [10] which minimizes f(x~ . . . . . Xq) cyclically with respect to the coordinate variables.

CAIE 32/2--F

410 D i m i t r i G o l e n k o - G i n z b u r g a n d V l a d i m i r K a t s

Table 1. Production speeds t'i~,. (uniform distribution)

Products Units Product 1 Product 2

Unit 1

Unit 2

Unit 3

v,~ = U[0.019R~ + 1.125R~:, 0.031R~ + 1.875R~] v.,, = U[0.038R~ + 2.250R~,~, 0.062R~ + 3.750R~2] t'm = U[0.075R, + 4.500R~, 0.125R~ + 7.500R~2] t '~ = U[0.038R,,~ + 0.750R:~, 0.062R,,~ + 1.250R:~] v~,,~ = U[0.075R,~ + 1.500R2~, 0.125R~ + 2.500R~2] t,~ = U[0.150R~ + 3.000R~, 0.250R~ + 5.000R~] t ,~ = U[0.056R~ + 0.375R~,, 0.094R~ + 0.625R~] vz~ = U[0.113R3~ + 0.750R~,,, 0.188R3~ + 1.250R~2] t,~ = U[0.225R3~ + 1.500R~,,, 0.375R3~ + 2.500R~]

v~2 = U[0.056R. + 0.375Ra, 0.094Rj~ + 0.625Rt.,] vm = U[0.113Rtl + 0.750R1.,, 0 .118R, + 1.250R1.,] v3~2 = U[O.225R, + 1.500Rt2, 0 .375R, + 2.500R~2] vt22 = U[0.038R;~ + 0.750R22, 0.062R2~ + 1.250R.,:] v.,.,,, = U[0.075R.,~ + 1.500R~, 0.125R.,~ + 2.500R.,,] vm = U[0.150R~ + 3.000R2~, 0.250R2~ + 5.000R,,~] t ,~ = U[0.019Rj~ + 1.125R~, 0.031R~ + 1.875R~.,] v~: = U[0.038Rj~ + 2.250R;. , 0.062R~ + 3.750R~z] v33: = U[0.075R~ + 4.500R~. 0.125Rs~ + 7.500R~]

Thus xj is changed first, then x2 and so forth through Xq. The process is then repeated starting with x~ again.

Thus, when using the coordinate descent method, the coordinate search is carried out only for independent values Vkc and Rka. Dependent values V,~ and R,a are corrected by means of (18) after performing each step of the search.

As to solution of reallocation problems (11", 12-14), it is carried out as follows. For each coordinate x~, 1 ~< i ~< (n - 1)(b + f ) , a constant increment h~ > 0 is pregiven. Value f (X) is calculated in two opposite points ( x ~ , . . . , x ~ - h , . . . . ,Xq) and (x~ . . . . . x ,+h~ . . . . . Xq) to determine the direction of the function's decrease. The search is undertaken along that direction, i.e., values f ( x j , . . . , x~+ Eh~ . . . . . Xq), E = -I-2, + 3 . . . . . are calculated. The iterative process terminates either at a local minimum point or on reaching a boundary point of set G. Note that the coordinate descent algorithm can be applied to a very broad class of NLP problems [10].

8. N U M E R I C A L E X A M P L E

The section's management is faced with the problem of manufacturing two types of products on three production units. Two types of resources are utilized in the course of manufacturing. Thus, n = 3 , b = 2 , f = 2 .

The section's parameters are as follows:

W j = 2700; W2= 500; T = 60;

R~= 135; R 2 = 8 .

Production speeds rjkc at the unit level are presented in Table 1. The resource parameters R,a, 1 ~< k < 3, 1 ~< d ~< 2, at the unit level are as follows:

Unit 1: Unit 2: R~i"= 10, R~I i n : 10, R~ "x = 100, R2rq "x = 110, R~'"= 1, R~'"= 1, R?~ ~ = 6 , R ~ ~ = 8 ,

Since reallocation problem (11", 12-14) will outline only average values of linear

~m~. = 0.1,

6m.~ = 6.0,

~i.,121 = 0 . 3 ,

rio,122 = 2.0,

Unit 3: R~in= 10, R~aX= 120, R~'"= 1, R~ ax = 8.

uses only maximal average speeds f,,kc(Rk~ . . . . . R~s) we coefficients for those speeds, i.e., values ~,kca:

?in2. = 0.2, ~,,3. = 0.3,

tim212 = 4.0, ~i,,3,2 = 2.0,

elm221 = 0.2, am321 = 0.1,

tio,m = 4.0, ~i,,,322 = 6.0.

