A positive linear discrete-time model of capacity planning and its controllability properties

-One of the most important concepts in production planning is that of the establishment of an overall or aggregate production plan. In this paper, the problem of establishing an aggregate production plan for a manufacturing plant is considered. A new dynamic discrete-time model of capacity planning utilizing concepts arising in positive linear systems (PLS) theory is proposed and its controllability property is analyzed. Controllability is a fundamental property of the system with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. Some new open problems regarding controllability of stationary and nonstationary PLS with linear constraints are posed in the paper. An optimal control problem for capacity planning is formulated and discussed. © 2004 Elsevier Ltd. All rights reserved. K e y w o r d s P r o d u c t i o n planning and control, Systems theory, Positive linear systems, Controllability, Optimal control. 1. I N T R O D U C T I O N In this paper, we consider the problem of establishing an aggregate production plan for a manufacturing plant. The basic issue is, given a set of production demands stated in some common unit, what levels of resources (such as inventory, regular time production, overtime production, labour, etc.) should be provided in each period? There has been a long history of academic research on aggregate planning, resulting in many mathematical programming (static) models and in a variety of heuristics [1,2]. However, as the firms attempt to implement manufacturing 0895-7177/04/$ see front matter (~) 2004 Elsevier Ltd. All rights reserved. Typeset by ~4j~AS-TEX doi: 10.1016/j.mcm.2003.03.010 218 L. CACCETTA et al. planning and control systems, they find deficiencies in these models and heuristics. We attempt to overcome some of these drawbacks by proposing a new dynamic model, utilizing concepts arising in positive linear systems (PLS) theory. The dynamic modeling approach (and, particularly, the optimal control approach) to the theory of the firm is motivated by three issues: (i) the need for policies, (ii) the contribution of deductive analysis, and (iii) the need to incorporate time. The state of the art of this area is well exposed in the monograph [3]. In this book, the authors discuss a number of continuous-time dynamic models and exploit the Pontryagin Maximum Principle developed for such models to determine the optimal policy. They do not consider positive systems as well as discrete-time models. However, discrete-time models seem to be somewhat more suitable to describe the firm's dynamics. The model we consider in this paper represents not only the system dynamics, but it contains a number of important parameters not included in the dynamic models described in [3]. Applying some recent results concerning PLS and developing further the theory, we analyze controllability properties of the simplified model. Controllability is a property of the system that shows its ability to move in space. It is a fundamental property with direct implications not only in dynamic optimization problems (such as those arising in inventory and production control) but also in feedback control problems. On the basis of the proposed model, we formulate and discuss a discrete-time optimal control problem for capacity planning. The paper is organized as follows. In Section 2, we present the model. Positivity properties of the model are identified in Section 3. In Section 4, we extend the definitions for reachabllity and controllability of stationary (time-invariant) positive systems to nonstationary (time-variant) positive systems and study the reachability and controllability properties of the (simplified) model for capacity planning proposed in Section 2. An optimal control problem formulated on the basis of the model is considered in Section 5. Finally, in Section 6, the conclusions are stated.