CLOSED-LOOP PLANNING OF MULTISTAGE ... - CiteSeerX

Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

CLOSED-LOOP PLANNING OF MULTISTAGE PRODUCTION UNDER RESOURCE CONSTRAINTS Jean-Claude Hennet, Isabelle Barth` es

LAAS-CNRS, 7, Ave. du Colonel Roche, 31077 Toulouse C´edex, FRANCE e-mail : [email protected], ibarthes @laas.fr

Abstract. The objective of this paper is to address two important issues in multistage production planning : integration of production capacity constraints and responsiveness to demand and supply changes. A natural, although non classical way to tackle these problems is to build a closed-loop production policy based on a state representation of the manufacturing system. The positive invariance approach provides a methodology, a closed-form control law and efficient LP-based algorithms to solve the considered state feedback constrained control problem. Keywords. Manufacturing System, Production Planning, MRP, Positive Invariance, Linear Programming

1. INTRODUCTION In spite of their extensive use in Industry, MRP (Material Requirement Planning) and MRP II (Manufacturing Resource Planning) systems are often criticized for their poor performance and limited ability to generate feasible production plans. It is shown in this paper that, using a dynamic model of the production plant, multistage planning problem can be solved in an integrated closed-loop way respecting capacity constraints. To some extent, MRP techniques aim at implementing a reactive control policy using stock level measurements. But this goal is partly missed, because multistage production requirements are generally computed from the output of the APP (Aggregate Production Plan), which is solved in openloop, using oversimplified capacity constraints and uncertain data on forecasted demand. One of the main reasons for inadequacy of real production to real demand is the lack of integration in the planning process of the detailed production features, including product structure and resource capacity constraints. The most classical production planning scheme consists of solving the mediumterm APP problem and deriving from it a detailed production plan (see e.g. [2]). This scheme has been improved, for example in the OPT technique (Op-

timized Production Technology) [5], by merging the critical constraints of the detailed planning problems into the MPS (Master Production Schedule) [3]. Such a merging is automatically achieved using the discrete-time dynamic model presented in section 2. In addition, the solution of the APP is highly dependent upon future demand and production data. Replanning and receding (or rolling) horizon techniques are the most currently proposed methods for tackling this problem. But such updating techniques call in question the validity and even the meaning of the planning problem and above all, they are very heavy to implement. Therefore, the difficulty to accurately predict future production and demand considerably limits the validity of openloop production plans. In practice, some system feedback often has to be introduced to provide the production schedule with responsiveness. This often leads to local production adjustments, in particular the net change approach in MRP, with no global view of the production schedule. Safety stocks and hedging policies may provide the system with some degree of responsiveness, but they are unable to permanently respond to demand changes, production disturbances and machine failures. The Kanban technique can be seen as a feedback control technique, based both on

Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

production capacity, real stocks and real demands measurements. It is reactive and robust, but limited to manufacturing systems with a simple structure and a regular demand pattern. Rather than a reactive planning technique, it can be seen as an attempt to suppress the planning stage, sometimes with the drawback of shortsightedness. A natural outcome of this analysis is to try to apply feedback control techniques to multistage production planning problems. The difficuly of this approach typically relates to the following features: • a large number of variables, • the existence of different production delays, which complicate the dynamical model of the system, • many constraints, particularly on resource and stock capacity, • important levels of disturbances and uncertainties. As shown in this paper, a positive invariance interpretation of predictive control laws directly provides a closed-form feedback control law, able to satisfy all the problem constraints and requirements. The extended state approach proposed in section 4 allows for reformulating the stochastic constrained control problem as an invariance problem for a well-chosen region of the state space. The main advantage of the invariance approach for linear systems and polyhedral regions, is that the whole control problem can be formulated as a set of linear relations and solved by Linear Programming. Notations. By convention, inequalities between vectors and inequalities between matrices are componentwise. The absolute value |v| of vector v is defined as the vector of the absolute value of its components. A (componentwise) nonnegative matrix is a matrix having all its components nonnegative. Matrix I denotes the identity matrix, matrix O the zero matrix, vector 0 and vector 1 respectively the zero and the unity vector, all with the appropriate dimension. The advance operator, q, applies to vector sequences (vk ) under the form: qvk = vk+1 . The delay operator is the inverse operator, denoted q −1 , such that: q −1 vk = vk−1 .