For all production units and all products set dk,. = 5. In order to illustrate the work of the two-level control algorithm outlined in Section 6 we will

perform the main steps of that algorithm.

Man-machine production systems 411

At t = 0 (step 6) reallocation problem (11 *, 12-14) to determine values V~,, Rka and Tk,, is solved. The problem is as follows: to minimize

Vtl 0.1R. + 6.0RI2

V2~ J = Min Max

~Vk,.R~,,', k 0.2R2~ + 4.0Rz2

V3,

0.3R31 + 2.0R32

subject to

Vi2 + 0.3Rit + 2.0RI2'

V2: + 0.2R21 + 4.0Rzfl

V32 + 0.1R31 + 6.0R32'

". (20)

VIi + V21 + I/3, = 2700, (21)

V12 + V22 + V32 = 500, (22)

R . + R21 + R31 ~< 135, (23)

Ri2 + R22 + R32 ~< 8, (24)

10 ~ R,, ~ 100, (25)

10 ~ R21 ~ 110, (26)

10 ~< R31 ~< 120, (27)

1 ~< R,2 ~< 6, (28)

1 ~< R2z <~ 8, (29)

1 ~< R3z ~< 8. (30)

Using the coordinate descent method outlined in Section 7 results in obtaining solutions as follows:

V. = 976; V2~ + 524; 1/3, = 1200;

V~z= 112; V22=307; V32=81;

Rll = 12; R2t = 46; R31 = 77;

RI2=4; R22=2; R3 ,=2 .

Assume that product 1 will be manufactured first. Thus, using (16) we obtain:

T . = 48.06, T21 = 37.83, T3t = 54.88,

Tl2 = T:2 = T3: = 60.

Note that objective J = 48.4 is less than T = 60. This enables the section to meet the deadline on time.

After realizing optimal reallocation problems (20-30) at t = 0, each production unit obtains both resources and target amounts to start manufacturing. We will outline below control actions at the unit level. Unit 1, t = 0

Using (9) results in determining production speed v3. = U[18.90, 31.5], i.e., j = 3. The first inspection point till, when manufacturing product 1, satisfies [see (10)]: 18.9.t,,, + 25.2(48.06 - tl,,) = 976 =~ t , . = 37.3. Unit 2, t = 0

Using relations (9) and (10) results in obtaining speed v32~ = U[12,90, 21.50] and inspection point tm = 29.5. Unit 3, t = 0

412 Dimitri Golenko-Ginzburg and Vladimir Kats

Product ion speed to manufac ture produc t 1 is v33~ = U[20.33, 33.88]; value 13H = 42.5. Since 29.5 = min[37.3, 29.5, 42.5] unit 2 has to be inspected first, at momen t t2,t = 29.5. Unit 2, t = 29.5

Sampling the r andom number ~ = U(0, 1) = 0.03, results in simulating the actual output o f p roduc t 1, observed at t = 29.5:

V2,(29.5) = t2,,'{12.90 + ~[21.5 - 12.9]} = 387.66.

Using (9) results in determining product ion speed v32, = U[12.90, 21.50]. The next inspection point t2,2 = 31.2. Since t2|2 - - t211 ----" 31.2 - 29.5 = 1.7 < d21 =~ t2,2 = t2H + 5 = 34.5. Since 34.5 = min[37.3, 34.5, 42.5] unit 2 has to be inspected first, at t = 34.5. Unit 2, t = 34.5

Sampling the r andom number ~t = 0.05 results in simulating V2,(34.5) = 387.66 + 5.{12.90 + 0.05121.5 - 12.9]} = 454.31. Since 454.31 + f3,~-(37.83 - 34.5) = 454.31 + 17.2.3.33 = 511.59 < V2, = 524, s topping rule (b) at Step 1 (see Section 4) holds. Unit 2 is unable to manufac ture produc t 1 on time by even introducing the highest speed, v32,. Thus emergency is declared, and decision-making is carried out at the section level.

