Fuzzy Control of Production Planning in Distributed Manufacturing Systems

Copyright © IFAC Distributed Intelligence Systems, Virginia, USA, 1991

FUZZY CONTROL OF PRODUCTION PLANNING IN DISTRIBUTED MANUFACTURING SYSTEMS T. L. Ward* and P. A. S. Ralston** *Department of Industrial Engineering, University of Louisville , Louisville, KY 40292, USA **Department of Engineering Mathematics and Computer Science, University of Louisville, Louisville, KY 40292, USA

Abstract. A fuzzy system for production planning in a multi-item manufacturing fa cility is discussed. The mathematical model of the fuzzy control of aggregate production planning is formulated and extended in such a way as to permit it to be applied to the multi-item case. This extension is discussed in terms of its appli cation to operating production systems, its robustness, and the interaction between manufacturing centers. Keywords. Fuzzy control; fuzzy systems; inventory control; linear quadratic control; management systems; production control.

[SJ developed heuristic control laws. There have been optimal control theory formulations of the HMMS model in both continuous time [9, 10J and discrete time [l1J. Taubert [12J converted the HMMS objective function to a 20 dimension response surface and used a search method to find the minimum cost. The HMMS model treats either single item,

INTRODUCTION Manufacturing production involves integrated systems of people, materials, equipment, and energy. An important part of the control of such systems, production planning, has the function of setting the overall level of manufacturing output [1 J. During the first half of this century the control of production planning was relatively informal. Strategy was set at an upper organizational level, for example by the manufacturing superintendent. Implementation was at lower levels , for example at that of the first line supervisor or foreman. Such human organizations were, in the sense of A. H. Levis [2], distributed in tell igence systems . About the middle of the cent ury formal production planning systems began to be investigated, created, and implemented. These were typically aggregate planning systems . Thus, at least with regard to production planning, utilized intelligence became more concentrated. Now, with the emergence of distributed computer-integrated manufacturing, it becomes important to utilize distributed production planning systems once again , but ones that are based on mathematical models and testable hypotheses.

or aggregate multi-item production; there have been explicit extensions to multi-item production [13, 14, 15J. Lee and subseq uent workers [16 , 17 , IS, 19, 20J formulated production planning as a goal programming problem. Rinks [2 1, 22J applied fuzzy set theory to the HMMS model. Although not exp licitly extensions of the HMMS model , there have been a number of formulations that are free of the assumption of quadratic cost functions; a general discussion will be found in Bensoussan, Crouhy, and Proth [23J. Mathematical programming has been applied to the general cost model [24). There have been some initia l steps toward the creation of product ion planning systems that can serve distributed manufacturing systems. These a re sometimes called multiplelevel , or hierarchical systems. Zangwell [25) and others have proposed' the use of dynamic programming. Steinberg and :'\apier [26J have given a constrained network formulati on. Afentakis , Gavish, and Karmarkar [2i) gi\'e a branch-andbound algorithm using Lagrangian relaxation. \1ost workers agree that existing production planning models are not being implemented in practice. Various explanation have been advanced. Jon es [S) suggested that it is difficult to force .. the managers's understanding of the relationships in his firm to the strict mathematical forms required by the optimizing techniques." Taubert [12) apparent ly feels that the decision makers distrust the quality of the model itself. Graves [28) notes that "production schedulers claim (with some just ifi cat ion) that their scheduling setting is not only unique, but sufficient ly different from any other setting to require a problem-specific solut ion. " Rinks [22J says that the solution '·techniques and their associated cost models simply do not adequately represent the realities of their operations. ,.

PRODUCTION PLANNING In 1952 Simon [3J app lied simple servomechanism theory to obtain a solution that sat isficed, but did not optimize the production planning problem. Starting in the mid-50s Holt, ~lodigliani, ~luth, and Simon [4, 5, 6J developed what has come to be called th e H\n-IS model of production planning. The H\nIS model is notable in two respects . First, by applying the calculus of variations to quadratic cost functions, an optimal solution was obtained. Second, t he so luti on was demonstrated using real , rather than contrived , data. There ha\'e been numerous modifications , extensions, and alternate solutions. Bowman [i J replaced the coefficients in the H\I\IS linear control laws with management coejJicifnts obtained by the analysis of the behavior of actual human managers. Jones

97

It is clear that there is a gap between production planning theory and practice. Neither devising new algorithms nor selling existing algorithms to management has been productive. The dynamics of the production planning environment itself mu~t be considered. Genetic algorithms which can optimize system behavior in response to particular en\'ironment s are ideal candidates to close the gap between theory and practice. In his application of fu zzy logic control to production planning, Rinks [2 1. 22]used linguistic variables. of the sort proposed by Zadeh [29]. for cost. inventory, production, and work force. He operated on these with fuzzy if-then rules in a manner like that used by l\lamdani and Assilan [30]

ment during the I-th month , IVt is the number of employees in the work force during the I-th month , and the resulting Hl\Il\IS decision rules are