2. THE PRODUCTION SYSTEM MODEL 2.1 The MIMO model The evolution of production and stocks is described by a discrete-time model. All the production leadtimes are supposed multiple of the elementary time period. Each time period is indexed by k, with k = 0, 1, .... To each product i , i = 1, ..., N is associated at period k: • an external demand dik , such that, in general, dik = 0 if i is not a final product,

• a production level uik , generating the output uik at period k + θi ; by definition, θi is the production lead-time for product i, • the delivery level at period k, zik , • the stock level at the end of period k, sik . In the considered production structures, it is assumed that there is a one-to-one correspondence between products and activities: each production activity has several input products and one output product. According to a given manufacturing recipe, the production of one unit of product i requires the combination of components j = 1, ..., N in quantities πji , for j = 1, ..., N . The production process is thus described by: • the input matrix Π = ((πil )), • the output matrix, which is either the identity matrix I in the static form describing the bill of materials, or matrix diag(q −θi ), representing the products lead times. For assembly systems, product structures are arborescent and can be decomposed into stages. Such production structures can be represented by Gozinto graphs [9], which typically characterize the field of application of MRP techniques. Fig.1 represents an elementary assembly structure (a) and its associated Gozinto graphs (b). The nodes of the Gozinto graph represent the prod2X

1

3X

1

2

2

2

3

3

3

(a)

(b)

Fig. 1. An elementary production transition ucts, and can be partitioned into levels. Level 0 is for final products, level l for products which are components of products of levels stricly less than l and of at least one level l − 1 product. Production stages are associated to node levels in the gozinto graph. By ordering products in the (non strict) decreasing order of their level, the input matrix Π becomes uppertriangular with zeros on its main  0 π12 . . .

 .. . . . . . . . diagonal: Π =  . .

π1N .. .

. πN −1,N 0 ... ... 0

  . 

Some connections between MRP and Input-Output models have been explored in [6] and related works. In order to produce the output uik at period k +θi , the components j = 1, ..., N are required in quantities πji uik at time k. The stock evolution equation for each product i takes the form: sik = si,k−1 + ui,k−θi − zik . (1)

Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

In equation (1), all the variables should be nonnegative, from their physical meaning.The first objective of the plan is to satisfy the demand as much and as soon as possible. Therefore, as long as the right-hand side of equality (1) remains nonnegative, the optimal PN delivery variable zik takes the value: zik = j=1 πij ujk + dik . To take this requirement into account, together with the possibility of backorders, the following modified balance equation can be written: yik = yi,k−1 + ui,k−θi −

N X

πij ujk − dik . (2)

with mean values d¯k = d, δ¯k = 0. A realistic assumption is that the disturbance vector δk is bounded and may take any value in a polyhedral domain defined by inequalities: −δ ≤ δk ≤ δ with δ ≤ d.

(8)

These constraints can be re-written as follows : δk ∈ R[L, λ] = {δ ; Lδ ≤ λ} · ¸ · ¸ I δ with L = , λ= . −I δ

(9) (10)

j=1

The relation between yik and sik will be clarified in the sequel, through relations (3).