In order to under take resource and target a m o u n t reallocation, all actual outputs o f p roduc t 1 manufac tured by other units, have to be observed at momen t t = 34.5. Unit 1, t = 34.5

Sampling the r a n d o m number, ~ = 0.57, results in simulating

V,,(34.5) = 34.5.[18.90 + 0.57(31.50 - 18.90)] = 899.83.

Unit 3, t = 34.5 Sampling ct = 0.84 results in simulating

V31(34.5) = 34.5.[20.33 + 0.84(33.88 - 20.33)] = 1094.06.

The opt imal reallocation problem to be solved at the section level is as follows: minimize (20) subject to (21-30) where constraint (21) is modified to V. + V2, + V3~ = 2700 - 454.31 - 899.83 - 1094.06 = 251.8, while other constraints remain unchanged. Value T is changed for 60 - 34.5 = 25.5. The results are as follows:

New target amounts

V,I = 131.3; V21 = 115.9; V31 = 4.6;

V,2 = 72.2; V22 = 88.3; I"32 -- 339.5.

New resource capacities

R,~ = 53; R 2 1 = 71; R 3 1 = 11;

Rj2 = 2; R 2 2 = 1; R32 = 5.

Note that since J = 11.27 < 25.5 the two-level system is able to accomplish manufac tur ing on time. Accord ing to step 5 o f the two-level algorithm, the products ' order is changed, i.e., p roduc t 2 has to be manufac tured first. Using (16) results in obtaining T,,. as follows:

7'12 = 42.70; 7"22 = 45.48; T32 = 59.25;

~ j = ~ , = ~ , = 6 0 .

Later on new resource capacities Rkd, together with new target amounts Vkc and planning horizons T,,., are passed to the product ion units. The process o f manufac tur ing proceeds at the unit level, as has been outlined above.

M a n - m a c h i n e p roduc t ion sys tems 413

9. E X P E R I M E N T A T I O N

The efficiency of the newly developed control model can be illustrated by applying it to a production system under random disturbances, with variable speeds and inspection points. The system comprises a section and six production units. Five different products are manufactured by using four types of resources. Thus, n = 6, b = 5 and f = 4. The total available resources are: R~ = 135; R2 = 8; R3 = 100; R4 = 10. Values Rka are presented in Table A1 of the Appendix. The section's production plans Wc are: W~ = 2700; W2 = 500; W3 =200; W4= 1500; W5 = 1000. Random values ajk,,t are presented in Tables A2-A5 of the Appendix, together with an example.

The computer program to simulate the two-level control algorithm is written in C + + Borland language on an IBM PC. The experimental design is given in Table 2. Two parameters were varied: the due date T and the minimal time span dkc (taken as equal for all production units k and products c). The closeness to the due date A,,. has been set Ak, = 2 for all values k and c.

Thus, in order to validate the two-level control algorithm, a total of 18 combinations (6 x 1 x 1 x 3 x 1) were considered. For each combination, 100 simulation runs were performed. The number of simulation runs was determined by applying classical estimation theory [14]. Given the error in estimating an outcome measure by its average value, the confidence coefficient and the sample standard deviation (obtained from the preliminary sample, of a size not less than 30), we can determine the sample size of simulation runs. We have taken 100 simulation runs, since that sample size fits all the system's parameters that we have estimated via simulation.

The outcome parameters are as follows:

N--the average number of control points per production unit within the planning horizon; /~--the probability of meeting the due date on time, i.e., the probability of accomplishing

the section's target We, 1 ~< c ~< 5, within the planning horizon; j - - the average index of all production speeds that have been introduced within the

planning horizon, i.e., in the course of one simulation run; /Ve--the average number of emergencies per production unit within the planning horizon;

No~--the average number of overall emergencies at the section level; To--the average computational time of one simulation run (on a PC486).

The summary of results is presented in Table 3. The following conclusions can be drawn from the summary:

1. Increasing the minimal time span d results both in decreasing the probability/~ of meeting the section's target on time and in decreasing the average number of control points 57.

2. The computational time Tc depends mainly on the average number of emergencies ~-e. That is because each emergency call results in solving a reallocation problem at the section level, and this takes more computational time than other control actions. For example, performing a simulation run for T = 65 and d = 5 takes 10.8s on the average, of which simulating reallocation problems take 9.3 s versus 1.5 s for all other control actions within the planning horizon.