Pt IVt

J(1t-l , Ft. W t_l ) g(1t-l. Wt-rl

(3) (-l)

where Ft is a linear combination of fi" 5t + I , . . . • 5t+ s [-l . 8], the forecasts of the numbers of units ordered in the (I + 11)th period . These 5 are an X-period ahead forecast made at the beginning of the I-th period. \Vhen such a formulation is \'iewed as a feedback control system, the input to the system is the order sequence {S,}. The controller (the decision rules) produces manipulated sequences {Pt} and {IV,} as inputs to the plant. In the Hl\IMS model the cost functions are quadratic, the overall formulation is that of a linear quadratic control problem, and J(.) and g(.) are linear decision rules (e.g .. see [40]).

to control a steam engine. Rinks membership functions and rules were subjectiw creations. Nevertheless, he obtained results that compared favorably to both prior and subsequent heuristic treatments of the Hl\l~S data. There is a general feeling that the performance of fuzzy logic controllers can be improved by improving the quality of the membership functions and the rule bases that manipulate them. In recent res ults. Buckley [31] has given a sort of central limit theorem for fuzzy controllers that shows when linear control rules are employed. and when the number of rules grow , the defuzzified outputs become linear functions of the inputs to the fuzzy controller. Ying , Siler, and Buckley [32] have shown examples in which the performance of fuzzy and nonfuzzy linear PI controllers were almost the same for linear processes, and in which the fuz zy controller produced superior resu lts for a nonlinear process. Karr , l\leredith , and Stanley [33] at the U. S. Bureau of 1\!ines Tuscaloosa Research Center have constructed a fuzzy logic controller for the control of liquid level and are using genetic algorithms to increas e p erformance by learning new membership functions. It should be noted that many investigators recommend that tuning of the controller be done by changing the rule base rather than modifying the membership functions. Whether to apply genetic algorithms to optimize the rule base or the membership functions is an open quest ion. Yen [34] has described a hy brid architecture in which fuzzy inference rules are used to make plausible inference, and neural networks are viewed as modules that perform flexible classifications from low-level input data. Keller and Yager [35] show a structure in which the knowledge of the rules are explicitly encoded in the weights of the neural network . Jamshidi and Vadiee [36] suggest that artificial neural networks and fuzzy logic paradigms may be integrated to both learn and tune membership functions. and to learn , tune. and edit the inference rules. Kosko [37. 38 . 39] has used neural adaptive vector quantization to train adapti\'e fuzzy controllers.

APP Using Fuzzy Sets Rinks [21, 22] used manager protocols (decision rules) written in simple language statements like " If the sales forecast is high , then production should be not low," to build fuzzy algorithms. One algorithm related input the variables of sales forecast X , period 1-1 work force le\'el Y , and period I - 1 inventory level Z to the first output variable, period t change in work force level IV. Another algorithm related the three input variables to the second output variable, period I production level P. He was able to closely approximate the results of much more powerful optimizing techniques for the aggregate planning and scheduling problem when applied to the classic HMMS paint factory data. Although crisp mathematical models of aggregate planning abound, implementation has been sparse [8,12 , 22,28]. Managers use their own rules which may not guarantee optimality, but do well and are robust. Rinks 's contribution was to use fuzzy set theory to build a fuzzy controller for a process. The input variables analogous to the X, Y , and Z used here are Ft. IVt-l, and 1,-1 , and the output variables that correspond to the Wand P used here are t::,. Hit and P,. Building the Fuzzy Algorithm First, names of labels such as posilive very big or sorl oj low are selected to describe the values the input and output \'ariables take on. Rinks used 11 labels. These values are represented as fuzzy sets. These fuzzy sets are defined on the finite uni\'erses X. Y . Z. \V. and P. where these are the three input \'ariable and the two output variable universes respectively.

AGGREGATE PRODUCTION PLANNING Defined defined defined defined defined

The aggregate production planning (.\ rp) formulation depends on the particular manufacturing sys tem being modeled. The Hl\ll\lS formulation minimizes the expected cost E{C} = E{C(It . Pt.Wt- l . \l'tl}

(1 )

+ Pt - St

X Y Z

W

P

may may may may may

be be be be be

fuzzy fuzzy fuzzy fuzzy fuzzy

sets sets sets sets sets

XI , Xz . ·· .. X n , 'y~ , }'; ....• }~ ,

ZI, Zz ....• Z,.