Using decomposition (7) of dk , the demand tracking problem is transformed into a regulation problem by introducing the translated control vector: vk = uk − u∗

2.2 The nominal stock levels The existence of safety stocks is a classical technique to react to demand fluctuations. The nominal stock levels are usually related to the probability distributions of final demands. They can be computed so as to achieve the best trade-off between expected inventory costs and expected stockout costs over the planning horizon. A nominal safety stock level, s∗i ≥ 0 can be considered for each product. The actual stock level can be deduced from equation (2) as follows :  if yik ≥ −s∗i  sik = yik + s∗i sik = 0 if yik ≤ −s∗i (3)  with a backorder of − yik − s∗i . Equation (2) can be re-written in a vector form, using the delay operator q −1 : (1 − q −1 )yk = T (q −1 )uk − dk

(4)

with by definition T (q −1 ) = diag(q −θi ) − Π (5) N

(11)

where u∗ is the steady-state nominal control vector associated with the mean demand vector d: u∗ = [diag(q −θi ) − Π]−1 d = (I − Π)−1 d (12) Assuming that the production system can be represented by an assembly graph, matrix Π can be ordered as an upper-triangular non-negative matrix with zeros on its main diagonal. Then, clearly, matrix (I − Π)−1 is well-defined and non-negative, and since vector d is also non-negative, the following lemma can be stated. Lemma 1 Under the assumption that the production structure admits an assembly graph representation, vector u∗ is well-defined by relation (12) and nonnegative. Using the change of variable (11), equation (4) is re-written in the form: (1 − q −1 )yk = T (q −1 )vk − δk

(13)

2.4 The constraints of the problem

The system delay is defined by: θ = max θi . i=1

As an example, consider the elementary production structure of figure 1, with lead-times θ1 = 2, θ2 = 3, θ3 = 3. The system delay is θ = 3 and  −2  q 0 −2 T (q −1 ) =  0 q −3 −3  . (6) 0 0 q −3 2.3 The steady-state nominal control It is clear from the MIMO representation (4) that the production planning problem can be seen as a control problem with yk as the output vector, dk as an external input vector and uk as the control vector. Assuming the demand stationary, the external demand vector can be decomposed as follows: d k = d + δk

(7)

From (11), nonnegativity of the production vector uk can be achieved by imposing the constraint : vk ≥ −u∗

(14)

For primary and intermediate products, the nonnegativity constraint on stocks is: yik ≥ −s∗i ∀k ∈ N .

(15)

The possibility of having backorders is normally restricted to final products. For these products, constraint (15) may be relaxed. Production resources may be manufacturing units, machines, work teams, raw materials, pallets, ... Aggregated production capacity constraints are associated with the R resources, and similarly, stock capacity constraints are associated with the P storage zones. Variable uik represents the production release order for product i at time k. Production is

Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

actually running at periods k, ..., k + θi − 1, and delivered at period k + θi . Assuming a constant use of resources along the production cycle, resource capacity constraints take the following form, for r = 1, ..., R, k = 1, ... : θi N X X mri ui,k−l ≤ Mkr (16) i=1 l=1

mri is the amount of ressource r currently needed to produce one unit of product i. Mkr is the capacity of resource i at period k. By convention, uk = 0 for k = −θ, ..., −1. If backorders are not allowed, storage capacity constraints take the form : N N X X p p ni yik ≤ Nk − npi s∗i (17) i=1

i=1

npi is the capacity at stock p required for one unit of product i. 3. RECEDING HORIZON APPROACHES The detailed plan seeks to optimize the sequence of detailed production decisions from period 0 to the horizon time T under a deterministic demand pattern. Because of the linearity of relation (13) and of constraints (14), (15), (16) and (17), the optimal planning problem can be solved by Linear Programming and implemented according to the receding horizon procedure, with only the first control vectors actually applied to the system. A possible to be minimized is: PT linear PI cost criterion + − + pi yik ). Variables J = k=1 i=1 (ci uik + hi yik + − yik , yik are respectively the positive and the negative part of yik . ci is the unit production cost, hi is the unit storage cost of product i and pi the unit relative shortage cost. If demand and supply predictions are accurate over the planning horizon T , this approach provides the optimal transient solution of the planning problem. But the problem has to be frequently run, to respond to disturbances. The predictive control approach is also a finite horizon approach, which can be implemented as a receding horizon policy. Its main objective is to drive the system towards its nominal operating conditions, characterized by: ½ yk = 0 (vector of safety stocks: sk = s∗ ≥ 0) . vk = 0 (nominal production vector: uk = u∗ ≥ 0) It is able to anticipate the future effects of past decisions and disturbances. The basic criterion in predictive control is the θ-steps minimum variance criterion: J = E(kyk+θ k2 |k) (18) In this framework, the transient response for 0 ≥ k < θ − 1 is not optimized. In this sense, this approach is complementary from the preceding optimal approach, which rather focuses on the transient trajectory. To determine a prediction equa-