3. Increasing the due date T usually results both in increasing the average outcome probability /~ and in decreasing the average index of production speeds. Note, however, that lower speeds may, in turn, decrease value/~. Thus, for certain cases, when increasing value T is followed by a drastic decrease of production speeds, value/~ may remain unchanged or may even decrease. In the summary, such a case can be observed when increasing the due date T from 75 to 80.

Table 2. The experimental design

Parameters Values given in the experiment Number of levels

Due date T 65; 70; 75; 80; 85; 90 6 Number of speeds m 3 I Distribution of a,,,,~ Uniform I Minimal time span d 5; 6; 7 3 Closeness to the due date A 2 I

414 Dimi t r i G o l e n k o - G i n z b u r g and Vlad imi r Ka t s

Table 3. Summary of results

Average index of all

Average Probability production Average Average Outcome number of of meeting speeds ~ Average number of computational

parameters control the section's within the number of overall time of one Due Time points I',7 target on planning emergencies emergencies astimulation

date T span d per unit time ~ horizon Iq~ per unit N~ 1"~ (in s)

65 5 13.02 0.490 2.805 6.930 0.270 10.8 6 11.68 0.460 2.817 6.090 0.330 10. I 7 10.71 0.380 2.814 5.440 0.280 8.9 5 13.22 0.610 2.167 3.310 0.000 5.9

70 6 12.25 0.600 2.163 3.020 0.020 5.5 7 11.54 0.590 2.152 3.140 0.010 5.7 5 11.47 0.720 1.707 1.610 0 3.7

75 6 10.66 0.700 1.733 1.720 0 3.9 7 10.17 0.690 1.696 1.750 0 3.8

80 5 10.90 0.710 1.274 1.250 0 3.3 6 10.57 0.680 1.257 1.580 0 3.7 7 10.11 0.670 1.219 1.500 0 3.6

85 5 9.48 0.940 I. 186 0.510 0 2.3 6 9.09 0.920 1.154 0.630 0 2.5 7 8.68 0.890 1.148 0.530 0 2.3

90 5 7.86 0.990 1.100 0.300 0 2.0 6 7.86 0.980 1.097 0.250 0 2.0 7 7.74 0.980 1.092 0.260 0 1.9

Example. In the course of simulating the section with due date T = 80, minimal time span d = 5 and closeness to the due date A = 2, we undertake in the average 10.9 inspections per production unit. The probability of meeting the section's target on time is 0.71, while the average index of production speeds is 1.274. For each production unit in the average 1.25 emergencies have been called. No emergencies have been called at the section level.

10. C O N C L U S I O N S A N D F U T U R E R E S E A R C H

The following conclusions can be drawn from the study:

1. The developed two-level control model can be applied to a broad spectrum of man-machine production system under random disturbances: these include various not fully automated industrial plants, building systems, mining, metallurgy, agriculture, etc. For such systems, the output can be measured only at preset inspection points as it is impossible, or too costly, to measure it continuously.

2. Control actions are introduced by decision-makers on both levels. At the unit level, control actions determine both the routine inspection (control) point and the speed to proceed with until the next point; at the section level, decision-making centers both on optimal resource manoeuvring and target amount reassignment.

3. The developed control algorithm is easy to handle; it can be implemented on a PC. Simulating a section of a medium size (six production units are manufacturing five different products by using four types of resources) takes little computational time.

4. Besides introducing control actions, the developed two-level algorithm enables calculation, via simulation, of the average expenses for each combination of A, d and T. One has to attach to all speeds the processing costs per time unit for each production unit and each product, together with the costs of performing a single control and with the penalty cost for not accomplishing the section's amount at the due date. Thus, a combination of parameters A, d and T can be determined to minimize the expenses on accomplishing the section's output plan.

5. This paper is a further development of our previous publications where two-level production systems under random disturbances have been considered. Unfortunately, we could not compare the newly-developed control model with the former ones, since the latter are not generalized. Each of them fits only certain manufacturing scenarios and covers only specific control actions. Those models cannot be applied to a production system with multiple resources and products. Similar studies by other authors have not been published elsewhere.

6. The developed two-level control model can be introduced into existing manufacturing scenarios, namely, the JIT systems with uncertainty [8, 12]. This is an important class of JIT systems for which the number of publications remains very scanty.