11'1. II'z. .... 11',. and PI' Pz, .. .. P,.

Just as a crisp value is used for a variable in a traditional mathematical algorithm, fuzzy values represented as fuzzy sets x;. etc. are used in fuzzy algorithms. (These fuzzy values are gi\'en names or labels.) The fuzzy sets. for example X. have membership functions. for example X(Xj) . etc .. defined over the finite range of the universes . for example j = 1, . ... 111' All universes were scaled to the [-1 , 1] interval. Exponential functions of Ostergaard [41 J were used to model membership functions.

subject to

It = It-I

on on on on on

(2)

\\·here: I, is the number of units of inwntory minus the number of units on back order at the during and at the end of the I-th month. Pt is the number of units produced during the I-th month . St is the actual number of units ordered for ship-

98

The values for the number of divisions in each universe nI, 112. 113' 11 • • n5 are all the same. The 40 rules representing the managers ' protocol were written as conditional statements: IF (Xi

A~D

The fuzzy output represented the algorithm's conclusion or decision. This output was made crisp by the maximum rule - namely, the crisp output was that \'alue of the output \'ariable having the maximum membership value. If the membership function did not have a unique maximum. other heuristic rules were used to chose the crisp value of the decision \'ariable (IV or P).

Y; A:\D Zi) THE!\, I\·i.

for the work force algorithm and 40 rules of the same form. IF (Xi

A~D

Y i A!'>'D Zi)

THE~

Pi'

Results and Comparison

for the prod uction algorithm. The su bscript i varies from 1 to 40. The \'alues for X. Y, Z, IV , and P could be any of these 11 . p, q, r. or s fuzzy sets defined for input of output.

Rinks obtained results [21] that compared favorably with H:\!:\1S linear decision rule model and later [22 ] shO\\'ed the fuzzy algorithm to be robust with respect to cost structures. Zimmermann [42] cites a study by Bowman [7] to show that the results obtained by Rinks compare favorably with actual management performance (40% higher costs) in the same case, and with the results that would ha\'e been obtained using the method of management coefficients (25'7c higher costs ). Rinks used 11 membership functions and corresponding linguistic terms. Although not all terms were used with each fuzzy \'ariable, there was the possibility for 11· = 14641 cells in the product space X x Y x Z x W. Since each rule occupies a point in that space, only 40 of the 14,641 cells are occupied. It is possible that a richer rule base might result in imprO\'ed performance.

but are those associated with the i-th rule. The 40 rules are connected by "OR ELSE." For example, A~D

IF (X,

Y,

A~D

Zrl THE\, IV,

OR ELSE IF (X2 AND Yi AND Z2) TI-IE\, P2 OR ELSE etc. To shorten the presentation. only the work force algorithm will be explained. A similar procedure is done for the production algorithm. Each rule represents a fuzzy point in a mapping from the Cartesian product space X x Y x Z to W. The fuzzy point is a subset of the Cartesian product space X x Y x Z x W. The collection of fuzzy points for each rule generates the overall rule relation. Each rule gives a fuzzy point Xi x li X Zi x IV. called Ri.

Multi-Item Fuzzy Production Control The HMMS production planning problem is an aggregate one. That is , if more than one item is being produced , the sales, production rates , and work force are aggregated and it is not possible to soke directly for the optimum plans for each item. If aggregate planning is us ed , then there must be a breakdown into individual item plans as a separate step.

The "or else" is modeled by the union operator. The overall relation is then

The H}'[MS model has been explicitly extended to the multiitem case by a number of workers [13, 43 , 44]. Each item, the i-th item for example, has its own inventory, lit , production level , Pit, sales, Sit, and work force , IVit . The problem then becomes to minimize the expected cost

40

R

= U(x. x Y;

X

(5)

Zi x Wi)

i=1

The membership function for the Cartesian product is computed by the minimum operation. The membership function for overall relation R is computed using membership functions of the inputs and output

=

R(Xj,Yk,Z/,W m )

subject to

40

V (Xi(Xj) 1\ }i(Yk) 1\ Z;(z/) 1\ IFi(w

(11)

