A Bi-Modal Scheme for Multi-Stage Production and Inventory Control

Transcription

1 Automatica 39 (3, pp A Bi-Modal Scheme for Multi-Stage Production and Inventory Control Jean-Claude Hennet LAAS - CNRS 7, Avenue du Colonel Roche, 377 Toulouse Cédex 4, FRANCE PHONE: ( , FAX: ( , hennet@laasfr Abstract This study considers a multi-stage multi-item production plant with its supply chain and customer environment The production, supply and inventory plan is optimized on a dual-mode basis, under two different information patterns The short-term plan relies on firm orders received from customers On the contrary, the long-term plan is based on predicted demands represented by random sequences In this study, the role of the long-term plan is mainly to impose a final condition set to the short-term plan Keywords Production planning, Multi-stage production, Linear Programming, State feedback, Model Predictive Control Introduction A major difficulty in production planning relates to the choice of the time-horizon As a medium-range tactical tool for a company, a production plan should provide reference production values over time not only to satisfy the current order list from customers, but also to anticipate future orders which have not yet been placed Accordingly, the planning time horizon is generally selected longer than the horizon for which the data are considered frozen Future demands which are expected or imperfectly known at the time of computation are replaced by their forecasts (see eg (Thomas and McClain, 993 and the references therein Due to the heterogeneity of its input data, the planning problem may then be decomposed into two subproblems: one which is deterministic over a short-range horizon and one which is stochastic and covers a longer time horizon However, such a decomposition generates an additional problem: when and how to link up and sequence the solutions of the two problems? In agreement with this observation, this paper proposes to solve the infinite horizon planning problem in two parts:

2 Automatica 39 (3, pp (i resolution of a finite horizon production planning problem under deterministic demand and a terminal constraint For the considered product structures, this problem can be formulated and solved as a Linear Program (ii construction of a stationary production and supply policy to optimally react to bounded random variations of demand for end products, through a robust closed-loop inventory control policy Under a rolling horizon implementation of the plan, only the finite horizon solution is applied, but the terminal constraint, associated with feasibility of the closed-loop policy, guarantees system stability and optimality with respect to the infinite horizon problem Such a stability condition combined with a finite-horizon rolling policy is somewhat similar to the one encountered in MPC (model predictive control schemes (Mayne et al, The closed-loop control approach used in this paper recovers most of the classical results on optimal inventory policies for multi-echelon systems (Clarke and Scarf, 96, (Federgruen and Zipkin, 984, (Porteus, 99 It is based on similar assumptions in terms of production costs, assumed proportional to quantities, and in terms of lead-times, supposed fixed or random, but independent of the quantities produced or ordered In this respect, the main originality of the approach proposed is that it considers general multiproduct structures directly, through vector and matrix representations, rather than as an extension of the single period single item model The remainder of the paper is organized as follows Section presents the equations and constraints defining the model Then, section 3 describes the optimization problem and focuses on the resolution of the infinite horizon planning problem It is proposed to construct an invariant domain in which the system trajectory can be maintained under any admissible demand vector trajectory This invariant domain is then used as a target domain in the short range planning problem studied in section 4 Section 5 presents an example to illustrate the approach, and the final section contains some concluding remarks The production and supply model Most of the literature on inventory theory under stochastic demand and fixed lead-times (Porteus, 99 first solves the single-stage case, extends the classical base stock policy result to series multi-echelon structures, using the decomposition proposed in (Clarke and Scarf, 96, and then to assembly systems, by constructing an equivalent series system (Rosling, 989 In particular, this approach allows to determine conditions on initial stocks under which an assembly system can be transformed into an equivalent series system (Federgruen, 993 More complex multi-stage product structures can be represented by acyclic graphs which are neither trees nor anti-trees The purpose of this study is to construct a dynamic model for such systems and to design a production and inventory control policy directly from this model Typically, the considered product structure involves primary products, ordered to suppliers, intermediate products internally produced and consumed, and final products delivered to customers The production planning problem is considered at the aggregate level where products are aggregated into product types and time-periods are much longer than set-up times Under such assumptions, demand,

