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By Thi Thu Thuy Huynh.

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For the FOQ policy, we need to solve for the points in time when available inventory drops to zero. { } R (t ) = £ R ( s ) −1 −1 ⎧⎛ nˆ −1  ⎞ ⎫ = £ ⎨⎜ Q ∑ e − sTn − Dˆ ( s ) ⎟ s −1 ⎬ . 34) which evaluated by the Residue Theorem will be  1 β + i∞ ⎛ ⎞ nQ / w − Dˆ ( w) ⎟ e wt dw = ⎜ ∫ = − ∞ β w i 2π i ⎝ ⎠ ⎛  ⎞ ⎛ nQ ⎞ = Res ⎜ − ∑ ⎜ Dˆ ( w)e wt ⎟ = ⎟ w=0 ⎝ w ⎠ residues ⎝ ⎠ ∞ ⎛ ∞ ( wt ) k ⎞ = nQ − ∑ ⎜ ∑ d j w j ∑ ⎟ = 0. k! 35) As an example, when requirements increase linearly and cumulative requirements therefore increase quadratically, cumulative requirements  behave according to Dˆ ( s ) = as −3 , where a is the slope of the linearly increasing requirements.

3 Solutions to Non-Negativity Conditions for Available Inventory with Requirements as Discrete Events As shown above, the L4L policy provides an immediate explicit expression for the production on all levels. For the other two policies, this is not equally simple. 30 In the FOQ case we need to solve for the latest as possible batch times T0 , T1 , T2 , … , such that available inventory R(t) is kept non-negative. The solution in the time domain is simple for the individual item, given ˆ (t ) at the times when there are steps t , t , t ,… the requirements D 0 i 1 2 Examining, for successive values of n, arg max Dˆ (ti −1 ) ≤ nQ < Dˆ (ti ) ≥ 0 , there will be a unique index i ti ) ( assigned to each n, which we denote in .

If the solution P2 is infeasible, then the external demand cannot be satisfied. Because backlogs are allowed for end items, we can reduce a production amount of end items. We now turn to the Reduction procedure. 3 Reduction Procedure For a given infeasible period t, we consider reducing a quantity qi , t +τ i of the production Pi , t +τ i of each item i at period t + τ i . 8). 17) with NPV Decrease defined by: ΔNPV = NPVbefore reduction − NPVafter reduction , calculated by ΔNPV = r ∑τ E ( ( D T n = tl + in (( T ∑τ n = tl + i +1 E − − i M ⎛ ⎞ − ⎜ ci q + K iν i1 + ∑ wk cˆkt ⎟ e − ρ t k =1 ⎝ ⎠ Di ( t +τ i ) − ( Pi ( t +τ i ) − q) −r ⎡ E ⎢⎣ + ) − Pin ) − ( Di ( n −1) − Pi ( n −1) ) e − ρ nΔ (( D i ( tl +τ i ) ) − (D − i ( t +τ −1) − ( Pi ( tl +τ i +1) − q) − Pi ( tl +τ i −1) ) − (D − i ( n −1) ) − )e − ρ ( t +τ i ) ) ⎤ − − ( Pi ( n −1) − q ) ) e − ρ nΔ ⎥ .

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