ENG  Vol.6 No.12 , November 2014
Optimal Aggregate Production Plans via a Constrained LQG Model
Abstract: In this paper, a single product, multi-period, aggregate production planning problem is formulated as a linear-quadratic Gaussian (LQG) optimal control model with chance constraints on state and control variables. Such formulation is based on a classical production planning model developed in 1960 by Holt, Modigliani, Muth and Simon, and known, since then, as the HMMS model [1]. The proposed LQG model extends the HMMS model, taking into account both chance-constraints on the decision variables and data generating process, based on ARMA model, to represent the fluctuation of demand. Using the certainty-equivalence principle, the constrained LQG model can be transformed into an equivalent, but deterministic model, which is called here as Mean Value Problem (MVP). This problem preserves the main properties of the original model such as convexity and some statistical moments. Besides, it is easier to be implemented and solved numerically than its stochastic version. In addition, two very simple suboptimal procedures from stochastic control theory are briefly discussed. Finally, an illustrative example is introduced to show how the extended HMMS model can be used to develop plans and to generate production scenarios.
Cite this paper: Filho, O. (2014) Optimal Aggregate Production Plans via a Constrained LQG Model. Engineering, 6, 773-788. doi: 10.4236/eng.2014.612075.

[1]   Holt, C.C., Modigliani, F., Muth, J.F. and Simon, H.A. (1960) Planning Production, Inventory and Work Force. Prentice-Hall, Upper Saddle River.

[2]   Yildirim, I., Tan, B. and Karaesmen, F. (2005) A Multiperiod Stochastic Production Planning and Sourcing Problem with Service Level Constraints. OR Spectrum, 27, 471-489.

[3]   Hackman, S., Riano, G., Serfozo, R., Huing, S., Lendermann, P. and Chan, L.P. (2002) A Stochastic Production Planning Model. Technical Report, The Logistic Institute, Georgia Tech and National University of Singapore, Singapore.

[4]   Silva Filho, O.S. (2012) Optimal Production Plan for a Stochastic System with Remanufacturing of Defective and Used Products. In: Mejía, G. and Velasco, N., Eds., Production Systems and Supply Chain Management in Emerging Countries: Best Practices, Springer, Vol. 1, 167-182.

[5]   Higgins, P., Le Roy, P. and Tierney, L. (1996) Manufacturing Planning and Control: Beyond MRP II. Chapman & Hall, London.

[6]   Sahinidis, N.V. (2004) Optimization under Uncertainty: State-of-the-Art and Opportunities. Computer and Chemical Engineering, 28, 971-983.

[7]   Mula, J., Poler, R., Garcia-Sabater, J.P. and Lario, F.C. (2006) Models for Production Planning under Uncertainty: A Review. International Journal of Production Economics, 103, 271-285.

[8]   Cheng, L., Subrahmanian, E., and Westerberg, A.W. (2004) A Comparison of Optimal Control and Stochastic Programming from a Formulation and Computational Perspective. Computers and Chemical Engineering, 29, 149-164.

[9]   Silva Filho, O.S. and Cezarino, W. (2004) An Optimal Production Policy Applied to a Flow-shop Manufacturing System. Brazilian Journal of Operations and Production Management, 1, 73-92.

[10]   Silva Filho, O.S. (2012) An Open-Loop Approach for a Stochastic Production Planning Problem with Remanufacturing Process. In: Andrade Cetto, J., Ferrier, J.-L., Pereira, J.M.C.D. and Filipe, J., Eds., Informatics in Control, Automation and Robotics, Springer, 174, 211-225.

[11]   Singhal, J. and Singhal, K. (2007) Holt, Modigliani, Muth, and Simon’s Work and Its Role in the Renaissance and Evolution of Operations Management. Journal of Operations Management, 25, 300-309.

[12]   Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (2008) Time Series Analysis: Forecasting and Control. 4th Edition, Wiley, Hoboken.

[13]   Bertesekas, D.P. (2007) Dynamic Programming and Stochastic Control. Volume 1. Athena Scientific, Belmont, USA.

[14]   Lassere, J.B., Bes, C. and Roubelat, F. (1985) The Stochastic Discrete Dynamic Lot Size Problem: An Open-Loop Solution. Operations Research, 3, 684-689.

[15]   Pekelman, D. and Rausser, G.C. (1978) Adaptive Control: Survey of Methods and Applications. In: Applied Optimal Control, TIMS Studies in the Management Science, Vol. 9, North-Holland, 89-120.

[16]   Bryson, A.E. and Ho, Y. (1975) Applied Optimal Control: Optimization, Estimation and Control. Hemisphere Publishing Corporation, Washington DC.

[17]   Papoulis, A. and Pillai, S.U. (2002) Probability, Random Variables, and Stochastic Process. 4th Edition, McGraw-Hill.

[18]   Graves, S.C. (1999) A Single-Item Inventory Model for a Nonstationary Demand Process. Manufacturing & Service Operations Management, 1, 50-61.

[19]   Kleindorfer, P.R. (1978) Stochastic Control Models in Management Science: Theory and Computation. Applied Optimal Studies in the Management Science, North-Holland, 9, 69-88.

[20]   Silva Fo, O.S. and Ventura, S.D. (1999) Optimal Feedback Control Scheme Helping Managers to Adjust Industrial Resources of the Firm. Control Engineering Practice, 7, 555-563.

[21]   Isermann, R. (1981) Digital Control System. Springer-Verlag, Heidelberg.

[22]   Shen, R.F.C. (1994) Aggregate Production Planning by Stochastic Control. European Journal of Operational Research, North-Holland, 73, 346-359.

[23]   Gershwin, S., Hildebrant, R., Suri, R. and Mitter, S.K. (1986) A Control Perspective on Recent Trends in Manufacturing Systems. IEEE Control Systems Magazine, 6, 3-15.