3 Automatica 39 (3, pp production and inventory quantities at each period take large integer values which can be accurately approximated by real numbers (see eg (Bitran and Tirupati, 993 Practical implementation of results of the aggregate plan then require a disaggregation process, not studied in this paper The bill of materials According to a given manufacturing recipe, the production of one unit of product i requires the combination of components j n in quantities π ji, for j n In the considered production structure, each production activity has several input products but only one output product The bill of materials can thus described by: the input matrix Π π il R n n, the output matrix, which is the identity matrix I R n n Such product structures typically characterize the field of application of MRP techniques Classically, it is assumed that the product structure has no cycle It can then be decomposed into levels : level products are the p end products, numbered from to p Then, intermediate and primary products are numbered in the increasing order of their level The level of product i, for i p n is the maximal number of stages to transform product i into a final product Matrix Π is lower triangular under such a level-consistent ordering of products Let d be the nonnegative vector of external demands over a given time period The gross production/supply vector, u, necessary to satisfy this demand is related to d through the following Leontief-type equation (Veinott, 969: I Π u d ( The structure of the net production matrix I Π implies that its inverse exists and is nonnegative Thus, vector u is uniquely defined by : Note that matrix I Π integer, as it is generally the case, then I Π u I Π d ( is the sum of the powers of matrix Π If all the terms of Π are nonnegative is also a matrix of nonnegative integers And thus, since the entries of d are nonnegative integers, the entries of vector u are also nonnegative integers The bill of materials defined by ( provides a static view of the system material requirements over a given time-horizon In the following section, a dynamic model will be constructed to formulate the aggregate planning problem, taking into account product lead-times, inventory levels and capacity constraints The dynamic model In addition to the quantitative description of the process by its bill of materials, the model of the system must include the logical aspects of inputs availability There are two basic methods to guarantee the availability of inputs as a pre-condition to production : inventories and synchronization of orders In practice, few processes run on a pure make-to-stock or make-to-order basis On the one hand, the need for stocks appears 3

4 Automatica 39 (3, pp very clearly to limit delays or lost sales when considering the random aspects of demands for final products and lead-times But reactivity and low costs also impose to adjust orders to demands in time and quantity This is a reason why nowadays, most practical production control systems, including Kanban and MRP, use a combination of the two basic methods A long-term time-horizon, denoted T L is considered Time is discretized into periods of equal lengths The planning time horizon is divided into elementary time-periods k, with k T L From the demand prediction viewpoint, the horizon is decomposed into parts Demands are supposed perfectly known from time-period to a short-range date, denoted T S, with T S T L and predicted after this date, on the basis of constant expected values and bounded random variations With each product-type i i n is associated a lead-time, denoted i It is defined as the difference of delivery time (finishing time minus release time ( starting time for a production order of product i All the manufacturing lead-times are supposed fixed, independent of the quantities ordered (within the limits of production capacity constraints and multiple of the elementary time-period However, as in the works of (Nahmias, 979 and (Ehrhardt, 984, extension of the results to random lead-times are possible using convolutions of demand distributions with lead-time distributions list The maximal lead-time over all the products is: n max i i (3 The internal and external variables associated with product i at period k are described in the following The external demand, d ik, is such that, in normal conditions, d ik if i is not a final product The decision to produce a quantity u ik of product i is taken at period k The corresponding manufacturing activity is supposed to start at period k and to be delivered at period k i, the integer i being the production lead-time for product i, The delivery level at period k is denoted z ik The stock level on hand at the end of period k is s ik In order to obtain a quantity output u ik of product i available at period k i, components j n are n takes the required in quantities π ji u ik at time k The stock evolution equation for each product i form: s ik s i k u i k i n π i j u jk z ik (4 j The initial conditions of equations (4 are the current stock levels on hand, s i, and past production orders All the variables involved in equation (4 should be nonnegative If product i is a final product and if its current external demand at period k (including a possible backorder from previous periods cannot u i i u i be satisfied due to capacity constraints, then a backorder, noted δ ik is created for the following period 4

5 ! Automatica 39 (3, pp Assuming that demands should be satisfied as much and as soon as possible, the delivery variable z ik takes the value: z ik d ik δ i k Replacing z ik in (4 by its expression (5, one obtains : δ ik (5 r ik r i k u i k i n j π i j u jk d ik (6 with r ik s ik δ ik s ik δ ik (7 For final products, inventory levels s ik, can be obtained from the new variables, r ik, as follows: s ik r ik if r ik s ik if r ik with a backorder of r ik (8 Consider the delay operator, q written in a vector form : with r k s k δ k Matrix T q, such that for any time series h k, q r k r k T q h k h k is called the dynamic net production matrix It is defined by: Equations (6 can be u k d k (9 T q Diag q i Π ( where Diag V i represents the diagonal matrix with diagonal terms V V n Equation (9 defines a dynamic input-output model which can be seen as a discrete-time counterpart of the Laplace transform representation used in (Grubbström and Ovrin, 99 to interpret MRP rules As in (Hennet and Barthès, 998, the dynamic input-output model is described as a delay system Matrix Π is the input matrix, and Diag q i is the dynamic output matrix, which transforms the vector of orders at period k, u k into a sequence of vectors of deliveries at periods k k In the sequel, matrix T q is decomposed as follows: T q T q T q T ( where T Π T j Diag t j i with t j if and only if i i j The system delay,, is the maximal lead-time (3 j t j i in any other case ( 5