7. Under emergency conditions the control algorithm periodically solves the resource and target

Man-machine production systems 415

amount reallocation problem. Solution can be obtained by using coordinate descent methods that are very attractive because of their easy implementation.

8. Future research can be undertaken as follows:

(a) in the future, the developed system has to be extended to three levels: production unit-section-factory (company). The main problem is to coincide control actions on different levels to optimize the objectives at the upper level;

(b) extensive experimentation has to be undertaken to examine various criteria in optimization problems. For example, for a general three-level control model, there are two alternative criteria:

- - to maximize the probability of the slowest unit meeting its deadline on time, and --to maximize the probability that the product with the least delivery performance can be

accomplished on time. The criterion's efficiency can be evaluated via simulation.

Acknowledgements--This research has been partially supported by the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University of the Negev. The authors are very grateful to the three anonymous referees for their helpful comments.

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in production planning, operations research in production planning and control. Proceedings of a Joint GermantU.S. Conference, Hagen, Germany, 25-26 June, 1992, 1993, p. 105.

4. Gabbay, H., Multi-stage production planning. Management Science, 1979, 25(11), 1138. 5. Golenko-Ginzburg, D., A two-level production control model with target amount rescheduling. Journal of the

Operational Research Society, 1990, 41(11), 1021. 6. Golenko-Ginzburg, D. and Sims, J., Controlling a two-level multi-product system. Proceedings of the 1990 Pacific

International Conference on Manufacturing, Vol. 1, Sydney-Melbourne, 16-21 Dec., 1990, p. 228. 7. Golenko-Ginzburg, D. and Sinuany-Stern, Z., Hierarchical control of semi-automated production systems. Production

Planning and Control, 1993, 4(4), 361. 8. Harrison, A., Just-In-Time Manufacturing in Perspective. The Manufacturing Practitioner Series, Prentice Hall, New

York, 1992. 9. Lefkowitz, I., Hierarchical control in large-scale industrial systems. Large Scale Systems, ed, Y. Y. Haimes. North

Holland, Amsterdam, 1982. 10. Luenberger, D. G., Introduction to Linear and Nonlinear Programming. Addison-Wesley, Massachusetts, 1973. 11. Mesarovic, M. D., Macko, D. and Takahara, Y., Theory of Multilevel Hierarchical Systems. Academic Press, New

York, 1970. 12. Pervozvanskiy, A. A. and Sheynis, I. Ye., Kanban system as a manufacturing control system with feedback. Journal

of Computer and Systems Science International, 1994, 32(4), 153. 13. Sawik, T., Hierarchical production planning and scheduling in a flexible manufacturing system. Presented at the l lth

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(Appendix Overleaf)

416 Dimit r i G o l e n k o - G i n z b u r g and Vladimir K a t s

A P P E N D I X

Initial Data of the System's Description

Table AI. Upper and lower bounds for resource capacities R,,I

Resources

Units

Resource # 1

Rk~mi. R ~

Resource # 2 Resource # 3 Resource # 4

R~Zm~° R,:m.. P~3.m Rk3... R...,. R.,..,

Unit 1 10 100 Unit 2 I 110 Unit 3 10 120 Unit 4 10 100 Unit 5 10 110 Unit 6 10 120

6 10 100 8 l0 110 8 10 120 6 10 100 8 10 119 8 10 120

Table A2. Random values a~,~ for resource # 1 (d = 1, uniform distribution)

Resource I Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

Product 1 j = 1 0.061-0.101 0.122-0.203 0.182-0,304 0.061-0.101 0.122-0,203 0,182-0.304 j = 2 0.0684). 113 0.135--0.225 0.20~0.338 0.068-0.113 0.135-0.225 0.2034).338 j = 3 0.075-0.125 0.150-0,250 0.225-0.375 0.075-0.125 0.150-0.250 0.225-0,375

Product 2 j = I 0.182-0.304 0.122-0,203 0,061-0.101 0.182-0.304 0.122-0.203 0.061-0.101 ] = 2 0.203-0.338 0.135-0.225 0,068-0.113 0.203-0.338 0.135--0.225 0,068-0. I 13 j = 3 0.225--0.375 0.15ff4).250 0.075-0.125 0.225-0.0375 0.15000.250 0.075-0,125