(6)

m ))

for i = 1, .... n . Given the cost functions and a knowledge of the sales {Si,} for each of the n items owr some planning horizon {I = 1. ... , T}. it is possible to calculate an optimum set of prod uction levels {Pit} and work forces {lI'it} O\'er all T periods. lJ nlike the H:\!~!S linear decis ion rules for the aggregate case. there have not been formal calculations of single I-th period values of Pit and lVit . E\'en heur istic methods for the single period case are missing from the lit erature. But it is known that linear control rules are pos sible when the cost functions are quadratic [4 0] (see [13] for a discu ss ion of n-item costs that follow the H:\l:\rS model and that are quadratic). Of course working solutions are obtained in actual production facilities. There are several such solutions that are in the spirit of H:\l:\!S . One has already been alluded to. That is. the aggregate problem is solH'd and then broken down into individual item plans. A second. not unrelated approach. is to prepare 11 individual plans under the assumption that the item plans are disjoint and then to aggregate the plans adjusting them on the basis of human judgement. What is needed is a heuristic that treats all 11 items together and produces a single period plan with no adjustment.

i=1

wherej = 1. .... n', k = 1. ... ,n2, I = 1. ... , n3. and m = 1..... n4' The fuzzy output IF was obtained using the compositional rule of inference. The membership function for output W(w m ) is computed as nl

IF{lL'm)

n2

n3

= V V V [ X(~')) 1\

}' (Yd 1\ Z(2d

j=1 k=11=1

1\ R( r), n1

"2

Yk.

n3

( 7)

2/. lC m )]

40

V V V V [X(r)) 1\ y(Yd 1\ Z(2d )=1 k=l 1=1 i=1

I\Xi(X))

2

1\

}i(Yk) 1\ Zi(2/)

[Wi(u'm)

1\

(~yqX))

1\

C~Y'(Yk) 1\ }i(Yd])

1\

(2[Z(zd

1\

1\ Wi(U' m

Zi(zd])]

1\

)]

(8)

Xi(X))])

(9)

99

The second working solution described above, that of preparing n individual disjoint plans, can be approached by means of the fuzzy solution described above. This would involve n sets of rules defined over n seperate (3 + 1) = 4 di· mensional product spaces (the universes of the three inputs F" I" W' _I , and the one universe of the output P, or IV,). The individual rules would have the form

REFERENCES

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(12)

where R;j is the j-th W rule for the i-th item. In order to acknowledge that the item production systems and cost functions are not disjoint. 2n sets of rules (n for {P,,} and 11

[2J A. H. Levis, "Human organizations as distributed intelligence systems ," in Distributed Intelligence Systems: Methods and Applications, (D. Mladenov, ed.), (Oxford) , pp. 5-11, International Federation of Automatic Control, Pergamon, June 1988. Selected papers from the IFAC/I~IACS symposium. Varna, Bulgaria.

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=

X lj

X

xX2j

Ylj

X Zlj

X

Y2j

X Z2j

X

Ynj

X Znj X W;j

[3J H. A. Simon, "On the application of servomechanism theory in the study of production control," Econometrica, vol. 20, pp. 247-268, 1952.

x·· . xX nj

(13)

[4] C. C. Holt, F. Modigliani , and H. A. Simon, "A linear

where R j is the j-th IV rule (in this case, yielding IV;). The performance of the 2n sets of rules over the two (3n + 1) product spaces could be compared to the calculated performance [13 , 43, 44J over a T period planning horizon. An initial subset of rules could be obtained based on man agement protocols as described above. These would be th e rules that would lead to the disjoint solutions. Additional rules that would play the role of the human judgement that would adjust the disjoint plans could be obtained , albeit with considerably more difficulty if done on an hoc basis. A systematic approach is required. It has already been noted that the Rinks rule base although formidable in human terms (40 rules each for P, and IV,) seems to be quite sparse in comparison to other fuzzy logic control systems. In order to build the rule base quickly a nd efficiently, and to have it capable of being adapted to changing production system environments, some sort of machine learning is required. Either an adapti"e vector quantizer using differential competitive learning [37, 38 , 39], or genetic algorithms [33, 45J have been used, or are being used for smaller problems and can be expected to serve in this case.

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DISCUSSION AND CONCLUSION

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Rinks [21J has applied fuzzy set theory to the control of production planning on an aggregate basis . A mathematical model that embraces that approach and that can be extended to n-item production is set out here. An initial rule set of the sort used by Rinks can be based on a manager 's protocol that is appropriate to each manufacturing center or item. The union of these initial rule sets will provide n disjoint single-item production plans. \lachine learning will enrich the rule set so as to introduce interaction between item plans. The diagnosis and evaluation of operating production planning systems based on fuzzy logic operations of the rule base can be facilitated by rule traces of each single period production plan [46J. The robustness of planning using fuzzy control can be examined identifying the vulnerable rules or by removing them entirely [39J. The effects of interaction between manufacturing centers can be accessed by alternately adding and removing the "enrichment"· rules added to the initial disjoint sets of rules. The "alue of the information provided by a shift from aggregate production planning to n-item production planning can be examined by seperately evaluating the cost functions using both the initial rule set and the enriched rule set .

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