6 #! # # # Automatica 39 (3, pp The following matrices will also be used in the sequel : or, in a more explicit form, U j T l defined for j (3 l j U I Π U l Diag U l i with Ui l if and only if l " i U l j From the definitions above, the following property can be derived : i in any other case (4 U j Diag i (5 j 3 The state model A possible technique to characterize the delay system (9 is, as in (Rosling, 989, by conctructing aggregated variables representing the echelon inventory position of each item i, which is the sum of the inventory on hand and the past orders which have not yet been delivered Here, a state representation is proposed instead The state vector is defined by: x k r k u k &( (6 u k The system dynamics are described by the discrete-time state equation : x k*,+ x k - d k - u k (7 with +/ I T T I I &( 3- I &( 4- T I &( 4 The demand pattern The vector of external demands, d k appears as an external input in equation (9 - For k T S, d k is deterministic It is computed from real orders or from decisions - For k T S T L, d k is supposed random but stationary It is decomposed as follows: d k d w k (8 6

7 ! Automatica 39 (3, pp with mean values E 5 d k6 d E 5 w k6 Such a sharp decomposition relies not only on real data but also on the deliberate choice to freeze the first part of the demand trajectory in order to plan the short-range activities in detail Such a choice is consistent with the fact that the short-range part of the aggregate plan is classically considered as the basic ingredient to develop the Master Production Schedule (Fogarty and Hoffmann, 983 For k 7 T S, the disturbance vector w k is supposed random and may follow any probability distribution, stationary or time-varying The only restrictive (but realistic assumption on w k is that it is bounded : ω " w k " ω! k T S (9 The polyhedron defined by the constraints on w k will be put in a standard form to facilitate the subsequent developments At any period k T S, this vector may take any value in the polyhedral domain R5 L λ 6, with L R m n, m n, rank L n, λ a nonnegative vector in R m : w k R5 L λ 6 8 ω ; Lω " λ with L :9 I I ; λ :9 ω ω ; ( 5 The problem constraints Due to their physical meaning, the control variables of the problem, denoted u ik should all be nonnegative : u ik! i n k =< ( Nonnegativity constraints on stocks are: s ik! k >< ( Nonnegativity constraints on backorders for final products: δ ik! k >< (3 Production resources may be manufacturing units, machines, work teams, robots, pallets, Aggregated production capacity constraints are associated with the R resources Variable u ik represents the production release order for product i at time k The corresponding production lot is actually manufactured at periods k k i, and delivered at period k i Assuming a constant use of resources along the production cycle, resource capacity constraints take the following form: m r i n i i l m r i u i k l " M r for r R k (4 is the amount of resource r currently needed to produce one unit of product i M r is the capacity of resource i, supposed to be the same at any period k Past order vectors u k for k are known as system data The set of equations (4 can be re-written in a more compact form : l? U l u k 7 l " M (5

8 Automatica 39 (3, pp with matrix U l defined by (4,? capacity vector, M T 5 M M m ri the resource-activity matrix per period and M the resource Similarly, stock capacity constraints are associated with the P storage zones Storage capacity constraints take the form : n p i n i n p i s ik " N p (6 is the capacity at stock p required for one unit of product i As for production constraints, storage capacity constraints can be re-written in a more compact vector form : By definition, < capacity vector, N T 5 N N P6 < s k " N (7 8 n ip is the matrix of storage capacity required per unit of products, and N is the stock 3 Long-term optimization The current cost of the ordering policy at period k is the sum of production costs, backorder costs and inventory costs, that is: χ x k u k n c i u ik h i s ik i p i b i δ ik (8 c i, b i and h i respectively denote the unit production cost, the unit backorder cost and the unit inventory cost All the variables involved in (8 should be nonnegative integers As in most classical stochastic inventory models, set-up costs are not considered in the criterion However they can be considered afterwards (heuristically by modifying the ordering policy through introduction of minimal ordering levels The short-term part of the criterion can also be modified to include set-up costs, since the corresponding subproblem is deterministic and set-up costs can be treated more easily in the deterministic context A possible criterion to be minimized over the long-term planning horizon is the mean value of the conditional expected total cost: where u 8 u u u TL Minimize ua T L x u T L E 5 TL χ r k u k B6 (9 k The current state vector x k defined by (6 is supposed perfectly known at the beginning of period k Such an assumption is the discrete-time analogous of the classical continuous review assumption under which an order point system may be implemented (see eg (Fogarty and Hoffmann, 983 In the proposed framework, an order point policy corresponds to a closed-loop control u k x k On the contrary, only a-priori information is assumed available with respect to the sequence of demand vectors d k The a-priori information is supposed perfect for k T S, and it takes the form of a given set-bounded random law defined by (8, (9 for k T S T L 8