tion in a simple way, matrix T (q −1 ) is decomposed as follows: T (q −1 ) = T0 + q −1 T1 + ... + q −θ Tθ where T0 = −Π, (Tj ) = diag(tji ) with tji = 1 if and only if θi = j, tji = 0 in any other case, ∀j = 1, ..., θ. Assuming now that the set points are reached within θ periods, the expected value of future control vectors is null : vˆk+1 |k = . . . = vˆk+θ = 0. Defining, for j = 1, ..., θ, ½ j θ X ei = 1 if j ≤ θi j , Ej = Tl = diag(ei ) with eji = 0 if j > θi l=j

the θ-step ahead output prediction at the beginning of period k is : θ X yˆk+θ|k = yk−1 + (I − Π)vk + Ej vk−j . (19) j=1

Under the minimum variance predictive scheme, the optimality condition is yˆk+θ|k = 0. The corresponding control vector at period k is : θ X vk = −(I − Π)−1 (yk−1 + Ej vk−j ). (20) j=1

If the control vector at time k0 , given by (20), is feasible, that is if all the constraints (14), (15), (16) and (17) are satisfied at each instant, this control law is clearly optimal at any time k ≥ θ. Several techniques have been proposed to integrate constraints in a predictive control scheme. One possibility for control constraints is to superimpose a saturation operator upon the expression (20) of vk . More complete and satisfactory treatments of constraints can be found in the literature on predictive control (see e.g. [8], [1], [?]). They consider quadratic criteria more general than (18), in particular in the classical frameworks of GPC (Generalized Predictive Control) and MPC (Model Predictive Control) . In any case, the treatment of constraints considerably complicates the predictive control approach, as long as a receding horizon scheme is involved for practical implementation. It will now be shown that a generalized form of the control law (20) can be efficiently applied as a closed-loop control law respecting the constraints.

4. THE D−INVARIANCE APPROACH 4.1 The extended state model The control law (20) obtained by the predictive control approach can be seen as a state feedback law, with respect to the following state vector xk = [yk−1 vk−1 · · · vk−θ ]T

(21)

Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

More general state feedback laws can be investigated, to combine the demand tracking requirement with output and control constraints : vk = [F G1 · · · Gθ ]xk

(22)

The dynamic closed-loop state equation associated to a feedback (22) is as follows : xk+1 = Axk + Dδk with

 I + T0 F F O . . .

  A=  

T1 + T0 G1 · · · · · · Tθ + T0 Gθ G1 ··· ··· Gθ I O ··· O . . .. . . .

.

.

. ···

O

.

. O

I

(23) 

 −I 

    , D =     

O . . . . . .

  .  

O

O

System (23) is subject to the set of linear constraints (14), (15), (16) and (17), written in the state form xk ∈ R[Q, ρ] = {x; Qx ≤ ρ} · with Q =  Q1 =

N O  .  .. O

O M1 .. . ···

··· O · · · Mθ . .. . .. O M1

Q1 Q2



· ¸ ρ , ρ = ρ1 , 2 

  , Q2 = 

 

and ρ1 =

¸

ν µ  . ,  ..  µ

(24)

−I O · · · O . .   O −I . . ..  . . .  . .. .. . O O · · · O −I



ρ2 =

s∗  u∗  .  .. u∗

   , 



 . 

