The term inventory-production theory is not well defined. It com­ prises e. g. such models like cash balance models, production smoothing models and pure inventory models. We shall here mainly be concerned with stochastic dynamic problems and shall give exact definitions in the next section. Most of our work will concentrate on cash balance models. However, production smoothing situations and pure inventory problems will also be investigated. Since we are faced in principle with dynamic stochastic situa­ tions a dynamic programming approach would be appropriate. This approach, however, due to computational restraints, is limited to only but the simplest models. Therefore, in practice, one ruduces stochastics just in taking forecasts of demand and then treating the problem as a deterministic optimization problem. In addition one often introduces certain safety stocks to safeguard the system from possible forecasting errors. In general, this proce­ dure is suboptimal. However, there exists one particular situa­ tion when a separation in a forecasting procedure and a subse­ quent optimization of the remaining deterministic model is not suboptimal. This is known as the linear-quadratic model, i. e. a model having linear system equations and a quadratic cost crite­ rion. For this type of model H. A. Simon ~3J and later H. Theil [25J have shown that the above separation property holds. In fact, Simon's and Theil's results are nothing else but what has later and more generally become known to control engineers as Kalman's famous separation principle.

Inhalt

1. The general model.- 2. The linear-quadratic model.- 2.1 Finite horizon case.- 2.2 Least square forecasts.- 2.2.1 Least square property of rk(i).- 2.2.2 Recursive calculation of rk(i).- 2.3 An ideal situation.- 2.4 Infinite horizon case.- 2.5 Appendix to Chapter 2.- 2.5.1 State space representation and Separation Theorem.- 2.5.2 Optimal policies for ARMA-processes.- 3. The linear non-quadratic model 3.- 3.1 The general linear non-quadratic model.- 3.2 The general solution.- 3.3 Special cost functions.- 3.3.1 Piecewise linear costs 4.- 3.3.2 Probability constraints.- 3.3.3 A production smoothing problem.- 3.4 Special stochastic demand sequences.- 3.4.1 Non-correlated demand.- 3.4.2 Exponentially correlated demand.- 3.5 A direct approach solving a LNQ-problem.- 3.5.1 Piecewise linear costs.- 3.5.2 Production smoothing problem.- 3.6 Appendix to Chapter 3: The normality condition.- 4. Comparison with optimal Dynamic Programming solutions.- 4.1 Piecewise linear costs (no set-up costs P=Q=0).- 4.1.1 Dynamic Programming solution.- 4.1.2 Numerical results.- 4.2 Piecewise linear costs (including set-up costs: P and/or Q?O).- 4.2.1 Optimal solution.- 4.2.2 Numerical results.- 4.3 Piecewise linear costs - Gauss-Markov case.- 4.3.1 Dynamic Programming solution.- 4.3.2 Numerical results.- 5. Comparison with deterministic approximations.- 5.1 White noise case.- 5.1.1 Numerical results.- 5.2 Gauss-Markov case.- 5.2.1 Numerical results.- 5.3 Appendix to Chapter 5: Derivation of the deterministic policy.- 6. Comparison with AHM-Inventory Models.- 6.1 No-set-up cost case (P=0).- 6.1.1 The LNQ-approach.- 6.1.2 Comparison of demand distributions.- 6.2 Set-up cost case (P+0).- 6.2.1 Derivation of an optimal S (T)-policy.- 6.2.2 Numerical results.- 7. Summary and concluding remarks.- Literature.