9 F Automatica 39 (3, pp Then, the average cost criterion (9 can be decomposed as follows: A 5 * TS T x u L T L χ x k u k E χ x k u k B6B k k T S (3 From the long-term viewpoint, the cost of the transient trajectory is negligible since T L 77 T S But what is important is the state reached at period T S Therefore, the problem is decomposed into a short term deterministic problem with a constrained final state and a long term stochastic problem with chosen initial conditions In particular, the set of admissible initial conditions selected for the long term problem can be selected so as to guarantee stability and optimality of the stationary infinite horizon inventory policy T L 3 Minimization of the infinite horizon expected average cost The infinite horizon average cost criterion can be re-written as follows: A x TS u lim T DC T E 5 TS* T χ x k u k B6 (3 k T S * In this section, it is assumed that the system evolves in a region of the state space where constraints are naturally satisfied by the unconstrained trajectory Then, it will be shown in the sequel that the optimal unconstrained trajectory actually satisfies all the problem constraints, provided that the initial state of the system belongs to the domain R5 Q ρ 6 From the works (Clarke and Scarf, 96, (Rosling, 989, (Federgruen, 993 and others, it is well known that the optimal solution of the unconstrained problem under the continuous review assumption is a base stock policy for each product, the optimal stock levels, SEi, for i n, depending on the convolution of the demand for end products over the lead-time durations In the single item case, the optimal base stock level is given by the closest integer to S such that: * S b c b h where F * denotes the -fold convolution of the demand distribution function, for the lead-time value (Porteus, 99 Furthermore, to take into account the existence of set-up costs, a minimal ordering level s can be defined so that the base stock policy is transformed into an s S defined as follows: u k if r k l uk l s u k max S r k u k l l if r k l uk l s In the multi-item case, optimality of base stock policies with respect to criterion (3 has been demonstrated and the optimal inventory levels SEi can be computed by an extension of Clarke and Scarf s procedure (Federgruen and Zipkin, 984; Rosling, 989 The vector of optimal inventory levels is then defined by: SE 5 SE T SE T N 6 (3 9

10 ! # Automatica 39 (3, pp In order to achieve stationarity of the expected inventory levels under stationary demand conditions, the dynamic system (9 must satisfy: T q ue d (33 where ue is the expected control vector Its stationarity implies, from ( and (3: ue I Π d (34 It can be noted that the nominal control vector for the dynamic model in stationary conditions takes the same form as the gross production/supply vector of the bill of materials ( The long-range control problem is thus to determine a closed-loop control law, u k x k and a set of admissible initial states Ω in order to minimize the infinite horizon expected average cost (3 subject to external disturbances, capacity constraints and nonnegativity conditions This is a constrained disturbance attenuation problem around a nominal trajectory defined by the nominal state SE and the nominal control vector ue The logics of the base stock policy are thus re-interpreted in the control terminology 3 The constrained disturbance attenuation problem In the class of stabilizing linear state-feedback closed-loop control laws u k f x k E 5 d k6b for k T S, the stochastic system is stationary and a necessary condition for stationarity of the control sequence u is: This condition is satisfied for E 5 u k6 f E 5 x k6 E 5 d k6b (35 E 5 u k6 ue E 5 x k6 xe The optimal long-term mean values of x k and u k optimize the functional χ x u under the system constraints Assuming, as it is natural, that the optimal set-point xeg ue is feasible with respect to the set of constraints, it is proposed to solve the long-term control problem by constructing a positively invariant domain Ω HJI containing xe as an interior point, such that any state trajectory started in Ω can be maintained in Ω under any admissible disturbance by an appropriate control sequence For each product i i n, a translated inventory/backorder level can be defined by : SE ue ue &( y ik r ik SEi (36 In the stage of regulation around nominal values for production and inventories, it can be assumed that the demand for final products can be satisfied without any backorder, that is under the constraints : y ik SEi i n (37