Product 3 j = 1 0,061-0,101 0.122-0,203 0,182-0.304 0,061-0.101 0,122-0,203 0.182-0.31M j = 2 0,068-0.113 0.135-0.225 0.2034).338 0.068-0.113 0.135-0,225 0,203-0.338 j = 3 0.075-0.125 0.1504).250 0,225--0.375 0.075-0.125 0.150-0.250 0.225-0.375

Product 4 j = I 0.182-0.304 0.122-0.203 0.061-0.101 0.182-0.304 0.122-0.203 0.061-0.101 j = 2 0.203-0.338 O. 135-0.225 0.0684). I 13 0.203-0.338 O. 135-0.225 0.068-0. I 13 j = 3 0.225--0.375 O. 15~0.250 0.075-0.125 0.2254).375 O. 150-0.250 0.075-0.125

Product 5 j = 1 0.061-0.101 0.122-0.203 0.182-0.304 0.061-0.101 0.122-0.203 0.182-0.304 j = 2 0.068-0.113 O. 135-0.225 0.203-0.338 0.068-0.113 O. 135--0.225 0.203-0.338 j = 3 0.075--0.125 O. 1504).250 0.225-0.375 0.075-0.125 O. 150-0.250 0.225-0.375

Table A3. Random values aj~,d for resource # 2 (d = 2, uniform distribution)

Resource 2 Unit l Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

Product I j = 1 3.645-6.075 2.430-4.050 1.215-2.025 3.645-6,075 2.430-4.050 1.215-2,025 j = 2 4.050°6.750 2,700-4,500 1.35002.250 4.050-6,750 2.700-4,500 1.350-2.250 j = 3 4,500-7.500 3.000-5,000 1.50002.500 4.500-7.500 3.000-5.000 1.500-2.500

Product 2 j = I 1.215-2.025 2,430-4.050 3.645-6.075 1,215-2,025 2,430-4.050 3.645-6.075 j = 2 1.35002.250 2.700--4.500 4,050-6.750 1.350-2.250 2.70(04.500 4,050-45,750 j = 3 1.50002.500 3.000-5.000 4.500-7.500 1.50002.500 3.000-5.000 4.500-7.500

Product 3 j = 1 3.645-6.075 2.430-4.050 1.215-2,025 3.645-6.075 2.430-4.050 1.215-2.025 j = 2 4.050-6,750 2.70(b4,500 1.35002.250 4.050~.750 2.700-4.500 1.35002.250 j = 3 4.500--7.500 3.000-5.000 1.50002.500 4.500-7.500 3.00005.000 1.50002.500

Product 4 j = I 1,215-2,025 2.430-4.050 3.645-6.075 1.21 5-2,025 2.430-4.050 3,645-6,075 j = 2 1.35002.250 2,700~.500 4.050-6,750 1.350-2.250 2.700-4.500 4.050-6.750 j = 3 1,500-2.500 3.000-5.000 4.50007.500 1.500-2.500 3.00005.000 4,500-7.500

Product 5 j = I 3.645-6.075 2.430--4.050 1.215-2.025 3,645-6.075 2.430-4,050 1.215-2.025 j = 2 4.050-6.750 2.700-4.500 1.35002.250 4.05(b6.750 2.7(gk4,500 1.350--2,250 j = 3 4,500-7.500 3,000-5.000 1.50002.500 4.500-7.500 3.000-5,000 1.500-2.500

M a n - m a c h i n e p roduc t ion systems

Table A4. Random values a~,,~ for resource # 3 (d = 3, uniform distribution)

417

Resource 3 Unit I Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

Product 1 j = 1 0.061-0.101 0.122-0.203 0.182-0.304 0.061-0.101 0.122-0.203 0.182-0.304 j = 2 0.068-0. I 13 0.1354).225 0.20341.338 0.068-0.113 0.135-0.225 0.203-0.338 j = 3 0.0754). 125 0.150-0.250 0.225-0.375 0.075-0.125 0.150-0.250 0.225-0.375

Product 2 j = 1 0.182-0.304 0.122-0.203 0.061-0.101 0.182-0.304 0.122-0.203 0.061-0.101 j = 2 0.203-0.338 0.135-0.225 0.068-0.113 0.203-0.338 0.135-0.225 0.068-0.113 j = 3 0.225-0.375 0.1504).250 0.0754).125 0.225-0.375 0.15ff4).250 0.0754).125