The D−invariance approach is a generalization of the positive invariance approach to systems subject to bounded disturbances [4]. By definition, a domain Ω is D−invariant with respect to system (23) if and only if any trajectory of (23) with initial state in Ω is entirely contained in Ω. This approach is able to provide local solutions to constrained control problems by imposing the following conditions: (1) The zero state lies in the interior of Ω, (2) Ω is D−invariant with respect to the controlled system, meaning that any state trajectory starting in Ω remains in Ω, for any disturbance vector δk in R[L, λ]. (3) Ω ⊂ R[Q, ρ], where R[Q, ρ]] describes the polyhedron of constraints. A set Ω which satisfies these three conditions is a set of admissible initial states for the constrained closed-loop system (23).

4.2 The direct approach Among the three conditions of section 4.1, the first and third ones are directly satisfied by the choice:

Ω = R[Q, ρ]. However, in general, it is not possible to get the second condition satisfied for a given polyhedron. To test if R[Q, ρ] can be made positively invariant with respect to system (23), it suffices to test the conditions of the following lemma (see e.g. [4]). Lemma 2 A necessary and sufficient condition for D−invariance of R[Q, ρ] with respect to system (23) and for any disturbance vector δk in R[L, λ], is the existence of control matrices F, G1 , ..., Gθ and of two nonnegative matrices H and J such that : HQ = QA (25) JL = QD Hρ + Jλ ≤ ρ

(26) (27)

These conditions can be directly tested by Linear Programming. A possible objective function to be minimized is the contraction rate ² obtained when replacing condition (27) by : Hρ + Jλ ≤ ²ρ

(28)

Note that for L, λ defined by (10), an arbitrary choice of matrix J satisfying (26) can be made without changing the solution. For matrix Q having dimension (q × (N + 1)θ) (with typically q = 2(N + 1)), the number of variables of the LP is q 2 + (N + 1)2 θ + 1. This number is relatively important for complex processes, but it only depends on the production structure, and problems of that size can easily be solved by current LP codes. Invariance conditions are then satisfied if the optimal value satisfies ²∗ ≤ 1. Then, the solution F, G1 , ..., , Gθ defines an feasible control law, through relation (22). However, in general, the direct approach, characterized by the choice : Ω = R[Q, ρ], may not yield a feasible solution. Other possible choices for Ω should then be investigated. 4.3 The indirect approach System xk+1 = Axk is stabilizable. Then, for this deterministic system, the constrained control problem can always be solved locally by constructing a polyhedron Ω = R[S, s] which satisfies the three conditions of section 4.1. This property is also valid for system (23), subject to bounded disturbances, as stated as follows. Proposition 1 D−invariance of certain well-chosen compact polyhedral domains of the state space, R[S, s], can always be achieved with respect to system (23), under an appropriate choice of the gain matrices F, G1 , ..., Gθ of the control law (22). Proof Consider the predictive control law (19)defined by the following gain matrices : F = −(I − Π)−1 , Gi = −(I − Π)−1 Ei for i = 1, ..., θ.

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Proc. 9th Symposium of the IFAC on INformation COntrol in Manufacturing (INCOM’98), Nancy (France), 24-26 Juin 1998, Vol.II, pp.485-490

A class of associated domains is de· ¸ symmetrical · ¸ Σ σ fined by: S = −Σ , s = σ , with 

I E1  O −(I − Π)

  Σ=  .  ..

O

O

···



··· O

··· Eθ ··· O   .. ..  −(I − Π) . . ,  ..  . O O −(I − Π)

  α

α .   σ =  ..  .  ..  α

N

and α a positive vector in < . By application of Lemma 2, D−invariance of R[S, from the  s] derives  O ··· ··· O

.    I O · · · ..  existence of the matrix H =  . ,  . ... ...  . O ··· I O

which satisfies HΣ = ΣA, and from the choice of any positive vector α ∈