11 # M? Q # #? # M R R T Automatica 39 (3, pp Storage capacity constraints (7 can be reformulated in terms of variables y k as follows: < y k " ν with ν N < SE (38 Using the decomposition (8, the problem can be transformed into a regulation problem by introducing the translated control vector: v k u k ue with v ik uei (39 Capacity constraints (5 can be reformulated in terms of variables v k as follows: l U l v k l " µ with µ M diag i ue (4 Using the changes of variables (36, (39, equation (9 is re-written in the following form: q y k T q v k w k (4 Then, at time k, the past evolution of the system can be summarized by the new state vector : X k 5 y Tk vtk v Tk 6 T (4 The system dynamics are described by the discrete-time state equation : X k*,+ X k - w k - v k (43 System (43 is subject to the set of linear constraints (, (, (5 and (7 This set of constraints may be written in the in the following state form: and Q with Q :9 Q Q ; X k R5 Q ρ6 8 x;qx " ρ (44 ρ :9 ρ ρ ; L M NONPN M NONPN Q U Q U M NONPN M Q and ρ ν µ µ U U &( dim Q q K n dim ρ q &( Q ρ su uu uu T M NONPN M T M M M SR NONON &( The constrained control problem aims at stabilizing the expected state vector while guaranteeing that the state vector trajectory belongs to the feasible domain over the planning time horizon, which may be considered infinite Three requirements are used to characterize the control problem : &(

12 # M! M! Automatica 39 (3, pp meeting the demand requirements at each period and stabilizing the system in a region around the nominal conditions XEV, corresponding to the nominal vector of stock levels SE, and to the nominal vector of activity levels, ue, satisfaction of all the constraints along the system trajectory : X k R5 Q ρ 6 k T S 3 Attenuation of the effect of the bounded persistent disturbance vector w k R5 L λ 6 closed-loop control law v k f X k k 7 T S, using a The purpose of the long-range control problem is now to determine a control law v k f X k to be applied for k T S, which minimizes the infinite horizon expected average cost (3 subject to constraints (4, 43, 44, and such that for k 7 T S, the sequence w k is a sequence of random vectors contained in R5 L λ 6 The class of control laws now investigated is further restricted to linear state feedback control laws, in the form: v k 5 F G G 6 X k (45 This restriction will be justified in the sequel by showing that this class contains optimal controls for some domain of initial states, W The dynamic closed-loop state equation associated to (45 is : X k* X+ cl X k - w k (46 with + cl NPNONZNONPN I Y T F T Y T G NPNONZNONPN F G M NONPN M M NPNON M I T Y T G G I &( 33 The polyhedral positive invariance approach Positive invariance of a domain Ω with respect to system (46 means that any state trajectory starting in Ω remains in Ω, for any disturbance vector w k in R5 L λ 6 The positive invariance approach is able to provide local solutions to constrained control problems by imposing the positive invariance of a domain Ω in the state space satisfying the following conditions (Vassilaki et al, 988 : The zero state lies in the interior of Ω, Ω is positively invariant with respect to the controlled system, 3 Ω H R5 Q ρ 6, where R5 Q ρ 6 describes the polyhedron of constraints (44

13 Automatica 39 (3, pp To maintain the linearity of the approach, the candidate domain is supposed to be a polyhedron, Ω R5 S η6 with η 7 (componentwise Positivity of vector η guarantees that requirement is met In order to design the controller, the idea is to first achieve unconstrained optimisation of A x TS f for X TS x TS xe[, then to determine the feasibility domain of the optimal control The input-output model (4 leads to the following steps prediction equation : y k* y k T v k* \ T T v k* \ T T v k \ T T v k T v k w k* l l Expectations are taken for both sides of equality (47 and the following convergence conditions are first assumed (they will be proven in the sequel: (47 E 5 v k* 6 E 5 v k* 6 (48 Then, the model predictive condition E 5 y k* 6 (49 yields the control law defined by : with matrices U l for l y k U v k U v k U v k E 5 w k* l6 (5 l defined as in (4 The control law derived from condition (5 is: v k ] I Π y k I Π l U l v k l ^ I Π E 5 w k* l6 (5 l This control law takes the form of a state feedback (45 defined by the following gain matrices: F _ I Π G i _ I Π U i for i (5 It can be noted that if the entries of matrix Π are nonnegative integers, then the entries of all the gain matrices F and G i are nonpositive integers Then, if initial conditions and disturbances are integers, controls and state variables take integer values at any time period The purpose of control law (5 is to exactly compensate for cumulated differences with respect to nominal stock levels, taking into account past production orders not yet delivered The properties of the state feedback control law defined by gain matrices (5 will be studied first in terms of optimality with respect to criterion (3, then in terms of positive invariance 33 Optimality of the control law Proposition Within the a-priori information pattern for demands and the perfect state information pattern, the state feedback control law defined by gain matrices (5 is optimal with respect to criterion (3 for the unconstrained system with dynamic equation (7 3