Product 3 j = I 0.061-0.101 0.122-0.203 0.182-0.304 0.061-0.101 0.1224).203 0.182-0.304 j = 2 0.0684). 113 0.135-0.225 0.203-0.338 0.068-0.113 0.135~.225 0.203-0.338 j = 3 0.075-0.125 0.15ff4).250 0.225-0.375 0.075-0.125 0.150-0.250 0.2254). 375

Product 4 j = I 0.182-0.304 0.122-0.203 0.061-0. I 01 0.182-0.304 0.122-0.203 0.061 4). 101 j = 2 0.203-0.338 0.1354).225 0.068-0.113 0.203-0.338 0.135-0.225 0.0684). 113 j = 3 0.2254).375 0.1504).250 0.0754). 125 0.225-0.375 0.1504).250 0.075-0.125

Product 5 j = 1 0.0614).101 0.122-0.203 0.182-0.304 0.061-0.101 0.122-0.203 0.1824).304 j = 2 0.068-0.113 0.13~0.225 0.203-0.338 0.068-0. I 13 0.135-0.225 0.2034).338 j = 3 0.075-0.125 0.15ff41.250 0.2254).375 0.075-0.125 0.150-0.250 0.225-0.375

Table A5. Random values a,r,a for resource # 4 (d = 4, uniform distribution)

Resource 4 Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6

Product 1 j = I 3.645-6.075 2.430-4.050 1.215-2.025 3.645-6.075 2.430-4.050 1.21 5-2.025 j = 2 4.050-6.750 2.700-4.500 1.350-2.250 4.05~6.750 2.700-4.500 1.350-2.250 j = 3 4.500-7.500 3.000-5.000 1.500-2.500 4.500-7.500 3.000-5.000 1.500-2.500

Product 2 j = 1 1.215-2.025 2.430~1.050 3.645-6.075 1.215-2.025 2.430-4.050 3.64~6.075 j = 2 1.350-2.250 2.700~.500 4.050~.750 1.350-2.250 2.700-4.500 4.050-6.750 j = 3 1.500-2.500 3.000-5.000 4.500-7.500 1.500-2.500 3.000-5.000 4.500-7.500

Product 3 j = 1 3.645-6.075 2.430-4.050 1.215-2.025 3.645-6.075 2.430-4.050 1.215-2.025 j = 2 4.05ff4i.750 2.700-4.500 1.350-2.250 4.050-6.750 2.70(I-4.500 1.350-2.250 j = 3 4.500-7.500 3.000-5.000 1.500-2.500 4.500-7.500 3.000-5.000 1.500-2.500

Product 4 j = I 1.215-2.025 2.430-4.050 3.645-6.075 1.215-2.025 2.430-4.050 3.645-6.075 j = 2 1.350-2.250 2.700-4.500 4.050~.750 1.350-2.250 2.70ff4.500 4.050~.750 j = 3 1.500-2.500 3.000-5.000 4.500-7.500 1.500-2.500 3.000--5.000 4.500-7.500

Product 5 j = 1 3.645-6.075 2.430-4.050 1.215-2.025 3.645-6.075 2.430-4.050 1.215-2.025 j = 2 4.05~6.750 2.700-4.500 1.350-2.250 4.050-6.750 2.700~.500 1.350-2.250 j = 3 4.500-7.500 3.000-5.000 1.500-2.500 4.500-7.500 3.000-5.000 1.500-2.500

Example. Let us de te rmine the m a x i m a l speed (j = 3) of unit # 5 (k = 5) to m a n u f a c t u r e p roduc t # 2 (c = 2), i.e., speed v352. A s s u m e tha t in the course o f resource real locat ion unit 5 utilizes resources with capacit ies Rs~ = 30, R52 = 6, R53 = 20 and R~ = 4. Us ing the nota t ion , we ob ta in speed v352 as follows:

v35z = ~ (a3s2a'Rs,t) = [U(0.150, 0.250)] '30 d = l

+ [u(3.0, 5.0)].6 + [u(o.150, 0.250)].20

+ [U(3.0, 5.0)].4 = U(37.5, 62.5)

with the ave rage va lue t~352 = 50.0 and with the lower boun d v~52 = 37.5.