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49) Assuming a zero safety stock, the FOQ policy implies that Q is ordered at Ti +1 , whenever Q ≥ D(Ti +1 ) − D (Ti ) . Total production will now have the transform; P ( s ) = Q ∞ ∑e − sTi = Q + Qe − sT1 + Qe − sT1 − sT2 + Qe− sT1 − sT2 − sT3 + ... 50) i =0 Because the Ti are independent, we may drop the index i: E ⎡e ⎣ − sT j ⎤ = E ⎡e − sTk ⎤ = E ⎡e − sT ⎤ . 52) T so that expected total production obeys: ( E ⎡⎣ P ( s ) ⎤⎦ = Q 1+E ⎡⎣e − sT1 ⎤⎦ + E ⎡⎣e − sT1 ⎤⎦ E ⎡⎣ e− sT2 ⎤⎦ + ... Q Q = = . 53) Let ν (t ) denote the setup frequency (Molinder, 1996, p.

If, after these moves, period 1 is still infeasible, then we apply a step consisting of forward shifts. Otherwise, an improvement step is considered. Forward shifts Production shifts from periods t = 1, 2, …, T - 1 are considered in that order. Portions of the production which affect an infeasible t are moved from period t + τ i to later target periods tl + τ i . For a given infeasible period t, we consider moving a quantity qi , t +τ i of the production Pi , t +τ i of each item i to later target periods tl + τ i .

This chapter aims to examine the fundamental equations of MRP Theory into in order to obtain closed-form expressions for the time development of the system, when standard ordering policies of MRP are applied. The fundamental equations of MRP Theory have been developed in several earlier papers, beginning with Grubbström and Ovrin (1992) and in some earlier unpublished studies. These equations are balance equations in the frequency domain explaining the development of total inventory, available inventory, backlogs and allocations.

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