14 R!! Automatica 39 (3, pp Proof The a-priori demand information pattern includes the condition E 5 w k6 control law (5 satisfies, for k T S :! k T S By construction, the I Π v k y k U l v k l l (53 and at period k, I Π v k* ` y k l U l v k l* (54 Replacing y k by its expression in (4 and taking mathematical expectations at period k for both terms of (54, one obtains: I Π E 5 v k* 6 y k Πv k T l v k l l v k U l* l vk l (55 Note from definition (4 that U T and T l U l* U l for l, to derive from (53: I Π E 5 v k* 6 And since matrix I Π is nonsingular, E 5 v k6! k 7 T S Given X k 5 y Tk vtk v Tk equation (46 yields, under the predictive condition (49, 6 T and applying v k 5 F G G 6 X k for k T S, the closed-loop system E 5 X k6! k ˆk with ˆk s (56 Condition (56, obtained under a control law such that E 5 v k6 expected cost function : E 5 χ x k u k B6 minimal value for this control a k yields the minimal possible value of the ˆk Therefore, the infinite horizon expected average cost takes its 33 Construction of the invariant domain Condition (56, obtained in the proof of Proposition, shows that the proposed state feedback control law assigns as the multiple eigenvalue of the closed-loop system This property will now be used to construct positively invariant domains of the closed-loop system (7 under the state-feedback control defined by gain matrices (5 Proposition Under the state feedback control law defined by gain matrices (5, the unconstrained system with dynamic equation (7 and w k R5 L λ6 defined by : k s admits as positively invariant sets the symmetrical polytopes R5 Φ η 6 Φ 9 Σ Σ ; and η T 5 σ T σ T 6 with σ T 5 α T α T 6 (57 4

15 # M M R b M R # M M R b R! M R b R Automatica 39 (3, pp with Σ and α a positive vector in R N which satisfies : NPNON NPNON I U M NPNON M I Πc I Πc M NPNON M α i max ω i ω i U M I Πc &( (58 i n (59 Proof Under the MPC law defined by gain matrices (5, a matrix similar to the closed-loop matrix takes the following form: H NONONZNPNONZM M NPNON I M This property can be shown by constructing the nonsingular matrix Σ given by (58, which satisfies: Vector σ defined as in (57, with α a positive vector in R N, satisfies : NONON I &( HΣ Σ+ cl (6 Hσ " σ (6 The two properties (6, (6 show positive invariance of the polytope R5 S η6 with respect to the undisturbed closed-loop system (Bitsoris, 988 Moreover, condition (59 on vector α allow to replace inequality (6 by: Hσ,d Jdλ " σ (6 where d Jd is the matrix of the absolute values of the components of a matrix J which satisfies : J Σ- (63 It is a classical result in positive invariance of polyhedral sets (see eg (Hennet and Dórea, 994 that conditions (6, (6, (63 guarantee positive invariance of the symmetrical polyhedron R5 Φ η 6 with respect to the disturbed closed-loop system (46 a It is worth noticing that the invariant control law which has just been constructed is explicit and thus very quickly computed The first two steps of the design scheme of Section 33 have been achieved through 5

16 Automatica 39 (3, pp R5 Φ η6 The third step corresponds to a correct sizing of the positively invariant domain R5 Φ η 6 The inclusion condition R5 Φ η 6 H R5 Q ρ 6 corresponds to satisfaction of the constraints for any the choice Ω point in R5 S η 6 It can be tested through the following property Property : Inclusion of polyhedra (Hennet, 989 A necessary and sufficient condition to achieve R5 Φ η 6 H R5 Q ρ 6 is the existence of a nonnegative matrix P such that : PΦ Q (64 Pη " ρ (65 Due to symmetry of the polyhedron R5 Φ η 6 defined in Proposition, existence of a solution P to (64, (65 is equivalent to the existence of a matrix R such that: RΣ Q (66 d Rd σ " ρ (67 And since, by construction, matrix Σ defined by (58 is invertible, equality (66 has the unique solution R QΣ and the set of constraints (66, (67 reduces to: d QΣ d σ " ρ (68 If conditions (59 and (68 cannot be simultaneously satisfied, the problem cannot be solved by the proposed technique due to a disturbance amplitude too high On the contrary, if there exists a vector α which satisfies (59 and such that for σ constructed as in (57, condition (67 is satisfied, then the domain R5 Φ η 6 is a candidate solution Ω which can be used as an invariant domain for the closed-loop stochastic system and as a target domain for the open-loop deterministic system From both viewpoints, it is important to construct R5 Φ η 6 as large as possible Such a requirement can be achieved by maximizing some measure on vector α, for instance a positive linear combination of its components : Maximize ef subject to constraints (59 and (65 g i α i (69 i g g g n Note that resolution of this linear program is the only computational task required to solve the state feedback control problem Thus, in spite of the apparent complexity of the mathematical analysis, the invariant domain associated with the closed-loop stationary policy can be easily computed Verification of the positive invariance property imposes that the initial state of the system should belong to the positively invariant domain R5 Φ η6 The fact of first transferring the system state into this domain is the basic condition of application of the proposed closed-loop policy The reachability problem applies to the first planning periods and may rely on the precise demand and production data which are generally available within such a short-range horizon An open-loop scheme may then be used to transfer the current state of the system to the target invariant domain, while minimizing storage, backorder and production costs 6

17 ! #!!! Automatica 39 (3, pp The open-loop planning trajectory The purpose of the open-loop planning problem is to minimize the total production, backorder and storage costs of the system trajectory from its current state, denoted X to a state X TS which belongs to the target domain, R5 Φ η6 The demand vectors d k are supposed perfectly known for k T S, where T S is the horizon of perfect demand conditions The initial state of the system, X, can be any point of the feasible domain, R5 Q ρ 6 In particular, if past orders and current inventory levels are unsufficient, backorders may be generated for final products (products numbered from to p Then, the control scheme is constrained by the reachability condition: X TS R5 Φ η 6 (7 The open-loop planning problem can be solved as the solution, u u TS of the following Linear Programming Problem : subject to : Minimize A T S x u s ik δ ik s i k δ i k T S TS u i k i n c i u ik h i s ik k i n p i b i δ ik (7 π i j u jk d ik for j i p k T S s ik s i k u i k π i j u jk for j i p n k T S n i i l m r i u! i k l " M r r R k T S n n p i i s! ik " N p p P k T S &( s TS δ TS SE u ΦX TS " η with X TS TS ue u ik i n u TS * ue i n s ik! i n δ ik! i p s δ and u k given for k h i k T S It can be noted that since the value of T S is imposed by practice, there is no guarantee that the reachability condition (7 can be satisfied In the non feasible case, the basic LP (7 can be relaxed, for instance by replacing (7 by a soft constraint through goal programming If demand and production capacity are compatible enough, it can be expected that, under a rolling horizon implementation of the aggregate plan, problem (7 will become feasible after several updates If the reachability condition (7 can be met, resolution of problem (7 by an LP software is straightforward for variables with real values In the framework of aggregate planning, real values are then rounded off to the closest integers with an acceptable precision The system trajectory is then computed and the corresponding production orders can be implemented 7

18 # # Automatica 39 (3, pp Example An small-size example with 5 types of products and 3 levels is described by the following dynamic net production matrix: T q The state vector has dimension 5 It is, for k, x k q q q 3 q q &( r k u k &( The initial inventory levels are given by: u k s 5 6 and there is no past production orders not yet delivered: u u 5 6 The vector of average demands per period is given by: d Then, using (34, the vector of nominal productions per period is : The selected base stock levels are given by: T ue SE Uniformly distributed random sample demand trajectories have been generated for products and, around their nominal values under the maximal deviation constraint (9, with ω ω i5 6 are represented on Figa and b over a time-horizon of 35 periods T They Figa Demand for product Figb Demand for product 8

19 Automatica 39 (3, pp The short-term horizon is T S 5 periods This means that the exact demand values are known until period 5 After this period, only their mean values and maximal deviation are known The unit cost parameters in the expression (8 of χ x k u k are: c c 9 c 3 5 c 4 3 c 5 3 h 5 h 4 h 3 h 4 h 5 b b 8 All the problem variables are supposed real and rounded off to some decimal, assuming that the quantities are measured in hundreds or thousands of units The curves of Fig describe the evolution of production variables u ik and arithmetic stocks (variables r ik of the 5 products, Quantities are supposed real under the two control modes, with a transition between these modes when the target set is reached, at period Figa Production Figb Inventory Figc Production Figd Inventory Fige Production 3 Figf Inventory 3 9

20 Automatica 39 (3, pp Figg Production 4 Figh Inventory Figi Production 5 Figj Inventory 5 Fig Production and Inventory Curves 6 Conclusion In this study, a typical production planning problem has been analyzed and solved using a control oriented approach It has been shown that, for this problem, the classical limitations of the application of control theory to large scale systems subject to many constraints can be overcome through the use of the positive invariance approach for polyhedral sets The planning problem has been decomposed in time into a short-term deterministic problem and a long-term stochastic problem The latter problem has been treated as a constrained disturbance attenuation problem around a regime with bounded random variations around a constant set point, matching the forecast demand pattern A solution to this problem has been found by combining the advantages of model predictive control and positive invariance It takes the form of a linear state feedback control law, with base stock levels and equilibrium production levels as state and control set points The domain of application of this control law is positively invariant It has been chosen as the target domain for the short-range production plan If the target domain can be reached within the time horizon over which the model is deterministic, then the main role of the short-range production plan is to transfer at minimal cost the current state of the system to the domain of feasibility of the closed-loop regime This problem can be solved efficiently by adding terminal conditions to the classical LP formulation of the production planning problem under deterministic demands and capacity constraints

21 Automatica 39 (3, pp Depending on the considered manufacturing system and on the market in which it operates, different uses of the proposed methodology can be considered The two stages of the control scheme may be implemented when a stationaryconstant average demand pattern is expected for the future If this is not the case, particularly if the market is subject to important trends, uncertainties and/or fluctuations, a rolling (or receding horizon production policy may be implemented, due to its ability to react to external changes Nevertheless, the proposed methodology can still be useful in this context It is believed that the effectiveness of the short-range production plan and the consistency of the whole production policy of the firm is improved by the computation of a target domain and its integration as a final condition in the short range planning problem Aknowledgement The author wants to thank the editor, the associate editor and the reviewers in charge of the paper for their very helpful corrections and comments References Bitran, GR and D Tirupati (993 Hierarchical production planning In: Handbooks in Operations Research and Management Science, vol 4, SC Graves, AHG Rinnooy Kan, PH Zipkin Eds North- Holland pp Bitsoris, G (988 Positively invariant polyhedral sets of discrete-time linear systems International Journal of Control 47, 73 6 Clarke, AJ and H Scarf (96 Optimal policies for a multiechelon inventory problem Management Science, Vol6, pp Ehrhardt, R (984 (s,s policies for a dynamic inventory model with stochastic lead-times Operations Research, Vol3, pp-3 Federgruen, A (993 Centralized planning models for multi-echelon inventory systems under uncertainty In: Handbooks in Operations Research and Management Science, vol 4, SC Graves, AHG Rinnooy Kan, PH Zipkin Eds North-Holland pp Federgruen, A and P Zipkin (984 Computational issues in an infinite-horizon, multi-echelon inventory model Operations Research, Vol3, pp Fogarty, DW and TR Hoffmann (983 Production and inventory management South-Western Publishing Company Grubbström, RW and P Ovrin (99 Intertemporal generalization of the relationship between material requirement planning and input-output analysis Intl J Production Economics, vol 6, pp3-38

22 Automatica 39 (3, pp Hennet, J-C (989 Une extension du lemme de farkas et son application au problème de régulation linéaire sous contraintes CR Ac Sciences, t38, I, pp45-49 Hennet, J-C and CET Dórea (994 Invariant regulators for linear systems under combined input and state constraints In: Proceedings of the 33th IEEE Conference on Decision and Control Vol Lake Buena Vista, Florida pp 3 35 Hennet, JC and I Barthès (998 Closed-loop planning of multi-level production under resource constraints In: Proceedings of the IFAC Symposium INCOM 98, Nancy, (France pp Mayne, DQ, JB Rawlings, CV Rao and POM Scockaert ( Constrained model predictive control: Stability and optimality Automatica, Vol36, pp Nahmias, S (979 Simple approximations for a variety of dynamic lead-time lost sales inventory models Operations Research, Vol7, pp94-94 Porteus, EL (99 Stochastic inventory theory In: Handbooks in Operations Research and Management Science, vol, DP Heyman, MJ Sobel Eds North-Holland pp Rosling, K (989 Optimal inventory policies for assembly systems under random demand Operations Research, Vol37, No4, pp Thomas, LJ and JO McClain (993 An overview of production planning In: Handbooks in Operations Research and Management Science, vol 4, SC Graves, AHG Rinnooy Kan, PH Zipkin Eds North- Holland pp Vassilaki, M, JC Hennet and G Bitsoris (988 Feedback control of linear discrete-time systems under state and control constraints Int Journal of Control, vol 47, pp77-35 Veinott, AFJr (969 Minimum concave-cost solution of leontief substitution models of multi-facility inventory systems Op Research, Vol7, No, pp6-9